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Beam characteristics for various sizes of annular aperture on scanning electron microscope
Ultramicroscopy 88 (2001) 151–170
Beam characteristics for various sizes of annular aperture on scanning electron microscope Tohru Ishitani*, Mitsugu Sato, Hideo Todokoro Instruments, Hitachi Ltd., 882 Ichige, Hitachinaka-shi, Ibaraki-ken 312-8504, Japan Received 6 November 2000; received in revised form 2 March 2001
Abstract Using an analogy between light optics and electron optics, we have calculated beam characteristics such as the beam profile and the optical transfer function for several sizes of annular and circular apertures on a scanning electron microscope (SEM). It has been found that an annular aperture improves the image quality with regard to several kinds of image resolution and the depth of focus at the price of good low-frequency (n) contrast. In contrast with conventional circular-aperture SEM images, a combination of a low-n-pass filtered, circular-aperture SEM image with a high-n-pass filtered, annular-aperture SEM image has the potential to enhance the image quality in terms of both the image resolution and the depth of focus. r 2001 Elsevier Science B.V. All rights reserved. PACS: 07.78.+s; 42.15; 42.25; 42.30 Keywords: Scanning electron microscope; Depth of the focus; Resolution; Annular aperture
1. Introduction Semiconductor integrated devices advance continuously, whereby critical-dimensions are shrinking and aspect-ratio features are standing out. Here, the demands for a scanning electron microscope (SEM) with a higher image resolution and a greater depth of focus are increasing in the fields of device metrology and device-defect analysis. With regard to the SEM optical systems, much effort has been directed to minimize the beam spot by optimizing the size of the beam-limiting aperture so that the effect of lens aberrations can be reduced. So far as we know, the apertures to date are all circular. In contrast, it has been known in light optics that obstructing the central part of the aperture (i.e., using annular aperture) narrows the principal maxima (i.e., main lobe) of the system’s point spread function, and thus enhances the resolution in the Rayleigh sense [1–3]. However, this effect is accompanied by the following: (1) a decrease of the brightness, since there is an increasing loss of light energy in the image, and (2) a reduced contrast in the low-frequency (n) region due to a considerable rise in the level of the side lobes of the point spread function, since a large amount of light is diffracted from its proper geometrical position. *Corresponding author. Tel.: +81-29-276-6353; fax: +81-29-276-6354. E-mail address: [email protected] (T. Ishitani). 0304-3991/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 1 ) 0 0 0 8 4 - 5
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The present paper makes clear how the results obtained in light optics are analogically applicable to electron optics. The beam characteristics on SEM system with various sizes of the annular aperture are calculated to discuss both the beam resolution and the depth of focus as well as their beam intensity profiles and the corresponding optical transfer functions (OTF). Besides, various types of both the beam resolution and the depth of focus are considered to better understand the beam characteristics. It is shown that a combination of a low-n-pass filtered, circular-aperture SEM image with a high-n-pass filtered, annular-aperture SEM image has a potential to enhance the image quality both in terms of the image resolution and the depth of focus. Here, it is implicitly assumed that only the beam profile determines both the image resolution and the depth of focus, and so the beam resolution is equal to the SEM image resolution.
2. Principal features of the electron beam focused through circular and annular apertures To elucidate the principal features of the electron beam focused through circular and annular apertures (or pupils), we consider a very simple optical system, i.e., an aberration-free optical system only subject to diffraction. Fig. 1(a) shows three kinds of pupils, i.e., large and small circular pupils (LCP and SCP) of radius a and a=21=2 (E0:71a), respectively, and an annular pupil (ANP) having inner and outer radii of ba and a, respectively, with b ¼ 0:71. Here, b is called the obstruction ratio and is some positive number less than unity. The opening area of SCP is equal to that of ANP, and each of them is one half that of LCP. Figs. 1(b 1 and 2) show beam intensity profiles observed on the Gaussian plane (z ¼ 0) in the systems (with LCP, SCP, and ANP) and their normalized profiles, respectively. The corresponding OTF curves tðn Þ are shown in Fig. 1(c) (see Appendix A.3). The ið0; rÞ profiles for circular apertures (LCP and SCP) are known as the Airy pattern, which has the first zero at 2r=DLCP E1:22 and 2r=DSCP E1:22, respectively. Here, DLCP ð DÞ ¼ l=a and DSCP ¼ DLCP =0:71;
ð1aÞ
l½nm ¼ 1:226=V½volts1=2 ;
ð1bÞ
n * ¼ nD:
ð1cÞ
l is the wavelength of the electron waves, a is the focusing beam semi-angle for LCP, and V is the electron accelerating potential on the observation plane. According to the Rayleigh criterion (refer to Appendix A.2), the beam resolution (referred to as the Rayleigh resolution RRayleigh in the present study) is equal to the Airy disc radius rAiry ð¼ 0:61DÞ. On the ið0; rÞ profile for ANP, the profile of principal maxima is relatively finer than that for LCP, although the z-direction axial intensity ið0; 0Þ is one quarter that of LCP. This result suggests that the SEM image resolution can be improved by ANP. This improvement is indicated also by the tðn Þ curves, which show the value of n0:1 satisfying tðn0:1 Þ ¼ 0:1 as 1.61 for LCP and 1.75 for ANP. Here, the value of D=n0:1 is the OTF resolution ROTF ð¼ 1=nÞ (refer to Appendix A.3). The tðn Þ curve for LCP a ANP is also plotted in Fig. 1(c) for later discussion. Fig. 1(d) shows normalized z-axial intensity profiles iðz; 0Þ in the system with LCP, SCP, and ANP. Using these profiles we obtain each intensity depth of the focus Dzint , which is defined as the z-range satisfying iðz; 0Þ=imax X0:8 (see Appendix A.5). It is observed that the Dzint value for ANP is larger than that for LCP. Several other definitions on both the beam resolution R and the depth of focus Dz are given in Appendix A. Table 1 summarizes those definitions and the corresponding values obtained in the aberration-free systems. In the following sections calculation procedures for obtaining iðz; rÞ and tðn Þ in a
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Fig. 1. Comparison of the beam characteristics of aberration-free systems among three types of apertures: (a) the large-circular aperture (LCP), the small-circular aperture (SCP), and the annular aperture (ANP) of b ¼ 1=21=2 ðE0:71Þ, (b 1) beam intensity profiles observed on the Gaussian plane (z ¼ 0) in the systems with LCP, SCP, and ANP, (b 2) the corresponding normalized curves of the beam intensity profiles shown in figure (b 1) (c) the corresponding OTF curves tðn Þ, and (d) the normalized z-axial intensity profiles iðz; 0Þ in these systems (see Appendix A.3).
realistic SEM system with aberrations and with a finite electron source are described, the dependency of the beam resolution and the depth of focus on b is derived from the calculations and plotted, and potential advantages of an SEM system with the circular/annular apertures are discussed.
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Table 1 Several definitions of the image resolution and the depth of focus in the SEM systems Type
Symbol
Definition
Aberr.-free system (circular aperture)
Resolution 1. Conventional 2. Rayleigh 3. OTFa 4. Information
Rconv RRayleigh ROTF Rinf
Beam diameter r value corresponding to the first-zero of ib ðrÞ D=n0:1 value satisfying tðn0:1 Þ ¼ 0:1 Spatial length per bit in IPCb
d 0.61D (Airy disc) 0.62D at n0:1 ¼ 1:61 0.51D at S=N ¼ 10
Depth of the focus 1. Conventional 2. Intensity 3. Information
Dzconv Dzint Dzinf
z-range satisfying daberr:@free ðzÞpdmin z-range satisfying iðzÞ=imax X0:8 z-range satisfying Rinf ðzÞ=Rinf;min p1:1
d=aðE0:61D=aÞ 1:02D=a 0:84D=a at S=N ¼ 10
a b
1 is the optical transfer function. 2 is the information-passing capacity.
Fig. 2. Aberration-laden optical imaging system with an annular exit aperture of the obstruction ratio b.
3. Theoretical expressions using wave optics Fig. 2 shows an optical imaging system with aberrations and with an annular exit aperture (or pupil) of obstruction ratio b. The system is rotationally symmetric and its optical axis and radial distance are taken as the z and r coordinate axes, respectively. The z origin is referred to as the Gaussian image plane. An aperture of the objective lens contributes to the exit pupil. An intensity distribution of electron wave in the image of a point electron-source called the point-spread function (PSF) of the optical system has been already calculated for a circular aperture in other studies [4–7]. Extending these approaches to the annular aperture, we obtain the PSF at point Qðz; rÞ in an observation plane near the Gaussian plane as ib ðz; rÞ ¼ Gjub ðz; rÞj2 ;
ð2Þ
where ub ðz; rÞ is the dimensionless wave’s complex amplitude expressed by ub ðz; rÞ ¼ C term þ iS term; C term ¼ 2
Z
ð3Þ
1
J0 ð2pru=DÞu cos½2pðAu2 þ Cu2 þ Bu4 Þdu;
ð4aÞ
J0 ð2pru=DÞu sin½2pðAu2 þ Cu2 þ Bu4 Þdu;
ð4bÞ
b
S term ¼ 2
Z
1 b
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A ¼ ð1=2Þza=D; B ¼ ð1=4ÞCs a3 =D; C ¼ ð1=2ÞCc ðDV=VÞa=D; G ¼ jp s2p =ðlLp
op Þ
2
155
ð5Þ ð6Þ
;
J0 is the first kind of Bessel function, Cc and Cs the chromatic- and the spherical-aberration coefficients (defined on the image side), respectively; a the focusing semi-angle of the outer beam, kð¼ 2p=lÞ the wave number of the electron wave, jp the power density of the electron wave irradiating on the pupil, sp ¼ pa2 the area of the pupil of b ¼ 0, Lp op the distance from the pupil to the observation plane (Lp op ba and r), DV the voltage spread of the electrons, and V the accelerating potential of the beam on the observation plane. Here, the potential origin is referred to as the electron source. Note that when jp is constant, the value of ib ð0; 0Þ is proportional to a4 , not a2 . This relation supports iSCP ð0; 0Þ=iLCP ð0; 0Þ ¼ 1=4 as shown in Fig. 1(b 1). To simplify the calculation, the term G is set to 1. The variables of B and C given in Eq. (5) are used in the latter discussion to express the spherical and chromatic aberrations, respectively. A frequency response function tðnÞ for an incoherent optical system is obtained by calculating either the Fourier transform of ib ðz; rÞ with subsequent normalization of tð0Þ ¼ 1 or the autocorrelation of the pupil function. Using the dimensionless pupil function PðuÞ with u2 ¼ x2 þ Z2 and n ¼ nD, the latter calculation yields Z Z % * Þ2 þ Z2 g1=2 Þ dx dZ; Pðfðx þ n * Þ2 þ Z2 g1=2 Þ Pðfðx@n ð7Þ tðn * Þ ¼ ð1=Yb Þ s
where PðuÞ ¼ Sd ðuÞ exp ½2piðAu2 þ Cu2 þ Bu4 Þ; Sd ðuÞ ¼
at bpmp1; at 0pmob;
ð8Þ ð9Þ
Yb ð¼ 1@b2 Þ is a fraction of the opening area of the annular pupil, which corresponds to a transmissivity of the electron wave power, and 1=Yb in Eq. (7) works as the normalization factor leading to tð0Þ ¼ 1. The % expression of PðuÞ denotes the complex conjugate of PðuÞ. The integration of Eq. (7) is carried out only in the hatched regions as shown in Fig. 3. For the aberration-free optical system (i.e., B ¼ C ¼ 0) with a circular aperture (b ¼ 0), the tðn Þ function on the Gaussian plane is simply given by tðn * Þ ¼ ð1=pÞð2y@sin 2yÞ;
ð10Þ
where y ¼ cos@1 ðn * =2Þ with 0pn * p2:
Fig. 3. Domain for the integration of Eq. (7).
