Material contrast in SEM: Fermi energy and work function effects

ARTICLE IN PRESS Ultramicroscopy 110 (2010) 242–253

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Material contrast in SEM: Fermi energy and work function effects Jacques Cazaux n Department of Physics, Faculty of Sciences, BP 1039, 51687 Reims Cedex 2, France

a r t i c l e in f o

a b s t r a c t

Article history: Received 13 April 2009 Received in revised form 14 November 2009 Accepted 1 December 2009

‘Is it possible to assign various grey levels of a scanning electron microscope (SEM) image to different components of a given sample? Among other instrumental effects, the answer is not only a function of the respective secondary electron emission (SEE) yields of the components, d, but also of the angular fraction of the secondary electrons (SE)s being collected, ka and of a possible voltage contact effect between sample and detector, kf. Expressed as a function of EF, Fermi energy, and f, work function of the components of interest, equations of spectral, (@d/@Ek), and angular, (@d/@a) distributions of the emitted SEs permit to evaluate ka and kf for Au and Si. It has been established that collected SE spectra, @da/@Ek, are distorted with respect to the emitted and fraction ka is material dependent for a solid angle of detection O1 less than 2p (or maximum semi-apex angle amax o 901) In particular, for coaxial detections around the normal incident beam the detected fraction of SEs from Au, ka(Au), is slightly larger than that for Si, ka(Si). For simple geometries in the vacuum gap, similar investigations show that parameter kj is also larger for gold than for n-doped Si as well as for p-doped Si with respect to n-doped Si. Then Au is always quite brighter than n-doped Si in the SEM images while a doping contrast, C, due to a work function effect may reach  15% for a Si p/n junction with Np  1016 and Nn  1015 cm  3. The present analysis may be extended to some metals such as Ag, Cu, Pb, Pd, Pt, and Zn that are expected to appear brighter than Si(n) and Ge in the SEM images. The influence of specimen surrounding in the vacuum gap and of detection conditions are outlined. The limitations of present approach are discussed and a strategy is suggested for the investigation of electronic devices where these components are in reduced number and are known a priori. & 2009 Elsevier B.V. All rights reserved.

Keywords: Secondary electron emission Scanning electron microscopy Image simulation Material and doping contrast Semiconductors and semiconducting devices

1. Introduction In scanning electron microscopy (SEM), a very ambitious goal would be the identification of various components of a specimen from their grey levels in the corresponding image, i.e. compositional identification. A less ambitious goal would be the assignment of each grey level in the SEM image to each known component in order to map their position in a x, y space, which is important for the investigation of microelectronic devices; where the components are in reduced number and are known from their elaboration process. For this more modest objective, a possible strategy would be use to publish secondary electron yield data, d(E1), of each component in order to assign brighter regions to the most emitting compound and the darkest regions to the least emitting compound. Recently this strategy has been successfully applied to a quartz/Cr system in order to explain the observed inversion contrast when the incident beam energy is changed, but many questions remain open with respect to the generalization of these very specific results [1]. The first question concerns the

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nature of the signal being detected: Is the image only formed with true secondary electrons (SE)? Details of the answer have been indicated elsewhere [2]. In SEM instruments, all the emitted SEs cannot be collected and only a fraction of them, kd, are. The correlated second question is ‘Is this fraction, k, the same for all the components or is it material dependent? The goal of the present contribution is an attempt to answer this second question by choosing a binary system composed of silicon and gold. The following approach is based on the use of analytical expressions involving the work function, f, (or electron affinity, w) and the Fermi level position of the related materials so that the obtained results may be generalized to other metals and semiconductors. The first aspect of the present investigation (Section 2) only concerns the evaluation of the fraction, ka, for various solid angles of detection, O1, less than 2p steradians. As a consequence of Fermi levels alignment, the second aspect (Section 3) includes the effects of contact potentials induced by the shift of the vacuum level at the SE emission point with respect to that of the SE detector. Because of the change in Fermi level position as a function of nature and concentration of dopants in semiconducting devices, this second aspect leads to estimate a possible contribution of a contact potential effect to doping contrast. In addition, the

ARTICLE IN PRESS J. Cazaux / Ultramicroscopy 110 (2010) 242–253

2. Detected SE intensities and spectral distribution for a partial angular collection 2.1. Starting hypotheses In the present investigation the postulated experimental arrangement is based on a flat sample irradiated at normal incidence with an incident electron beam of energy E1 and the SE emission presents a ‘z’-axis symmetry around normal to sample. A detector collects the intensity of the true SE, Id, being emitted between amin and amax (where a is the SE emission angle with respect to this normal), and the solid angle of detection, O1, is defined by the angular interval of a from amin to amax. Sample holder and entrance aperture of detection system (here after: detector) are set to ground so that the surroundings of the sample are free of any applied electric or magnetic field and straight SE trajectories are expected from the emission point to the entrance of the detector system excepted when the influence of Fermi level alignment on vacuum levels is considered (Section 3). Lastly, the present results obtained for gold and silicon are extended to other metals and semiconductors when the position of their Fermi level and the value of their work function (or energy affinity) are known. 2.2. SE energy distribution In SEM and in related techniques, the intensity of true SE being collected, Id, by a detector of solid angle of detection O1 follows: Z 50 eV Z O3 dEk ð@2 d=@O @Ek Þ dO ð1Þ Id =I3 ¼ 0

0

In Eq. (1), I1 is the incident beam intensity and the fraction of SE being collected, ka, over a partial solid angle of detection, O1 o2p, may be written in the form ka ¼ Id =Ie ¼ dd =d

ð2Þ

In Eq. (2), Ie corresponds to the total collection of the emitted SE, O1= 2p, and dd is Id/I1 when O1 o2p. The goal of the present subsection is to evaluate ka for Au and Si using analytical expressions describing their angular and energy distributions. The analytical expression used here for gold spectral distribution, @d/@Ek, is that of Chung and Everhart [30]: @d=@Ek ¼ F 0 Ek =ðEk þ fÞ4

100

∂δ/∂Ek (in %)

Au

ð3Þ

80

Exp Au

60

C & E; 4 eV C & E; 5.3 eV

40 20 0 0

2

4

6

8

10

12

Ek (eV)

100 Si

Exp. 0.6 keV

80 ∂δ/∂Ek (in %)

influence of work function on the SEE yield of the materials of interest is also considered, via the SE escape probability. To the author’s knowledge, the first investigation is new in SEM [3] as well as in SE spectral distribution experiments [4], because it was implicitly admitted that the measurement methods were material-independent and without distortion even for measurements performed over narrow collection angles, a. The shift of the SE spectral distribution due to contact potential effects, second investigation, is known from a long time as well as in SE spectral measurements [4] as in SEM [5–8]. The originality of present approach is an investigation based on the material dependence of the SE angular distribution, which has never been considered. Obviously driven by its impact on the semiconductor industries, there is an abundant literature about doping contrast in SEM and various explanations have been suggested to explain why p-doped regions of a p–n junction appear brighter than n-doped regions [9–29]. For this last point, the goal of the present contribution is not to suggest another explanation but it is just to evaluate numerically the weight of a possible contribution of voltage contrast to this effect from a detailed calculation based on angular, @d/@a, and energy, @d/@Ek, distributions of SEs emitted from Si.

