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The direct force on electromigrating hydrogen isotopes
Journal of Alloys and Compounds 288 (1999) 1–6
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The direct force on electromigrating hydrogen isotopes a, b A. Lodder *, K. Hashizume a
Faculty of Sciences, Division Physics and Astronomy, Free University, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands b Department of Advanced Energy Engineering Science, Kyushu University, Kasuga, Fukuoka 816 -8580, Japan Received 22 October 1998
Abstract The present understanding of the magnitude of the direct force on hydrogen and its isotopes deuterium and tritium is discussed in the light of new experimental data and existing theoretical formulations. By this it is proven, that the direct valence of hydrogen can be significantly different from the unscreened jellium value of unity. Furthermore, a more detailed interpretation of a resistivity dependent direct valence is given, which clarifies the controversy around that subject. Selfconsistent calculations of the electronic charge density at and around an interstitial, at all positions along its electromigration jump path, are required to come to a more complete understanding of the measured results. 1999 Elsevier Science S.A. All rights reserved. Keywords: Hydrogen isotopes; Direct valence; Electromigration
1. Introduction Electromigration occurs when a metal is subjected to an electric field. The random diffusive motion of atoms then gets a preferential direction. The force on an atom induced by the electric field consists of two components, a direct force F direct and a wind force F wind . The direct force is due to a net charge of the migrating atom, while the wind force is due to scattering of the current-carrying electrons off the atom. Both forces are proportional to the electric field and can be characterized by a valence [1] F 5 (Zdirect 1 Zwind )eE 5 Z*eE.
(1)
The effective valence Z* is the measurable quantity. Although consensus consists about the form of Eq. (1), the present understanding of the two contributions is quite different. The wind force, being proportional to the current density, is found to be inversely proportional to the sample resistivity r, independently of the complexity of the theoretical formulation used in the description of the total force. One conventionally writes Z* as K Z* 5 Zdirect 1 ], r
(2)
in which the temperature independent proportionality *Corresponding author. E-mail address: [email protected] (A. Lodder)
constant K is determined by the scattering of the electrons by the migrating atom. In addition, a microscopic quantum-mechanical expression for the wind force is available [2], which traces back to the pioneering work of Bosvieux and Friedel [3]. This expression has been applied recently for ab initio calculations of the wind valence of interstitial atoms such as hydrogen [4], and of substitutionally migrating atoms [5,6], in numerous FCC and BCC metals. In contrast to the consensus concerning the wind force, and its computational accessibility, the magnitude of the direct force has been the subject of a long-standing controversy. Either one is in favour of complete screening, so that only the wind force remains, or one defends that the direct force is effectively unscreened. The controversy started in 1962, being induced by the screening prediction of Bosvieux and Friedel [3], and has not come to an end completely yet [7,8]. Unfortunately, an attempt to decide on this issue by measuring the driving force on hydrogen in the transition metals V, Nb, and Ta turned out to be not conclusive [9,10]. Sorbello [7] even expresses the possibility that the conventional distinction of two contributions to the driving force, Eq. (1), eventually may turn out to be less meaningful than it has been assumed to be up to now. In the meantime, however, new efforts have been made on the experimental side [11–17]. These efforts were partially challenged by a recent prediction, that complete screening holds only for the asymptotic electron-impurity model, while in real samples the screening is hindered by the
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overwhelmingly present electron–phonon interaction [18]. So a new look at the present status of our understanding is justified. The present status is characterized by the fact that, while conventionally Zdirect is assumed to be a constant, a relaxation-time dependent direct valence [19] and a resistivity-dependent expression [10] Zi Zdirect 5 ]]]2 1 1 (b /r )
(3)
have been proposed, in the latter Zi and r being the unscreened ionic valency and the contribution to the sample resistivity due to electron–phonon scattering respectively, while b is an adjustable parameter. The relaxation-time dependent expression was based on an old analysis valid to the lowest order in the electron-impurity scattering potential only, and turned out to be unsuccessful in describing the data available [9]. The resistivity-dependent expression, Eq. (3), is based on a more complete analysis [18]. According to that expression Zdirect can become much smaller than the unscreened valency at low temperatures, because then r becomes small and the denominator can become much larger than unity, while at higher temperatures, in fact at temperatures at which electromigration experiments are done, Zdirect will approach Zi . It can be considered as a compromise between the two extremes defended in the literature [8], and it has been presented so [18]. It indeed describes the available data, while no Zdirect values larger than Zi enter, as they do in a constant Zdirect analysis. However a question remains concerning the meaning of the parameter b in Eq. (3), while the data did not discriminate convincingly between the two possible descriptions. First, in Section 2, the new results for hydrogen migra-
tion in pure Pd and two alloys with Ag will be analysed. It will turn out to be necessary to reinterpret the jellium based Eq. (3), in order to make it applicable to real metallic systems. In Section 3 the electromigration of hydrogen and its isotopes in the transition metals V, Nb, and Ta will be analysed and discussed. The paper ends with a summary and concluding remarks. As far as the units are concerned, the parameters K and b, and the resistivity r will be expressed in mV cm.