ð11Þ
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The tðn Þ curves for various b values are shown in the later Fig. 8(a). The maximum n value, which corresponds to the cut-off frequency, is 2. When l and Lp op are constant, the cut-off frequency is higher as the aperture size a (ELp op a) is larger . For an optical system with a finite-size and spatially incoherent source, its response function ts ðn Þ is modified as ts ðn * Þ ¼ Sðn * Þtðn * Þ:
ð12Þ
Here, Sðn Þ is the Fourier transform function of a source’s image intensity profile sðrÞ. When the sðrÞ profile is a Gaussian type with a standard deviation s as sðrÞ ¼ ½1=fð2pÞ1=2 sgexp ½@ð1=2Þðr=sÞ2 ;
ð13Þ
the source size dg defined as a full-width at half-maximum (FWHM) yields dg ¼ 2ð2 ln 2Þ1=2 s½E2:35s. The Sðn Þ function for the Gaussian sðrÞ profile yields % * Þ2 with Sðn * Þ ¼ exp½@2ðpsn
s% ¼ s=D
and
n * ¼ nD:
ð14Þ
Last, consider a source with an energy distribution NðDVÞ. Since the electron with its energy deviation DV focuses geometrically at z ¼ Cc ðDV=VÞ, i.e., A ¼ @C, the beam current distribution !ıp ðz; rÞ and the frequency response function tp ðn Þ for the source with the NðDVÞ distribution are obtained by integrating the NðDVÞ-weighted functions of ib ðz; rÞ and tðz; n Þ with respect to DV as Z þN !ıb;p ðz; rÞ ¼ NðDVÞib ðzðDVÞ; rÞdDV; ð15Þ @N
tp ðn * Þ ¼
Z
þN
NðDVÞtðzðDVÞ; n * ÞdDV:
ð16Þ
@N
Here, tðz; n Þ is the z-dependent function of tðn Þ obtained from Eq. (7). As to the NðDVÞ distribution, the Gaussian distribution is typically assumed here. NðDV; sv Þ ¼ ½1=fð2pÞ1=2 s g exp ½@ð1=2ÞðDV=sv Þ2 :
ð17Þ
The energy spread (FWHM) is expressed by the form DVFWHM ½¼ 2ð2 ln 2Þ1=2 sn E2:35sn . The C value (defined in Eq. (5)) at DV ¼ DVFWHM is expressed as CFWHM for later discussion.
4. Calculations and discussion 4.1. Aberration-free optical system ðB ¼ C ¼ 0Þ For an aberration-free optical system with a point source, the wave amplitude on the Gaussian imageplane (i.e., z ¼ 0) is calculated from Eq. (3) as [1–3]
ub ð0; rÞ ¼ 2
Z 0
1
J0 ð2pru=DÞu du@2
Z
b
J0 ð2pru=DÞudu;
0
¼ f2J1 ð2pr=DÞ=ð2pr=DÞg@b2 f2J1 ð2pbr=DÞ=ð2pbr=DÞg:
ð18Þ
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Then, the wave intensity is expressed as ib ð0; rÞ ¼ ½f2J1 ð2pr=DÞ=ð2pr=DÞg@b2 f2J1 ð2pbr=DÞ=ð2pbr=DÞg2 :
ð19Þ
Especially, the central intensity at r ¼ 0 is given by ib ð0; 0Þ ¼ ð1@b2 Þ2 :
ð20Þ
The curves of ib ð0; rÞ vs. r and their normalized curves for several b values together with an inset of enlarged portions are shown in Figs. 4 (a) and (b), respectively. Note that, if the electron irradiance of the aperture is held constant, the total power through the aperture decreases as ð1@b2 Þ½¼ Yb and the central maximum intensity decreases as ð1@b2 Þ2 with the increase in b. The minima (zeros) of the intensity have a value of zero at r-values given by J1 ð2pr=DÞ ¼ bJ1 ð2pbr=DÞ
with
ra0;
ð21Þ
while the maxima occur at r values given by J2 ð2pr=DÞ ¼ b2 J2 ð2pbr=DÞ with
ra0:
ð22Þ
Fig. 4. (a) Curves of ib ð0; rÞ vs. r for several b values together with an inset of enlarged portions and (b) their normalized curves.
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It is apparent from Eq. (19) that as b-1, ib ð0; rÞ=ib ð0; 0Þ-J02 ð2pr=DÞ. Since the first zero of the J0 ðxÞ function lies at x ¼ 2:40, the first zero of the intensity for b ¼ 1 occurs at the value given by 2r=DðE2:40=pÞE0:76 compared with 2r=DE1:22 [first zero of J1 ð2pr=DÞ for b ¼ 0. The relative intensity values of the secondary maxima to its principal maximum at r ¼ 0 becomes higher as b increases. For example, when b ¼ 0:6, the first secondary maximum (at 2r=DE4:65=pE1:48) has a value of 12.0% of the principal maximum compared to a value of 1.75% (at 2r=DE5:14=pE1:63) for a circular The minima, maxima, and their corresponding intensities ib ð0; rÞ and encircled-powers aperture. Rr 0 2prib ð0; rÞdr are given for b ¼ 0@0:9 (at intervals of 0.1) in the textbook [3]. It is interesting to observe the intensity distribution at large values of r and b. The circular-aperture distribution consists of rings of maxima and minima, and the successive maxima decrease monotonically, whereas the annularaperture distributions consist of not only the maxima and minima but also of a periodic ring group structure. The distribution appears to be divided into ring groups. The details have been discussed in the textbook [3]. For an aberration-free SEM system with a circular aperture (b ¼ 0), the Rayleigh criterion indicates that the image resolution is equal to the r-value of 0:61Dð¼ rAiry Þ corresponding to the first-zero of the intensity profile iðrÞ (refer to Appendix A.2). Similarly, by an approximate application of the Rayleigh criterion to a system showing aberrations with a circular/annular aperture, the image resolution (referred to as the Rayleigh resolution RRayleigh in the present study) is obtained. It follows that as b ¼ 0-1, the RRayleigh value is improved from 0.61D to 0.38D. Note that as the b value increases, the RRayleigh value expresses less specifically the image resolution because of increasing successive maxima of the ib ðz; rÞ profile as seen in Fig. 4(b), or diminishing tðn Þ value in the low-n ð> 0Þ region as seen in the later Fig. 8(a). Next, let us calculate the intensity profile in the direction of the z-axis. The terms of C term and S term of Eqs. (4a) and (4b) are simplified as C term ¼ ð1=A * Þðsin A * @sin A * b2 Þ;
ð23aÞ
S term ¼ @ð1=A * Þðcos A * þ cos A * b2 Þ;
ð23bÞ
where A * ¼ 2pA ¼ pza2 =l ¼ pza=D:
ð24Þ
Introducing a new parameter w (in wave-phase) for representing A ð1@b2 Þ=2, we obtain the ib ðz; 0Þ form as ib ðz; 0Þ ¼ ð1@b2 Þ2 ½ðsin wÞ=w2 :
ð25Þ
Fig. 5 shows normalized curves of ib ðz; 0Þ vs. z for several b values on both the aberration-free system ðB ¼ C ¼ 0Þ and a typical spherical-aberration system of (B ¼ 0:5 and C ¼ 0). All curves for the aberration-free system have minima (i.e., zero intensities) at w ¼ np, where n is an integer. The first minima of n ¼ 71 correspond to az=D ¼ 72=ð1@b2 Þ, i.e., zE7ð3:3 rAiry =aÞ=ð1@b2 Þ. Note that, when b ¼ 0, the first minimum on the z-axis is 3:3=a times larger than that on the r-axis for the circular aperture. The portion of the wave quantity between the central maximum (at w ¼ 0) to the first minimum (at w ¼ p) in the direction of the z-axis is calculated from Z p 2 2 MðpÞ ¼ ð1@b Þ ½ðsin wÞ=w2 dw 0
¼ ð1@b2 Þ2
Z
0
2p
ðsin 2wÞ=ð2wÞdð2wÞ ¼ 1:433ð1@b2 Þ2
ð26Þ
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Fig. 5. Normalized curves of ib ðz; 0Þ vs. z for several b values; (a) an aberration-free system (B ¼ C ¼ 0) and (b) a typical sphericalaberration system of B ¼ 0:5 and C ¼ 0.
