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Spin dependent screening and Auger neutralization of singly-charged noble gas ions in metals
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 256 (2007) 24–29 www.elsevier.com/locate/nimb
Spin dependent screening and Auger neutralization of singly-charged noble gas ions in metals J.I. Juaristi b
a,b,*
, M. Alducin
c
a Departamento de Fı´sica de Materiales, Facultad de Quı´mica UPV/EHU, Apartado 1072, 20080 San Sebastia´n, Spain Unidad de Fı´sica de Materiales, Centro Mixto CSIC-UPV/EHU, Facultad de Quı´mica UPV/EHU, Apartado 1072, 20080 San Sebastia´n, Spain c Donostia International Physics Center (DIPC), P. Manuel de Lardizabal 4, 20018 San Sebastia´n, Spain
Available online 16 January 2007
Abstract The screening of Ne+ and Ar+ ions embedded in a paramagnetic free electron gas is calculated using density functional theory within the local spin density approximation, analyzing the spin polarization of the screening cloud. As a second step, the Auger neutralization rates for these ions in a free electron gas are calculated, paying special attention to their dependence on the spin direction of the electron excited in the Auger process. It is observed that the spin polarization of the screening cloud and that of the Auger rates are related. Comparison with previous calculations performed for He+ shows that the spin polarization of the excited electrons is lower in the case of Ne+ and Ar+ than in the case of He+. This result is consistent with the experimental data obtained for the spin polarization of the emitted electrons in the deexcitation of metastable He*, Ne* and Ar* atoms in front of metal surfaces. Ó 2006 Elsevier B.V. All rights reserved. PACS: 34.50.Dy; 79.20.Rf Keywords: Electron emission; Ions; Auger neutralization; Spin polarization; Metals
1. Introduction The study of spin effects in the interaction of spinpolarized atomic particles interacting with paramagnetic metal surfaces constitutes an active field of research. For instance, the case of He+ ion or He* metastable atom projectiles has been widely studied both experimentally [1–3] and theoretically [4–10]. Special attention has been paid to the spin polarization of the electrons emitted in the neutralization/deexcitation of these projectiles. Experimentally, the emitted electrons show an overall spin polarization of about 30% parallel to the spin polarization of the incoming projectile and weakly dependent on the target *
Corresponding author. Address: Departamento de Fı´sica de Materiales, Facultad de Quı´mica UPV/EHU, Apartado 1072, 20080 San Sebastia´n, Spain. Tel.: +34 943 015396; fax: +34 943 015270. E-mail addresses: [email protected] (J.I. Juaristi), wapalocm@sq. ehu.es (M. Alducin). 0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.087
surface At high emission energies, the spin polarization of the emitted electrons is higher (60%–70%) [1–3]. From now on, we denote as " (#) the direction of the spin parallel (antiparallel) to the direction of the spin polarization of the projectile. Some recent works devoted to the study of the screening and Auger neutralization of a spin-polarized He+ ion embedded in a free electron gas (FEG) have been able to explain these experimental facts. First of all, the theoretical results show a high spin polarization (>70%) of the electrons excited in the Auger process [7,8]. The reason for this is twofold. On the one hand the screening of the spin-polarized He+ ion is dominated by spin-" electrons. On the other hand, in the case of exciting spin-# electrons, since the electron to be captured is also spin-#, it exists a destructive interference between two indistinguishable processes, with the subsequent reduction of the probability for this channel. The interference is absent in case of excitation of spin-" electrons since the two electrons participating in
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29
the Auger process have different spin directions. This interference was shown to be the most important reason for the predominant excitation of spin-" electrons [7,8]. The high spin polarization that results from the Auger process is reduced to about the measured 30% due to the creation of an unpolarized cascade of low energy secondary electrons [9,10]. The strong spin dependence of the Auger process is reflected in that of the electrons emitted at high energies. In [1] the spin polarization of electrons emitted in the deexcitation of Ne(3P2) and A(3P2) metastable atoms was also reported. It was found that the measured spin polarizations in these cases are significantly lower than in the He(23S) case. Motivated by this fact, here we present a study of the spin-dependent screening and the Auger neutralization of spin-polarized Ne+ and Ar+ ions embedded in FEG. The paper is organized as follows. In Section 2 we present the model used, in Section 3 we show and discuss the results obtained and in Section 4 we summarize the main conclusions of the work. Atomic units (a.u.) are used unless it is otherwise stated.
