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On the increase of the intensity ratio of doubly charged ions to singly charged ions for liquid gold and copper ion sources
Ultramicroscopy 89 (2001) 69–74
On the increase of the intensity ratio of doubly charged ions to singly charged ions for liquid gold and copper ion sources Yasuhito Gotoh*, Hiroshi Tsuji, Junzo Ishikawa Department of Electronic Science and Engineering, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan Received 18 July 2000; received in revised form 24 November 2000
Abstract The reason for the increase of R21 which is defined by the intensity ratio of the doubly charged ions to the singly charged ions, was studied. Based on the conventional field evaporation theory, an increase in the electric field and=or the source temperature is considered to be attributable. We took the fact that R21 turned to decrease at the higher current regime into consideration and examined whether the change of R21 due to change in the field or temperature would finally meet the criterion for the decrease of R21 : It was concluded that an increase of the source temperature may be a possible reason for the increase of R21 : r 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Although the ion formation mechanism of liquid-metal ion sources (LMISs) is believed to be field evaporation [1], no quantitative argument for this field evaporation has been made so far. This would be mainly because the conventional field evaporation theory does not predict the experimentally obtained charge state distribution of the emitted ions from LMISs. One of the reasons for the difficulties in establishing the model of LMISs is such that only a few systematic investigations, especially from the experimental standpoint, have been performed. We have investigated mass spectra of some element metal ion sources [2] and also alloy ion sources [3] as a function of the source current (total current that *Corresponding author. Tel.: +81-75-753-5355; fax: +8175-751-1576. E-mail address: [email protected] (Y. Gotoh).
flows from the ion emitter), Is : It was found that the intensity ratio of the doubly charged ions to singly charged ions, R21 ; decreased with an increase in Is at a higher current regime. Some researchers claim that the decrease of R21 is attributed to the shielding of the electric field FP due to huge space charge. In such a situation, we attempted to reproduce such a change by a computer simulation [4]. The change in the liquid shape is dominated by Poisson’s equation, fluid dynamic equation, and equation between the electric field and current density. We performed the simulation with a simplified structure: concentric sphere configuration of an extractor and ion-emitter without the jet-like protrusion. Poisson’s equation was solved in a manner similar to that done by Miskovsky and Cutler [5]. The fluid dynamic equation given by Kingham and Swanson [6] was used. In this calculation, we adopted the image hump model [7] as an ion formation mechanism. The solution of the above equations,
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 1 1 8 - 8
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however, did not agree with the experimental results: only monotonical increase of FP was obtained and R21 had no chance to decrease. While we could explain the decrease of R21 ; giving a restriction to the magnitude of kinetic energy term [2]. The results suggested that less significant dependence of the current density J on FP is necessary, unlike the conventional field evaporation theory. We confirmed this criterion with the analysis using different approaches [8]. As will be described later, we calculated the relation required for the decrease of FP from the equation of space charge limited current (SCLC) with small charge approximation [9] and the fluid dynamic equation. Together with these results, we experimentally observed the increase of R21 at very low current regime between 1–30 mA for Au and Cu ion sources. Fig. 1 shows the relation between R21 and Is for Au and Cu ion sources [2]. Some researchers reported similar results with Al ion sources [10,11]. Also, Sn ion source showed a tiny maximum in R21 at a very low current regime [12]. The increase of R21 primarily implies the increase of FP : At the low current regime, the theoretical calculation described above agrees with the experimental result, but we do not find any possibility for the field to decrease. Consequently, something should be changed to give the turning point of FP from an increase to a decrease. We tried to show such a condition in which the change
in R21 finally results in satisfying the criterion for decrease of R21 at a higher current regime. As a result, it was suggested that the temperature increase could be a possible reason for this effect.