2. Hydrogen migration in pure Pd and alloys with Ag In Fig. 1 the effective valence is plotted as a function of 100 /r for hydrogen migration in pure Pd and the alloys Pd 77 Ag 33 and Pd 50 Ag 50 , systems which have been investigated experimentally by Pietrzak et al. [16,17]. The bold lines show the constant–Zdirect fit according to Eq. (2). The corresponding Zdirect and Kc values are given in Table 1. The label c has been added to the slope parameter, referring to the kind of fit. One sees an increased negative slope on adding silver to palladium, represented by increasingly negative Kc values. For pure silver a Kc value lower than 2100 is expected [4]. The thinner lines in Fig. 1 follow from a fit, using Eq. (3) in Eq. (2), and taking Zi 5 1. These three lines obviously end at the value Z* 5 1 in the origin. In Table 1 the parameters according to this fit are indicated by b and KZ i 51 . The conclusion is clear. The latter fit, using the jellium value Zi 5 1 for the unscreened direct valence, fails. For Pd(H) one still does not see anything disturbing, but this is certainly due to the fact that the bold straight line already ends at a Z* value close to unity, at 1.05. But for the alloys the fit is no good. So, either the idea of a resistivity-dependent direct force is
Fig. 1. The effective valence Z* of hydrogen in pure Pd and two alloys with Ag, plotted as a function of 100 /r.
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Table 1 The fit parameters according to three different fits, a constant–Zdirect fit, a Zi 5 1 fit leading to values for the parameters b and KZ i 51 and a fit using b51 / 2rmin in Eq. (3), leading to the parameters Zib and Kb System
Zdirect
Kc
b
KZ i 51
Zib
Kb
b ( 5 1 / 2rmin )
Pd Pd 77 Ag 33 Pd 50 Ag 50
1.05 0.73 0.79
23.81 211.82 229.4
3.65 12.32 18.93
22.17 212.01 228.1
0.96 0.65 0.72
20.54 27.85 223.8
5.75 9.52 13.64
wrong, or one has to consider the origin of Eq. (3) more consequently than the way it was done in the earlier fit [10]. We will show that the latter option is the correct one. However, since the subject has been quite controversial [20–24], and it apparently still is [7,19,25,27], we support our view by a brief historical account. As mentioned in Section 1, the prediction by Bosvieux and Friedel [3] of a screened direct force on an interstitial impurity formed the start of a controversy. Since, for a long time, experiments were too difficult to decide, Kumar and Sorbello introduced a linear response approach in the field, which gives an exact formal expression for the driving force, in the hope of finding a conclusive answer [28]. Many efforts were made in order to evaluate the formal expression for the electron-impurity model or, equivalently, the impurity-in-a-jellium model. This model system is supposed to simulate a metallic system with a low concentration of interstitial impurities. Most of the treatments ended with an expression, which contained the (interstitial) impurity potential only to lowest order. As far as the wind force is concerned this did not form a problem, because the lowest order expression for this scattering property has the same form as the exact expression. The latter reduces to the lowest order result simply by replacing a t-matrix element with a matrix element for the potential v. That is why there never has been a problem with the wind force. However, the screening contribution to the direct force, evaluated to lowest order, was small compared to the unscreened force. In spite of the fact, that in the same period an exact treatment was published supporting Bosvieux and Friedel’s result [24], the matter remained undecided. One of the arguments used was, that for a migrating atom, undergoing a dynamical process, the screening is not effective, so that the direct force acts on the bare nucleus only. This may be true to a certain extent, but we did not find an estimate of this dynamical effect in the literature. Therefore we indicate how this can be done. Confining ourselves to hydrogen, one can use its zero-point-motion energy of about 0.025 eV, or an activation energy of the order of 0.1 eV. Taking these energies equal to 1 / 2mv 2 , one finds velocities of 2310 3 m / s and 4310 3 m / s, respectively, to be compared with the Fermi velocity of a metallic electron of about 10 6 m / s. Using the activation energy in an energy-time uncertainty relation, and interpreting this time as a jump time, one finds an upper limit
of the velocity of about 4310 4 m / s over a jump path of ˚ In conclusion, the ion velocity is at least two orders 2.5 A. of magnitude smaller than the electron velocity, so that the screening reduction due to the dynamical effect is small. Interestingly, this argument against screening applies both to lowest order and exact treatments, because the force calculated in the electron-impurity model is independent of the position of the impurity. At present a newly derived exact result is available, which again forms a support of Bosvieux and Friedel [18]. In presenting this result it was emphasized strongly, that it only holds for the electron-impurity model. It could be derived rather straightforwardly after advantage was taken of a simplification of the formal linear-response expression, which was not discerned fully in earlier work. Old lowest order results were reproduced without failure, but the exact result led to screening [26]. Objections were raised [20,21], which were mainly based on obsolete arguments developed in the emotional discussion carried on at the end of the 1970s [8], but it formed no problem to refute these points [22,23]. However, it was pointed out, that the simple electron-impurity model is not a valid representation of the samples investigated experimentally. At and above room temperature electron-phonon interaction is dominantly present, forming a serious hindrance for the screening. As soon as this was realized, it was phrased, that the exact screening result, holding exactly for the dilute electron-impurity model system, is fading away in its relevance to the real systems encountered in an experimental situation [18]. By that the controversy can be considered as being clarified. In fact, Eq. (3) forms the bridge between the two extreme views. The question remains, how to use it. First, it is based on an approach, in which free electrons interact with migrating impurities and with phonons. That is why Zi , representing the unscreened ionic valency, for hydrogen isotopes is equal to unity. Further, the theory does not provide means to calculate the parameter b by way of ab initio methods, so that one has to make a choice. In the first attempt to describe the experimental data, Zi was taken to be 1, and b served as a fitting parameter [10]. Surprisingly, a good fit could be obtained for all three transition metals considered, namely the metals V, Nb, and Ta, but the author was not too critical concerning the magnitude of b. For Nb, for example, b was found to be equal to 53 mV cm, while the sample resistivity values lay in the range of 16 to 27 mV
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cm, the lowest value corresponding to a temperature of 276 K. So, for this fit the direct valence is 0.08 at 276 K, which implies a huge screening, contrary to the expectation expressed above. Regarding this expectation, and the experimental results shown in Fig. 1, we conclude that a value for Zi has to be found by a fitting procedure, while taking for b a reasonable value. We have tried various values in a range, in which b /rmin is not larger than 1 / 2, rmin being the smallest resistivity of the experimental set. For b /r 5 0.5, Eq. (3) gives a direct valence equal to 4 / 5Zib , in which the unscreened valence now has been denoted as Zib . The latter quantity has to be determined by a fit. The result of this fit for the three systems is shown by the dotted lines in Fig. 1, and the parameters are given in Table 1. Firstly, it is seen that this new fit is as good as the constant–Zdirect fit is. Secondly, the result that the Kb values are less negative than the Kc values is consistent with the negative slope of the Zdirect function of Eq. (3). Thirdly, the differences between the two fits are small, and lead only to some difference in the (unscreened) valency. In fact, Zib will always be found to be somewhat smaller than the fitted Zdirect , and the difference decreases by taking smaller b values. Finally, and most importantly, this analysis of the experimental results demonstrates evidently, that metallic effects have to be considered in explaining the deviation of the valency values from the protonic valency of 1. It also appears justified to use a constant– Zdirect fit, if one leaves open the possibility that the real ‘unscreened’ valence may be slightly smaller than the value found by such a fit. Let us now turn to another recently obtained set of data.