243

Exp. 0.2 keV Theo.

60 40 20 0 0

2

4

6

8

10

12

Ek (eV) Fig. 1. Spectral distribution, @d/@E, for SE issued from Au (a) and Si (b). Lines: calculations; symbols: experimental results—see text for details.

and is frequently used for metals. In Eq. (3), F0 is a normalization factor fitting the maximum at 100%. The energetic position of this maximum is at  f/3, where f is the work function. Calculated distributions, or lines, are shown in Fig. 1a. They correspond to f =4 eV (approximate fit to experimental results) and f  5.3 eV ([31] and Table 1) and are compared to experimental results and symbols obtained by Henke et al. from Al Ka X-ray excitation (Fig. 7 in [32]). With a similar normalization factor to 100% at maximum, F00 , the corresponding expression for semiconductors is of the form [32] @d=@Ek ¼ F 00 Ek =ðEk þ wEG Þ2 ðEk þ wÞ5=2

ð4Þ

Fig. 1b, line, shows an example of this calculated distribution for Si with w (electron affinity)  3.2 eV and EG (band gap energy)  1.12 eV. These values are kept for further calculations because of their rather good agreement with experiments, or symbols, that were performed at an incident beam energy, E1, of  0.6 keV under ultra-high vacuum conditions on a clean Si(1 0 0) surface, where the doping level has not been indicated [33]. The agreement is less for lower beam energies such as E1  0.2 keV where the spectral distribution is broader because SEs are generated closer to the surface and they may escape into vacuum after a reduced number of inelastic events (see Section 4.3). For the present purpose, the most important point is the large difference between the spectral distribution of SE emitted from a metal such as gold, where the most probable energy, MPE, is at  1.35 eV and that for a semiconductor, such as silicon, where the MPE is at  0.75 eV. The spectral distribution of SE issued from

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Table 1 The work function [31] and the Fermi energy [41] of some conductors. Element

Ag

Al

Au

Be

Ca

Cd

Cs

Cu

Ga

f (eV) EF (eV)

4.52–4.74 5.48

4.06–4.26 11.63

5.1–5.47 5.51

4.98 14.14

2.87 4.68

4.08 7.46

2.14 1.58

4.53–5.10 7.

4.32 10.35

In

K

Li

Mg

Na

Pb

Pd

Pt

Zn

2.36 3.23

4.25 9.37

5.22–5.6

5.12–5.93

f (eV) EF (eV)

4.09 8.6

2.29 2.12

2.93 4.72

3.66 7.13

insulators is narrower than that of SE issued from metals [34,35] and even that of SE issued from semiconductors because the spectral distribution of SE issued from insulators obeys [32,36] @d=@Ek ¼ F 000 Ek =ðEk þ wÞ3

1

ð5Þ

Si

Au

since it has a MPE at around  w/2 and less than 0.5 eV.

0.25 0.5 1 1.35 (MPE) 0.5 2 4 8

2.3. SE angular detection The SE emission into vacuum is governed by a refraction effect at the specimen/vacuum interface and this refraction effect is a function of the kinetic energy of SEs in the specimen, ES, and of their inner angle of incidence, b. Their corresponding kinetic energy in vacuum and a their corresponding angle of emission into vacuum, with respect to the normal, are given in terms of Ek by the useful law OES sin b ¼ OEk sin a

4G1=2 ½1þ G1=2 2

0.25 0.5 1 0.75 (MPE) 2 4 8

0 -0.5

-1

0.5

0

1

ð6Þ

This refraction effect is specific to each energy, Ek, and angle of emission, a, and it is at origin of the angular distribution of SE emitted into vacuum, @d/@O (or @d/@a). A more detailed expression can be derived from the transmission probability of SE across the surface potential barrier, T(S/V) written in the form [36–38] TðS=VÞ ¼

3.63–4.9 9.39

with G ¼ 1þ

ES Ek Ek cos2 a

10 0.25 0.5 1 1.35 (MPE) 2 4 5 8

ð7Þ

Eq. (7) applies for metals, semiconductors and insulators, with the only difference being in the numerical values of ES Ek, which can be derived from ES  Ek =EF + f for metals (EF: Fermi energy) or from ES  Ek = w (for semiconductors and insulators). Then ES  Ek is 9.5–10 eV for Au and it is only 3.2 eV for Si. Fig. 2a illustrates the corresponding numerical applications using a transmission probability normalized to 100% in the forward direction, with Au on the left side and Si on the right side, for Ek energies ranging from 0.25 up to 8 eV and passing by the most probable energy, MPE, at 1.35 eV for Au and 0.75 for Si. These angular distributions are based on an isotropic distribution for inner SEs arriving at the sample/vacuum interface after random walks resulting from many elastic and inelastic collisions. Such a simplification leads to T(S/V)=(@d/@a) and it is hope to be valid for incident beam energies, E1Z, E1max (E1max: energy where d = dmax) but such a simplification is more questionable at lower beam energies E1. 2.4. Relative intensities of detected SE The differences between spectral and angular distributions of SEs issued from Au and Si have been established; the next step is to evaluate from Eq. (1) the differences in the detected intensities. Here, such an evaluation is performed in a numerical form for a coaxial detector characterized by its solid angle of collection angle O1 or its semi-apex angle a ranging from amin to amax. This numerical approach consists of dividing the SE kinetic energy interval from 0 to 50 eV, Eqs. (3) or (4), into adjacent energy

0.25 0.5 1 0.75 (MPE) 2 4 8

0 -90

-60

-30

0 αmax (°)

30

60

90

Fig. 2. (a) Angular distribution (@d/@a) of SE escaping for Au (left) and Si (right) for SE kinetic energies, Ek, ranging from 0.25 eV up to 8 eV: polar co-ordinates normalized to unity in the forward direction. The results are derived from numerical applications of expression T(S/V), Eq. (7), combined with an isotropic distribution of inner electrons arriving at the sample/vacuum interface. amax is the maximum opening angle for coaxial detection; and (b) from Eq. 8, numerical integration of the above angular distribution, for increasing values of amax at selected kinetic energies, Ek, Ek =0.25 to 8 eV and to the most probable energy, MPE, of the two elements of interest.

windows of width DEk =0.1 eV and in dividing the angular distribution, Eq. (7), from a =0 to 901, into a series of annuli of discrete adjacent angles, ai, of angular width Dai =215. Then for a given kinetic energy, Ek 70.05 eV, the fraction, Dd(Ek)/d(Ek), of the collected SE corresponds to the discrete sum, from amin to amax, of the form

