Appl. Phys. A 73, 663–666 (2001) / Digital Object Identifier (DOI) 10.1007/s003390100957

Applied Physics A Materials Science & Processing

Theoretical description of the Fano-effect in the angle-integrated valence-band photoemission of paramagnetic solids J. Min´ar1,∗ , H. Ebert1 , L.H. Tjeng2 , P. Steeneken2 , G. Ghiringhelli3 , O. Tjernberg4 , N.B. Brookes5 1 Department Chemie, Physikalische Chemie, Universität München, Butenandtstr. 5–13, 81377 München, 2 Solid State Physics Laboratory, Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 3 Dipartimento di Fisica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy 4 Material Physics, Royal Institute of Technology, 100 44 Stockholm, Sweden 5 European Synchrotron Radiation Facility, Boîte Postale 220, 38043 Grenoble Cedex, France

Germany AG Groningen, The Netherlands

Received: 23 May 2001/Accepted: 4 July 2001/Published online: 5 October 2001 –  Springer-Verlag 2001

Abstract. A theoretical description of the Fano-effect in the angle-integrated valence-band photoemission (VB XPS) of paramagnetic solids is presented that is based on the one-step model of photoemission and relativistic multiple scattering theory. Applications to fcc-Cu and fcc-Ag led to a very satisfying agreement with recent experimental data that show the Fano-effect, i.e., a finite spin polarisation of the photoelectrons is found for excitation with circularly polarised radiation. As can be demonstrated by model calculations, this finding is caused by the presence of spin–orbit coupling. To allow for a more detailed discussion of the spectra, a simplified description of the Fano-effect is presented that treats spin–orbit coupling as a perturbation.

based on the one-step model of photoemission and relativistic multiple scattering theory. To some extent this is complementary to the approach developed by Thole and van der Laan [12]. While the work of these authors is based on an atomic description of the underlying electronic structure, an itinerant behavior of the electrons will be assumed here. To discuss the resulting spectra in some detail, we perform model calculations by treating the spin–orbit coupling as a perturbation, thereby arriving at a very transparent explanation for the observed spin polarization.

PACS: 71.15.Rf; 71.20.Be; 82.80.Pv; 71.70.Ej

1.1 Fully relativistic description

The Fano-effect, the creation of spin-polarized electrons in an excitedstatewhenusingcircularlypolarizedlightforexcitation, was predicted by Fano [1, 2] to occur for free atoms as a consequence of spin–orbit coupling and the selection rules connected with circularly polarized radiation. This was demonstrated by Heizmann et al. for free Cs atoms [3] and also for polycrystalline Cs films [4]. Later, the Fano-effect, which is sometimes called optical spin orientation, was observed in many different systems, such as semiconductors and absorbates [5, 6]. Recently Ghiringhelli et al. [7] observed the Fano-effect in the spin-resolved, angle-integrated valence-band X-ray photoemission spectroscopy (VB XPS) of transition metals. A corresponding theoretical description for the experiment on paramagnetic solids was first presented by Koyama and Merz [8] on the basis of a model that relies on the artificial decomposition of the photoemission process into three steps. Later theoretical work [9, 10] was based on a more advanced one-step model. In addition, a fully relativistic description of spin- and angle-resolved photoemission has been developed during the last few years by several groups [9–11]. In this paper we provide a detailed analysis of the Fanoeffect in the angle-integrated VB XPS of paramagnetic solids, ∗ Corresponding

author. (Fax: +49-89/2180-7584, E-mail: [email protected])

1 Theoretical framework

As mentioned above a rather sophisticated fully relativistic description of spin- and angle-resolved photoemission has been developed during the last few years by several groups [9–11]. It is based on the one-step model and represents the electronic Green’s function by making use of the multiple scattering or Korringa–Kohn–Rostoker (KKR) formalism. By dealing with the multiple-scattering problem in real space, a very simple expression could be derived by Ebert and Schwitalla [11] for the spin-resolved, angle-integrated VB XPS spectra of ferromagnetic solids. Within this approach one can express the photocurrent I(E, k, σ; ω, q, λ) for photons with frequency ω and polarization λ by
−m s −m s I(E, m s ; ω, q, λ) ∝ Im CΛ CΛ  ×