Fig. 3. The measured effective valence of hydrogen isotopes in V and Ta, and the lines according to the constant Zdirect fit. The dotted lines in the left panel represent Zib fits.
3. Discussion of the data for hydrogen isotopes in V, Nb, and Ta The results for electromigration of hydrogen and its isotopes deuterium and tritium in niobium are shown in Fig. 2 [11,12]. The recently measured Z* points for hydrogen and tritium in vanadium [15] are displayed in Fig. 3a, while Fig. 3b shows the results for the three hydrogen isotopes in tantalum [13,14]. The bold lines in the panels a to c of Fig. 2 represent the constant–Zdirect fits, the thin lines again show failing Zi 5 1 fits, and the dotted lines correspond to Zib fits as outlined in Section 2. The straight lines in Fig. 3 represent constant–Zdirect fits. For V, in addition, the dotted lines are given for Zib fits. The parameters following from the two relevant fits are given in Table 2. Let us first discuss the results for Nb. The result of Verbruggen and Griessen for Nb(H) was quite remarkable
Fig. 2. The effective valence of H, D and T in Nb, plotted as a function of 100 /r. In panel d only the constant–Zdirect fits are reproduced, the dashed line holds for T, and the experimental points carry the symbols used in the panels a to c.
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Table 2 The fit parameters according to two different fits, the conventional constant–Zdirect fit, and a fit using b 5 1 / 2rmin in Eq. (3), leading to the parameters Zib and Kb System
Zdirect
Kc
Zib
Kb
b ( 5 1 / 2rmin )
Source
V(H) V(T) Nb(H)(4) Nb(H)(6) Nb(D) Nb(T) Ta(H) Ta(D) Ta(T)
1.2 0.6 0.53 0.75 0.59 0.76 1.7 0.9 1.2
16 36 28 24 27 25 224 211 219
1.1 0.5 0.48 0.68 0.53 0.69 1.53 0.81 1.05
23 43 32 28 31 29 216 26.7 213
12.7 12.7 9.2 9.3 9.2 9.3 8.9 8.9 8.9
Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref.
at that time, because their fitted Zdirect value of 0.44 implied considerable screening [9]. That is why in the fit using Eq. (3) the author was not too critical concerning the magnitude of the parameter b [10]. Electron-phonon effects were not really in the picture yet. Since leastsquares fits can be obtained for b values lying in different distinct intervals, the best fit was obtained for a large b value, which, of course, turned out to represent large screening. But according to our discussion given in Section 2, large b values are no longer considered as physically acceptable, and the Zi 5 1 fits in Fig. 2a–c, ending at Z* 5 1, apparently fail again. That is why the corresponding parameters are not given in Table 2. Looking more closely at Fig. 2a one sees that two experimental sets are shown, a set of four points, which in fact has been extracted from 10 measured points [11], and a more recently published set of six points [12]. The complete set of three lines corresponds to the more recent set. For the set of four points, the diamonds, only the constant–Zdirect result is shown by the dashed line. Looking at the table, one sees that both sets lead to Zdirect values considerably smaller than unity, but both values, 0.53 and 0.75, are larger than the very small value of 0.44 found by Verbruggen and Griessen. The smaller Zib values can be read from the fourth column of Table 2. In conclusion, experimental evidence exists for a Zdirect value distinctly smaller than unity, but the precise value is not certain, although the value of Verbruggen and Griessen agrees within the experimental error with the value corresponding to the four-points set. If one takes the latter observation as pointing in the direction of a real property, one arrives at interesting isotope effects. While the Zdirect values increase with the mass of the isotope, the Kc values decrease. Their sum, in fact the Z* values, increase slightly in the resistivity range in which the measured points lie, which can be seen in Fig. 2d. As far as the wind force is concerned, this implies that in Nb the lighter isotope is the stronger scatterer. Looking at Table 2, for Ta holds the same if one disregards Ta(D), while for V the opposite holds. The calculated isotope effect for the wind force is small, but it agrees with the observed effect for V and Nb, but for Ta an opposite effect
[15] [15] [11] [12] [11] [12] [13] [13] [14]
is found [4]. As far as the direct valency is concerned, no ab initio theoretical description is available yet, which is applicable to real metallic systems. The results for V are hard to understand, because, if one looks at the Kc or Kb values, they imply tritium to be twice as strong a scatterer as hydrogen is. It is indeed true, that for electrons on a certain extremal orbit on the Fermi surface of Pd, the deuteron has been found to be a much stronger scatterer than hydrogen is [29]. But for a property in which the entire Fermi surface is involved, which also applies for the wind force, such an effect is certainly washed out [6]. Further, the largest difference between the measured Z* values is 0.3, at the 100 /r value of about 4, while the differences at the three other measuring points are much smaller. It looks, as if the effects compensate each other more or less. The result of a twice as large Zdirect for H compared with T is a direct consequence of the much smaller slope of the fitted line for H, which would mean that H is a much weaker scatterer than T is. In itself such a relationship cannot be excluded logically, because a large direct valence could be reconciled with weak scattering, but a real description for it is not at all available. In addition, the calculated isotope effect in V is small. As for Ta, the analysis of the measured results leads to parameters for deuterium, which look anomalous. However, if one looks at Fig. 3b in the resistivity range of the measured points, the Z* values decrease with increasing isotope mass. In that region only the slope of the fitted line for deuterium behaves anomalously. We are approaching the borderline of speculation, but, since no adequate description is available, one could reason the other way around as well. By considering the measured slopes for H and T as being anomalous, the result for D would point in the direction of smaller slopes for H and T. This possibility would resolve another problem concerning the results for Ta. For H it would lead to a smaller value of Zdirect than the present value of 1.7, which is hard to understand. The Zib value is indeed smaller, but values larger than unity require new concepts in the theory anyhow. We conclude this section by the observation that the measured results for the hydrogenated metals show the