Ddi ðEk Þ=dðEk Þ ¼ TðS=VÞi 2pðsin ai ÞDai =NðEk Þ

ð8Þ

In Eq. (8), N(Ek) is a normalization factor corresponding to the same angular sum but extended from 0 to 901. From Eq. (8), Fig. 2b shows the non-normalized evolution of this sum,

ARTICLE IN PRESS J. Cazaux / Ultramicroscopy 110 (2010) 242–253

245

100

80

y = -2.6 ln (x) + 73.5

Au 60

Au

60

80 ∂δα/∂Ek

y = -2.74 ln (x) + 48 fα

45 40 y = -1.8 ln (x) + 24

Ref 0-90°

0-15° (6.5%)

0-30° (22%)

0-45° (44.5%)

0-60° (71%)

15-30° (15.5%)

1.35

30

60 45-90° (55.5%) 40

1.4

20 y = -0,6 ln (x) + 7.3

15

20

0 0

2

4

6

8

10

0

Ek (eV)

0

1.2

2

4

6

8

10

EK (eV) 80

100

y = -3.4 ln (x) + 71 Si

80

y = -3.2 ln (x) + 45

∂δα/∂Ek

fα (%)

Si

60

60

45

40 y = -1,94 ln (x) + 22.5 y = -0.63 ln (x) + 6.7

ref 100% 0-30 (21%) 0-60 (68%) 15-30(15%)

60 40

30

20

~0.75

0-15 (6,5%) 0-45 (43%) 45-90 (57%)

0.85

20 15

0 0

2

4

6

8

0

10

Ek (eV)

2

4

6

8

10

Ek (eV)

numerical integration, for increasing values of amax as a function of a for Au (left) and Si (right) and for SE kinetic energies ranging from 0.25 eV up to 8 eV. Substitution of the double integral of Eq. (1) by a double series of discrete values leads to tedious calculations that may be partly simplified by selecting specified angular intervals from amin =01 to amax = 151, 301, 451 and 601, and some kinetic energies, Ek, Ek =0.1, 0.25, 0.75, 1, 1.5, 2, 4, and 8 eV. From this selection it is rather easy to find empirical expressions for the weighting factor, fa, to apply over all kinetic energy ranges. Illustrations of such a procedure are shown in Fig. 3a for Au and in Fig. 3b for Si. With parameters ‘a’ and ‘b’ being a function of amax, and of the material of interest, these empirical expressions are of the form ð9Þ

In Eq. (9), parameter ‘b’ is the weighting factor corresponding to Ek = 1 eV and parameter ‘a’ describes the change in fa as a function of Ek. A multiplication of the initial spectral distribution, @d/@Ek, by this weighting expression allows one to obtain the expected spectral distribution, @da/@Ek, relative to a given value of amax and of the material of interest. The results are shown in Fig. 4a for Au and in Fig. 4b for Si. Examples of the expected spectral distribution @da/@Ek obtained for annular detections, amin a0, are easily obtained from a channel by channel subtraction, and are also shown in the same figures for 151 o a o301 and 451o a o901. Next a sum from 0 to 50 eV of channels, each of 0.1 eV width, allows one to obtain the SE fraction of intensity being collected with respect to a total collection, ka, ka = dd/d. The corresponding numerical results are

Fig. 4. Calculated spectral distribution of detected intensities, qda/qEk, for different intervals of angles of detection (amin, amax) with respect to the reference, corresponding to amin = 01 and amax = 901. The reference spectral distribution is shown in Fig. 1a for Au (f = 4 eV) and in Fig. 1b for Si. The percentages of collected intensities with respect to a total collection, ka, are indicated in caption. Note the shifts of the most probable energy (MPE), vertical arrows, as a function of the detection interval, amin, amax.

120

kα (Au)/ kα (Si)

100 80 kα (%)

Fig. 3. Fitting procedure used to determine the weighting factors, fa, of Eq. (9). (a) Au and (b) Si. Symbols: results obtained for selected kinetic energies, Ek, and maximum angles of detection, amax; lines: empirical equations corresponding to different values of amax.

fa ð%Þ ¼ a lnðEk Þ þ b

0 ~0.6

60 Au 40

Si Au/Si

20 0 0

30

60

90

αmax ( ) Fig. 5. Also indicated in caption of Fig. 4, percentages of collected intensities with respect to a total collection, ka, where the angular detection interval increases from amax = 01 to amax = 901. Ratio, ka(Au)/ka(Si), is also shown.

indicated in captions of Fig. 4a and b where they are given in percentage. The same ka values are also displayed in Fig. 5 for Au and Si, where it may also be seen that they are material dependent with ka(Au) always exceeding ka(Si), but only by a few percent – probably 10% at maximum – despite their difference in the spectral and energetic distributions. This slight

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difference can be qualitatively explained by the fact that the angular distribution of Au, Fig. 2a, is narrower than that of Si, Fig. 2b, because of the significant difference of ES Ek values (  9.5–10 eV for Au and only 3.2 eV for Si) via the SE refraction effect. However, this significant change mainly concerns a low kinetic energy region, 0 oEk o1 eV, that is around the MPE of Si but is below that of Au. The same explanation may also be developed to parameters ‘a’ and ‘b’ of Eq. (9) where one can observe that parameter ‘b’ corresponding to Ek =1 eV is, for Au, always larger than for Si: 7.3%, 24%, 48%, and 73.5% instead of  6.7%, 22.5%, 45%, and 71% (for amax =151, 301, 451, and 601, respectively) but the distortion factor ‘a’ is less for Au than for Si. Depending on the specific values of amin and amax and being material dependent, another important and new result is the distortion effect of the spectral distributions leading, in particular, to slight shifts of the most probable energies. Then, spectral SE measurements based on a partial angular detection have to be suspected of such a distortion. This is the case of spectral distributions shown in Fig. 1a and b where the selection of a better fit has not been performed.

3. Work function effects 3.1. Fermi levels’ alignment and shift of SE spectral distribution When two conductors are set in contact, contact voltages occur as a consequence of the alignment of Fermi levels via the crossed transfer of conduction electrons. Then the vacuum levels, which are the levels of zero electron kinetic energy just outside the surface at each point, change from place to place, and a SE that is being emitted from one point of a specimen experiences changes when it approaches either the detector or another point of an heterogeneous sample. When two differently doped regions of a semiconductor are considered the Fermi level on n side, EFn, (measured with respect to the top of valence band) is given by [39–41] EFn ¼ EG þ ðkB TÞlnðNn =NC Þ

ð10Þ

For Si, EG (band gap energy) is  1.12 eV, kB is the Boltzmann constant, T is the absolute temperature and kBT is  0.025 eV at room temperature, RT. Nn is the n-doping concentration and NC is a parameter nearly equals to 0.22  1019 cm  3 for Si at RT. EFn is 0.19 eV below the bottom of conduction band at a distance larger than 5 kBT and Eq. (10) applies, for an impurity concentration Nn of  1015 cm  3, non-degenerated situation. A similar equation holds for the p side where the Fermi level, EFp, is above the top of the valence band, when a non-degenerated situation is also considered. When the two parts of the p–n junction are set in contact a built-in potential, VJ, is established via Fermi levels alignment. This potential is given by [39–41] qVJ ¼ EFn EFp ¼ ðkB TÞ lnðNn Np =Ni2 Þ