Since MðNÞ ¼ ðp=2Þð1@b2 Þ2 , the value of MðpÞ=MðNÞ approaches 1:433=ðp=2ÞE0:92, being independent of b. With the definition that the intensity depth of focus Dzint is the z-range satisfying the condition of ib ðz; 0Þ=ib ð0; 0ÞX0:8, the w-region is given by jwjp0:51p and the Dzint expression by Dzint ¼ 1:02ðD=aÞ=ð1@b2 Þ:
ð27Þ 2
Note that as b ¼ 0-1, the Dzint value increases as 1=ð1@b Þ. Here, the value of w½¼ A ð1@b2 Þ corresponds to the wave-phase difference between the outer and inner geometrical beams converging at the observation point Qðz; 0Þ. The Rayleigh’s l=4 rule [3] discloses that a quarter-wave of primary spherical aberration reduces the intensity at the Gaussian image point by 20%; in other words, the Strehl ratio for this aberration is 0.8. (Here, the Strehl ratio of an image or a system is defined as the ratio of the intensities at the center of image in a plane with and without the aberration.) A variant of this definition is that an aberration-laden or aberration-free wave front, which lies between two concentric spheres that are spaced a quarter-wave apart, will give a Strehl ratio of approximately 0.8. We see that the region of jwjp0:51p mentioned above obeys this variant of Rayleigh’s l=4 rule. Typical isometric ib ðz; rÞ figures at b ¼ 0 and 0.6 in an aberration-free system are plotted in Figs. 6(a) and (b), respectively. Lengths of the arrows on the z-axis and the Gaussian plane correspond to RRayleigh and Dzint , respectively. As b ¼ 0-0:6, we see that Dzint ða=DÞ ¼ 1:0-1:6 and 2RRayleigh =D ¼ 1:22-0:95. In Fig. 6 (a) for the circular aperture, the arrows 1 and 10 correspond to geometric trajectories of most-outer incoming and outgoing beams, respectively. In Fig. 6 (b) for the annular aperture, on the other hand, the incoming beams go straight between the arrows 1 and 2, while the outgoing trajectories do between the arrows 10 and 20 . The isometric ib ðz; rÞ figures cannot be reduced from geometric beam trajectories. To see the dependency on b of RRayleigh , Dzint , and Yb values, their values normalized at b ¼ 0 are plotted as a
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Fig. 6. Isometric ib ðz; rÞ figures for the aberration-free system: (a) the circular aperture (b ¼ 0) and (b) the annular aperture (b ¼ 0:6). Lengths of the arrows on the Gaussian plane and the z-axis correspond to RRayleigh and Dzint , respectively. The arrows 1 and 10 correspond to geometric trajectories of most-outer incoming and outgoing beams, respectively.
function of b in Fig. 7 together with the values of response resolution ROTF , information resolution Rinf and information depth of the focus Dzinf (at the signal-to-noise ratio S=N ¼ 10), which are discussed later. 4.2. Spherical aberration optical system ðBa0; C ¼ 0Þ As a typical example of the optical systems having spherical aberrations, the normalized ib ðz; 0Þ profiles for B ¼ 0:5 are shown for several b values in Fig. 5 (b). Note that the optimum observation plane is displaced to a z position given by A½¼ fð1=2Þzag=DE@Bð1 þ b2 Þ, compared to A ¼ 0 (i.e., z ¼ 0) for the aberration-free system. Here, the least confusion disc in the geometric optics with b ¼ 0 is formed at A ¼ @1:5B (i.e., z ¼ @0:75Cs a2 ) [8,9]. 4.3. Response function tðn Þ The tðn Þ curves for several b values on the aberration-free system are shown in Fig. 8(a). As b ¼ 0-1, the tðn Þ curve is gradually lowered in the low-n ð> 0Þ region and is raised in the high-n region. Rising of the tðnÞ curve in the high-n region improves the OTF resolution ROTF defined as D=n0:1 to satisfy
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161
Fig. 7. Normalized values of RRayleigh , ROTF , Dzint , and Yb plotted as a function of b.
tðn0:1 Þ ¼ 0:1 (Appendix A.3). The normalized ROTF values for several b values have already been plotted in Fig. 7. The resolution of ROTF improves steadily with the increase in b, but large values of b > 0:74 cause the tðn Þ values to reduce to less than 0.1 in the middle-n region. For a chromatic-aberration system of CFWHM ¼ 021 at intervals of 0.25, the calculated tp ðn Þ curves for b ¼ 0 and 0.6 are plotted in Fig. 8(b). It is observed that the chromatic aberrations effect on the tp ðn Þ curves is less for b ¼ 0:6 than for b ¼ 0. The reason is that electron waves of the energy distribution with a larger DVFWHM value converge in a larger depth of focus, but the profile of iðz; rÞ along the z-axis is more broad for a larger b value than for b ¼ 0, as shown in Fig. 6. As to the source size, Fig. 9 shows the curves of Sðn Þ vs. n for several dg =D values of the Gaussian source profile. When dg =D ¼ 0:172, for example, the ts ðn Þ value is reduced by 10% at n ¼ 1. 4.4. Information resolution Rinf and information depth of the focus Dzinf As a new approach to evaluate SEM images from the viewpoint of image information, we have proposed information resolution, Rinf , defined as a spatial length per bit in information-passing capacity (IPC) [7]. Its calculation procedures are briefly described in Appendix A.4. For the aberration-free system the curves of Rinf =D vs. b for several S=N values are plotted in Fig. 10. Here, all these calculations are carried out under a constant power of the electron waves transmitted through the aperture. The dependence on b of the Rinf values (for a typical S=N of 10) is compared with those of the RRayleigh and ROTF values in Fig. 7. The increase in b toward 1 decreases (or improves) both the RRayleigh and the ROTF values, but does increase (or deteriorates) the Rinf value. The reason is that the Rinf value is defined as an average diameter of the circle per bit in information-passing capacity (IPC), not as the ultimate resolution as obtained from the high frequency nu to satisfy Gðnu Þ ¼ 1 in Eq. (A.12) in Appendix A.4. The Rinf value varies with the z-position of the observation plane. Like the definition of Dzint , the information depth of focus Dzinf is defined as the z-range satisfying the condition of Rinf ðzÞ=Rinf;min p1.1. Typical curves of Rinf =D vs. za=D are shown in Figs. 11(1a), (2a) and (3a) for optical systems of (b; BÞ ¼ ð0; 0Þ, (0.5, 0) and (0.5, 0.5), respectively. The corresponding normalized curves are shown in Figs. 11(1b), (2b) and (3b), respectively. All figures show that the Dzint value increases with the increase in
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Fig. 8. tðn Þ curves for several b values: (a) the aberration-free system, and (b) the chromatic-aberration system of CFWHM ¼ 021 at intervals of 0.25.
Fig. 9. Curves of Sðn Þ vs. n for several dg =D values of the Gaussian source profile.
S=N. The values of (a=DÞDzinf at S=N ¼ 10 for the corresponding systems are 0.83, 1.0, and 1.0, respectively. Then, it is found that the spherical aberrations of Bðp1Þ hardly affect the Dzinf values. The normalized Dzinf values at S=N ¼ 10 for the aberration-free system are also plotted in Fig. 7 as a function
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163
Fig. 10. Curves of Rinf =D vs. b for several S=N values for the aberration-free systems.
of b. It is shown that the Dzinf value increases with the increase in b, of which b-dependency is rather similar to that for the Dzint value. With a point source of Gaussian energy distribution, geometric trajectories of the electrons emitted from the source focus on the z-axis with the Gaussian distribution Nðz; sz Þ expressed by Eq. (17), where sz ¼ Cc ðsn =VÞ. Like the definition of Dzint , i.e., the z-range satisfying ib ðz; 0Þ=ib ð0; 0Þ X0.8 for the aberration-free optical system, we introduce a geometrical depth of focus Dzn defined as the Dz-range satisfying Nð0:5Dz; sz Þ=Nð0; sz Þ ½¼ expf@ð1=2Þð0:5Dz=sz Þ2 gX0:8. This results in Dzn ¼ 1:34sz ð¼ 1:14 CFWHM D=aÞ. Thus, the intensity or information depth of focus Dzj (j ¼ int or inf ) for sv a0 is approximated to (Dz2j;s¼0 þ Dz2n Þ1=2 , where Dzj;s¼0 is the Dzj value for sn ¼ 0. The Dzint form, for example, are expressed by 1:02ðD=aÞf1=ð1@b2 Þ2 þ ð1:12 CFWHM Þ2 g1=2 . This expression shows that the CFWHM parameter heavily affects the Dzint value for the system with a smaller b.
5. SEM system using a combination of circular and annular apertures Let us consider an SEM system using a combination of circular and annular apertures to improve the image quality. The system is simplified as aberration-free and the apertures are LCP (b ¼ 0) and ANP (b ¼ 0:71) as already shown in Fig 1 (a). The corresponding tðn Þ curves shown in Fig. 1(c) cross each other at the point Pðn ¼ nc Þ, associating with tLCP ðn ÞXtANP ðn Þ at n pnc and vice versa at n Xnc . This crossing predicts that a combination of the low-n -pass filtered image taken with the LCP-SEM and the high-n -pass filtered image taken with the ANP-SEM provides a good quality image just taken with an imaginary ‘‘LCP a ANP’’-SEM. The tðn Þ curve for ‘‘LCP a ANP’’ is also shown in Fig. 1(c). Here, remember that the electron wave power transmitted through ANP is ð1@b2 Þ times weaker than LCP. To equalize the tLCP ðnc Þ value with the tANP ðnc Þ one, the electron wave power irradiating ANP should be 1=ð12b2 Þ times higher than LCP, or the electron irradiating time on the sample for ANP should be 1=ð12b2 Þ times longer than LCP so as to result in no difference in the electron dose between both images. Another way is a computer image processing, which enhances high-n ðn Xnc Þ components of the Fourier-transformed ANP-image so as to satisfy tANP ðnc Þ ¼ tLCP ðnc Þ but does not directly improve the difference in SN ratio at n ¼ nc between the images for ANP and LCP. Note that on practical applications the image quality is strongly dependent on the specimen structure expressed with ts ðn Þ in Eq. (A.12). Let us move on an aberration-laden SEM system. Table 2 describes typical beam conditions and beam optical parameters set with small and large apertures, of which radii ra are a and 1.4a to take the beam semi-
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Fig. 11. (a) Curves of Rinf =D vs. za=D for the optical systems with (b; B)=(0, 0), (0.5, 0), and (0.5, 0.5), and (b) their normalized curves.
angle a as 10.8 and 15.12 mrad, respectively. Figs. 12(a) and (b) show the tðn Þ curves for the small and large apertures, respectively, with those for the corresponding center-obstructed aperture (i.e., annular aperture) of b ¼ 0:6. The curves for the case of B ¼ dg =D ¼ 0 are also plotted in the same figures to look at the CFWHM ’s contribution to them. It is observed that the CFWHM ’s contribution at b ¼ 0 is more than at b ¼ 0:6 as predicted. Here, the n -axis in Fig. 12(b) is scaled as same as that in Fig. 12(a), and so the former axis is 2.8 (=2 1.4) at maximum. The curves for the small-circular and large-annular apertures are plotted again in Fig. 12(c). These figures supports that the ‘‘LCP a ANP’’-SEM has a potential to provide good quality images for the combination of small LCP and large ANP as well as the combination of both small LCP and ANP. The b value should be appropriately determined in consideration of the reduction and enhancement of the tðn Þ curve in the corresponding low and high-n regions, the decrease of electron wave power, and the resultant image quality.
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Fig. 12. tðn Þ curves for apertures of b ¼ 0 and 0.6 on the aberration-laden SEM system with (B, CFWHM , dg =D)=(0.091, 0.639, 0.076) together with those of which one or two parameters are set to zero: (a) Aperture radius ra =a; (b) Aperture radius ra=1.4a; (c) Aperture radius ra=a and 1.4a.