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same trend as the results of Verbruggen and Griessen. The Zdirect values for H in Nb are in both cases remarkably smaller than unity, the values for H in V and Ta are larger than unity, and the Zdirect value for H in Ta is the largest one. It can be considered as a challenge to describe these findings quantitatively.
Acknowledgements One of the authors (A.L.) wishes to thank Dr. R. Pietrzak for providing measured results for the effective valence in tabular form, prior to the date of publication.
References 4. Summary and conclusions The measured resistivity dependence of the effective valence Z* for H in Pd and two alloys with Ag has been shown to prove that the direct valence of the hydrogen can be significantly smaller than unity. These results have led also to a conceptually satisfactory interpretation and application of the prediction of a resistivity-dependent direct valence. The main conclusion in this respect is, that the conventional fit of measured resistivity dependent Z* values leads to a useful value of the direct valence, provided that one has to keep open the possibility that the real unscreened direct valence is about 10% smaller. The measured Z* behaviour for H in V, Nb and Ta, and the corresponding parameters Zdirect and K, the latter describing the wind valence, show the same trend as the results obtained by Verbruggen and Griessen. The measured isotope effect in the wind valence in V and Nb agree with ab initio calculated results, obtained by accounting for a more intense zero point motion of the lighter isotope, but the predicted effects are smaller than the measured ones. Important questions remain, requiring both theoretical and more experimental effort. Since direct valences have been measured which are significantly different from unity, metallic charge density effects appear to be important. A quantitative description of this requires efforts beyond the jellium model. Selfconsistent charge density calculations are possible, and have been carried out extensively for substitutional impurities in metallic hosts in the past 15 years, but the description of dilute interstitial alloys has not shown any development at all since some first preliminary activity [30]. The barriers encountered have been razed in the meantime [31], and the problems met in the understanding of results for interstitial electromigration form a new challenge to calculate and visualize the charge density around an interstitial impurity. This activity should not only be carried out for the impurity at its initial position, but at all positions along its jump path similarly as it now is done routinely for the wind force [5,6]. As far as experiment is concerned, interesting isotope effects have been measured, but in some cases the effects are very large and hard to understand. An independent measurement would help to make clear on which aspects further theoretical work should be focused.
[1] P.S. Ho, T. Kwok, Rep. Progr. Phys. 52 (1989) 301. [2] R.S. Sorbello, A. Lodder, S.J. Hoving, Phys. Rev. B 25 (1982) 6178. [3] C. Bosvieux, J. Friedel, J. Phys. Chem. Solids 23 (1962) 123. [4] J. van Ek, A. Lodder, J. Phys. Condens. Matter 3 (1991) 7331. [5] J.P. Dekker, A. Lodder, J. van Ek, Phys. Rev. B 56 (1997) 12167. [6] J.P. Dekker, A. Lodder, J. Phys. Condens. Matter 10 (1998) 6687. [7] R.S. Sorbello, in: H. Ehrenreich, F. Spaepen (Eds.), Solid State Physics, Vol. 51, Academic Press, San Diego, 1998, p. 159. [8] J. van Bk, A. Lodder, Defect Diff. Forum 115–116 (1994) 1. [9] A.H. Verbruggen, R. Griessen, Phys. Rev. B 32 (1985) 1426. [10] A. Lodder, J. Phys. Condens. Matter 3 (1991) 399. [11] K. Hashizume, Y. Kawabata, K. Matsumoto, N. Tsutsumi, M. Sugisaki, Defect Diff. Forum 95–98 (1993) 329. [12] K. Hashizume, K. Fujii, M. Sugisaki, Fusion Techn. 28 (1995) 1179. [13] K. Hashizume, Y. Kawabata, K. Fujii, M. Sugisaki, J. Alloys Comp. 215 (1994) 71. [14] K. Hashizume, K. Fujii, M. Sugisaki, Defect Diff. Forum 143–147 (1997) 1689. [15] K. Fujii, K. Hashizume, Y. Hatano, M. Sugisaki, J. Alloys Comp. 282 (1999) 38. [16] R. Pietrzak, R. Szatanik, M. Szuszkiewicz, Defect Diff. Forum 143–147 (1997) 951. [17] R. Pietrzak, R. Szatanik, M. Szuszkiewicz, J. Alloys Comp. 282 (1999) 130. [18] A. Lodder, Solid State Commun. 79 (1991) 143. [19] R.S. Sorbello, C.S. Chu, J. Phys. Chem. Solids 52 (1991) 501. [20] R. Landaner, Solid State Commun. 72 (1989) 867. [21] R.S. Sorbello, Solid State Commun. 76 (1990) 747. [22] A. Lodder, Solid State Commun. 73 (1990) 611. [23] A. Lodder, Solid State Commun. 79 (1991) 147. ` [24] L. Turban, P. Nozieres, M. Gerl, J. Physique 37 (1976) 159. [25] H. Ishida, Phys. Rev. B 51 (1995) 10345. [26] A. Lodder, Physica A 158 (1989) 723. [27] A. Lodder, J.P. Dekker, in: H. Okabayashi, S. Shingubara, P.S. Ho (Eds.), Stress Induced Phenomena in Metallization. Fourth International Workshop (Tokyo, Japan, 1997), American Institute of Physics, Woodbury, NY, 1998, p. 315. [28] P. Kumar, R.S. Sorbello, Thin Solid Films 25 (1975) 25. [29] P.M. Oppeneer, A. Lodder, R. Griessen, J. Phys. F: Met. Phys. 18 (1988) 1743. [30] S. Ellialtioglu, H. Akai, R. Zeller, P.H. Dederichs, Ann. Int. Symp. Electr. Str. Met. All. 15 (1985) 99. [31] J.P. Dekker, A. Lodder, R. Zeller, A.F. Tatarchenko, Phys. Rev. B 54 (1996) 4531.
L
The direct force on electromigrating hydrogen isotopes a, b A. Lodder *, K. Hashizume a