ð11Þ

where Np is the p-impurity concentration, Ni is the intrinsic concentration of carriers and is 4.6  109 cm  3 for Si at RT, and q is  1.6  10  19 C. Then qVJ is  0.67 eV for Nn 1015 cm  3 and Np 1016 cm  3 and it increases 0.058 eV per decade of Nn or Np. Fig. 6a shows the band structure scheme (top part) and the correlated changes of vacuum potentials (bottom part) of a p–n junction and metal such as gold in contact to a sample holder itself in contact with a detector. When f (sample) is larger than f (det.), the spectral distribution of emitted SEs, @df/@Ek, is accelerated towards the detector, and the corresponding increase of SE kinetic energy, f (sample)–f(det.), is shown in Fig. 6b (left). When f(sample) is less than f(det.), Fig. 6b(right), the detected spectral distribution is shifted towards the opposite

direction and it presents a sharp cut-off when the experimental arrangement is assimilated to a point emission source and a concentric hemispherical detector (central part of Fig. 6b). The energy shifts of @d/@Ek, related to contact potential effects, may reach up to 3 eV as deduced from the differences of tabulated work functions values shown in Table 1 for some metals [31]—values combined to the corresponding Fermi energies for numerical applications of Eq. (7) [41]. The possible occurrence of these shifts has been known for a long time in SE spectral measurements, where the use of spherical collectors has also been outlined for a non-distorted spectral acquisition [4], and the same shift effect has been a subject of great attention in the calibration of electron spectrometers in surface analysis [42]. In SEM the most striking effect is a sharp cut-off of @d/@Ek leading to a decrease in the collected intensities when f(sample) is less than f(det). A numerical integration of @d/@Ek from f(det.) to 50 eV allows one to evaluate the ratio, kf, between detected intensities with and without a work function effect. Fig. 7 shows the results obtained, from Eq. (3) for @d/@Ek, for gold when f is chosen to be  4 eV for fit with experimental results of Fig. 1 and 5.3 eV for the mean tabulated value in Table 1. Here, Df also represents the energy difference between vacuum level and Fermi level for semiconductors and results obtained for differently doped silicon samples are also shown for Si (n) (n-doping concentration  1015 cm  3) and Si (p) (p-doping concentration  1016 cm  3) and also Si (p + ) (p-doping concentration  1018 cm  3). Postulating that the change in SE kinetic energy from the sample to the detector does not influence the response function of this detector, the most striking result is the influence of f(det.) on ratio kf. As expected from this postulated geometry, this ratio is 100% when f(det.) is less than f(sample) but it rapidly decreases when f(det.) progressively increases above f(sample). Being f(sample) dependent, the starting point of this decrease differs for two different ‘‘species’’ of gold, f  4 and 5.3 eV. The same result holds between Si(n) and Si(p) when f(det.) is above 3.2 eV and one may expect a possible contribution of the contact potential effect to contrast doping. 3.2. Distortion of SE trajectories into vacuum for a parallel geometry In a SEM, the point sources are scarcely surrounded by concentric potential surfaces and calculations of SE trajectories are basically a problem of electron optics in the vacuum gap where the role of magnetic fields or of biased electrodes has been taken into account [2,43]. Such calculations are specific to each instrument and operating mode but an isotropic emission is often postulated for evaluating the detected intensities. In contrast, in the present approach the angular and the energy distributions are carefully taken into account for such an evaluation but the specimen surroundings are simplified. These surroundings are assimilated to a plane parallel geometry with a uniform electric field, Fe =  DV/w, in the vacuum gap of width w between sample and detector. In such a simple situation the SE equations of motion are [44,45] m@2 x=@t2 ¼ 0

m@2 z=@t2 ¼ qFe

ð12Þ 19

If EV = qDV= f(det.)  f(sample) and qo0 ( 1.6  10 C), an SE trajectory of initial energy Ek and angle of emission a (to the normal) follows: z=w ¼ ðEV =4Ek Þðx=wsin aÞ2 þðx=wÞcot a

ð13Þ

In Eq. (13), the second term, (x/w) cot a, corresponds to straight SE trajectories in a free field space and the first term describes parabolic positive or negative deflections, depending

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247

levels

Vacuum

χ (Si) χ (Si)

φ (Au)

φ (holder)

φ (det.)

EG (Si)

EF

EF EG (Si)

n

p

Au

holder

detector (ref) V (x)

φ(

Au)/q ΔV (pn)

φ(Au)/q

∂δ/∂Ek

φ (det.)- φ (sample)<0

φ ( hol.)

0

∂δ/∂Ek

φ (det.)- φ (sample)>0

Fig. 6. (a) Oversimplified band structure scheme showing the consequence of the Fermi levels’ alignment to the vacuum levels for Si(n), Si(p) and Au (full line: f = 4 eV; and dashed line f =5.3 eV) being set on a metallic sample holder and in front of a grounded detector; (b) left. Increase of kinetic energy of SE going from sample to detector when f(sample) is larger than f(det), (b) right. Decrease in kinetic energy of SE going from sample to detector when f(sample) is less than f(det). The corresponding shift leads to a sharp cut-off when the experimental arrangement is associated with a point emission source surrounded by a concentric hemispherical detector, central part.

upon the sign of Df = f(det.)  f(sample). Fig. 8a illustrates a few SE trajectories for a angles around a maximum angle of collection for straight trajectories, amax = 2215: results from numerical applications of Eq. (13) with reduced co-ordinates, z/w and x/w, and for different values of (EV/Ek). On the right side, f(det.)  f(sample) is negative and some SEs, emitted at angles larger than amax, i.e. a = 301, can be collected mainly when their initial kinetic energy is Ek r9Df9. On the left side, f(det.)  f(sample) is positive and some SEs, emitted at initial angles less than amax, i.e. a = 201, cannot be collected even when Ek Z  9Df9. As intuitively expected, these examples show that the field effects also influence all trajectories of low energy SEs and then they change the SE collection efficiency of the detector, even when f(det.)  f(sample) is negative. A purely numerical procedure has been used with the help of mathematical software, EXCELTM, for a quantitative evaluation of

the change in detected intensities. For a given maximum angle amax (tan amax =x1/w) and selected [f(sample)  f(det.)]/Ek values, this procedure consists of first calculating the SE trajectories, Eq. (13), in order to determine, by interpolation, the value of the effective maximum angle of collection, aeff, corresponding to an impact point permitting the corresponding SE to be just collected, x= x1 at z =w. Next, the corresponding relative change in the solid angle of detection, DO/O1, is evaluated for the same selected values of [f(sample)  f(det.)]/Ek and an interpolation procedure is next applied in a way similar to that followed in Section 2.4 as illustrated in Fig. 3. The calculated change in DO/O1 as a function of [f(sample)  f(det.)]/Ek, or Df/Ek, is illustrated in Fig. 8b for various nominal angles amax ranging from 101 up to 451; where one may observe a nearly common linear evolution of DO/O1 for amax angles less than 301 at positive Df. For amax = 451 there is the saturation atDO/O1  3.5 because Oeff  2p(or aeff  p/2) at Df  4.