2. Criterion for decrease of R21 Prior to the analysis of the increase of R21 ; we should consider the criterion for the decrease of R21 : We derived the criterion with the following equations. The first one is the SCLC [9] c pffiffiffiffi ð1Þ FL2 FP2 ¼ J V ; e0 FP are the Laplace and Poisson where FL and pffiffiffiffiffiffiffiffiffiffiffi fields, c ¼ 43 2m=e; e0 is the permittivity of vacuum, J is the current density, and V is the applied voltage. The second is the fluid dynamic equation at the top of the cylindrical jet: 1 2 1 J 2 2g SF ¼ e0 FP ¼ r þ ¼ K þ ST ; ð2Þ 2 2 eNv ra where r is the mass density, Nv is the atomic density, and ra is the apex radius. SF ; K; and ST represent the electric field stress, kinetic energy per unit volume, and surface tension stress, respectively. From these equations, we can derive the criterion for decrease of R21 or FP : Details of the derivation are given elsewhere [8], and we explain them here briefly. In Eq. (1), the Laplace field can be expressed by V FL ¼ ; ð3Þ bra where b is a coefficient weakly dependent on ra : Differentiating Eq. (1) into which Eq. (3) has been substituted, and neglecting the change in FP ; we obtain the relation between J; V; and ra : 3dV dra dJ ¼2 þ : 2V J ra
ð4Þ
When we increase the applied voltage (dV > 0), the right-hand side of Eq. (4) should be positive. The critical relation is thus given by dV=V ¼ 0: By differentiating Eq. (2), we obtain Fig. 1. Experimentally obtained relation between R21 and Is :
dJ SF dFP ST dra ¼ þ : J K FP 2K ra
ð5Þ
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Y. Gotoh et al. / Ultramicroscopy 89 (2001) 69–74
has the form
"
1=2
Q1 s01 FP J ¼ eNs n exp kB T
#
h i 1=2 ¼ J0 exp s1 FP ; ð7Þ
where e is the electronic charge, Ns is the surface atomic density, n is the vibration frequency of the surface atom, Q1 is the difference of the potential between an atom in a solid and in vacuum, J0 is the current density preexponential, s01 is a coefficient, and s1 ¼ s01 =kB T: J0 includes the temperature term but here we assume no temperature change, thus we can write J simply by the above equation. Since Fig. 2. Power of the critical relation as a function of ST =K:
Substituting this equation into Eq. (4), the critical equation finally becomes, 1þð3ST =KÞ=ð4þST =KÞ
J ¼ FP
¼ FPN :
ð6Þ
Fig. 2 shows the power N as a function of ST =K: We have some different situations with the magnitude of ST =K in Eq. (2): (a) static model suggested by Forbes [13]; (b) dynamic model suggested by Kingham and Swanson [6]; and (c) Intermediate model between (a) and (b) suggested by us [2]. We assigned these as cases III, I, and II [8], according to the order from the higher current to the lower current. The cases I, II, and III correspond to ST =K ¼ 0; 1; and N; and these cases are shown in the figure as ‘K–S’, ‘G’, and ‘F’. At the higher current regime, the dependence of J on FP should be lower than the critical relation. N shows monotonical increase with ST =K but is between 1 and 4 for any ST =K values.
dJ 1 1=2 dFP ¼ s1 F P ; J 2 FP
ð8Þ
the relation between J and FP varies with FP : Based on the conventional model, however, change in FP is negligible, and no change in 1=2 temperature is assumed, thus s1 FP would be constant. Under such a situation, we consider the change in the sign of dFP : Since the ST =K changes from N to 0 with an increase in the source current, the current increase is shown as a motion from right to left in Fig. 2. In the conventional model, which does not assume change in temperature, this motion becomes almost lateral as shown in Fig. 3(a). If we start from the point where dFP > 0; which is shown by the upper arrow in the figure, we do not find any possibility to turn to dFP o0: If we start from the point which may cross the critical relation as shown in the lower arrow, the change in FP would first be a decrease and then an increase. Both of these do not agree with the experimental results. Thus, a constant temperature assumption does not show any agreement with the experimental results. Here, we introduce the effect of the change in temperature. If the temperature changes, the dependence of J on FP changes. 1=2
3. Model of increase in R21 at a low current regime
dJ s0 @FP Q1 s01 FP @kB T ¼ 1 : þ J kB T 2kB T F 1=2 kB T
ð9Þ
P
3.1. Qualitative discussion The formulation of J due to field evaporation theory based on the image hump model
According to the conventional theory, FP is almost 1=2 equal to the evaporation field thus Q1 s01 FP B0: If we assume that the temperature increase is proportional to the ion current, the operational
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with the changes in kB T and FP are
dJ 1 s01 1=2 @FP 1=2 @kB T ¼ FP þ ðQ1 s01 FP Þ ; J kB T 2 kB T FP dR21 1 ¼ kB T R21 0
s21 1=2 @FP 1=2 @kB T 0 F þ ðV2 f s21 FP Þ : kB T 2 P FP ð10Þ The above equations imply that the effects of increase both in temperature and in field result in 1=2 increase of R21 or J; because Q1 s01 FP and V2 1=2 0 f s21 FP are both positive for Au and Cu. Assuming Au, in which the operational field would be about 52 V nm1 and the operational temperature of about 1400 K ð0:12 eVÞ; these equations can be written as dJ ¼ 0:69@FP þ 2:8@kB T; J dR21 ¼ 1:3@FP þ 32@kB T; R21
Fig. 3. Change in dFP as the source current increases: (a) at constant temperature, and (b) at increasing temperature with current.