25
the screening is preferably due to spin-" electrons, as shown in [6–8]. In a similar way, here the Ne+ (Ar+) ion is modelled by populating with only two electrons the 2p (3p) KS state corresponding to spin-# electrons. This means that for Ne+ there are nine bound electrons, five of them are spin-" and four of them spin-#. In case of Ar+ there are seventeen bound electrons, nine of them spin-" and eight of them spin-#. Note that in all the considered cases, the screening induced electronic density in the continuum integrates to unity. In a second step, we calculate the Auger capture rate (AC), i.e. the probability per unit time that a spin-# electron from the conduction band fills the unoccupied bound state of the ion. The energy released in the transition is transferred to a second electron which gets excited in the continuum. The AC rate C can be calculated in first order perturbation theory for excitation of electrons with spin parallel (C") and antiparallel (C#) to the direction of polarization of the projectile (C ¼ C" þ C# ). The former (C") in which the electrons participating in the Auger process have antiparallel spin is calculated as follows [7,13]: X X Z X drdr0 u# ðrÞ u" ðr0 Þ C" ¼ 2p 3 a u#1 2 occ: u"2 2 occ: u"3 62 occ:
2. Theory The theoretical framework used to calculate the screening of the Ne+ and Ar+ ions in the FEG is the same as that used previously for the case of He+ [7] and it is only summarized here. The ion is treated as a static charged impurity embedded in the FEG with background density n0 and we define the customary one electron average radius rs from 1=n0 ¼ 4pr3s =3. Density functional theory (DFT) and the local spin density (LSD) approximation are used to calculate the electronic properties of the system. The Kohn–Sham (KS) wave functions ½uri ðrÞ are solutions with eigenvalue eri of the standard KS equations [11] with the parametrization of [12] for the exchange correlation potential. The index r runs over the spin orientations " and #. The electron density nðrÞ is obtained as: X X nðrÞ ¼ juri ðrÞj2 : ð1Þ r¼";# i2occ:
The electron density for just spin-" (spin-#) electrons n" (r) [n# (r)] is defined by limiting the sum over occupied states to the required spin component. Furthermore, we define the induced density for each spin orientation as: Dnr ðrÞ ¼ nr ðrÞ n0 =2;
ð2Þ +
with r = ", #. In previous works [6–10] the He ion, in which there is only one bound electron, has been modelled by populating just one of the two bound KS states of the He+/FEG system (there is one for each spin orientation). Furthermore, the spin of the electron bound to the projectile is fixed ("). As a consequence, the one electron KS potential, that includes the spin-dependent exchangecorrelation part and the electronic charge that surrounds the He+ ion are different for each spin orientation. Indeed,
2 vðr; r0 Þu"2 ðr0 Þu#1 ðrÞ d e#1 þ e"2 e#a e"3 ;
ð3Þ
where vðr; r0 Þ ¼ 1=jr r0 j is the Coulomb potential respon";# sible for the decay and u";# are the KS wave i ðrÞ and ei functions and eigenvalues, respectively. The empty bound state is represented by its KS wave function u#a ðrÞ and its KS eigenenergy e#a . The evaluation of C# is different because the two electrons participating in the Auger process have parallel spin direction (spin-#). Therefore, the AC may take place via two indistiguishable processes that interfere. The probability C# reads: X X 1 Z X drdr0 u# ðrÞ u# ðr0 Þ C# ¼ 2p 3 a 2 u#1 2 occ: u#2 2 occ: u#3 62occ:
Z vðr; r0 Þu#2 ðr0 Þu#1 ðrÞ drdr0 u#a ðrÞ u#3 ðr0 Þ 2 vðr; r0 Þu#1 ðr0 Þu#2 ðrÞ d e#1 þ e#2 e#a e#3 :
ð4Þ
There are two ingredients that contribute to the spin dependence of the AC rate, i.e. to the difference between C" and C#: the spin-dependent screening that accounts for the differences u"i ðrÞ 6¼ u#i ðrÞ and the indistinguishability of electrons with equal spin that leads to the interference in the calculation of C#. For the case of He+ it was shown that these two effects give rise to a high polarization of the AC (>70% in the metallic density range) and that the interference was the major responsible for this [7,8]. In the following, using the above mentioned formalism, we study how these conclusions are modified for the cases of Ne+ and Ar+.
26
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29
3. Results In Fig. 1 we show the radial density induced in the continuum r2 Dnrc ðrÞ by the spin-polarized Ne+ and Ar+ ions embedded in a FEG of different densities. The left (right) panels correspond to Ne+ (Ar+). The ions are located at the origin, r = 0. In order to illustrate the effect of the spin-dependent perturbation in the medium, we distinguish between the induced electron density with spin-" (Dn"c ðrÞ, by solid lines) and spin-# (Dn#c ðrÞ, by dashed lines). In case of Ne+ the induced density is very similar for the two spins, i.e. the induced spin polarization is low. For Ar+ larger differences are observed between the curves for the different spins. The spin polarization of the screening cloud can be more directly quantified by comparing the integrated induced density for each spin direction (r ¼"; #):
Qrc
¼ 4p
Z
1
dr r2 Dnrc ðrÞ:
ð5Þ
0
Note that Q"c þ Q#c ¼ 1. The values of Q"c for Ne+ and Ar+ are shown in Table 1 [14] compared to those obtained for He+ in [7]. Some general ideas can be extracted from these results. At low densities, the spin polarization induced by the Ne+ and Ar+ ions is much lower than that induced by the He+. This reflects the larger energy difference between the excited 3S triplet and 1S singlet states of He in vacuum as compared to the energy splitting between the 3P and 1P excited states of Ne and Ar. At higher densities nontrivial effects appear in the screening characteristics. For He+ the spin polarization of the screening cloud shows the strongest variation with density and a monotonic decreasing behaviour with increasing density. In the case of Ne+ the spin polarization is always low and almost independent of the medium electronic density. Finally, for Ar+ the spin polarization first increases when decreasing density, it has a maximum around rs = 3 and finally decreases at higher densities. Note that in the high density range Ar+ is the ion that induces the highest spin polarization of the screening cloud. 0.04
0.04 Ne, r s = 2 a.u.
0.02
0.02 0.01 0
-0.02
-0.02
Ne, r s = 3 a.u.
Ar, r s = 3 a.u.
0.02 r 2 Δn σ c (a.u.)
r 2 Δn σ c (a.u.)
0
-0.01
0.02 0.01 0
0.01 0
-0.01
-0.01
Ne, r s = 4 a.u.
0.015
Ar, r s = 4 a.u.
0.015
0.01
r 2 Δn σ c (a.u.)
r 2 Δn σ c (a.u.)