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100

z/w

80

αmax = 22°5

Si (n) 60

Si (p) Si (p+)

40

Au (5.3 ev) 20

φ (sample) <φ (det.)

0 2

3

22°5; V = 0 (ref) 22°5; qV = E 22°5; qV = 2 E 22°5 ; qV = 4 E 30°; qV = E φ (sample) >φ (det.)

22°5 ; -qV = E 22°5; -qV= E/2 20°; -qV = E/2 17°5; -qV = E/2

Au (4 eV)

4

5

6

7

φ (det.) (eV)

-1

Fig. 7. Evolution of ratio, kf as a function of the detector’s work function, f(det.), when a point emission source surrounded by a concentric hemispherical detector is postulated. Results concern Au (f  4 eV); Au (f  5.3 eV) and differently doped Si samples: Si(n):n-doping concentration  1015 cm  3); Si (p), p-doping concentration  1016 cm  3; Si (p + ) , p-doping concentration  1018 cm  3. Note the significant influence of the work function effect when f(det.) varies around 4.5 eV, horizontal arrow.

0 x/w

1

0.5

ΔΩ/Ω° 4

10° 12°5 20° 22°5 30° 40° 45°

αeff = ~90° 3 2 1 0

-2

0

2 [φ (sample)- φ (det.)]/Ek

4

6

6 Au

5

0 <α< 30°

4 fφ

If f(det.) is larger than f (sample), then Df is negative; as a result the decrease in DO/O1 starts to be linear and, at around Df   0.95Ek, it suddenly falls down to DO/O1= 0 when f(sample) approaches f(det.), because the corresponding SEs are all deflected back to the specimen and they cannot be collected. Another mathematical step consists of changing the evolution of DO/O1= f(Df/Ek) into ff = f(Ek) where ff is the work function factor influencing the detected spectral distributions of Section 2.3 and of Fig. 3c. An example of this change is illustrated in Fig. 8c for Au at amax angles amax less than 301. Finally the detected spectral distributions including the work function effect, @df/@Ek, may be obtained from a multiplication by ff, of the expected spectral collection previously obtained for Df = 0. Fig. 9a shows results obtained for Au when Df is + 1, +2, and + 3 eV, for amax less than 301, and Fig. 9b shows similar results for Si. Normalized to 100% at the maximum, the reference distribution corresponds to Df = 0 and the origin of kinetic energies is kept at emission points near from the sample surfaces so that they have to be shifted by Df for a representation of @df/@Ek at detector level. With respect to the case of Section 3.1; where the constant potential surfaces are concentric hemispheres, the most important change of a parallel geometry is the significant increases of @df/@Ek at low kinetic energies Ek. Such increases were qualitatively expected from the inspection of Fig. 8a (right), with the possible collection of low energy SEs initially emitted at angles a larger than amax. In addition, such increases are larger for Au than for Si because of the narrower initial angular distribution shown in Fig. 2a. A key result is the larger ratio, kf, between detected intensities with and without work function effects, obtained from a numerical integration of @df/@Ek from f(det.) to 50 eV. These ratios, kf, are indicated between parentheses in captions of Figs. 9a and b. Again for amax less than 301, Fig. 10a shows results obtained for Au when Df is  1, 2, and  3 eV. Fig. 10b shows similar results for Si. Normalized to 100% at the maximum, the reference distribution corresponds again to Df =0 and the origin of kinetic energies are kept at emission points near the sample surface to show more clearly the distortion of spectral distributions. With respect to the case of Section 3.1 where the constant potential surfaces are concentric hemispheres, there is again the cut-off at Ek =  Df followed by a significant decrease in @df/@Ek at initial kinetic energies Ek larger than  Df. Such decreases were also qualitatively expected from the lack of collecting SEs initially

0

-0.5

3

∇

kφ (%)

φ (det.)

1

kφ~1+0.4 ( φ/Ek)

1 eV 2 eV 3 eV Fit 1 eV Fit 2 eV Fit 3 eV

2 1 0

1

2

3

4

5

Ek (eV) Fig. 8. (a) Examples of the work function effects on the distortion of a few SE trajectories, Eq. (13). Maximum angle of collection for straight trajectories, amax = 2215 and different values of [f(sample)  f(det.)]/Ek indicated in captions where Ek is E and Df = [f(sample)  f(det.)] is –qV. Note that if f(sample), on the right side, is larger than f(det.) some SE emitted at angles such as a = 3014 amax may be collected. If f(sample), on the left side, is less than f(det.) some SE emitted at a =201o amax cannot be collected; (b) change in DO/O1 as a function of [f(sample)  f(det.)]/Ek for various angles amax. Note the linearity of this change for Df 40 and its independence on a when 01o amax o301: a remark facilitating interpolation processes between selected points (symbols). At right, note also that DO/O1 changes in hundreds of percent; (c) calculated evolution of the work function factor, ff, derived from (b), influencing the detected spectral distributions (Section 2.3 and of Fig. 3c): example for Au at 01o amax o 301, where the lines correspond to a purely mathematical interpolation process empirically described by kf  1+ 0.4(Df/Ek).

emitted angles less than amax, Fig. 8a left. In addition, it may be established from the evaluation of the ratio, kf, that these decreases are larger for Si than for Au because of its initial SE spectral distribution characterized by a most probable energy less than 1 eV and a narrow FWHM, comparing the reference curves of Figs. 10a and b. For Au (f = 4 eV), Si (n), Si(p) and Si (p+ ), the influence of kf on the work function value is summarized in Fig. 11 for a parallel

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249

100

∂δφ/∂Ek 300

Au 0 <αmax<30

80

Au 0<αmax< 30

- 1 eV (69%)

0 eV (ref.)