On the practical use of ANP, problems might arise from the factors such as supporters for the central stop of ANP, deviation from the symmetrical shape of ANP, asymmetric charging of the aperture or the sample, stigmatic fields of the system. The reason is that asymmetricness or stigmatism of these factors deteriorates the interfered profiles of interest. Unwanted electron-wave scattering from the aperture’s edge also weaken the interfered profiles. Highly manufacturing and operating techniques might be required.
6. Conclusion It has been known in light optics that obstructing the central part of the aperture (i.e., using annular aperture; ANP) narrows the principal maxima (i.e., main lobe) of the system’s point spread function, and
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Table 2 Optical characteristics of a typical aberration-laden SEM system Aperture radius ra Accelerating voltage Vo [kV] Virtual source size r [nm] Energy spread DVFWHM [V] Lens magnification M Cc [mm] Cs [mm] Beam semi-angle a [mrad] l [nm] D {=l/a} [nm] CFWHM {=(1/2)Cc(DVFWHM/Vo)a/D} B {=(1/4)Csa3/D} dg/D dd {=0.61D} [nm] dg {=Mr} [nm] dc {=(1/2)Cc(DVFWHM/Vo)a} [nm] ds {=(1/4)Csa3} [nm] d¼ ðdd2 þ dg2 þ dc2 þ ds2 Þ1=2 ½nm
1.4a
a 0.8 15 0.4 0.0204 0.95 1.16 10.8 0.0433 4.01 0.639 0.091 0.076
15.12 0.0433 2.87 1.253 0.350 0.107
2.45 0.31 2.57 0.37 3.58
1.75 0.31 3.59 1.00 4.13
thus enhances the resolution in the Rayleigh sense. However, this effect is accompanied by the followings; (1) a decrease of the brightness, since there is an increasing energy loss of light in the image, and (2) a reduced contrast in the low-n region due to a considerable rise in the level of the side lobes of the point spread function, since a large amount of light is diffracted from its proper geometrical position. The present paper has made clear that the results obtained in light optics are analogically applicable to electron optics. The beam characteristics on SEM system have been discussed from both viewpoints of three kinds of beam resolutions (RRayleigh , ROTF , and Rinf ) and three kinds of depths of the focus (DzRayleigh , DzOTF , and Dzinf ). When only the beam profile approximately determines the SEM image resolution, the beam resolution is equal to the SEM image resolution. Typical features of the beam characteristics are as follows: (1) All kinds of the depth of focus (i.e., DzRayleigh , DzOTF , and Dzinf ) are improved with the increase in b. The beam resolutions of ROTF and RRayleigh also are improved, being in contrast with the information resolution Rinf . Note that the RRayleigh value represents less specifically the beam resolution as the b value approaches to 1, since the magnitude of successive maxima (i.e., side lobes) of the intensity profile ib ðz; rÞ becomes more pronounced. Repeatedly speaking, the OTF resolution ROTF is certainly improved by using ANP with its proper obstruction-ratio b. (2) The conventional SEM with a circular-aperture provides absolutely a good-contrast image in the wide n-range. Besides, obstructing the central part of the aperture (i.e., using ANP) enhances high-n structures of the image contrast at the cost of low-n structures, and improves the ultimate image resolution (in RRayleigh and ROTF , not in Rinf ). Here, the b value should be appropriately determined by balancing the improvements with the reduction of electron wave power transmitted through the ANP. (3) On the SEM system with both a circular aperture and an annular aperture, a combination of the lown-pass filtered, circular-aperture SEM image with the high-n-pass filtered, annular-aperture SEM image has a potential to improve the image quality in both terms of any image resolution (i.e., Rconv , Rint , or Rinf ) and any depth of the focus (i.e., Dzconv , Dzint , or Dzinf ). Here, since the ANP reduces the transmitted electron wave power by (12b2 ) times, the electron wave power irradiating the ANP or the electron irradiating time on the sample for the ANP should be 1/(12b2 ) times magnified than the circular one so as to result in no
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difference in the electron dose between both images. Another way is the computer image processing (see the above section). All discussions are analogically applicable also for a scanning transmission electron microscope (STEM). On the practical use of ANP, problems might arise from the factors such as supporters for the central stop of ANP, deviation from the symmetrical shape of ANP, asymmetric charging of the aperture or the sample, stigmatic fields of the system. The asymmetricness or stigmatism deteriorates the interfered profiles of interest. Unwanted electron-wave scattering from the aperture’s edge also weaken the interfered profiles. Although the results are incomplete and imperfect, they are sufficient to show that ANP is useful to improve the beam characteristics such as resolution and depth of the focus.
Appendix A. Resolution and depth of the focus A.1. Conventional resolution (Rconv ) and conventional depth of the focus (Dzconv ) The conventional beam diameter dðERconv ; the conventional image resolution) is calculated by adding the diffraction image size dd to the Gaussian image size dg , the chromatic disk size dc , and the spherical disk sizes ds each quadratically as follows: d ¼ ðdg2 þ dd2 þ dc2 þ ds2 Þ1=2 ;
ðA:1Þ
where dg ¼ Mrs ; dd ¼ 0:61D
ðA:2Þ with
D ¼ l=a;
ðA:3Þ
dc ¼ kc Cc ðDVFWHM =VÞa;
ðA:4Þ
ds ¼ ks ð1=2ÞCs a3 ;
ðA:5Þ
l½nm ¼ 1:226=V½volts1=2 :
ðA:6Þ
Here, M is the lens magnification, rs the virtual source size, Cc and Cs the chromatic- and the spherical aberration coefficients (defined on the image side), respectively, a the beam focusing semi-angle, DVFWHM the effective voltage spread of the electrons, V the voltage of the beam at the image plane. Here, the potential origin is referred to the source. Both fitting parameters of kc and ks are constants, typically assumed to be 0.5 [6]. The beam depth of focus is simply given by Dzconv ¼ d=a;
ðA:7Þ
where d is the beam diameter and a is the focusing semi-angle at the sample (see Fig. 13). These conventional parameters of Rconv and Dzconv are applicable only for the systems with a circular aperture. A.2. Rayleigh resolution (RRayleigh ) According to Rayleigh’s criterion, two images of equal intensity are regarded as just resolved when the principal intensity maximum of one coincides with the first intensity minimum of the other, as shown in Fig. 14. For a diffraction-limited system with a circular aperture, the limit of resolution RRayleigh (referred to
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Fig. 13. Conventional beam-depth of the focus.
Fig. 14. Overlapped Airy patterns showing the Rayleigh’s criterion.
as the Rayleigh resolution) is equal to the Airy disc radius given by RRayleigh ¼ 0:61Dð¼ rAiry : Airy disc radiusÞ:
ðA:8Þ
We assume that Rayleigh’s criterion is applicable also to the systems with an annular aperture of the obstruction ratio of b (see Fig. 1(b)). As the b value increases, however, the RRayleigh value represents less specifically the image resolution because of highly increased successive maxima of the intensity profile ib ð0; rÞ as shown in Fig. 4. A.3. OTF resolution (ROTF ) The optical transfer function (OTF) t(n) is an image-quality measure that reflects an optical image amplitude response to sinusoidal patterns. Here, tðnÞ can be given either by a Fourier transform of the beam current distribution, or by autocorrelation of the pupil function PðnÞ given from Eq. (10). The image resolution is conventionally defined as the reciprocal of the spatial frequency at which tðnÞ has fallen to 0.1.
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Table 3 Information contents (per pD2 ) Hs and normalized information resolutions Rinf =D for various S=N values on the aberration-free system S/N
1
2
5
10
20
50
100
HS [bits] Rinf =D
2.79 1.20
4.99 0.895
9.89 0.636
15.46 0.509
22.81 0.419
35.23 0.337
46.53 0.293
This resolution is referred to as an OTF resolution, ROTF in the present study. OTF ¼
* 1=n0:1 ð¼ D=n0:1
with
n * ¼ Dn
and
D ¼ l=aÞ:
ðA:9Þ
This resolution is no more appropriate when the large spherical aberrations provide two or more n0:1 solutions for tðn0:1 Þ ¼ 0:1. A.4. Information resolution (Rinf ) and information depth of the focus (Dzinf ) Combining the image resolution with the image quality, the authors (T.I. and M.S.) have proposed the information resolution Rinf , which is defined as an average diameter of the circle per bit in informationpassing capacity (IPC) [7] Rinf ¼ 2D=Hs1=2 ¼ 2=ðprsÞ1=2 ; where H s ¼ p2
Z
ðA:10Þ
2
Gðn * Þn * dv * ;
ðA:11Þ
0
Gðn * Þ ¼ log2 ½1 þ jtðn * Þts ðn * Þj2 ðS=NÞ
with
n * ¼ Dn;
ðA:12Þ
rs ¼ Hs =ðpD2 Þ;
ðA:13Þ
S and N are the statistical mean power densities of the signal and of the noise, respectively; tðn Þ the OTF function of the optical system including the source size effect, and tðn Þ the specimen contrast, i.e., the Fourier transform of the specimen structure, which is assumed to be 1 in this study. Here, Hs is the mean information content of the optical image per basic area (defined by pD2 ), rs the IPC density. Table 3 shows the values of Hs and Rinf =D for various S=N values on the aberration-free system. Using the information resolution Rinf , we have introduced an information depth of focus (Dzinf ) defined as the z-range satisfying Rinf ðzÞ=Rinf;min p1:1 A.5. Intensity depth of the focus (Dzint ) Using the beam intensity on the optical axis (z-axis) iðzÞ, we define the intensity depth of the focus as the z-region satisfying the condition of iðzÞ=imax X 0.8. The Dzint value of 1.02 D=a is given for an aberrationfree optics (see text).
References [1] M. Born, E. Wolf, Principles of Optic, 7th Edition, Cambridge University Press, Cambridge, 1999, p. 461. [2] H. Kubota, Wave Optics, Iwanami Shoten, Tokyo, 1971, p. 278. (in Japanese).
170 [3] [4] [5] [6] [7] [8] [9]
T. Ishitani et al. / Ultramicroscopy 88 (2001) 151–170 V.N. Mahajan, Aberration Theory Made Simple, SPIE Optical Engineering Press, Washington, 1991 (Chapter 9). Z. Shao, A.V. Crewe, Ultramicroscopy 23 (1987) 169. M. Sato, J. Vac. Sci. Technol. B 9 (1991) 2972. M. Sato, Resolution, in: J. Orloff (Ed.), Handbook of Charged Particle Optics, CRC Press, New York, 1997 (Chapter 8). T. Ishitani, M. Sato, Ultramicroscopy 84 (2000) 199. P. W. Hawkes, E. Kasper, Principles of Electron Optics, Vol. 1, Academic Press, London, 1996, p. 350. T. Ishitani, J. Electron Microsc. 48 (1999) 617.