mode moves downward, for example, as shown in Fig. 3(b). We can start from the point where dFP > 0; and cross a point, which means the increase to decrease of R21 : To show the exact trace of this point, it is necessary to perform a detailed simulation, but we can qualitatively show how R21 first increases and then decreases. In order to obtain the relation of JpFP4 ; the temperature of 0.5–1:0 eV is necessary. 3.2. Quantitative discussion In this section, we consider the changes in a quantitative manner. The changes of J and R21
ð11Þ
where FP and kB T are in V nm1 and eV. The physical parameters of Au are taken from Tsong’s paper [14]. The increase in kB T will cause the decrease of FP : If the deformation of the liquid due to change in the operational parameters, is maintained so as to have the same current density, that is, dJ ¼ 0; @FP ¼ 0:41@kB T: In such a case, dR21 =R21 becomes 29@kB T: This means R21 can increase with less significant increase in J:
4. Another evidence of temperature increase 4.1. Intensity of molecular ions In order to confirm the elevation of the temperature, we investigated the intensity of molecular ions. The method of measurement is the same as that reported previously [2]. Here, we define Rdm as the current intensity ratio of singly charged dimer ions to singly charged monomer ions. Also, we define Rtm for trimer ions in a similar manner. Fig. 4 shows Rdm and Rtm as a function of Is together with R21 : Rdm and Rtm are
Y. Gotoh et al. / Ultramicroscopy 89 (2001) 69–74
73
dimer and trimer, respectively. If H Eb2 and 2H Eb3 do not differ so much, the behavior of Rdm and Rtm would be similar; at least, the binding energy for three atoms would be larger than two atoms, thus the ratio of ð2H Eb3 Þ=ðH Eb2 Þ would be approximately 2. The results shown in Fig. 4 can be interpreted as that the temperature becomes higher until the R21 decreases. This result suggests that the discussion in the previous section holds. 4.2. Increase of R21 and thermal conductivity Fig. 4. Rdm and Rtm as a function of Is together with R21 :
both very small at the current of 50 mA; but rapidly increase until Is becomes 100 mA; where R21 begins to decrease. At higher current, Rdm and Rtm are saturated. Although the mechanism for molecular ion formation has not been clarified, it would be possible to consider it quite similar to that of atomic ions: introduction of Born–Harber energy cycle to the molecular ions. We found that this model provides a qualitative explanation for the emitted ions [15,16]. Some experimental results on the energy distribution [17] suggests field ionization of the evaporated molecules, but the formation of molecular ions of several mA by field ionization requires a high molecular gas density; thus, it is not possible to determine whether the molecular ions are formed completely by field ionization. In any of these models, it is necessary to increase the temperature for an increase of Rdm : For the former model, Rdm and Rtm can be written as
H Eb2 Rdm ¼ n2 exp ; kB T
2H Eb3 Rtm ¼ n3 exp ; kB T
ð12Þ
where n2 and n3 are probabilities for the formation of molecules of dimer and trimer, H is the heat of sublimation, Eb2 and Eb3 are binding energies of
The increase of R21 at a lower current could be observed for the metals which possess high thermal conductivity. Cu and Au have 5–7 times higher thermal conductivity than Sn. Since the temperature is almost inversely proportional to the thermal conductivity [18], the increase in temperature for Cu and Au is slow with respect to the current increase. For those with lower thermal conductivity, the temperature rapidly increases with the current, so saturation may occur at very low current, resulting in a very tiny peak of R21 :
5. Conclusion The reason for the increase of the intensity ratio of doubly charged ions to the singly charged ions was discussed and it was suggested that the increase of R21 could be attributed to an increase in the temperature rather than the increase of the electric field. At the low current regime, the temperature would be low and the electric field would increase as the current increases. At a certain current, the temperature increase will stop because the current density begins to be lowered. This could be attributed to the fact that the dependence of the current density on the field becomes lower and satisfies the criterion for the decrease of the electric field. The present argument has been based on the imagehump model. Further investigation with a combination of charge-exchange model and post-ionization model can be conducted to confirm the present hypothesis.