Faculty of Sciences, Division Physics and Astronomy, Free University, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands b Department of Advanced Energy Engineering Science, Kyushu University, Kasuga, Fukuoka 816 -8580, Japan Received 22 October 1998
Abstract The present understanding of the magnitude of the direct force on hydrogen and its isotopes deuterium and tritium is discussed in the light of new experimental data and existing theoretical formulations. By this it is proven, that the direct valence of hydrogen can be significantly different from the unscreened jellium value of unity. Furthermore, a more detailed interpretation of a resistivity dependent direct valence is given, which clarifies the controversy around that subject. Selfconsistent calculations of the electronic charge density at and around an interstitial, at all positions along its electromigration jump path, are required to come to a more complete understanding of the measured results. 1999 Elsevier Science S.A. All rights reserved. Keywords: Hydrogen isotopes; Direct valence; Electromigration
1. Introduction Electromigration occurs when a metal is subjected to an electric field. The random diffusive motion of atoms then gets a preferential direction. The force on an atom induced by the electric field consists of two components, a direct force F direct and a wind force F wind . The direct force is due to a net charge of the migrating atom, while the wind force is due to scattering of the current-carrying electrons off the atom. Both forces are proportional to the electric field and can be characterized by a valence [1] F 5 (Zdirect 1 Zwind )eE 5 Z*eE.
(1)
The effective valence Z* is the measurable quantity. Although consensus consists about the form of Eq. (1), the present understanding of the two contributions is quite different. The wind force, being proportional to the current density, is found to be inversely proportional to the sample resistivity r, independently of the complexity of the theoretical formulation used in the description of the total force. One conventionally writes Z* as K Z* 5 Zdirect 1 ], r
(2)
in which the temperature independent proportionality *Corresponding author. E-mail address: [email protected] (A. Lodder)
constant K is determined by the scattering of the electrons by the migrating atom. In addition, a microscopic quantum-mechanical expression for the wind force is available [2], which traces back to the pioneering work of Bosvieux and Friedel [3]. This expression has been applied recently for ab initio calculations of the wind valence of interstitial atoms such as hydrogen [4], and of substitutionally migrating atoms [5,6], in numerous FCC and BCC metals. In contrast to the consensus concerning the wind force, and its computational accessibility, the magnitude of the direct force has been the subject of a long-standing controversy. Either one is in favour of complete screening, so that only the wind force remains, or one defends that the direct force is effectively unscreened. The controversy started in 1962, being induced by the screening prediction of Bosvieux and Friedel [3], and has not come to an end completely yet [7,8]. Unfortunately, an attempt to decide on this issue by measuring the driving force on hydrogen in the transition metals V, Nb, and Ta turned out to be not conclusive [9,10]. Sorbello [7] even expresses the possibility that the conventional distinction of two contributions to the driving force, Eq. (1), eventually may turn out to be less meaningful than it has been assumed to be up to now. In the meantime, however, new efforts have been made on the experimental side [11–17]. These efforts were partially challenged by a recent prediction, that complete screening holds only for the asymptotic electron-impurity model, while in real samples the screening is hindered by the
0925-8388 / 99 / $ – see front matter 1999 Elsevier Science S.A. All rights reserved. PII: S0925-8388( 99 )00113-9
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overwhelmingly present electron–phonon interaction [18]. So a new look at the present status of our understanding is justified. The present status is characterized by the fact that, while conventionally Zdirect is assumed to be a constant, a relaxation-time dependent direct valence [19] and a resistivity-dependent expression [10] Zi Zdirect 5 ]]]2 1 1 (b /r )
(3)
have been proposed, in the latter Zi and r being the unscreened ionic valency and the contribution to the sample resistivity due to electron–phonon scattering respectively, while b is an adjustable parameter. The relaxation-time dependent expression was based on an old analysis valid to the lowest order in the electron-impurity scattering potential only, and turned out to be unsuccessful in describing the data available [9]. The resistivity-dependent expression, Eq. (3), is based on a more complete analysis [18]. According to that expression Zdirect can become much smaller than the unscreened valency at low temperatures, because then r becomes small and the denominator can become much larger than unity, while at higher temperatures, in fact at temperatures at which electromigration experiments are done, Zdirect will approach Zi . It can be considered as a compromise between the two extremes defended in the literature [8], and it has been presented so [18]. It indeed describes the available data, while no Zdirect values larger than Zi enter, as they do in a constant Zdirect analysis. However a question remains concerning the meaning of the parameter b in Eq. (3), while the data did not discriminate convincingly between the two possible descriptions. First, in Section 2, the new results for hydrogen migra-
tion in pure Pd and two alloys with Ag will be analysed. It will turn out to be necessary to reinterpret the jellium based Eq. (3), in order to make it applicable to real metallic systems. In Section 3 the electromigration of hydrogen and its isotopes in the transition metals V, Nb, and Ta will be analysed and discussed. The paper ends with a summary and concluding remarks. As far as the units are concerned, the parameters K and b, and the resistivity r will be expressed in mV cm.