Beam characteristics for various sizes of annular aperture on scanning electron microscope Tohru Ishitani*, Mitsugu Sato, Hideo Todokoro Instruments, Hitachi Ltd., 882 Ichige, Hitachinaka-shi, Ibaraki-ken 312-8504, Japan Received 6 November 2000; received in revised form 2 March 2001
Abstract Using an analogy between light optics and electron optics, we have calculated beam characteristics such as the beam profile and the optical transfer function for several sizes of annular and circular apertures on a scanning electron microscope (SEM). It has been found that an annular aperture improves the image quality with regard to several kinds of image resolution and the depth of focus at the price of good low-frequency (n) contrast. In contrast with conventional circular-aperture SEM images, a combination of a low-n-pass filtered, circular-aperture SEM image with a high-n-pass filtered, annular-aperture SEM image has the potential to enhance the image quality in terms of both the image resolution and the depth of focus. r 2001 Elsevier Science B.V. All rights reserved. PACS: 07.78.+s; 42.15; 42.25; 42.30 Keywords: Scanning electron microscope; Depth of the focus; Resolution; Annular aperture
1. Introduction Semiconductor integrated devices advance continuously, whereby critical-dimensions are shrinking and aspect-ratio features are standing out. Here, the demands for a scanning electron microscope (SEM) with a higher image resolution and a greater depth of focus are increasing in the fields of device metrology and device-defect analysis. With regard to the SEM optical systems, much effort has been directed to minimize the beam spot by optimizing the size of the beam-limiting aperture so that the effect of lens aberrations can be reduced. So far as we know, the apertures to date are all circular. In contrast, it has been known in light optics that obstructing the central part of the aperture (i.e., using annular aperture) narrows the principal maxima (i.e., main lobe) of the system’s point spread function, and thus enhances the resolution in the Rayleigh sense [1–3]. However, this effect is accompanied by the following: (1) a decrease of the brightness, since there is an increasing loss of light energy in the image, and (2) a reduced contrast in the low-frequency (n) region due to a considerable rise in the level of the side lobes of the point spread function, since a large amount of light is diffracted from its proper geometrical position. *Corresponding author. Tel.: +81-29-276-6353; fax: +81-29-276-6354. E-mail address: [email protected] (T. Ishitani). 0304-3991/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 1 ) 0 0 0 8 4 - 5
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The present paper makes clear how the results obtained in light optics are analogically applicable to electron optics. The beam characteristics on SEM system with various sizes of the annular aperture are calculated to discuss both the beam resolution and the depth of focus as well as their beam intensity profiles and the corresponding optical transfer functions (OTF). Besides, various types of both the beam resolution and the depth of focus are considered to better understand the beam characteristics. It is shown that a combination of a low-n-pass filtered, circular-aperture SEM image with a high-n-pass filtered, annular-aperture SEM image has a potential to enhance the image quality both in terms of the image resolution and the depth of focus. Here, it is implicitly assumed that only the beam profile determines both the image resolution and the depth of focus, and so the beam resolution is equal to the SEM image resolution.
2. Principal features of the electron beam focused through circular and annular apertures To elucidate the principal features of the electron beam focused through circular and annular apertures (or pupils), we consider a very simple optical system, i.e., an aberration-free optical system only subject to diffraction. Fig. 1(a) shows three kinds of pupils, i.e., large and small circular pupils (LCP and SCP) of radius a and a=21=2 (E0:71a), respectively, and an annular pupil (ANP) having inner and outer radii of ba and a, respectively, with b ¼ 0:71. Here, b is called the obstruction ratio and is some positive number less than unity. The opening area of SCP is equal to that of ANP, and each of them is one half that of LCP. Figs. 1(b 1 and 2) show beam intensity profiles observed on the Gaussian plane (z ¼ 0) in the systems (with LCP, SCP, and ANP) and their normalized profiles, respectively. The corresponding OTF curves tðn Þ are shown in Fig. 1(c) (see Appendix A.3). The ið0; rÞ profiles for circular apertures (LCP and SCP) are known as the Airy pattern, which has the first zero at 2r=DLCP E1:22 and 2r=DSCP E1:22, respectively. Here, DLCP ð DÞ ¼ l=a and DSCP ¼ DLCP =0:71;
ð1aÞ
l½nm ¼ 1:226=V½volts1=2 ;
ð1bÞ
n * ¼ nD:
ð1cÞ
l is the wavelength of the electron waves, a is the focusing beam semi-angle for LCP, and V is the electron accelerating potential on the observation plane. According to the Rayleigh criterion (refer to Appendix A.2), the beam resolution (referred to as the Rayleigh resolution RRayleigh in the present study) is equal to the Airy disc radius rAiry ð¼ 0:61DÞ. On the ið0; rÞ profile for ANP, the profile of principal maxima is relatively finer than that for LCP, although the z-direction axial intensity ið0; 0Þ is one quarter that of LCP. This result suggests that the SEM image resolution can be improved by ANP. This improvement is indicated also by the tðn Þ curves, which show the value of n0:1 satisfying tðn0:1 Þ ¼ 0:1 as 1.61 for LCP and 1.75 for ANP. Here, the value of D=n0:1 is the OTF resolution ROTF ð¼ 1=nÞ (refer to Appendix A.3). The tðn Þ curve for LCP a ANP is also plotted in Fig. 1(c) for later discussion. Fig. 1(d) shows normalized z-axial intensity profiles iðz; 0Þ in the system with LCP, SCP, and ANP. Using these profiles we obtain each intensity depth of the focus Dzint , which is defined as the z-range satisfying iðz; 0Þ=imax X0:8 (see Appendix A.5). It is observed that the Dzint value for ANP is larger than that for LCP. Several other definitions on both the beam resolution R and the depth of focus Dz are given in Appendix A. Table 1 summarizes those definitions and the corresponding values obtained in the aberration-free systems. In the following sections calculation procedures for obtaining iðz; rÞ and tðn Þ in a
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Fig. 1. Comparison of the beam characteristics of aberration-free systems among three types of apertures: (a) the large-circular aperture (LCP), the small-circular aperture (SCP), and the annular aperture (ANP) of b ¼ 1=21=2 ðE0:71Þ, (b 1) beam intensity profiles observed on the Gaussian plane (z ¼ 0) in the systems with LCP, SCP, and ANP, (b 2) the corresponding normalized curves of the beam intensity profiles shown in figure (b 1) (c) the corresponding OTF curves tðn Þ, and (d) the normalized z-axial intensity profiles iðz; 0Þ in these systems (see Appendix A.3).
realistic SEM system with aberrations and with a finite electron source are described, the dependency of the beam resolution and the depth of focus on b is derived from the calculations and plotted, and potential advantages of an SEM system with the circular/annular apertures are discussed.
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Table 1 Several definitions of the image resolution and the depth of focus in the SEM systems Type
Symbol
Definition
Aberr.-free system (circular aperture)
Resolution 1. Conventional 2. Rayleigh 3. OTFa 4. Information
Rconv RRayleigh ROTF Rinf
Beam diameter r value corresponding to the first-zero of ib ðrÞ D=n0:1 value satisfying tðn0:1 Þ ¼ 0:1 Spatial length per bit in IPCb
d 0.61D (Airy disc) 0.62D at n0:1 ¼ 1:61 0.51D at S=N ¼ 10
Depth of the focus 1. Conventional 2. Intensity 3. Information
Dzconv Dzint Dzinf
z-range satisfying daberr:@free ðzÞpdmin z-range satisfying iðzÞ=imax X0:8 z-range satisfying Rinf ðzÞ=Rinf;min p1:1
d=aðE0:61D=aÞ 1:02D=a 0:84D=a at S=N ¼ 10
a b
1 is the optical transfer function. 2 is the information-passing capacity.
Fig. 2. Aberration-laden optical imaging system with an annular exit aperture of the obstruction ratio b.
3. Theoretical expressions using wave optics Fig. 2 shows an optical imaging system with aberrations and with an annular exit aperture (or pupil) of obstruction ratio b. The system is rotationally symmetric and its optical axis and radial distance are taken as the z and r coordinate axes, respectively. The z origin is referred to as the Gaussian image plane. An aperture of the objective lens contributes to the exit pupil. An intensity distribution of electron wave in the image of a point electron-source called the point-spread function (PSF) of the optical system has been already calculated for a circular aperture in other studies [4–7]. Extending these approaches to the annular aperture, we obtain the PSF at point Qðz; rÞ in an observation plane near the Gaussian plane as ib ðz; rÞ ¼ Gjub ðz; rÞj2 ;
ð2Þ
where ub ðz; rÞ is the dimensionless wave’s complex amplitude expressed by ub ðz; rÞ ¼ C term þ iS term; C term ¼ 2
Z
ð3Þ
1
J0 ð2pru=DÞu cos½2pðAu2 þ Cu2 þ Bu4 Þdu;
ð4aÞ
J0 ð2pru=DÞu sin½2pðAu2 þ Cu2 þ Bu4 Þdu;
ð4bÞ
b
S term ¼ 2
Z
1 b
T. Ishitani et al. / Ultramicroscopy 88 (2001) 151–170
A ¼ ð1=2Þza=D; B ¼ ð1=4ÞCs a3 =D; C ¼ ð1=2ÞCc ðDV=VÞa=D; G ¼ jp s2p =ðlLp
op Þ
2
155
ð5Þ ð6Þ
;
J0 is the first kind of Bessel function, Cc and Cs the chromatic- and the spherical-aberration coefficients (defined on the image side), respectively; a the focusing semi-angle of the outer beam, kð¼ 2p=lÞ the wave number of the electron wave, jp the power density of the electron wave irradiating on the pupil, sp ¼ pa2 the area of the pupil of b ¼ 0, Lp op the distance from the pupil to the observation plane (Lp op ba and r), DV the voltage spread of the electrons, and V the accelerating potential of the beam on the observation plane. Here, the potential origin is referred to as the electron source. Note that when jp is constant, the value of ib ð0; 0Þ is proportional to a4 , not a2 . This relation supports iSCP ð0; 0Þ=iLCP ð0; 0Þ ¼ 1=4 as shown in Fig. 1(b 1). To simplify the calculation, the term G is set to 1. The variables of B and C given in Eq. (5) are used in the latter discussion to express the spherical and chromatic aberrations, respectively. A frequency response function tðnÞ for an incoherent optical system is obtained by calculating either the Fourier transform of ib ðz; rÞ with subsequent normalization of tð0Þ ¼ 1 or the autocorrelation of the pupil function. Using the dimensionless pupil function PðuÞ with u2 ¼ x2 þ Z2 and n ¼ nD, the latter calculation yields Z Z % * Þ2 þ Z2 g1=2 Þ dx dZ; Pðfðx þ n * Þ2 þ Z2 g1=2 Þ Pðfðx@n ð7Þ tðn * Þ ¼ ð1=Yb Þ s
where PðuÞ ¼ Sd ðuÞ exp ½2piðAu2 þ Cu2 þ Bu4 Þ; Sd ðuÞ ¼
at bpmp1; at 0pmob;
ð8Þ ð9Þ
Yb ð¼ 1@b2 Þ is a fraction of the opening area of the annular pupil, which corresponds to a transmissivity of the electron wave power, and 1=Yb in Eq. (7) works as the normalization factor leading to tð0Þ ¼ 1. The % expression of PðuÞ denotes the complex conjugate of PðuÞ. The integration of Eq. (7) is carried out only in the hatched regions as shown in Fig. 3. For the aberration-free optical system (i.e., B ¼ C ¼ 0) with a circular aperture (b ¼ 0), the tðn Þ function on the Gaussian plane is simply given by tðn * Þ ¼ ð1=pÞð2y@sin 2yÞ;
ð10Þ
where y ¼ cos@1 ðn * =2Þ with 0pn * p2:
Fig. 3. Domain for the integration of Eq. (7).