200

∂δφ/∂Ek

250

0 eV

1 eV (121%) 2 eV (142%)

150

- 2 eV (50%)

60

- 3 eV (37%) 40

3 eV (161%) 100

20

50

0 0

Δφ

2

4

0 0

2

4

6

8

6

10

Ek (eV)

10

100 Si

∂δφ/∂Ek

0 eV (ref) -0.2 eV (93.5 %)

80

250

-0.4 eV (86.5%)

100%) Si

0.2 eV (103%)

∂δφ/∂Ek

200

8

Ek (eV)

0.4 eV (106.5 %) 0. 6 eV (110%)

150

-0.6 eV (79%)

60

-0.8 eV (71%) -1 eV (65%)

40

-2 eV (41%)

0.8 eV (113%) 1 eV (116,5%)

100

-3 eV (27%)

20

2 eV (133 %) 3 eV(149.5 %)

50

0 0

2

4

6

8

10

Ek (eV)

0 0

2

4

6

8

10

Ek (eV) Fig. 9. Detected spectral distributions including the work function effect, qdf/@Ek, for Au, (a), and Si, (b), when Df is +1, + 2, and + 3 eV, where amax is less than 301. The kinetic energy scales correspond to the initial kinetic energies at the emission points and the spectra have to be shifted of Df (horizontal dashed arrows) for the final kinetic energies at the detector level. Normalized to 100% at maximum, the reference distribution corresponds to Df = 0 and other curves correspond to its multiplication by a factor, ff, derived from calculations similar to that shown in Fig. 8c. Factor kf between detected intensities with and without work function effects is indicated in captions. For a more detailed evaluation of ff for doped Si, additional curves are also shown in Fig. 9b for Df =0.2, 0.4, 0.6, and 0.8 eV.

Fig. 10. Similar to Fig. 9 detected spectral distributions including the work function effect, @df/@Ek, for Au, (a), and Si, (b), when Df =  1,  2, and  3 eV, where amax is less than 301. Again the kinetic energy scales correspond to initial kinetic energies at emission points and the whole spectra have to be shifted by Df(horizontal dashed arrows) for the final kinetic energies at the detector level. (b) also shows additional curves for Df =  0.2,  0.4,  0.6, and  0.8 eV.

150 αmax< 30° Si (n) Si (p)

100

Si (p+) Au (4eV)

kφ

geometry of amax less than 301. Similar to Fig. 7, the problem is the value of detector’s work function, which is generally unknown. Fortunately present calculations show that the evolution of kf values as a function of f(det.) are nearly parallel to each other, so that the ratio [kf(Au)/kf(Si)] is nearly independent from the contact potential of detector to ground. A comparison between Figs. 11 and 7 illustrates the strong influence of the specimen surroundings on the collected SE intensities. For more realistic surroundings of SEM samples, eventually involving biased collectors or biased specimen holders, the SEs’ trajectories can be derived from electron optic calculations with boundaries conditions adding the difference of work functions to that of applied voltages. The main mechanism responsible for the spectral distortion of the collected spectra, also reported for the cathode–lens system [46], is an increased collection efficiency for a positively biased detector, Everhart–Thornly (ET) type electron detector, (or negatively biased sample holder) for low energy SEs

50

0 2

3

4 5 φ (det.) (eV)

6

7

Fig. 11. Similar to Fig. 7 but for a parallel geometry, evolution of the ratio, kf, as a function of the detector’s work function, f(det.). The numerical values, obtained for amax o 301, have been indicated in captions of Figs. 9 and 10 and they concern Au (f  4 eV) and differently doped Si samples previously considered in Fig. 7: Si(n):n-doping concentration  1015 cm  3), Si (p): p-doping concentration  1016 cm  3, and Si (p + ): p-doping concentration  1018 cm  3.

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initially emitted at large angles with respect to the attractive field direction. 3.3. Work function effect on SEE yields From definitions of ka (Eq. (2)) and of kf (Section 3.1), the ratio between the SE detected intensity and the incident beam intensity, Eq. (1), may be written in the form

simplified description of the three-step process giving rise to SEE emission: SE generation, SE transport, and finally SE escape. For the constant loss model, the corresponding final expression is

d ¼ ½AE3 =Ese a1 ð1-ea Þ

ð16Þ

If successively applied to gold and Si(n), this equation allows one to obtain the ratio of the detected intensities of these two elements:

where A is the SE escape probability; Ese is the mean energy required to produce a SE able to escape into vacuum and then have a kinetic energy Ek above vacuum level, a is Rse/s with s: SE exponential SE attenuation length and Rse: incident electron range for generating a SE and often postulated to be of the form Rse = CmE1n (Cm: material constant; exponent n: 1.3on o1.8). For the parabolic model the corresponding final expression is

Id ðAuÞ=Id ðSiÞ ¼ ½ka ðAuÞ=ka ðSiÞ ½kf ðAuÞ=kf ðSiÞ ½dðAuÞ=dðSiÞ

d ¼ ½AE3 =Ese  ½3=ð23kÞ a1 ½D þF

Id =I3 ¼ ka kf d

ð14Þ

ð15Þ

This final result is a function of the ratio [d(Au)/d(Si)]. Fig. 12 shows a compilation of experimental results (symbols) obtained on various gold samples [47–49], Fig. 12a, and various silicon samples [49–51], Fig. 12b. In addition, results derived from the Monte Carlo simulations are also shown [52]. For a better visualization of experimental deviations between these various experiments, lines corresponding to the use of two models have been added. The two models, parabolic [36,45] and constant loss models [53,54, discussed in detail elsewhere [36], are based on a

2 Exp. 1 Exp. 2 Exp. 3 M.C. Cal Model 1 Model 2

Au

δ

1.5

0.5

0 1

2

3

with D= 1 2k+ 2ka  1 2a  2 and F= 2a  1(1 ka  1)e  a and again a = Rse/s. In the context of the present paper, the most important parameter is the escape probability, A, directly influencing the magnitude of the yield at any primary energy, E1. This escape probability is a direct consequence of the transmission probability at the specimen/vacuum interface, T(S/V) of Eq.(7), which is related to refraction effects, Eq. (6). Then parameter A can be estimated from the inner critical solid angle for the SE emission into the vacuum, Ol, normalized to 2p and weighted by the transmission probability, T(01), at normal inner incidence (b =01), [36,45]: A  Tð03 Þð1cos bl Þ:

1

0

ð17Þ

4

5

E° (keV)

ð18Þ

In Eq. (18) the critical inner angle, bl, is derived from Eq. (6) and it is given by sin bl = O(Ek/ES). Fig. 13 illustrates the change in A as a function of EF + f (or w) for different values of the SE kinetic energies into vacuum, Ek. As shown in Fig. 1, the selected kinetic energies correspond to the most probable energies of Au: 1.35 eV (f = 4 eV) and  1.8 eV (f =5.3 eV) and of Si: 0.75 and 1 eV. From Fig. 13, it is clear that a decrease in 1 eV for f induces a relative increase in the escape probability, DA/A, larger than 15%. Besides inherent uncertainties of the experiments, the difference in the yield of a given material results in such a correlation between f and A, and it explains a part of large deviations between various results obtained with Au, Fig. 12a, or obtained with Si, Fig. 12b. In fact these deviations have been reported for a long time and, among other parameters, they have been attributed to the prime

1.5 Exp. 1 Si

20

Exp. 2 Exp. 3

1 δ

Exp. 4

0.75 eV 1 eV 1.35 eV 1.8 eV

15

Model 1 Model 2 A (%)

0.5

10

∇

0

A

0

1

2

3

4

5

5

E° (keV)

∇

Fig. 12. Yield results, d = f(E1), from different gold samples, (a), and different silicon samples, (b). Symbols correspond to the Monte Carlo simulations [52] and to experimental results: Exp. 1 [47], Exp. 2 [48], and Exp. 3 [49] for Au and Exp. 1 [50], Exp. 2 [51], and Exp. 3 [49] for Si. Lines, based on a fit at the maximum between experiments and calculations, correspond to the use of models: model 1, parabolic model, derived from Eq. (17) or model 2, constant loss model, derived from Eq. (16). Parameters of model 1 are: k= 0.05 and n= 1.35 for Au, k =0.375 and n= 1.7 for Si. For model 2, exponent n in Rse = CmE1n is also chosen to be 1.35 for Au and 1.7 for Si. From the fit at the maximum one obtains: A/Ese  5.5 keV  1 and s  1.7 nm for Si.