2. Hydrogen migration in pure Pd and alloys with Ag In Fig. 1 the effective valence is plotted as a function of 100 /r for hydrogen migration in pure Pd and the alloys Pd 77 Ag 33 and Pd 50 Ag 50 , systems which have been investigated experimentally by Pietrzak et al. [16,17]. The bold lines show the constant–Zdirect fit according to Eq. (2). The corresponding Zdirect and Kc values are given in Table 1. The label c has been added to the slope parameter, referring to the kind of fit. One sees an increased negative slope on adding silver to palladium, represented by increasingly negative Kc values. For pure silver a Kc value lower than 2100 is expected [4]. The thinner lines in Fig. 1 follow from a fit, using Eq. (3) in Eq. (2), and taking Zi 5 1. These three lines obviously end at the value Z* 5 1 in the origin. In Table 1 the parameters according to this fit are indicated by b and KZ i 51 . The conclusion is clear. The latter fit, using the jellium value Zi 5 1 for the unscreened direct valence, fails. For Pd(H) one still does not see anything disturbing, but this is certainly due to the fact that the bold straight line already ends at a Z* value close to unity, at 1.05. But for the alloys the fit is no good. So, either the idea of a resistivity-dependent direct force is
Fig. 1. The effective valence Z* of hydrogen in pure Pd and two alloys with Ag, plotted as a function of 100 /r.
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Table 1 The fit parameters according to three different fits, a constant–Zdirect fit, a Zi 5 1 fit leading to values for the parameters b and KZ i 51 and a fit using b51 / 2rmin in Eq. (3), leading to the parameters Zib and Kb System
Zdirect
Kc
b
KZ i 51
Zib
Kb
b ( 5 1 / 2rmin )
Pd Pd 77 Ag 33 Pd 50 Ag 50
1.05 0.73 0.79
23.81 211.82 229.4
3.65 12.32 18.93
22.17 212.01 228.1
0.96 0.65 0.72
20.54 27.85 223.8
5.75 9.52 13.64
wrong, or one has to consider the origin of Eq. (3) more consequently than the way it was done in the earlier fit [10]. We will show that the latter option is the correct one. However, since the subject has been quite controversial [20–24], and it apparently still is [7,19,25,27], we support our view by a brief historical account. As mentioned in Section 1, the prediction by Bosvieux and Friedel [3] of a screened direct force on an interstitial impurity formed the start of a controversy. Since, for a long time, experiments were too difficult to decide, Kumar and Sorbello introduced a linear response approach in the field, which gives an exact formal expression for the driving force, in the hope of finding a conclusive answer [28]. Many efforts were made in order to evaluate the formal expression for the electron-impurity model or, equivalently, the impurity-in-a-jellium model. This model system is supposed to simulate a metallic system with a low concentration of interstitial impurities. Most of the treatments ended with an expression, which contained the (interstitial) impurity potential only to lowest order. As far as the wind force is concerned this did not form a problem, because the lowest order expression for this scattering property has the same form as the exact expression. The latter reduces to the lowest order result simply by replacing a t-matrix element with a matrix element for the potential v. That is why there never has been a problem with the wind force. However, the screening contribution to the direct force, evaluated to lowest order, was small compared to the unscreened force. In spite of the fact, that in the same period an exact treatment was published supporting Bosvieux and Friedel’s result [24], the matter remained undecided. One of the arguments used was, that for a migrating atom, undergoing a dynamical process, the screening is not effective, so that the direct force acts on the bare nucleus only. This may be true to a certain extent, but we did not find an estimate of this dynamical effect in the literature. Therefore we indicate how this can be done. Confining ourselves to hydrogen, one can use its zero-point-motion energy of about 0.025 eV, or an activation energy of the order of 0.1 eV. Taking these energies equal to 1 / 2mv 2 , one finds velocities of 2310 3 m / s and 4310 3 m / s, respectively, to be compared with the Fermi velocity of a metallic electron of about 10 6 m / s. Using the activation energy in an energy-time uncertainty relation, and interpreting this time as a jump time, one finds an upper limit
of the velocity of about 4310 4 m / s over a jump path of ˚ In conclusion, the ion velocity is at least two orders 2.5 A. of magnitude smaller than the electron velocity, so that the screening reduction due to the dynamical effect is small. Interestingly, this argument against screening applies both to lowest order and exact treatments, because the force calculated in the electron-impurity model is independent of the position of the impurity. At present a newly derived exact result is available, which again forms a support of Bosvieux and Friedel [18]. In presenting this result it was emphasized strongly, that it only holds for the electron-impurity model. It could be derived rather straightforwardly after advantage was taken of a simplification of the formal linear-response expression, which was not discerned fully in earlier work. Old lowest order results were reproduced without failure, but the exact result led to