0.01
-0.01
0.005 0
0.01 0.005 0
-0.005
-0.005 -0.01 0
Ar, r s = 2 a.u.
0.03
r 2 Δn σ c (a.u.)
r 2 Δn σ c (a.u.)
0.03
-0.01 2
4
6 r (a.u.)
8
10
0
2
4
6
8
10
r (a.u.)
Fig. 1. Radial electron density r2 Dnrc ðrÞ induced in the continuum by a Ne+ ion (left panels) and a Ar+ ion (right panels) embedded in a free electron gas. Different panels correspond to different medium electronic densities. The ions are located at r = 0 and their spin polarization is spin-". The induced density with spin parallel to the spin polarization of the ion Dn"c is shown by solid lines and that with spin antiparallel Dn#c by dashed lines.
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29 Table 1 Contribution of Q"c to the screening cloud around the different ions for different values of the medium electronic density rs
1
2
3
4
5
He+ Ne+ Ar+
0.57 0.55 –
0.61 0.54 0.68
0.66 0.54 0.70
0.75 0.54 0.63
0.90 – 0.58
The spin polarization of the ions is spin-".
In Fig. 2 we show the Auger capture rates (C" and C#) for these ions as a function of rs. The results for He+ were already presented in [7] and we show them here for comparison. In all the cases, the value of C" is higher than the value of C#. This implies a larger probability for excitation of spin-" electrons. This result is a combination of two effects. On the one hand, due to the spin-dependent screening it is easier to excite spin-" electrons, because the probability of finding them in the region close to the ions is higher. On the other hand, the destructive interference between the two equivalent processes that appears in the evaluation of C# results in a further reduction of this rate. Except in the very low density region (where C" for He+ is larger than for Ar+), the values of the rates for the different ions follow the following order: Cr(Ar+) > Cr(He+) > Cr(Ne+). The highest values of the rates obtained for Ar+ can be understood taking into account that the energy position of the level at which the electron is captured (3p# for Ar+) is closer to the bottom of the conduction band than in the other two cases. The levels at which the electron is captured in case of Ne+ and He+ ions are very close in energy. Therefore, the difference in the rates in these two cases is due to the different characteristics of the wavefunctions that appear in the evaluation of the Auger rates. In order to quantify the spin dependence of the Auger process we define the spin polarization of the excitation nAC by the following expression:
nAC ¼
C" C# : C" þ C#
ð6Þ
This quantity is related to the average spin polarization of the electrons excited in the Auger process. The dependence of nAC on rs is represented in Fig. 3. Results obtained by considering the interference (LSD&INT) and neglecting it (LSD0) are shown. The latter represents the effect of the spin-dependent screening in the polarization of the Auger process. A clear relation can be observed between the dependence on the background electronic density of the spin polarization of the screening cloud induced by each ion (see Table 1) and that of the Auger rates (see Fig. 3). For He+ and Ne+ ions both quantities show a monotonic increasing behaviour with rs, whereas for He+ the variations are much stronger than for Ne+. However, Ar+ represents a special case in which both quantities show a non monotonic behaviour when varying density and a maximum around rs = 3. Nevertheless, an important result that can be obtained from the comparison between the LSD&INT and LSD0 is that the interference in the calculation of C# is the main responsible of the spin polarization of the Auger process. In the following, we discuss the reasons behind the special behavior of the spin polarization of AC as a function of the density for Ar+. As explained above, the spin-polarization of AC is mostly due to the interference term that reduces the value of C#. This reduction is more effective when the two electrons participating in the Auger process have similar initial energies, i.e. when e#1 and e#2 are similar. In general, since for low background electronic densities the conduction band is narrower, one may expect the interference term and the subsequent reduction of C#, to be more important the lower the density is. In fact, this is what
100
10 0
He LSD&INT
80
%
10-1 -2
Ar LSD&INT
60
AC
10
ξ
Auger capture rate (a.u.)
27
-3
40
Ne LSD&INT
10
He LSD0
20
-4
10
Ar LSD0
0 1
-5
10
1
1.5
2
2.5
3 rs
3.5
4
4.5
5
Fig. 2. rs dependence of the Auger capture rate undergone by He+, Ne+ and Ar+ ions. Thick lines (thin lines) represent the AC rate in which a spin-" (spin-#) electron is excited: C"(C#). Solid lines are the results for He+, dash–dotted lines are the results for Ne+ and dashed lines are the results for Ar+.
Ne LSD0
2
3 rs
4
5
Fig. 3. Spin polarization of the Auger capture rate nAC , defined in Eq. (6) as a function of the density parameter rs. Thick lines (LSD&INT) are the results that include the interference in the calculation of C#. Thin lines (LSD0) are obtained neglecting the interference in the calculation of C#. Solid lines are the results for He+, dash–dotted lines are the results for Ne+ and dashed lines are the results for Ar+.
28
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29
is obtained for He+ and Ne+ ions for which nAC increases when lowering the density, but not for Ar+. In order to gain more information on the screening characteristics, we focus on the density of states in momentum space induced by the impurity in the continuum Dqr ðkÞ [15]: 1X d ð2l þ 1Þ drl ðkÞ; ð7Þ Dqr ðkÞ ¼ p l dk where k is the electron momentum, l is the angular momentum in a partial wave expansion and drl ðkÞ are the phase shifts of the KS radial wave functions. Since only spin-# electrons participate in the evaluation of C#, in Fig. 4 we show Dq# ðkÞ for Ne+ (top panel) and Ar+ (bottom panel) and different electronic densities. Both panels are in the same scale in order to better show the difference between the two cases. In case of Ar+, at high electronic densities (rs = 2–3) we observe that Dq# ðkÞ is negative at small k values, but takes large and positive values close to the Fermi wavevector. This implies that in the screening process electrons are removed from the bottom of the conduction band and that most of the screening electrons are located in a narrow range of k values close to the Fermi level. As a con6
Ne
5
Δρ (k)
4 3 2 1 0 -1
Ar
5
Δρ (k)
4 3 2 1 0 -1 0
0.2
0.4 0.6 k (a.u.)