φ

0 2

4

6 8 EF+φ (or χ) (eV)

10

12

Fig. 13. Calculated influence of the sum EF + f(or w), on escape probability, A, as derived from Eq. (18). These calculations correspond to SE kinetic energies indicated in the caption. For Ek = 1.8 eV, note the increase DA/A  15% induced by a decrease in f from  10.5 eV down to  9.5 eV.

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4. Discussion 4.1. SEM contrast for Au/Si and related systems One main goal of the present paper was to answer the question: from the grey levels of a SEM image, is it possible to determine the spatial distribution of each component of a given sample? From the above developments it has been seen, Fig. 5, that [ka(Au)/ka(Si)] was slightly larger than unity. From Fig. 11 parallel geometry with amax less than 301, it has also been seen that kf(Au) was significantly larger than kf(Si) of Si (n). Finally a comparison between Figs. 12a and b shows that d(Au) is always larger than d(Si) in the energy range of interest (E14E1max for a correct use of spectral distributions derived from Eqs. (3) and (4)): d(Au) is 1o d(Au)o1.5 while 0.4 o d(Si) o1 at E1= 1 keV; 0.7 o d(Au)o1.2 and d(Si)o0.5 at E1 =3 keV. Consequently, all three contributions of Eq. (15) tend to make Au significantly brighter than Si(n) in the SEM images but uncertainties in the respective yield values do not permit a numerical evaluation of the expected contrast. Also gold probably appears to be brighter than Si(p + ) but kf(Si) of Si(p + ) is nearly equal to kf(Au) when amax is o301 and the contrast between the two is less than between Au and Si(n). The present investigation is based only on the use of Fermi energy and work function of Au. Then arguments used for Au may be applied to other metals of similar Fermi energies and work functions, the two being closely related, even if it is not exactly the same. Table 1 shows that Ag, Cu, Pb, Pd, Pt, and Zn present such similarities with Au and the corresponding values of ka and kf are larger than those of Si. The value of their SEE yield, d, may differ from each others because of their specific SE generation and SE transport, but being EF + f dependent, their escape probability, A, is expected to be nearly the same. In addition, most of them have medium or large Z values so the contribution of SE of type II, SEs generated by backscattered electrons, to d, is, for these metals, certainly larger than for Si: an analysis verified by a detailed comparison of results of a compilation [56]. The present analysis is also supported by the visualization of sub-monolayers of Ag on Si surfaces in a off-axis detection system; where a high negative bias of the sample increases the kf(Ag)/kf(Si) ratio, via the deflection of low energy SEs, towards the detector [57]. The reverse arguments may be used for light alkaline metals, Li, Na and K; where ka and kf as well as their SEE yield are less than that of Si. Unfortunately the experimental verification is difficult because these alkaline metals are spontaneously oxidized in air, also under UHV conditions. This oxidation also concerns light metals such as Al or Mg and the corresponding oxides are insulating materials having far larger SEE yields than any metal [36]. Consequently inorganic insulators, MgO, Al2O3,etc., probably appear brighter than other components of SEM images despite the difficulty to evaluate kf because of the uncertain Fermi level

positions in the band gap. Besides their larger SEE yields, oxides have a w value less than 1 eV leading to ka values certainly larger than that of any metal, because their angular and energy distributions, Eq. (5), are narrower than that of other materials [46]. From the experimental point of view the investigated insulating oxides would have a thin film form backed by a conductive substrate having a thickness less than incident electron range to prevent charging effects [36,45]. Concerning semiconductors, other than Si, similar calculations have been performed for Ge (results not shown). Experimental SE spectral distribution for Ge in Eq. (4) are in excellent agreement for w = 4 eV [32] and ka(Au)/ka(Ge) has been founded to be larger than 115% for amax o301. Parameter kf(Ge) is also of the same order of magnitude as kf(Si) and, despite a larger Z value, the SEE yield of Ge may be compared to that of Si [56]. Consequently the above considerations on the expected contrasts between Si and metals such as Ag; Cu; Pb; Pd; Pt, and Zn may be transposed to contrasts between Ge and the same metals. 4.2. Doping contrast When p/n junctions are considered, ratios such as ka(Si-p)/ ka(Si-n) are the same for the two sides of these junctions. The ratios of the detected intensities, derived from Eq. (14), take a simplified form when a same SEE yield value, d, is postulated: Id ðSipÞ=Id ðSinÞ ¼ kf ðSipÞ=kf ðSinÞ

ð19Þ

This ratio may be then easily derived from Fig. 7 for hemispherical potential surfaces and from captions of Fig.11b for a parallel geometry. Keeping in mind that f represents here the distance between Fermi level and vacuum level, it is now easy to evaluate the contrast changes between the two differently doped regions and then of different f values with respect to a detector of work function f(det.). From the usual definition of contrast [35], the simplified expression is: Cð%Þ ¼ 100½Id ðSipÞId ðSinÞ=½Id ðSipÞ þ Id ðSinÞ

ð20Þ

Fig. 14 shows this contrast change for a hemispherical geometry compared to that of a parallel plane geometry. Expressed as a function of f(det.), the contrast changes concern a Si p/n junction with Nn  1015 cm  3 and Np  1016 cm  3 while, derived from Section 3.1, the corresponding work functions’

30 C (p/n) PG αmax< 30°

C (p+/p) PG C (p/n) CH

20

C (p+/p) CH

C (%)

importance of the surface conditions of the target material for instance, with the strong influence of thin absorbed monolayers on work functions and SE escape probabilities [4]. The case of a niobium sample submitted to various vacuum surface treatments provides an excellent example of this effect [55] and a large compilation of SEE yield data [56] shows the relative divergences between different experimental results even performed in instruments dedicated to SEE measurements and quite always operated under ultra-high vacuum conditions. One conclusion is that the increase in f decreases the SEE yield but this decrease is partly compensated by the increase in ka and of the kf work function effect.