Λ Λ ,µ=µ




Λ1 Λ2

τΛ0 01 Λ2 (E) ×

×


Λ

−


Λ

 qλ 0  tΛ  Λ (E )MΛ Λ 1

∗

qλ 0  tΛ  Λ (E )MΛ Λ 2


Λ Λ Λ1

 0  0∗  tΛ  Λ (E )IΛ Λ Λ tΛ Λ (E ) , 1 qλ

(1)

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where ω, q, λ specify the photon energy, wave vector and polarization of the radiation and E = E  − ω is the energy of the initial states. The quantities Cλm s are Clebsch–Gordan coefficients with the relativistic quantum number Λ = (κ, µ), qλ and MΛΛ are the corresponding matrix elements. The electronic structure of the system is represented by the single site n nm t-matrix tΛΛ  (E) and by the scattering path operator τΛΛ (E). (1) has been successfully applied to ferromagnetic metals and alloys [11]. Obviously, (1) can be applied straightforwardly to calculate the spin-resolved photocurrent of paramagnetic (i.e., non-magnetic) solids using circularly polarized photons. To allow for a direct comparison of the resulting theoretical spectra with experiments, one has to account for various intrinsic and apparative broadening mechanisms in a proper way. To represent the various lifetime effects, a Lorentzian broadening was applied with an energy-dependent width Γ(E) = 0.01 + 0.02(E − E F )2 eV, where E F is the Fermi energy. Additional Gaussian broadening was applied using σ = 0.4 eV. 1.2 Perturbational treatment of spin–orbit coupling The scheme sketched above, which is based on four-component Dirac formalism, deals with all the relativistic effects in a rigorous way. However, it is not very transparent and does not allow for simple interpretation of the resulting spectra. To overcome this problem a simpler non-relativistic theory is introduced here. The starting point of this approach is the following expression for the photocurrent [13]:   final (r1 , E  )† X qλ (r1 ) I(E, k, m s ; ω, q, λ) ∝ d3r1 d3r2 φkm s final  × Im G(r1 , r2 , E)X × qλ (r2 )φkm s (r2 , E ) ,

(2) where the initial valence states are represented by the Green’s final function G(r1 , r2 , E) and the final state φkm is assumed to be s a time-reversed LEED state [13]. Originally, (2) was meant to deal with non-magnetic solids, and the influence of spin–orbit coupling was ignored. In our case the influence of spin–orbit coupling on the valence states is treated as a perturbation by SOC = 1 ξlσ [14]. This leads representing it by the operator H 2 to the 2 × 2 Green’s function matrix G (for more details see Ebert et al. [15]): G = G 0 + G 0 Hˆ SOC G 0 + G 0 Hˆ SOC G 0 Hˆ SOC G 0 + . . .

(3)

As confirmed by model calculations (see below), in our case the Fano-effect is a first-order effect with respect to the spin–orbit coupling. Inserting the expressions final qλ into (2) one obtains a very genfor G, φkm and X s eral expression that allows one to deal with the spin- and angle-resolved photoemission in solids. Although the Fanoeffect can also be studied in great detail for the angleresolved case, we are interested here only in the angleintegrated mode. In particular, this implies that one has to average the expression for I(E, k, σ, m s ; ω, q, λ) with respect to the wave vector k of the photoelectron. For the spin-resolved photocurrent Iλm s of a paramagnetic solid, one

obtains



λ λ∗ M Lm Iλm s ∝ M Lm Im τ L 1 L 2 (E) [tl (E)]2 s L 1ms s L 2ms +




L

L1 L2

λ M Lm M LSOC M Lλ∗L 4 s L 1ms 2 ms L 3ms

 × Im τ L 1 L 2 (E)τ L 3 L 4 (E) ,

L3L4

(4) λ where M Lm   is a matrix element of the non-relativistic s L ms electron–photon interaction operator for the non-relativistic quantum numbers L = (l, m l ). The angular part of these matrix elements immediately implies for circularly polarized radiation (λ = ±1) the selection rules l − l  = ±1, m l − SOC m l = λ and m s − m s = 0. In (4) M Lm   is a matrix elems L ms ent of the spin–orbit coupling operator HSOC (for more details see Min´ar et al. [16]). Restricting the l character of the initial states to l = 2, i.e., d electrons, one ends up with a very simple expression:  2  2
l 12
l 1 2 ∆Iλ ∝ (2l + 1) −m λ m 1 000 m1 l
 × m 2 Im τ2m 1 2m 2 τ2m 2 2m 1 m2

∝




  Slλ Im τt2g 4 τeg + τt2g .