0.8
1
Fig. 4. Spin-# component of the induced density of states in the continuum Dq# ðkÞ as a function of the momentum k for a Ne+ ion (upper panel) and a Ar+ ion (lower panel) embedded in a free electron gas. The spin polarization of the ions is spin-". The density parameter of the free electron gas is rs = 2 (solid lines), rs = 3 (dashed lines), rs = 4 (dash–dotted lines) and in case of Ar+, rs = 5 (dotted line).
sequence, the electrons participating in the Auger process are very likely to have similar energies which makes the interference in C# to gain importance. Briefly, in (rs = 2–3) range, due to the screening of the Ar+ ion an effective reduction of the bandwidth is produced with the subsequent reduction of C# and increase of spin-polarization of AC. For lower electronic densities (rs = 4–5) the screening electrons are located in two different regions of k space: close to the Fermi level and to the bottom of the band. As a consequence, though the conduction band is narrower than for high densities, the electrons are distributed in a wider k range. Hence, the interference is indeed less effective than in the high density range. Therefore, it is this competition between the actual reduction of the conduction bandwidth at low densities and the effective reduction of the band at high densities due to screening what gives rise to the non monotonic behavior of nAC on rs for Ar+. We note, that this effective reduction of the bandwidth at high densities also affects the magnitude of the spin dependent screening. This is reflected in the non monotonic behavior with the background density of the spin polarization of the screening cloud (see Table 1) and in the LSD0 results (see Fig. 3) for Ar+. In case of Ne+, as shown in the figure, no effective reduction of the band due to screening is observed for any value of the electronic density. Consequently, the spin polarization of the AC follows the usual increase when lowering electronic density. Finally, we mention than in case of He+ as it was shown in [7], the effective reduction of the band due to screening is more effective at low densities, i.e. the above mentioned two effects work together in the same direction. As a consequence, for He+ we obtain the strongest increase of the spin polarization of AC when lowering density. Experimental studies [1] of the deexcitation of He(23S), Ne(3P2) and Ar(3P2) atoms in front of a Pd(1 1 0) surface show an average polarization of the ejected electrons much lower than that reported here for the electrons excited in the Auger process. As it was shown for the case of He [9,10], this is due to the creation of an unpolarized cascade of secondary electrons, due to electron–electron scattering processes at the surface, that reduce the spin polarization of emitted electrons. Nevertheless, our results for the dependence of the spin polarization of the excited electrons on the particular projectile, are consistent with the measured data. In both cases, He induces the highest spin polarization of the electron yield, Ar is the intermediate case and Ne induces the lowest spin polarization. We also note, that for rs = 2, which is a typical metallic density, the calculated spin polarization of excited electrons for He is around a factor 2.26 larger than for Ne and a factor 1.57 larger than for Ar. These values compare well with the corresponding experimental 2.11 and 1.58 ratios. 4. Conclusions In this work we have studied the spin polarization of the electrons excited in the Auger neutralization of
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29
spin-polarized Ne+ and Ar+ ions embedded in an electron gas. We have shown that both the spin polarization of the screening density induced by these ions and that of the excited electrons follow the same qualitative behavior when varying the density. However, the main ingredient for the spin polarization of the excited electrons is the interference term that appears in the calculation of C#. In general, this interference term is more effective at low densities due to the narrowing of the conduction band. This general behavior is not fulfilled in case of Ar+ due to an effective reduction of the bandwidth that is caused by screening at high densities. Comparison of these results with previous calculations performed for He+ shows that, in agreement with the experimental data, the spin polarization of the electrons excited in the Auger neutralization is lower for Ne+ and Ar+ than for He+. In particular, Ne+ is responsible of the lowest degree of spin polarization. Acknowledgements Partial support by the University of the Basque Country UPV/EHU (Grant No. 9/UPV 00206.215-13639/2001), the Eusko Jaurlaritza, the Spanish MCyT (Grant No. FIS2004-06490-CO3-00) and the European Community 6th Network of Excellence NANOQUANTA (NMP4CT-2004-500298).
29
References [1] M.W. Hart, M.S. Hammond, F.B. Dunning, G.K. Walters, Phys. Rev. B 39 (1989) 5488. [2] D.L. Bixler, J.C. Lancaster, F.J. Kontur, P. Nordlander, G.K. Walters, F.B. Dunning, Phys. Rev. B 60 (1999) 9082. [3] J.C. Lancaster, F.J. Kontur, G.K. Walters, F.B. Dunning, Phys. Rev. B 67 (2003) 115413. [4] L.A. Salmi, Phys. Rev. B 46 (1992) 4180. [5] N. Bonini, G.P. Brivio, M.I. Trioni, Phys. Rev. B 68 (2003) 035408. [6] M. Alducin, R. Dı´ez Muin˜o, J.I. Juaristi, A. Arnau, J. Electron Spectrosc. Relat. Phenom. 137–140 (2004) 401. [7] M. Alducin, R. Dı´ez Muin˜o, J.I. Juaristi, Phys. Rev. A 70 (2004) 012901. [8] M. Alducin, Nucl. Instr. and Meth. B 232 (2005) 8. [9] J.I. Juaristi, M. Alducin, R. Dı´ez Muin˜o, M. Ro¨sler, Nucl. Instr. and Meth. B 232 (2005) 73. [10] M. Alducin, J.I. Juaristi, R. Dı´ez Muin˜o, M. Ro¨sler, P.M. Echenique, Phys. Rev. A 72 (2005) 024901. [11] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [12] (a) O. Gunnarsson, B.I. Lundqvist, Phys. Rev. B 13 (1976) 4274; (b) O. Gunnarsson, B.I. Lundqvist, Phys. Rev. B 15 (1977) 6006 (E). [13] P.J. Feibelman, E.J. McGuire, K.C. Pandey, Phys. Rev. B 15 (1977) 2202. [14] In case of Ar+ results for rs = 1 are not shown, because at this high density the 3p state is not bound. In case of Ne+ self-consistency was not achieved for rs = 5. This is related to a resonance in the density of states at the bottom of the conduction band, that in some iterations appears as a weakly bound 3s state. [15] R. Dı´ez Muin˜o, Nucl. Instr. and Meth. B 203 (2003) 8.