screening [26]. Objections were raised [20,21], which were mainly based on obsolete arguments developed in the emotional discussion carried on at the end of the 1970s [8], but it formed no problem to refute these points [22,23]. However, it was pointed out, that the simple electron-impurity model is not a valid representation of the samples investigated experimentally. At and above room temperature electron-phonon interaction is dominantly present, forming a serious hindrance for the screening. As soon as this was realized, it was phrased, that the exact screening result, holding exactly for the dilute electron-impurity model system, is fading away in its relevance to the real systems encountered in an experimental situation [18]. By that the controversy can be considered as being clarified. In fact, Eq. (3) forms the bridge between the two extreme views. The question remains, how to use it. First, it is based on an approach, in which free electrons interact with migrating impurities and with phonons. That is why Zi , representing the unscreened ionic valency, for hydrogen isotopes is equal to unity. Further, the theory does not provide means to calculate the parameter b by way of ab initio methods, so that one has to make a choice. In the first attempt to describe the experimental data, Zi was taken to be 1, and b served as a fitting parameter [10]. Surprisingly, a good fit could be obtained for all three transition metals considered, namely the metals V, Nb, and Ta, but the author was not too critical concerning the magnitude of b. For Nb, for example, b was found to be equal to 53 mV cm, while the sample resistivity values lay in the range of 16 to 27 mV
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cm, the lowest value corresponding to a temperature of 276 K. So, for this fit the direct valence is 0.08 at 276 K, which implies a huge screening, contrary to the expectation expressed above. Regarding this expectation, and the experimental results shown in Fig. 1, we conclude that a value for Zi has to be found by a fitting procedure, while taking for b a reasonable value. We have tried various values in a range, in which b /rmin is not larger than 1 / 2, rmin being the smallest resistivity of the experimental set. For b /r 5 0.5, Eq. (3) gives a direct valence equal to 4 / 5Zib , in which the unscreened valence now has been denoted as Zib . The latter quantity has to be determined by a fit. The result of this fit for the three systems is shown by the dotted lines in Fig. 1, and the parameters are given in Table 1. Firstly, it is seen that this new fit is as good as the constant–Zdirect fit is. Secondly, the result that the Kb values are less negative than the Kc values is consistent with the negative slope of the Zdirect function of Eq. (3). Thirdly, the differences between the two fits are small, and lead only to some difference in the (unscreened) valency. In fact, Zib will always be found to be somewhat smaller than the fitted Zdirect , and the difference decreases by taking smaller b values. Finally, and most importantly, this analysis of the experimental results demonstrates evidently, that metallic effects have to be considered in explaining the deviation of the valency values from the protonic valency of 1. It also appears justified to use a constant– Zdirect fit, if one leaves open the possibility that the real ‘unscreened’ valence may be slightly smaller than the value found by such a fit. Let us now turn to another recently obtained set of data.
Fig. 3. The measured effective valence of hydrogen isotopes in V and Ta, and the lines according to the constant Zdirect fit. The dotted lines in the left panel represent Zib fits.
3. Discussion of the data for hydrogen isotopes in V, Nb, and Ta The results for electromigration of hydrogen and its isotopes deuterium and tritium in niobium are shown in Fig. 2 [11,12]. The recently measured Z* points for hydrogen and tritium in vanadium [15] are displayed in Fig. 3a, while Fig. 3b shows the results for the three hydrogen isotopes in tantalum [13,14]. The bold lines in the panels a to c of Fig. 2 represent the constant–Zdirect fits, the thin lines again show failing Zi 5 1 fits, and the dotted lines correspond to Zib fits as outlined in Section 2. The straight lines in Fig. 3 represent constant–Zdirect fits. For V, in addition, the dotted lines are given for Zib fits. The parameters following from the two relevant fits are given in Table 2. Let us first discuss the results for Nb. The result of Verbruggen and Griessen for Nb(H) was quite remarkable
Fig. 2. The effective valence of H, D and T in Nb, plotted as a function of 100 /r. In panel d only the constant–Zdirect fits are reproduced, the dashed line holds for T, and the experimental points carry the symbols used in the panels a to c.
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Table 2 The fit parameters according to two different fits, the conventional constant–Zdirect fit, and a fit using b 5 1 / 2rmin in Eq. (3), leading to the parameters Zib and Kb System
Zdirect
Kc
Zib
Kb
b ( 5 1 / 2rmin )
Source
V(H) V(T) Nb(H)(4) Nb(H)(6) Nb(D) Nb(T) Ta(H) Ta(D) Ta(T)
1.2 0.6 0.53 0.75 0.59 0.76 1.7 0.9 1.2
16 36 28 24 27 25 224 211 219
1.1 0.5 0.48 0.68 0.53 0.69 1.53 0.81 1.05
23 43 32 28 31 29 216 26.7 213
12.7 12.7 9.2 9.3 9.2 9.3 8.9 8.9 8.9
Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref.