ð11Þ
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The tðn Þ curves for various b values are shown in the later Fig. 8(a). The maximum n value, which corresponds to the cut-off frequency, is 2. When l and Lp op are constant, the cut-off frequency is higher as the aperture size a (ELp op a) is larger . For an optical system with a finite-size and spatially incoherent source, its response function ts ðn Þ is modified as ts ðn * Þ ¼ Sðn * Þtðn * Þ:
ð12Þ
Here, Sðn Þ is the Fourier transform function of a source’s image intensity profile sðrÞ. When the sðrÞ profile is a Gaussian type with a standard deviation s as sðrÞ ¼ ½1=fð2pÞ1=2 sgexp ½@ð1=2Þðr=sÞ2 ;
ð13Þ
the source size dg defined as a full-width at half-maximum (FWHM) yields dg ¼ 2ð2 ln 2Þ1=2 s½E2:35s. The Sðn Þ function for the Gaussian sðrÞ profile yields % * Þ2 with Sðn * Þ ¼ exp½@2ðpsn
s% ¼ s=D
and
n * ¼ nD:
ð14Þ
Last, consider a source with an energy distribution NðDVÞ. Since the electron with its energy deviation DV focuses geometrically at z ¼ Cc ðDV=VÞ, i.e., A ¼ @C, the beam current distribution !ıp ðz; rÞ and the frequency response function tp ðn Þ for the source with the NðDVÞ distribution are obtained by integrating the NðDVÞ-weighted functions of ib ðz; rÞ and tðz; n Þ with respect to DV as Z þN !ıb;p ðz; rÞ ¼ NðDVÞib ðzðDVÞ; rÞdDV; ð15Þ @N
tp ðn * Þ ¼
Z
þN
NðDVÞtðzðDVÞ; n * ÞdDV:
ð16Þ
@N
Here, tðz; n Þ is the z-dependent function of tðn Þ obtained from Eq. (7). As to the NðDVÞ distribution, the Gaussian distribution is typically assumed here. NðDV; sv Þ ¼ ½1=fð2pÞ1=2 s g exp ½@ð1=2ÞðDV=sv Þ2 :
ð17Þ
The energy spread (FWHM) is expressed by the form DVFWHM ½¼ 2ð2 ln 2Þ1=2 sn E2:35sn . The C value (defined in Eq. (5)) at DV ¼ DVFWHM is expressed as CFWHM for later discussion.
4. Calculations and discussion 4.1. Aberration-free optical system ðB ¼ C ¼ 0Þ For an aberration-free optical system with a point source, the wave amplitude on the Gaussian imageplane (i.e., z ¼ 0) is calculated from Eq. (3) as [1–3]
ub ð0; rÞ ¼ 2
Z 0
1
J0 ð2pru=DÞu du@2
Z
b
J0 ð2pru=DÞudu;
0
¼ f2J1 ð2pr=DÞ=ð2pr=DÞg@b2 f2J1 ð2pbr=DÞ=ð2pbr=DÞg:
ð18Þ
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Then, the wave intensity is expressed as ib ð0; rÞ ¼ ½f2J1 ð2pr=DÞ=ð2pr=DÞg@b2 f2J1 ð2pbr=DÞ=ð2pbr=DÞg2 :
ð19Þ
Especially, the central intensity at r ¼ 0 is given by ib ð0; 0Þ ¼ ð1@b2 Þ2 :
ð20Þ
The curves of ib ð0; rÞ vs. r and their normalized curves for several b values together with an inset of enlarged portions are shown in Figs. 4 (a) and (b), respectively. Note that, if the electron irradiance of the aperture is held constant, the total power through the aperture decreases as ð1@b2 Þ½¼ Yb and the central maximum intensity decreases as ð1@b2 Þ2 with the increase in b. The minima (zeros) of the intensity have a value of zero at r-values given by J1 ð2pr=DÞ ¼ bJ1 ð2pbr=DÞ
with
ra0;
ð21Þ
while the maxima occur at r values given by J2 ð2pr=DÞ ¼ b2 J2 ð2pbr=DÞ with
ra0:
ð22Þ
Fig. 4. (a) Curves of ib ð0; rÞ vs. r for several b values together with an inset of enlarged portions and (b) their normalized curves.
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It is apparent from Eq. (19) that as b-1, ib ð0; rÞ=ib ð0; 0Þ-J02 ð2pr=DÞ. Since the first zero of the J0 ðxÞ function lies at x ¼ 2:40, the first zero of the intensity for b ¼ 1 occurs at the value given by 2r=DðE2:40=pÞE0:76 compared with 2r=DE1:22 [first zero of J1 ð2pr=DÞ for b ¼ 0. The relative intensity values of the secondary maxima to its principal maximum at r ¼ 0 becomes higher as b increases. For example, when b ¼ 0:6, the first secondary maximum (at 2r=DE4:65=pE1:48) has a value of 12.0% of the principal maximum compared to a value of 1.75% (at 2r=DE5:14=pE1:63) for a circular The minima, maxima, and their corresponding intensities ib ð0; rÞ and encircled-powers aperture. Rr 0 2prib ð0; rÞdr are given for b ¼ 0@0:9 (at intervals of 0.1) in the textbook [3]. It is interesting to observe the intensity distribution at large values of r and b. The circular-aperture distribution consists of rings of maxima and minima, and the successive maxima decrease monotonically, whereas the annularaperture distributions consist of not only the maxima and minima but also of a periodic ring group structure. The distribution appears to be divided into ring groups. The details have been discussed in the textbook [3]. For an aberration-free SEM system with a circular aperture (b ¼ 0), the Rayleigh criterion indicates that the image resolution is equal to the r-value of 0:61Dð¼ rAiry Þ corresponding to the first-zero of the intensity profile iðrÞ (refer to Appendix A.2). Similarly, by an approximate application of the Rayleigh criterion to a system showing aberrations with a circular/annular aperture, the image resolution (referred to as the Rayleigh resolution RRayleigh in the present study) is obtained. It follows that as b ¼ 0-1, the RRayleigh value is improved from 0.61D to 0.38D. Note that as the b value increases, the RRayleigh value expresses less specifically the image resolution because of increasing successive maxima of the ib ðz; rÞ profile as seen in Fig. 4(b), or diminishing tðn Þ value in the low-n ð> 0Þ region as seen in the later Fig. 8(a). Next, let us calculate the intensity profile in the direction of the z-axis. The terms of C term and S term of Eqs. (4a) and (4b) are simplified as C term ¼ ð1=A * Þðsin A * @sin A * b2 Þ;
ð23aÞ
S term ¼ @ð1=A * Þðcos A * þ cos A * b2 Þ;
ð23bÞ
where A * ¼ 2pA ¼ pza2 =l ¼ pza=D:
ð24Þ
Introducing a new parameter w (in wave-phase) for representing A ð1@b2 Þ=2, we obtain the ib ðz; 0Þ form as ib ðz; 0Þ ¼ ð1@b2 Þ2 ½ðsin wÞ=w2 :
ð25Þ
Fig. 5 shows normalized curves of ib ðz; 0Þ vs. z for several b values on both the aberration-free system ðB ¼ C ¼ 0Þ and a typical spherical-aberration system of (B ¼ 0:5 and C ¼ 0). All curves for the aberration-free system have minima (i.e., zero intensities) at w ¼ np, where n is an integer. The first minima of n ¼ 71 correspond to az=D ¼ 72=ð1@b2 Þ, i.e., zE7ð3:3 rAiry =aÞ=ð1@b2 Þ. Note that, when b ¼ 0, the first minimum on the z-axis is 3:3=a times larger than that on the r-axis for the circular aperture. The portion of the wave quantity between the central maximum (at w ¼ 0) to the first minimum (at w ¼ p) in the direction of the z-axis is calculated from Z p 2 2 MðpÞ ¼ ð1@b Þ ½ðsin wÞ=w2 dw 0
¼ ð1@b2 Þ2
Z
0
2p
ðsin 2wÞ=ð2wÞdð2wÞ ¼ 1:433ð1@b2 Þ2
ð26Þ
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Fig. 5. Normalized curves of ib ðz; 0Þ vs. z for several b values; (a) an aberration-free system (B ¼ C ¼ 0) and (b) a typical sphericalaberration system of B ¼ 0:5 and C ¼ 0.