Nuclear Instruments and Methods in Physics Research B 256 (2007) 24–29 www.elsevier.com/locate/nimb
Spin dependent screening and Auger neutralization of singly-charged noble gas ions in metals J.I. Juaristi b
a,b,*
, M. Alducin
c
a Departamento de Fı´sica de Materiales, Facultad de Quı´mica UPV/EHU, Apartado 1072, 20080 San Sebastia´n, Spain Unidad de Fı´sica de Materiales, Centro Mixto CSIC-UPV/EHU, Facultad de Quı´mica UPV/EHU, Apartado 1072, 20080 San Sebastia´n, Spain c Donostia International Physics Center (DIPC), P. Manuel de Lardizabal 4, 20018 San Sebastia´n, Spain
Available online 16 January 2007
Abstract The screening of Ne+ and Ar+ ions embedded in a paramagnetic free electron gas is calculated using density functional theory within the local spin density approximation, analyzing the spin polarization of the screening cloud. As a second step, the Auger neutralization rates for these ions in a free electron gas are calculated, paying special attention to their dependence on the spin direction of the electron excited in the Auger process. It is observed that the spin polarization of the screening cloud and that of the Auger rates are related. Comparison with previous calculations performed for He+ shows that the spin polarization of the excited electrons is lower in the case of Ne+ and Ar+ than in the case of He+. This result is consistent with the experimental data obtained for the spin polarization of the emitted electrons in the deexcitation of metastable He*, Ne* and Ar* atoms in front of metal surfaces. Ó 2006 Elsevier B.V. All rights reserved. PACS: 34.50.Dy; 79.20.Rf Keywords: Electron emission; Ions; Auger neutralization; Spin polarization; Metals
1. Introduction The study of spin effects in the interaction of spinpolarized atomic particles interacting with paramagnetic metal surfaces constitutes an active field of research. For instance, the case of He+ ion or He* metastable atom projectiles has been widely studied both experimentally [1–3] and theoretically [4–10]. Special attention has been paid to the spin polarization of the electrons emitted in the neutralization/deexcitation of these projectiles. Experimentally, the emitted electrons show an overall spin polarization of about 30% parallel to the spin polarization of the incoming projectile and weakly dependent on the target *
Corresponding author. Address: Departamento de Fı´sica de Materiales, Facultad de Quı´mica UPV/EHU, Apartado 1072, 20080 San Sebastia´n, Spain. Tel.: +34 943 015396; fax: +34 943 015270. E-mail addresses: [email protected] (J.I. Juaristi), wapalocm@sq. ehu.es (M. Alducin). 0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.087
surface At high emission energies, the spin polarization of the emitted electrons is higher (60%–70%) [1–3]. From now on, we denote as " (#) the direction of the spin parallel (antiparallel) to the direction of the spin polarization of the projectile. Some recent works devoted to the study of the screening and Auger neutralization of a spin-polarized He+ ion embedded in a free electron gas (FEG) have been able to explain these experimental facts. First of all, the theoretical results show a high spin polarization (>70%) of the electrons excited in the Auger process [7,8]. The reason for this is twofold. On the one hand the screening of the spin-polarized He+ ion is dominated by spin-" electrons. On the other hand, in the case of exciting spin-# electrons, since the electron to be captured is also spin-#, it exists a destructive interference between two indistinguishable processes, with the subsequent reduction of the probability for this channel. The interference is absent in case of excitation of spin-" electrons since the two electrons participating in
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29
the Auger process have different spin directions. This interference was shown to be the most important reason for the predominant excitation of spin-" electrons [7,8]. The high spin polarization that results from the Auger process is reduced to about the measured 30% due to the creation of an unpolarized cascade of low energy secondary electrons [9,10]. The strong spin dependence of the Auger process is reflected in that of the electrons emitted at high energies. In [1] the spin polarization of electrons emitted in the deexcitation of Ne(3P2) and A(3P2) metastable atoms was also reported. It was found that the measured spin polarizations in these cases are significantly lower than in the He(23S) case. Motivated by this fact, here we present a study of the spin-dependent screening and the Auger neutralization of spin-polarized Ne+ and Ar+ ions embedded in FEG. The paper is organized as follows. In Section 2 we present the model used, in Section 3 we show and discuss the results obtained and in Section 4 we summarize the main conclusions of the work. Atomic units (a.u.) are used unless it is otherwise stated.