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References [1] L.W. Swanson, A.E. Bell, Liquid metal ion sources, in: I.G. Brown (Ed.), Technology and Physics of Ion Sources, Academic Press, New York, 1989, p. 313 (Chapter 15). [2] Y. Gotoh, T. Kashiwagi, H. Tsuji, J. Ishikawa, Appl. Phys. A 64 (1997) 523. [3] Y. Gotoh, H. Tsuji, J. Ishikawa, Ultramicroscopy 73 (1998) 83. [4] Y. Gotoh, unpublished data. [5] N.M. Miskovsky, P.H. Cutler, Appl. Phys. A 28 (1982) 73. [6] D.R. Kingham, L.W. Swanson, Appl. Phys. A 34 (1984) 123. [7] T.T. Tsong, Atom-Probe Field Ion Microscopy, Cambridge University Press, Cambridge, 1990. [8] Y. Gotoh, H. Tsuji, J. Ishikawa, Rev. Sci. Instr. 71 (2000) 725.
[9] G.L.R. Mair, J. Phys. D 17 (1984) 2323. [10] A.E. Bell, G.A. Schwind, L.W. Swanson, J. Appl. Phys. 53 (1982) 4602. [11] Y. Torii, H. Yamada, Jpn. J. Appl. Phys. 21 (1982) L132. [12] Y. Gotoh, Master Thesis, Kyoto University, 1987, unpublished (in Japanese). [13] R.G. Forbes, Appl. Surf. Sci. 67 (1993) 9. [14] T.T. Tsong, Surf. Sci. 70 (1978) 211. [15] T. Kashiwagi, Master Thesis, Kyoto University, 1988, unpublished (in Japanese). [16] Y. Gotoh, H. Tsuji, J. Ishikawa, Jpn. J. Appl. Phys. 35 (1996) 3670. [17] R.J. Culbertson, G.H. Robertson, T. Sakurai, J. Vac. Sci. Technol. 16 (1979) 1868. [18] G.L.R. Mair, K.L. Aitken, J. Phys. D 17 (1984) L13.
On the increase of the intensity ratio of doubly charged ions to singly charged ions for liquid gold and copper ion sources Yasuhito Gotoh*, Hiroshi Tsuji, Junzo Ishikawa Department of Electronic Science and Engineering, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan Received 18 July 2000; received in revised form 24 November 2000
Abstract The reason for the increase of R21 which is defined by the intensity ratio of the doubly charged ions to the singly charged ions, was studied. Based on the conventional field evaporation theory, an increase in the electric field and=or the source temperature is considered to be attributable. We took the fact that R21 turned to decrease at the higher current regime into consideration and examined whether the change of R21 due to change in the field or temperature would finally meet the criterion for the decrease of R21 : It was concluded that an increase of the source temperature may be a possible reason for the increase of R21 : r 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Although the ion formation mechanism of liquid-metal ion sources (LMISs) is believed to be field evaporation [1], no quantitative argument for this field evaporation has been made so far. This would be mainly because the conventional field evaporation theory does not predict the experimentally obtained charge state distribution of the emitted ions from LMISs. One of the reasons for the difficulties in establishing the model of LMISs is such that only a few systematic investigations, especially from the experimental standpoint, have been performed. We have investigated mass spectra of some element metal ion sources [2] and also alloy ion sources [3] as a function of the source current (total current that *Corresponding author. Tel.: +81-75-753-5355; fax: +8175-751-1576. E-mail address: [email protected] (Y. Gotoh).