Since MðNÞ ¼ ðp=2Þð1@b2 Þ2 , the value of MðpÞ=MðNÞ approaches 1:433=ðp=2ÞE0:92, being independent of b. With the definition that the intensity depth of focus Dzint is the z-range satisfying the condition of ib ðz; 0Þ=ib ð0; 0ÞX0:8, the w-region is given by jwjp0:51p and the Dzint expression by Dzint ¼ 1:02ðD=aÞ=ð1@b2 Þ:
ð27Þ 2
Note that as b ¼ 0-1, the Dzint value increases as 1=ð1@b Þ. Here, the value of w½¼ A ð1@b2 Þ corresponds to the wave-phase difference between the outer and inner geometrical beams converging at the observation point Qðz; 0Þ. The Rayleigh’s l=4 rule [3] discloses that a quarter-wave of primary spherical aberration reduces the intensity at the Gaussian image point by 20%; in other words, the Strehl ratio for this aberration is 0.8. (Here, the Strehl ratio of an image or a system is defined as the ratio of the intensities at the center of image in a plane with and without the aberration.) A variant of this definition is that an aberration-laden or aberration-free wave front, which lies between two concentric spheres that are spaced a quarter-wave apart, will give a Strehl ratio of approximately 0.8. We see that the region of jwjp0:51p mentioned above obeys this variant of Rayleigh’s l=4 rule. Typical isometric ib ðz; rÞ figures at b ¼ 0 and 0.6 in an aberration-free system are plotted in Figs. 6(a) and (b), respectively. Lengths of the arrows on the z-axis and the Gaussian plane correspond to RRayleigh and Dzint , respectively. As b ¼ 0-0:6, we see that Dzint ða=DÞ ¼ 1:0-1:6 and 2RRayleigh =D ¼ 1:22-0:95. In Fig. 6 (a) for the circular aperture, the arrows 1 and 10 correspond to geometric trajectories of most-outer incoming and outgoing beams, respectively. In Fig. 6 (b) for the annular aperture, on the other hand, the incoming beams go straight between the arrows 1 and 2, while the outgoing trajectories do between the arrows 10 and 20 . The isometric ib ðz; rÞ figures cannot be reduced from geometric beam trajectories. To see the dependency on b of RRayleigh , Dzint , and Yb values, their values normalized at b ¼ 0 are plotted as a
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Fig. 6. Isometric ib ðz; rÞ figures for the aberration-free system: (a) the circular aperture (b ¼ 0) and (b) the annular aperture (b ¼ 0:6). Lengths of the arrows on the Gaussian plane and the z-axis correspond to RRayleigh and Dzint , respectively. The arrows 1 and 10 correspond to geometric trajectories of most-outer incoming and outgoing beams, respectively.
function of b in Fig. 7 together with the values of response resolution ROTF , information resolution Rinf and information depth of the focus Dzinf (at the signal-to-noise ratio S=N ¼ 10), which are discussed later. 4.2. Spherical aberration optical system ðBa0; C ¼ 0Þ As a typical example of the optical systems having spherical aberrations, the normalized ib ðz; 0Þ profiles for B ¼ 0:5 are shown for several b values in Fig. 5 (b). Note that the optimum observation plane is displaced to a z position given by A½¼ fð1=2Þzag=DE@Bð1 þ b2 Þ, compared to A ¼ 0 (i.e., z ¼ 0) for the aberration-free system. Here, the least confusion disc in the geometric optics with b ¼ 0 is formed at A ¼ @1:5B (i.e., z ¼ @0:75Cs a2 ) [8,9]. 4.3. Response function tðn Þ The tðn Þ curves for several b values on the aberration-free system are shown in Fig. 8(a). As b ¼ 0-1, the tðn Þ curve is gradually lowered in the low-n ð> 0Þ region and is raised in the high-n region. Rising of the tðnÞ curve in the high-n region improves the OTF resolution ROTF defined as D=n0:1 to satisfy
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Fig. 7. Normalized values of RRayleigh , ROTF , Dzint , and Yb plotted as a function of b.
tðn0:1 Þ ¼ 0:1 (Appendix A.3). The normalized ROTF values for several b values have already been plotted in Fig. 7. The resolution of ROTF improves steadily with the increase in b, but large values of b > 0:74 cause the tðn Þ values to reduce to less than 0.1 in the middle-n region. For a chromatic-aberration system of CFWHM ¼ 021 at intervals of 0.25, the calculated tp ðn Þ curves for b ¼ 0 and 0.6 are plotted in Fig. 8(b). It is observed that the chromatic aberrations effect on the tp ðn Þ curves is less for b ¼ 0:6 than for b ¼ 0. The reason is that electron waves of the energy distribution with a larger DVFWHM value converge in a larger depth of focus, but the profile of iðz; rÞ along the z-axis is more broad for a larger b value than for b ¼ 0, as shown in Fig. 6. As to the source size, Fig. 9 shows the curves of Sðn Þ vs. n for several dg =D values of the Gaussian source profile. When dg =D ¼ 0:172, for example, the ts ðn Þ value is reduced by 10% at n ¼ 1. 4.4. Information resolution Rinf and information depth of the focus Dzinf As a new approach to evaluate SEM images from the viewpoint of image information, we have proposed information resolution, Rinf , defined as a spatial length per bit in information-passing capacity (IPC) [7]. Its calculation procedures are briefly described in Appendix A.4. For the aberration-free system the curves of Rinf =D vs. b for several S=N values are plotted in Fig. 10. Here, all these calculations are carried out under a constant power of the electron waves transmitted through the aperture. The dependence on b of the Rinf values (for a typical S=N of 10) is compared with those of the RRayleigh and ROTF values in Fig. 7. The increase in b toward 1 decreases (or improves) both the RRayleigh and the ROTF values, but does increase (or deteriorates) the Rinf value. The reason is that the Rinf value is defined as an average diameter of the circle per bit in information-passing capacity (IPC), not as the ultimate resolution as obtained from the high frequency nu to satisfy Gðnu Þ ¼ 1 in Eq. (A.12) in Appendix A.4. The Rinf value varies with the z-position of the observation plane. Like the definition of Dzint , the information depth of focus Dzinf is defined as the z-range satisfying the condition of Rinf ðzÞ=Rinf;min p1.1. Typical curves of Rinf =D vs. za=D are shown in Figs. 11(1a), (2a) and (3a) for optical systems of (b; BÞ ¼ ð0; 0Þ, (0.5, 0) and (0.5, 0.5), respectively. The corresponding normalized curves are shown in Figs. 11(1b), (2b) and (3b), respectively. All figures show that the Dzint value increases with the increase in
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Fig. 8. tðn Þ curves for several b values: (a) the aberration-free system, and (b) the chromatic-aberration system of CFWHM ¼ 021 at intervals of 0.25.
Fig. 9. Curves of Sðn Þ vs. n for several dg =D values of the Gaussian source profile.
S=N. The values of (a=DÞDzinf at S=N ¼ 10 for the corresponding systems are 0.83, 1.0, and 1.0, respectively. Then, it is found that the spherical aberrations of Bðp1Þ hardly affect the Dzinf values. The normalized Dzinf values at S=N ¼ 10 for the aberration-free system are also plotted in Fig. 7 as a function
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Fig. 10. Curves of Rinf =D vs. b for several S=N values for the aberration-free systems.
of b. It is shown that the Dzinf value increases with the increase in b, of which b-dependency is rather similar to that for the Dzint value. With a point source of Gaussian energy distribution, geometric trajectories of the electrons emitted from the source focus on the z-axis with the Gaussian distribution Nðz; sz Þ expressed by Eq. (17), where sz ¼ Cc ðsn =VÞ. Like the definition of Dzint , i.e., the z-range satisfying ib ðz; 0Þ=ib ð0; 0Þ X0.8 for the aberration-free optical system, we introduce a geometrical depth of focus Dzn defined as the Dz-range satisfying Nð0:5Dz; sz Þ=Nð0; sz Þ ½¼ expf@ð1=2Þð0:5Dz=sz Þ2 gX0:8. This results in Dzn ¼ 1:34sz ð¼ 1:14 CFWHM D=aÞ. Thus, the intensity or information depth of focus Dzj (j ¼ int or inf ) for sv a0 is approximated to (Dz2j;s¼0 þ Dz2n Þ1=2 , where Dzj;s¼0 is the Dzj value for sn ¼ 0. The Dzint form, for example, are expressed by 1:02ðD=aÞf1=ð1@b2 Þ2 þ ð1:12 CFWHM Þ2 g1=2 . This expression shows that the CFWHM parameter heavily affects the Dzint value for the system with a smaller b.
5. SEM system using a combination of circular and annular apertures Let us consider an SEM system using a combination of circular and annular apertures to improve the image quality. The system is simplified as aberration-free and the apertures are LCP (b ¼ 0) and ANP (b ¼ 0:71) as already shown in Fig 1 (a). The corresponding tðn Þ curves shown in Fig. 1(c) cross each other at the point Pðn ¼ nc Þ, associating with tLCP ðn ÞXtANP ðn Þ at n pnc and vice versa at n Xnc . This crossing predicts that a combination of the low-n -pass filtered image taken with the LCP-SEM and the high-n -pass filtered image taken with the ANP-SEM provides a good quality image just taken with an imaginary ‘‘LCP a ANP’’-SEM. The tðn Þ curve for ‘‘LCP a ANP’’ is also shown in Fig. 1(c). Here, remember that the electron wave power transmitted through ANP is ð1@b2 Þ times weaker than LCP. To equalize the tLCP ðnc Þ value with the tANP ðnc Þ one, the electron wave power irradiating ANP should be 1=ð12b2 Þ times higher than LCP, or the electron irradiating time on the sample for ANP should be 1=ð12b2 Þ times longer than LCP so as to result in no difference in the electron dose between both images. Another way is a computer image processing, which enhances high-n ðn Xnc Þ components of the Fourier-transformed ANP-image so as to satisfy tANP ðnc Þ ¼ tLCP ðnc Þ but does not directly improve the difference in SN ratio at n ¼ nc between the images for ANP and LCP. Note that on practical applications the image quality is strongly dependent on the specimen structure expressed with ts ðn Þ in Eq. (A.12). Let us move on an aberration-laden SEM system. Table 2 describes typical beam conditions and beam optical parameters set with small and large apertures, of which radii ra are a and 1.4a to take the beam semi-
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Fig. 11. (a) Curves of Rinf =D vs. za=D for the optical systems with (b; B)=(0, 0), (0.5, 0), and (0.5, 0.5), and (b) their normalized curves.
angle a as 10.8 and 15.12 mrad, respectively. Figs. 12(a) and (b) show the tðn Þ curves for the small and large apertures, respectively, with those for the corresponding center-obstructed aperture (i.e., annular aperture) of b ¼ 0:6. The curves for the case of B ¼ dg =D ¼ 0 are also plotted in the same figures to look at the CFWHM ’s contribution to them. It is observed that the CFWHM ’s contribution at b ¼ 0 is more than at b ¼ 0:6 as predicted. Here, the n -axis in Fig. 12(b) is scaled as same as that in Fig. 12(a), and so the former axis is 2.8 (=2 1.4) at maximum. The curves for the small-circular and large-annular apertures are plotted again in Fig. 12(c). These figures supports that the ‘‘LCP a ANP’’-SEM has a potential to provide good quality images for the combination of small LCP and large ANP as well as the combination of both small LCP and ANP. The b value should be appropriately determined in consideration of the reduction and enhancement of the tðn Þ curve in the corresponding low and high-n regions, the decrease of electron wave power, and the resultant image quality.
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Fig. 12. tðn Þ curves for apertures of b ¼ 0 and 0.6 on the aberration-laden SEM system with (B, CFWHM , dg =D)=(0.091, 0.639, 0.076) together with those of which one or two parameters are set to zero: (a) Aperture radius ra =a; (b) Aperture radius ra=1.4a; (c) Aperture radius ra=a and 1.4a.