25
the screening is preferably due to spin-" electrons, as shown in [6–8]. In a similar way, here the Ne+ (Ar+) ion is modelled by populating with only two electrons the 2p (3p) KS state corresponding to spin-# electrons. This means that for Ne+ there are nine bound electrons, five of them are spin-" and four of them spin-#. In case of Ar+ there are seventeen bound electrons, nine of them spin-" and eight of them spin-#. Note that in all the considered cases, the screening induced electronic density in the continuum integrates to unity. In a second step, we calculate the Auger capture rate (AC), i.e. the probability per unit time that a spin-# electron from the conduction band fills the unoccupied bound state of the ion. The energy released in the transition is transferred to a second electron which gets excited in the continuum. The AC rate C can be calculated in first order perturbation theory for excitation of electrons with spin parallel (C") and antiparallel (C#) to the direction of polarization of the projectile (C ¼ C" þ C# ). The former (C") in which the electrons participating in the Auger process have antiparallel spin is calculated as follows [7,13]: X X Z X drdr0 u# ðrÞ u" ðr0 Þ C" ¼ 2p 3 a u#1 2 occ: u"2 2 occ: u"3 62 occ:
2. Theory The theoretical framework used to calculate the screening of the Ne+ and Ar+ ions in the FEG is the same as that used previously for the case of He+ [7] and it is only summarized here. The ion is treated as a static charged impurity embedded in the FEG with background density n0 and we define the customary one electron average radius rs from 1=n0 ¼ 4pr3s =3. Density functional theory (DFT) and the local spin density (LSD) approximation are used to calculate the electronic properties of the system. The Kohn–Sham (KS) wave functions ½uri ðrÞ are solutions with eigenvalue eri of the standard KS equations [11] with the parametrization of [12] for the exchange correlation potential. The index r runs over the spin orientations " and #. The electron density nðrÞ is obtained as: X X nðrÞ ¼ juri ðrÞj2 : ð1Þ r¼";# i2occ:
The electron density for just spin-" (spin-#) electrons n" (r) [n# (r)] is defined by limiting the sum over occupied states to the required spin component. Furthermore, we define the induced density for each spin orientation as: Dnr ðrÞ ¼ nr ðrÞ n0 =2;
ð2Þ +
with r = ", #. In previous works [6–10] the He ion, in which there is only one bound electron, has been modelled by populating just one of the two bound KS states of the He+/FEG system (there is one for each spin orientation). Furthermore, the spin of the electron bound to the projectile is fixed ("). As a consequence, the one electron KS potential, that includes the spin-dependent exchangecorrelation part and the electronic charge that surrounds the He+ ion are different for each spin orientation. Indeed,
2 vðr; r0 Þu"2 ðr0 Þu#1 ðrÞ d e#1 þ e"2 e#a e"3 ;
ð3Þ
where vðr; r0 Þ ¼ 1=jr r0 j is the Coulomb potential respon";# sible for the decay and u";# are the KS wave i ðrÞ and ei functions and eigenvalues, respectively. The empty bound state is represented by its KS wave function u#a ðrÞ and its KS eigenenergy e#a . The evaluation of C# is different because the two electrons participating in the Auger process have parallel spin direction (spin-#). Therefore, the AC may take place via two indistiguishable processes that interfere. The probability C# reads: X X 1 Z X drdr0 u# ðrÞ u# ðr0 Þ C# ¼ 2p 3 a 2 u#1 2 occ: u#2 2 occ: u#3 62occ:
Z vðr; r0 Þu#2 ðr0 Þu#1 ðrÞ drdr0 u#a ðrÞ u#3 ðr0 Þ 2 vðr; r0 Þu#1 ðr0 Þu#2 ðrÞ d e#1 þ e#2 e#a e#3 :
ð4Þ
There are two ingredients that contribute to the spin dependence of the AC rate, i.e. to the difference between C" and C#: the spin-dependent screening that accounts for the differences u"i ðrÞ 6¼ u#i ðrÞ and the indistinguishability of electrons with equal spin that leads to the interference in the calculation of C#. For the case of He+ it was shown that these two effects give rise to a high polarization of the AC (>70% in the metallic density range) and that the interference was the major responsible for this [7,8]. In the following, using the above mentioned formalism, we study how these conclusions are modified for the cases of Ne+ and Ar+.
26
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29
3. Results In Fig. 1 we show the radial density induced in the continuum r2 Dnrc ðrÞ by the spin-polarized Ne+ and Ar+ ions embedded in a FEG of different densities. The left (right) panels correspond to Ne+ (Ar+). The ions are located at the origin, r = 0. In order to illustrate the effect of the spin-dependent perturbation in the medium, we distinguish between the induced electron density with spin-" (Dn"c ðrÞ, by solid lines) and spin-# (Dn#c ðrÞ, by dashed lines). In case of Ne+ the induced density is very similar for the two spins, i.e. the induced spin polarization is low. For Ar+ larger differences are observed between the curves for the different spins. The spin polarization of the screening cloud can be more directly quantified by comparing the integrated induced density for each spin direction (r ¼"; #):
Qrc
¼ 4p
Z
1
dr r2 Dnrc ðrÞ:
ð5Þ
0
Note that Q"c þ Q#c ¼ 1. The values of Q"c for Ne+ and Ar+ are shown in Table 1 [14] compared to those obtained for He+ in [7]. Some general ideas can be extracted from these results. At low densities, the spin polarization induced by the Ne+ and Ar+ ions is much lower than that induced by the He+. This reflects the larger energy difference between the excited 3S triplet and 1S singlet states of He in vacuum as compared to the energy splitting between the 3P and 1P excited states of Ne and Ar. At higher densities nontrivial effects appear in the screening characteristics. For He+ the spin polarization of the screening cloud shows the strongest variation with density and a monotonic decreasing behaviour with increasing density. In the case of Ne+ the spin polarization is always low and almost independent of the medium electronic density. Finally, for Ar+ the spin polarization first increases when decreasing density, it has a maximum around rs = 3 and finally decreases at higher densities. Note that in the high density range Ar+ is the ion that induces the highest spin polarization of the screening cloud. 0.04
0.04 Ne, r s = 2 a.u.
0.02
0.02 0.01 0
-0.02
-0.02
Ne, r s = 3 a.u.