flows from the ion emitter), Is : It was found that the intensity ratio of the doubly charged ions to singly charged ions, R21 ; decreased with an increase in Is at a higher current regime. Some researchers claim that the decrease of R21 is attributed to the shielding of the electric field FP due to huge space charge. In such a situation, we attempted to reproduce such a change by a computer simulation [4]. The change in the liquid shape is dominated by Poisson’s equation, fluid dynamic equation, and equation between the electric field and current density. We performed the simulation with a simplified structure: concentric sphere configuration of an extractor and ion-emitter without the jet-like protrusion. Poisson’s equation was solved in a manner similar to that done by Miskovsky and Cutler [5]. The fluid dynamic equation given by Kingham and Swanson [6] was used. In this calculation, we adopted the image hump model [7] as an ion formation mechanism. The solution of the above equations,
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 1 1 8 - 8
70
Y. Gotoh et al. / Ultramicroscopy 89 (2001) 69–74
however, did not agree with the experimental results: only monotonical increase of FP was obtained and R21 had no chance to decrease. While we could explain the decrease of R21 ; giving a restriction to the magnitude of kinetic energy term [2]. The results suggested that less significant dependence of the current density J on FP is necessary, unlike the conventional field evaporation theory. We confirmed this criterion with the analysis using different approaches [8]. As will be described later, we calculated the relation required for the decrease of FP from the equation of space charge limited current (SCLC) with small charge approximation [9] and the fluid dynamic equation. Together with these results, we experimentally observed the increase of R21 at very low current regime between 1–30 mA for Au and Cu ion sources. Fig. 1 shows the relation between R21 and Is for Au and Cu ion sources [2]. Some researchers reported similar results with Al ion sources [10,11]. Also, Sn ion source showed a tiny maximum in R21 at a very low current regime [12]. The increase of R21 primarily implies the increase of FP : At the low current regime, the theoretical calculation described above agrees with the experimental result, but we do not find any possibility for the field to decrease. Consequently, something should be changed to give the turning point of FP from an increase to a decrease. We tried to show such a condition in which the change
in R21 finally results in satisfying the criterion for decrease of R21 at a higher current regime. As a result, it was suggested that the temperature increase could be a possible reason for this effect.
2. Criterion for decrease of R21 Prior to the analysis of the increase of R21 ; we should consider the criterion for the decrease of R21 : We derived the criterion with the following equations. The first one is the SCLC [9] c pffiffiffiffi ð1Þ FL2 FP2 ¼ J V ; e0 FP are the Laplace and Poisson where FL and pffiffiffiffiffiffiffiffiffiffiffi fields, c ¼ 43 2m=e; e0 is the permittivity of vacuum, J is the current density, and V is the applied voltage. The second is the fluid dynamic equation at the top of the cylindrical jet: 1 2 1 J 2 2g SF ¼ e0 FP ¼ r þ ¼ K þ ST ; ð2Þ 2 2 eNv ra where r is the mass density, Nv is the atomic density, and ra is the apex radius. SF ; K; and ST represent the electric field stress, kinetic energy per unit volume, and surface tension stress, respectively. From these equations, we can derive the criterion for decrease of R21 or FP : Details of the derivation are given elsewhere [8], and we explain them here briefly. In Eq. (1), the Laplace field can be expressed by V FL ¼ ; ð3Þ bra where b is a coefficient weakly dependent on ra : Differentiating Eq. (1) into which Eq. (3) has been substituted, and neglecting the change in FP ; we obtain the relation between J; V; and ra : 3dV dra dJ ¼2 þ : 2V J ra
ð4Þ
When we increase the applied voltage (dV > 0), the right-hand side of Eq. (4) should be positive. The critical relation is thus given by dV=V ¼ 0: By differentiating Eq. (2), we obtain Fig. 1. Experimentally obtained relation between R21 and Is :
dJ SF dFP ST dra ¼ þ : J K FP 2K ra
ð5Þ
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Y. Gotoh et al. / Ultramicroscopy 89 (2001) 69–74
has the form
"
1=2
Q1 s01 FP J ¼ eNs n exp kB T
#
h i 1=2 ¼ J0 exp s1 FP ; ð7Þ
where e is the electronic charge, Ns is the surface atomic density, n is the vibration frequency of the surface atom, Q1 is the difference of the potential between an atom in a solid and in vacuum, J0 is the current density preexponential, s01 is a coefficient, and s1 ¼ s01 =kB T: J0 includes the temperature term but here we assume no temperature change, thus we can write J simply by the above equation. Since Fig. 2. Power of the critical relation as a function of ST =K:
Substituting this equation into Eq. (4), the critical equation finally becomes, 1þð3ST =KÞ=ð4þST =KÞ
J ¼ FP
¼ FPN :
ð6Þ
Fig. 2 shows the power N as a function of ST =K: We have some different situations with the magnitude of ST =K in Eq. (2): (a) static model suggested by Forbes [13]; (b) dynamic model suggested by Kingham and Swanson [6]; and (c) Intermediate model between (a) and (b) suggested by us [2]. We assigned these as cases III, I, and II [8], according to the order from the higher current to the lower current. The cases I, II, and III correspond to ST =K ¼ 0; 1; and N; and these cases are shown in the figure as ‘K–S’, ‘G’, and ‘F’. At the higher current regime, the dependence of J on FP should be lower than the critical relation. N shows monotonical increase with ST =K but is between 1 and 4 for any ST =K values.