251

10 φ (Si-n)

φ (Si-p)

0 2

3

4 φ (det.) (eV)

5

6

Fig. 14. Doping contrast. Expressed as a function of f(det.), contrast changes for a Si p/n junction, C(p/n), with Nn  1015 cm  3 and Np  1016 cm  3, and for two differently p-doped regions, C(p/p + ) with Np  1016 cm  3 and Np  1018 cm  3. Results obtained for a parallel geometry, PG, and for concentric hemispherical surfaces, CH, as derived from Eq. (20) with the numerical values kf used for Figs. 7 and 11.

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values are: f(Si-n)= w + 0.2 eV =3.4 eV; f(Si-p)= f(Si-n)+0.67 eV. With breaks at around f(Si-n)and f(Si-p), the contrast variations are from C 5% up to  15% for a parallel geometry and from C =0% to 15% for a hemispherical geometry. Also shown in Fig. 14, a contrast change of about 1% per doping decade is expected between two differently p-doped regions: Np 1016 cm  3 and Np 1018 cm  3. Then the weight of a work function effect on doping contrast is significant and detectable. Nevertheless, corresponding to Df 0.068 eV per doping decade, the present calculated change of contrast on the p side,  1% per decade, is significantly below values published by D. Venables et al. [13] or experimental results published by Masenelli-Varlot et al. [58]. Consequently, additional contributions to doping contrast remain possible and the present investigation does not invalidate the other causes suggested in literature [9–29].

4.3. Complements The present investigation demonstrates the important role of constant potential lines in the evaluation of parameter kf,compare results of Fig. 7 to results of Fig. 11. A more precise evaluation requires the use of electron optical calculations, specific to the sample surroundings of each experimental arrangement [2,43], e.g. type of microscope, working distance, position, calibration and response function of detectors, etc. These calculations have to be combined with physical effects concerning angular and energy distributions of emitted SEs that are, as illustrated here, specific to each specimen composition. When the specimen holder and (or) the detector are biased the electron optical calculations of SE trajectories have to include the field effect due to work function difference to those of the applied field. In such a situation, the work function difference, limited to a few eV, is negligible for larger biases but the material dependence of kf remains, via the specific SE angular distribution and the collection of low energy SEs being emitted along directions oblique to the field direction, but experiencing large deflections towards this field direction. Despite some possible simplifications, such a global approach is obviously a challenging task, and comparison to SEM images is complicated by the initial assumption being scarcely satisfied [2]: only true SEs are collected by detectors. The present approach may be, in principle, applied to any incident beam energy, limited to physical aspects concerning angular and energy distributions of emitted SEs, but the lack of analytical expressions for SEs generated by low energy incident electrons (see Fig. 1b at E1= 0.2 keV) practically limits the present investigation to beam energies larger than E1max 0.5 to 1 keV, and above. Nevertheless the work function effect remains at any energy as it certainly remains in photoemission experiments at any photon energy even if detailed calculations are not presently possible. Similarly to charging mechanisms in AES and XPS [44,45] where there are complications due to SE emission from one point and retuning to another point, present investigation on p/n junctions omits such complications due to electrons emitted from the p side and arriving on the n side so present approach is probably limited to the early beginning of irradiation and it ignores dynamic effects of beam current, mag., scan speeds giving rise to some contrast reversals as reported by El Gomati et al. [22]. Finally there is a considerable problem in regards to an unknown detector work function and of the influence of change of work function on the SEE yield d of materials having the same composition, Fig. 12a for Au and Fig. 12b for Si. To overcome this key difficulty that is combined to the well-known problem of

contamination in SEM, the most unique strategy would be to clean the sample just before its visualization and to use standards of known composition set side by side to the investigated sample. Sample and standards have to be implemented simultaneously to an efficient cleaning process [59], in order to minimize deviations between the work functions (and then between the SEE yields) of sample compounds and of standards of the same composition. Next a simple comparison between the grey levels of the standards and of the sample may permit a spatial localization of the various components in the SEM images.

5. Conclusion The relationship between SE detected intensity, Id, and incident electron beam intensity, I1, can be expressed as (Eq. 14), Id/I1= kakfd; where ka is the angular fraction of detector collection and kf is a weighting factor depending upon the difference between the work function of the sample and the work function of the detector. Postulating a simplified geometry in the vacuum gap between sample and detector, factors ka and kf have been deduced from analytical expressions of spectral and angular distributions of secondary electrons issued from Au and Si. It has been found that the ratio Id(Au)/Id(Si-n), combined to the corresponding published yield data, was larger than 100% if a coaxial detection is used with a maximum angle of collection amax less than 301 and if the incident beam energy is larger than 0.5–1 keV. The present finding, important for the assignment to various components of semiconducting devices from the corresponding grey levels in their SEM image, has been extended to metals such as Ag; Cu; Pb; Pd; Pt, and Zn that are expected to appear brighter than Si(n) and Ge. Based on similar calculations postulating a parallel geometry in the vacuum gap and depending upon the work function value of the detector, f(det.), doping contrast between the two sides of a Si p–n junction has been evaluated to be C 5–15% between a n-doped region with Nn  1015 cm  3 and a p-doped region with Np  1016 cm  3. The corresponding intensity change is  15–20%, however an undetectable contrast change is expected between two p-doped regions with Np  1016 cm  3 and Np 1018 cm  3. In fact, the important role of constant potential lines in the evaluation of parameter kf has been demonstrated and larger contrasts may result from different geometries in the vacuum gap. Consequently, a work function effect, or voltage contrast, significantly contributes to doping contrast, howbeit other contributions remain possible; and pursuing this matter further, the work function effect has to be added to applied field effects in electron optical calculations of SE trajectories. These calculations would then be specific to the sample surroundings of each experimental arrangement and also specific to the specimen composition in order to take into account, as illustrated here, the physical effects concerning angular and energy distributions of emitted SEs. An additional finding, Section 2.2, is related to the spectral distortion of partially collected SEs when the solid angle of detection, O1, is less than 2p. This remark also holds for SE spectral measurement in dedicated instruments and the corresponding spectral SE measurements have to be suspected when a partial angular collection is operated. The corresponding evaluation of ka, established for various intervals of the emission angle a, is found to be material dependent. If the specimen holder is negatively biased (or the detector is positively biased), ka increases and the collected SE spectral distribution is mainly amplified in the very low energy range. The reason is a larger deflection towards the detector of SEs having less initial kinetic energies and these SEs may be collected even when they have

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been initially emitted along directions oblique to the applied field direction. Finally, a strategy based on the use of standards has been suggested, to assign the various grey levels of a SEM image to the various components of a given sample. These standards, set side by side to the investigated sample, have to be submitted to the same cleaning process as the specimen, in order to minimize deviations between the effective work functions of compounds of the same composition and to minimize contamination effects. Then a simple comparison between the grey levels of the standards and of the sample would permit a spatial localization of the various components. This strategy would be very useful for the SEM investigation of electronic devices where these components are in reduced number and are known a priori.

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