Ar, r s = 3 a.u.
0.02 r 2 Δn σ c (a.u.)
r 2 Δn σ c (a.u.)
0
-0.01
0.02 0.01 0
0.01 0
-0.01
-0.01
Ne, r s = 4 a.u.
0.015
Ar, r s = 4 a.u.
0.015
0.01
r 2 Δn σ c (a.u.)
r 2 Δn σ c (a.u.)
0.01
-0.01
0.005 0
0.01 0.005 0
-0.005
-0.005 -0.01 0
Ar, r s = 2 a.u.
0.03
r 2 Δn σ c (a.u.)
r 2 Δn σ c (a.u.)
0.03
-0.01 2
4
6 r (a.u.)
8
10
0
2
4
6
8
10
r (a.u.)
Fig. 1. Radial electron density r2 Dnrc ðrÞ induced in the continuum by a Ne+ ion (left panels) and a Ar+ ion (right panels) embedded in a free electron gas. Different panels correspond to different medium electronic densities. The ions are located at r = 0 and their spin polarization is spin-". The induced density with spin parallel to the spin polarization of the ion Dn"c is shown by solid lines and that with spin antiparallel Dn#c by dashed lines.
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29 Table 1 Contribution of Q"c to the screening cloud around the different ions for different values of the medium electronic density rs
1
2
3
4
5
He+ Ne+ Ar+
0.57 0.55 –
0.61 0.54 0.68
0.66 0.54 0.70
0.75 0.54 0.63
0.90 – 0.58
The spin polarization of the ions is spin-".
In Fig. 2 we show the Auger capture rates (C" and C#) for these ions as a function of rs. The results for He+ were already presented in [7] and we show them here for comparison. In all the cases, the value of C" is higher than the value of C#. This implies a larger probability for excitation of spin-" electrons. This result is a combination of two effects. On the one hand, due to the spin-dependent screening it is easier to excite spin-" electrons, because the probability of finding them in the region close to the ions is higher. On the other hand, the destructive interference between the two equivalent processes that appears in the evaluation of C# results in a further reduction of this rate. Except in the very low density region (where C" for He+ is larger than for Ar+), the values of the rates for the different ions follow the following order: Cr(Ar+) > Cr(He+) > Cr(Ne+). The highest values of the rates obtained for Ar+ can be understood taking into account that the energy position of the level at which the electron is captured (3p# for Ar+) is closer to the bottom of the conduction band than in the other two cases. The levels at which the electron is captured in case of Ne+ and He+ ions are very close in energy. Therefore, the difference in the rates in these two cases is due to the different characteristics of the wavefunctions that appear in the evaluation of the Auger rates. In order to quantify the spin dependence of the Auger process we define the spin polarization of the excitation nAC by the following expression:
nAC ¼
C" C# : C" þ C#
ð6Þ
This quantity is related to the average spin polarization of the electrons excited in the Auger process. The dependence of nAC on rs is represented in Fig. 3. Results obtained by considering the interference (LSD&INT) and neglecting it (LSD0) are shown. The latter represents the effect of the spin-dependent screening in the polarization of the Auger process. A clear relation can be observed between the dependence on the background electronic density of the spin polarization of the screening cloud induced by each ion (see Table 1) and that of the Auger rates (see Fig. 3). For He+ and Ne+ ions both quantities show a monotonic increasing behaviour with rs, whereas for He+ the variations are much stronger than for Ne+. However, Ar+ represents a special case in which both quantities show a non monotonic behaviour when varying density and a maximum around rs = 3. Nevertheless, an important result that can be obtained from the comparison between the LSD&INT and LSD0 is that the interference in the calculation of C# is the main responsible of the spin polarization of the Auger process. In the following, we discuss the reasons behind the special behavior of the spin polarization of AC as a function of the density for Ar+. As explained above, the spin-polarization of AC is mostly due to the interference term that reduces the value of C#. This reduction is more effective when the two electrons participating in the Auger process have similar initial energies, i.e. when e#1 and e#2 are similar. In general, since for low background electronic densities the conduction band is narrower, one may expect the interference term and the subsequent reduction of C#, to be more important the lower the density is. In fact, this is what
100
10 0
He LSD&INT
80
%
10-1 -2
Ar LSD&INT
60
AC
10
ξ
Auger capture rate (a.u.)
27
-3
40
Ne LSD&INT
10
He LSD0
20
-4
10
Ar LSD0
0 1
-5
10
1
1.5
2
2.5
3 rs
3.5
4
4.5
5
Fig. 2. rs dependence of the Auger capture rate undergone by He+, Ne+ and Ar+ ions. Thick lines (thin lines) represent the AC rate in which a spin-" (spin-#) electron is excited: C"(C#). Solid lines are the results for He+, dash–dotted lines are the results for Ne+ and dashed lines are the results for Ar+.