dJ 1 1=2 dFP ¼ s1 F P ; J 2 FP
ð8Þ
the relation between J and FP varies with FP : Based on the conventional model, however, change in FP is negligible, and no change in 1=2 temperature is assumed, thus s1 FP would be constant. Under such a situation, we consider the change in the sign of dFP : Since the ST =K changes from N to 0 with an increase in the source current, the current increase is shown as a motion from right to left in Fig. 2. In the conventional model, which does not assume change in temperature, this motion becomes almost lateral as shown in Fig. 3(a). If we start from the point where dFP > 0; which is shown by the upper arrow in the figure, we do not find any possibility to turn to dFP o0: If we start from the point which may cross the critical relation as shown in the lower arrow, the change in FP would first be a decrease and then an increase. Both of these do not agree with the experimental results. Thus, a constant temperature assumption does not show any agreement with the experimental results. Here, we introduce the effect of the change in temperature. If the temperature changes, the dependence of J on FP changes. 1=2
3. Model of increase in R21 at a low current regime
dJ s0 @FP Q1 s01 FP @kB T ¼ 1 : þ J kB T 2kB T F 1=2 kB T
ð9Þ
P
3.1. Qualitative discussion The formulation of J due to field evaporation theory based on the image hump model
According to the conventional theory, FP is almost 1=2 equal to the evaporation field thus Q1 s01 FP B0: If we assume that the temperature increase is proportional to the ion current, the operational
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with the changes in kB T and FP are
dJ 1 s01 1=2 @FP 1=2 @kB T ¼ FP þ ðQ1 s01 FP Þ ; J kB T 2 kB T FP dR21 1 ¼ kB T R21 0
s21 1=2 @FP 1=2 @kB T 0 F þ ðV2 f s21 FP Þ : kB T 2 P FP ð10Þ The above equations imply that the effects of increase both in temperature and in field result in 1=2 increase of R21 or J; because Q1 s01 FP and V2 1=2 0 f s21 FP are both positive for Au and Cu. Assuming Au, in which the operational field would be about 52 V nm1 and the operational temperature of about 1400 K ð0:12 eVÞ; these equations can be written as dJ ¼ 0:69@FP þ 2:8@kB T; J dR21 ¼ 1:3@FP þ 32@kB T; R21
Fig. 3. Change in dFP as the source current increases: (a) at constant temperature, and (b) at increasing temperature with current.
mode moves downward, for example, as shown in Fig. 3(b). We can start from the point where dFP > 0; and cross a point, which means the increase to decrease of R21 : To show the exact trace of this point, it is necessary to perform a detailed simulation, but we can qualitatively show how R21 first increases and then decreases. In order to obtain the relation of JpFP4 ; the temperature of 0.5–1:0 eV is necessary. 3.2. Quantitative discussion In this section, we consider the changes in a quantitative manner. The changes of J and R21
ð11Þ
where FP and kB T are in V nm1 and eV. The physical parameters of Au are taken from Tsong’s paper [14]. The increase in kB T will cause the decrease of FP : If the deformation of the liquid due to change in the operational parameters, is maintained so as to have the same current density, that is, dJ ¼ 0; @FP ¼ 0:41@kB T: In such a case, dR21 =R21 becomes 29@kB T: This means R21 can increase with less significant increase in J:
4. Another evidence of temperature increase 4.1. Intensity of molecular ions In order to confirm the elevation of the temperature, we investigated the intensity of molecular ions. The method of measurement is the same as that reported previously [2]. Here, we define Rdm as the current intensity ratio of singly charged dimer ions to singly charged monomer ions. Also, we define Rtm for trimer ions in a similar manner. Fig. 4 shows Rdm and Rtm as a function of Is together with R21 : Rdm and Rtm are
Y. Gotoh et al. / Ultramicroscopy 89 (2001) 69–74
73
dimer and trimer, respectively. If H Eb2 and 2H Eb3 do not differ so much, the behavior of Rdm and Rtm would be similar; at least, the binding energy for three atoms would be larger than two atoms, thus the ratio of ð2H Eb3 Þ=ðH Eb2 Þ would be approximately 2. The results shown in Fig. 4 can be interpreted as that the temperature becomes higher until the R21 decreases. This result suggests that the discussion in the previous section holds. 4.2. Increase of R21 and thermal conductivity Fig. 4. Rdm and Rtm as a function of Is together with R21 :