On the practical use of ANP, problems might arise from the factors such as supporters for the central stop of ANP, deviation from the symmetrical shape of ANP, asymmetric charging of the aperture or the sample, stigmatic fields of the system. The reason is that asymmetricness or stigmatism of these factors deteriorates the interfered profiles of interest. Unwanted electron-wave scattering from the aperture’s edge also weaken the interfered profiles. Highly manufacturing and operating techniques might be required.
6. Conclusion It has been known in light optics that obstructing the central part of the aperture (i.e., using annular aperture; ANP) narrows the principal maxima (i.e., main lobe) of the system’s point spread function, and
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Table 2 Optical characteristics of a typical aberration-laden SEM system Aperture radius ra Accelerating voltage Vo [kV] Virtual source size r [nm] Energy spread DVFWHM [V] Lens magnification M Cc [mm] Cs [mm] Beam semi-angle a [mrad] l [nm] D {=l/a} [nm] CFWHM {=(1/2)Cc(DVFWHM/Vo)a/D} B {=(1/4)Csa3/D} dg/D dd {=0.61D} [nm] dg {=Mr} [nm] dc {=(1/2)Cc(DVFWHM/Vo)a} [nm] ds {=(1/4)Csa3} [nm] d¼ ðdd2 þ dg2 þ dc2 þ ds2 Þ1=2 ½nm
1.4a
a 0.8 15 0.4 0.0204 0.95 1.16 10.8 0.0433 4.01 0.639 0.091 0.076
15.12 0.0433 2.87 1.253 0.350 0.107
2.45 0.31 2.57 0.37 3.58
1.75 0.31 3.59 1.00 4.13
thus enhances the resolution in the Rayleigh sense. However, this effect is accompanied by the followings; (1) a decrease of the brightness, since there is an increasing energy loss of light in the image, and (2) a reduced contrast in the low-n region due to a considerable rise in the level of the side lobes of the point spread function, since a large amount of light is diffracted from its proper geometrical position. The present paper has made clear that the results obtained in light optics are analogically applicable to electron optics. The beam characteristics on SEM system have been discussed from both viewpoints of three kinds of beam resolutions (RRayleigh , ROTF , and Rinf ) and three kinds of depths of the focus (DzRayleigh , DzOTF , and Dzinf ). When only the beam profile approximately determines the SEM image resolution, the beam resolution is equal to the SEM image resolution. Typical features of the beam characteristics are as follows: (1) All kinds of the depth of focus (i.e., DzRayleigh , DzOTF , and Dzinf ) are improved with the increase in b. The beam resolutions of ROTF and RRayleigh also are improved, being in contrast with the information resolution Rinf . Note that the RRayleigh value represents less specifically the beam resolution as the b value approaches to 1, since the magnitude of successive maxima (i.e., side lobes) of the intensity profile ib ðz; rÞ becomes more pronounced. Repeatedly speaking, the OTF resolution ROTF is certainly improved by using ANP with its proper obstruction-ratio b. (2) The conventional SEM with a circular-aperture provides absolutely a good-contrast image in the wide n-range. Besides, obstructing the central part of the aperture (i.e., using ANP) enhances high-n structures of the image contrast at the cost of low-n structures, and improves the ultimate image resolution (in RRayleigh and ROTF , not in Rinf ). Here, the b value should be appropriately determined by balancing the improvements with the reduction of electron wave power transmitted through the ANP. (3) On the SEM system with both a circular aperture and an annular aperture, a combination of the lown-pass filtered, circular-aperture SEM image with the high-n-pass filtered, annular-aperture SEM image has a potential to improve the image quality in both terms of any image resolution (i.e., Rconv , Rint , or Rinf ) and any depth of the focus (i.e., Dzconv , Dzint , or Dzinf ). Here, since the ANP reduces the transmitted electron wave power by (12b2 ) times, the electron wave power irradiating the ANP or the electron irradiating time on the sample for the ANP should be 1/(12b2 ) times magnified than the circular one so as to result in no
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difference in the electron dose between both images. Another way is the computer image processing (see the above section). All discussions are analogically applicable also for a scanning transmission electron microscope (STEM). On the practical use of ANP, problems might arise from the factors such as supporters for the central stop of ANP, deviation from the symmetrical shape of ANP, asymmetric charging of the aperture or the sample, stigmatic fields of the system. The asymmetricness or stigmatism deteriorates the interfered profiles of interest. Unwanted electron-wave scattering from the aperture’s edge also weaken the interfered profiles. Although the results are incomplete and imperfect, they are sufficient to show that ANP is useful to improve the beam characteristics such as resolution and depth of the focus.
Appendix A. Resolution and depth of the focus A.1. Conventional resolution (Rconv ) and conventional depth of the focus (Dzconv ) The conventional beam diameter dðERconv ; the conventional image resolution) is calculated by adding the diffraction image size dd to the Gaussian image size dg , the chromatic disk size dc , and the spherical disk sizes ds each quadratically as follows: d ¼ ðdg2 þ dd2 þ dc2 þ ds2 Þ1=2 ;
ðA:1Þ
where dg ¼ Mrs ; dd ¼ 0:61D
ðA:2Þ with
D ¼ l=a;
ðA:3Þ
dc ¼ kc Cc ðDVFWHM =VÞa;
ðA:4Þ
ds ¼ ks ð1=2ÞCs a3 ;
ðA:5Þ
l½nm ¼ 1:226=V½volts1=2 :
ðA:6Þ
Here, M is the lens magnification, rs the virtual source size, Cc and Cs the chromatic- and the spherical aberration coefficients (defined on the image side), respectively, a the beam focusing semi-angle, DVFWHM the effective voltage spread of the electrons, V the voltage of the beam at the image plane. Here, the potential origin is referred to the source. Both fitting parameters of kc and ks are constants, typically assumed to be 0.5 [6]. The beam depth of focus is simply given by Dzconv ¼ d=a;
ðA:7Þ
where d is the beam diameter and a is the focusing semi-angle at the sample (see Fig. 13). These conventional parameters of Rconv and Dzconv are applicable only for the systems with a circular aperture. A.2. Rayleigh resolution (RRayleigh ) According to Rayleigh’s criterion, two images of equal intensity are regarded as just resolved when the principal intensity maximum of one coincides with the first intensity minimum of the other, as shown in Fig. 14. For a diffraction-limited system with a circular aperture, the limit of resolution RRayleigh (referred to
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Fig. 13. Conventional beam-depth of the focus.
Fig. 14. Overlapped Airy patterns showing the Rayleigh’s criterion.
as the Rayleigh resolution) is equal to the Airy disc radius given by RRayleigh ¼ 0:61Dð¼ rAiry : Airy disc radiusÞ:
ðA:8Þ
We assume that Rayleigh’s criterion is applicable also to the systems with an annular aperture of the obstruction ratio of b (see Fig. 1(b)). As the b value increases, however, the RRayleigh value represents less specifically the image resolution because of highly increased successive maxima of the intensity profile ib ð0; rÞ as shown in Fig. 4. A.3. OTF resolution (ROTF ) The optical transfer function (OTF) t(n) is an image-quality measure that reflects an optical image amplitude response to sinusoidal patterns. Here, tðnÞ can be given either by a Fourier transform of the beam current distribution, or by autocorrelation of the pupil function PðnÞ given from Eq. (10). The image resolution is conventionally defined as the reciprocal of the spatial frequency at which tðnÞ has fallen to 0.1.
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Table 3 Information contents (per pD2 ) Hs and normalized information resolutions Rinf =D for various S=N values on the aberration-free system S/N
1
2
5
10
20
50
100
HS [bits] Rinf =D
2.79 1.20
4.99 0.895
9.89 0.636
15.46 0.509
22.81 0.419
35.23 0.337
46.53 0.293
This resolution is referred to as an OTF resolution, ROTF in the present study. OTF ¼
* 1=n0:1 ð¼ D=n0:1
with
n * ¼ Dn
and
D ¼ l=aÞ:
ðA:9Þ
This resolution is no more appropriate when the large spherical aberrations provide two or more n0:1 solutions for tðn0:1 Þ ¼ 0:1. A.4. Information resolution (Rinf ) and information depth of the focus (Dzinf ) Combining the image resolution with the image quality, the authors (T.I. and M.S.) have proposed the information resolution Rinf , which is defined as an average diameter of the circle per bit in informationpassing capacity (IPC) [7] Rinf ¼ 2D=Hs1=2 ¼ 2=ðprsÞ1=2 ; where H s ¼ p2
Z
ðA:10Þ
2
Gðn * Þn * dv * ;
ðA:11Þ
0
Gðn * Þ ¼ log2 ½1 þ jtðn * Þts ðn * Þj2 ðS=NÞ
with
n * ¼ Dn;
ðA:12Þ
rs ¼ Hs =ðpD2 Þ;
ðA:13Þ
S and N are the statistical mean power densities of the signal and of the noise, respectively; tðn Þ the OTF function of the optical system including the source size effect, and tðn Þ the specimen contrast, i.e., the Fourier transform of the specimen structure, which is assumed to be 1 in this study. Here, Hs is the mean information content of the optical image per basic area (defined by pD2 ), rs the IPC density. Table 3 shows the values of Hs and Rinf =D for various S=N values on the aberration-free system. Using the information resolution Rinf , we have introduced an information depth of focus (Dzinf ) defined as the z-range satisfying Rinf ðzÞ=Rinf;min p1:1 A.5. Intensity depth of the focus (Dzint ) Using the beam intensity on the optical axis (z-axis) iðzÞ, we define the intensity depth of the focus as the z-region satisfying the condition of iðzÞ=imax X 0.8. The Dzint value of 1.02 D=a is given for an aberrationfree optics (see text).
References [1] M. Born, E. Wolf, Principles of Optic, 7th Edition, Cambridge University Press, Cambridge, 1999, p. 461. [2] H. Kubota, Wave Optics, Iwanami Shoten, Tokyo, 1971, p. 278. (in Japanese).
170 [3] [4] [5] [6] [7] [8] [9]
T. Ishitani et al. / Ultramicroscopy 88 (2001) 151–170 V.N. Mahajan, Aberration Theory Made Simple, SPIE Optical Engineering Press, Washington, 1991 (Chapter 9). Z. Shao, A.V. Crewe, Ultramicroscopy 23 (1987) 169. M. Sato, J. Vac. Sci. Technol. B 9 (1991) 2972. M. Sato, Resolution, in: J. Orloff (Ed.), Handbook of Charged Particle Optics, CRC Press, New York, 1997 (Chapter 8). T. Ishitani, M. Sato, Ultramicroscopy 84 (2000) 199. P. W. Hawkes, E. Kasper, Principles of Electron Optics, Vol. 1, Academic Press, London, 1996, p. 350. T. Ishitani, J. Electron Microsc. 48 (1999) 617.