Ne LSD0
2
3 rs
4
5
Fig. 3. Spin polarization of the Auger capture rate nAC , defined in Eq. (6) as a function of the density parameter rs. Thick lines (LSD&INT) are the results that include the interference in the calculation of C#. Thin lines (LSD0) are obtained neglecting the interference in the calculation of C#. Solid lines are the results for He+, dash–dotted lines are the results for Ne+ and dashed lines are the results for Ar+.
28
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29
is obtained for He+ and Ne+ ions for which nAC increases when lowering the density, but not for Ar+. In order to gain more information on the screening characteristics, we focus on the density of states in momentum space induced by the impurity in the continuum Dqr ðkÞ [15]: 1X d ð2l þ 1Þ drl ðkÞ; ð7Þ Dqr ðkÞ ¼ p l dk where k is the electron momentum, l is the angular momentum in a partial wave expansion and drl ðkÞ are the phase shifts of the KS radial wave functions. Since only spin-# electrons participate in the evaluation of C#, in Fig. 4 we show Dq# ðkÞ for Ne+ (top panel) and Ar+ (bottom panel) and different electronic densities. Both panels are in the same scale in order to better show the difference between the two cases. In case of Ar+, at high electronic densities (rs = 2–3) we observe that Dq# ðkÞ is negative at small k values, but takes large and positive values close to the Fermi wavevector. This implies that in the screening process electrons are removed from the bottom of the conduction band and that most of the screening electrons are located in a narrow range of k values close to the Fermi level. As a con6
Ne
5
Δρ (k)
4 3 2 1 0 -1
Ar
5
Δρ (k)
4 3 2 1 0 -1 0
0.2
0.4 0.6 k (a.u.)
0.8
1
Fig. 4. Spin-# component of the induced density of states in the continuum Dq# ðkÞ as a function of the momentum k for a Ne+ ion (upper panel) and a Ar+ ion (lower panel) embedded in a free electron gas. The spin polarization of the ions is spin-". The density parameter of the free electron gas is rs = 2 (solid lines), rs = 3 (dashed lines), rs = 4 (dash–dotted lines) and in case of Ar+, rs = 5 (dotted line).
sequence, the electrons participating in the Auger process are very likely to have similar energies which makes the interference in C# to gain importance. Briefly, in (rs = 2–3) range, due to the screening of the Ar+ ion an effective reduction of the bandwidth is produced with the subsequent reduction of C# and increase of spin-polarization of AC. For lower electronic densities (rs = 4–5) the screening electrons are located in two different regions of k space: close to the Fermi level and to the bottom of the band. As a consequence, though the conduction band is narrower than for high densities, the electrons are distributed in a wider k range. Hence, the interference is indeed less effective than in the high density range. Therefore, it is this competition between the actual reduction of the conduction bandwidth at low densities and the effective reduction of the band at high densities due to screening what gives rise to the non monotonic behavior of nAC on rs for Ar+. We note, that this effective reduction of the bandwidth at high densities also affects the magnitude of the spin dependent screening. This is reflected in the non monotonic behavior with the background density of the spin polarization of the screening cloud (see Table 1) and in the LSD0 results (see Fig. 3) for Ar+. In case of Ne+, as shown in the figure, no effective reduction of the band due to screening is observed for any value of the electronic density. Consequently, the spin polarization of the AC follows the usual increase when lowering electronic density. Finally, we mention than in case of He+ as it was shown in [7], the effective reduction of the band due to screening is more effective at low densities, i.e. the above mentioned two effects work together in the same direction. As a consequence, for He+ we obtain the strongest increase of the spin polarization of AC when lowering density. Experimental studies [1] of the deexcitation of He(23S), Ne(3P2) and Ar(3P2) atoms in front of a Pd(1 1 0) surface show an average polarization of the ejected electrons much lower than that reported here for the electrons excited in the Auger process. As it was shown for the case of He [9,10], this is due to the creation of an unpolarized cascade of secondary electrons, due to electron–electron scattering processes at the surface, that reduce the spin polarization of emitted electrons. Nevertheless, our results for the dependence of the spin polarization of the excited electrons on the particular projectile, are consistent with the measured data. In both cases, He induces the highest spin polarization of the electron yield, Ar is the intermediate case and Ne induces the lowest spin polarization. We also note, that for rs = 2, which is a typical metallic density, the calculated spin polarization of excited electrons for He is around a factor 2.26 larger than for Ne and a factor 1.57 larger than for Ar. These values compare well with the corresponding experimental 2.11 and 1.58 ratios. 4. Conclusions In this work we have studied the spin polarization of the electrons excited in the Auger neutralization of
J.I. Juaristi, M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 24–29
spin-polarized Ne+ and Ar+ ions embedded in an electron gas. We have shown that both the spin polarization of the screening density induced by these ions and that of the excited electrons follow the same qualitative behavior when varying the density. However, the main ingredient for the spin polarization of the excited electrons is the interference term that appears in the calculation of C#. In general, this interference term is more effective at low densities due to the narrowing of the conduction band. This general behavior is not fulfilled in case of Ar+ due to an effective reduction of the bandwidth that is caused by screening at high densities. Comparison of these results with previous calculations performed for He+ shows that, in agreement with the experimental data, the spin polarization of the electrons excited in the Auger neutralization is lower for Ne+ and Ar+ than for He+. In particular, Ne+ is responsible of the lowest degree of spin polarization. Acknowledgements Partial support by the University of the Basque Country UPV/EHU (Grant No. 9/UPV 00206.215-13639/2001), the Eusko Jaurlaritza, the Spanish MCyT (Grant No. FIS2004-06490-CO3-00) and the European Community 6th Network of Excellence NANOQUANTA (NMP4CT-2004-500298).
29
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