both very small at the current of 50 mA; but rapidly increase until Is becomes 100 mA; where R21 begins to decrease. At higher current, Rdm and Rtm are saturated. Although the mechanism for molecular ion formation has not been clarified, it would be possible to consider it quite similar to that of atomic ions: introduction of Born–Harber energy cycle to the molecular ions. We found that this model provides a qualitative explanation for the emitted ions [15,16]. Some experimental results on the energy distribution [17] suggests field ionization of the evaporated molecules, but the formation of molecular ions of several mA by field ionization requires a high molecular gas density; thus, it is not possible to determine whether the molecular ions are formed completely by field ionization. In any of these models, it is necessary to increase the temperature for an increase of Rdm : For the former model, Rdm and Rtm can be written as
H Eb2 Rdm ¼ n2 exp ; kB T
2H Eb3 Rtm ¼ n3 exp ; kB T
ð12Þ
where n2 and n3 are probabilities for the formation of molecules of dimer and trimer, H is the heat of sublimation, Eb2 and Eb3 are binding energies of
The increase of R21 at a lower current could be observed for the metals which possess high thermal conductivity. Cu and Au have 5–7 times higher thermal conductivity than Sn. Since the temperature is almost inversely proportional to the thermal conductivity [18], the increase in temperature for Cu and Au is slow with respect to the current increase. For those with lower thermal conductivity, the temperature rapidly increases with the current, so saturation may occur at very low current, resulting in a very tiny peak of R21 :
5. Conclusion The reason for the increase of the intensity ratio of doubly charged ions to the singly charged ions was discussed and it was suggested that the increase of R21 could be attributed to an increase in the temperature rather than the increase of the electric field. At the low current regime, the temperature would be low and the electric field would increase as the current increases. At a certain current, the temperature increase will stop because the current density begins to be lowered. This could be attributed to the fact that the dependence of the current density on the field becomes lower and satisfies the criterion for the decrease of the electric field. The present argument has been based on the imagehump model. Further investigation with a combination of charge-exchange model and post-ionization model can be conducted to confirm the present hypothesis.
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References [1] L.W. Swanson, A.E. Bell, Liquid metal ion sources, in: I.G. Brown (Ed.), Technology and Physics of Ion Sources, Academic Press, New York, 1989, p. 313 (Chapter 15). [2] Y. Gotoh, T. Kashiwagi, H. Tsuji, J. Ishikawa, Appl. Phys. A 64 (1997) 523. [3] Y. Gotoh, H. Tsuji, J. Ishikawa, Ultramicroscopy 73 (1998) 83. [4] Y. Gotoh, unpublished data. [5] N.M. Miskovsky, P.H. Cutler, Appl. Phys. A 28 (1982) 73. [6] D.R. Kingham, L.W. Swanson, Appl. Phys. A 34 (1984) 123. [7] T.T. Tsong, Atom-Probe Field Ion Microscopy, Cambridge University Press, Cambridge, 1990. [8] Y. Gotoh, H. Tsuji, J. Ishikawa, Rev. Sci. Instr. 71 (2000) 725.
[9] G.L.R. Mair, J. Phys. D 17 (1984) 2323. [10] A.E. Bell, G.A. Schwind, L.W. Swanson, J. Appl. Phys. 53 (1982) 4602. [11] Y. Torii, H. Yamada, Jpn. J. Appl. Phys. 21 (1982) L132. [12] Y. Gotoh, Master Thesis, Kyoto University, 1987, unpublished (in Japanese). [13] R.G. Forbes, Appl. Surf. Sci. 67 (1993) 9. [14] T.T. Tsong, Surf. Sci. 70 (1978) 211. [15] T. Kashiwagi, Master Thesis, Kyoto University, 1988, unpublished (in Japanese). [16] Y. Gotoh, H. Tsuji, J. Ishikawa, Jpn. J. Appl. Phys. 35 (1996) 3670. [17] R.J. Culbertson, G.H. Robertson, T. Sakurai, J. Vac. Sci. Technol. 16 (1979) 1868. [18] G.L.R. Mair, K.L. Aitken, J. Phys. D 17 (1984) L13.