at that time, because their fitted Zdirect value of 0.44 implied considerable screening [9]. That is why in the fit using Eq. (3) the author was not too critical concerning the magnitude of the parameter b [10]. Electron-phonon effects were not really in the picture yet. Since leastsquares fits can be obtained for b values lying in different distinct intervals, the best fit was obtained for a large b value, which, of course, turned out to represent large screening. But according to our discussion given in Section 2, large b values are no longer considered as physically acceptable, and the Zi 5 1 fits in Fig. 2a–c, ending at Z* 5 1, apparently fail again. That is why the corresponding parameters are not given in Table 2. Looking more closely at Fig. 2a one sees that two experimental sets are shown, a set of four points, which in fact has been extracted from 10 measured points [11], and a more recently published set of six points [12]. The complete set of three lines corresponds to the more recent set. For the set of four points, the diamonds, only the constant–Zdirect result is shown by the dashed line. Looking at the table, one sees that both sets lead to Zdirect values considerably smaller than unity, but both values, 0.53 and 0.75, are larger than the very small value of 0.44 found by Verbruggen and Griessen. The smaller Zib values can be read from the fourth column of Table 2. In conclusion, experimental evidence exists for a Zdirect value distinctly smaller than unity, but the precise value is not certain, although the value of Verbruggen and Griessen agrees within the experimental error with the value corresponding to the four-points set. If one takes the latter observation as pointing in the direction of a real property, one arrives at interesting isotope effects. While the Zdirect values increase with the mass of the isotope, the Kc values decrease. Their sum, in fact the Z* values, increase slightly in the resistivity range in which the measured points lie, which can be seen in Fig. 2d. As far as the wind force is concerned, this implies that in Nb the lighter isotope is the stronger scatterer. Looking at Table 2, for Ta holds the same if one disregards Ta(D), while for V the opposite holds. The calculated isotope effect for the wind force is small, but it agrees with the observed effect for V and Nb, but for Ta an opposite effect
[15] [15] [11] [12] [11] [12] [13] [13] [14]
is found [4]. As far as the direct valency is concerned, no ab initio theoretical description is available yet, which is applicable to real metallic systems. The results for V are hard to understand, because, if one looks at the Kc or Kb values, they imply tritium to be twice as strong a scatterer as hydrogen is. It is indeed true, that for electrons on a certain extremal orbit on the Fermi surface of Pd, the deuteron has been found to be a much stronger scatterer than hydrogen is [29]. But for a property in which the entire Fermi surface is involved, which also applies for the wind force, such an effect is certainly washed out [6]. Further, the largest difference between the measured Z* values is 0.3, at the 100 /r value of about 4, while the differences at the three other measuring points are much smaller. It looks, as if the effects compensate each other more or less. The result of a twice as large Zdirect for H compared with T is a direct consequence of the much smaller slope of the fitted line for H, which would mean that H is a much weaker scatterer than T is. In itself such a relationship cannot be excluded logically, because a large direct valence could be reconciled with weak scattering, but a real description for it is not at all available. In addition, the calculated isotope effect in V is small. As for Ta, the analysis of the measured results leads to parameters for deuterium, which look anomalous. However, if one looks at Fig. 3b in the resistivity range of the measured points, the Z* values decrease with increasing isotope mass. In that region only the slope of the fitted line for deuterium behaves anomalously. We are approaching the borderline of speculation, but, since no adequate description is available, one could reason the other way around as well. By considering the measured slopes for H and T as being anomalous, the result for D would point in the direction of smaller slopes for H and T. This possibility would resolve another problem concerning the results for Ta. For H it would lead to a smaller value of Zdirect than the present value of 1.7, which is hard to understand. The Zib value is indeed smaller, but values larger than unity require new concepts in the theory anyhow. We conclude this section by the observation that the measured results for the hydrogenated metals show the
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same trend as the results of Verbruggen and Griessen. The Zdirect values for H in Nb are in both cases remarkably smaller than unity, the values for H in V and Ta are larger than unity, and the Zdirect value for H in Ta is the largest one. It can be considered as a challenge to describe these findings quantitatively.
Acknowledgements One of the authors (A.L.) wishes to thank Dr. R. Pietrzak for providing measured results for the effective valence in tabular form, prior to the date of publication.
References 4. Summary and conclusions The measured resistivity dependence of the effective valence Z* for H in Pd and two alloys with Ag has been shown to prove that the direct valence of the hydrogen can be significantly smaller than unity. These results have led also to a conceptually satisfactory interpretation and application of the prediction of a resistivity-dependent direct valence. The main conclusion in this respect is, that the conventional fit of measured resistivity dependent Z* values leads to a useful value of the direct valence, provided that one has to keep open the possibility that the real unscreened direct valence is about 10% smaller. The measured Z* behaviour for H in V, Nb and Ta, and the corresponding parameters Zdirect and K, the latter describing the wind valence, show the same trend as the results obtained by Verbruggen and Griessen. The measured isotope effect in the wind valence in V and Nb agree with ab initio calculated results, obtained by accounting for a more intense zero point motion of the lighter isotope, but the predicted effects are smaller than the measured ones. Important questions remain, requiring both theoretical and more experimental effort. Since direct valences have been measured which are significantly different from unity, metallic charge density effects appear to be important. A quantitative description of this requires efforts beyond the jellium model. Selfconsistent charge density calculations are possible, and have been carried out extensively for substitutional impurities in metallic hosts in the past 15 years, but the description of dilute interstitial alloys has not shown any development at all since some first preliminary activity [30]. The barriers encountered have been razed in the meantime [31], and the problems met in the understanding of results for interstitial electromigration form a new challenge to calculate and visualize the charge density around an interstitial impurity. This activity should not only be carried out for the impurity at its initial position, but at all positions along its jump path similarly as it now is done routinely for the wind force [5,6]. As far as experiment is concerned, interesting isotope effects have been measured, but in some cases the effects are very large and hard to understand. An independent measurement would help to make clear on which aspects further theoretical work should be focused.
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