(5)

l

In this expression we exploited the cubic symmetry of the systems studied here. In that case only two different elements of the multiple scattering path operator occur, i.e., τeg and τt2g . The sign function Slλ in (5) has for left (LCP) and right (RCP) circularly polarized light the following simple properties:  +1 d → p λ = +1 (LCP),   −1 d → f λ = +1 (LCP), Slλ = . (6)  −1 d → p λ = −1 (RCP), +1 d → f λ = −1 (RCP). Finally one can make use of the identity ∂G ∂E = −GG for the Green’s function to interpret the right-hand side of (5). Accordingly, this expression is essentially proportional to the energy derivative of the density of states n(E). 2 Results and discussion The relativistic approach sketched in the previous section has been applied so far only to deal with a number of ferromagnetic binary alloy systems [11]. For these systems it has been predicted that even for the angle-integrated mode magnetic circular dichroism should in general occur. Of course, (1) can be used directly to calculate the photo-current for paramagnetic solids as well. The left panel of Fig. 1 shows the results for the spin- and angle-integrated VB XPS spectra of pure fcc-Cu for a photon energy of 600 eV together with the corresponding experimental data [7]. In the right panel of Fig. 1 the spin-resolved spectra I+m s for excitation with left circularly polarized light (λ = +1) is shown. As in the case of the spin-integrated spectra the relativistic calculation also reproduces the experimental data for the spin

665 100

↑ I+ ↓ I+

60 40 20

4

exp. theory Intensity (a.u.)

Spin-difference (a.u.)

Intensity (a.u.)

80

exp. theory

6

exp. theory

2 0 -2

50

Spin-difference (a.u.)

100

↑ I+ ↓ I+

10

exp. theory

0

-10

-4 0

-6

-4 -2 0 Binding energy (eV)

-6

-4 -2 0 Binding energy (eV)

Fig. 1. Left: spin- and angle-integrated VB XPS spectrum of fcc-Cu for ↑ ↓ a photon energy of 600 eV. Right: spin difference ∆I+ = I+ − I+ of the photocurrent for excitation with left circularly polarised radiation. Solid line: theory; dashed line: experiment [7]. The left panel shows in addition ↑(↓) the theoretical spin-resolved spectra I+ ↑

↓

difference ∆I+ = I+ − I+ in a very satisfying way. The right panel of the Fig. 1 shows in particular that the spin difference amounts to about 5% (the same normalization has been used for both panels). A decomposition of the theoretical spectrum according to the l-character of the initial state shows that the d-contribution is by far the dominant one and that the spectrum indeed maps the corresponding density of states. To investigate the mechanism giving rise to the spin difference ∆I+ , two types of model calculations have been performed that were done in a relativistic way but with the spin–orbit coupling manipulated [17, 18]. In a first step the relative strength of the spin–orbit coupling x ξ was varied between 0 and 1. As can be seen from the left panel of Fig. 2 the shape of the spin difference ∆I+ hardly changes, and the amplitude increases more or less linearly with x ξ . This clearly demonstrates that the Fano-effect is a firstorder effect with respect to the spin–orbit coupling. These results clarify that the observed spin difference, ∆Iλ , can be seen as a rather direct measure for the spin–orbit coupling strength ξ. Consequently, one may expect, for example, for Ag, a spin difference ∆Iλ that is according to the ratio of the spin–orbit energy splittings, ∆E SOC , of the d electrons [∆E SOC (Cu) = 0.27 eV, ∆E SOC (Ag) = 0.52 eV] about a factor of 2 higher than for Cu. This has been demonstrated by experimental measurements [7] and our calculations based on (1). As one can see from Fig. 3 the dif4

2 0 x=0.0 x=0.2 x=0.4 x=0.6 x=0.8 x=1.0

-2 -4 -6

-4 -2 0 Binding energy (eV)

Spin-difference (a.u.)

Spin-difference (a.u.)

4

exact xy zz

2 0 -2 -4 -6

-4 -2 0 Binding energy (eV)

Fig. 2. Theoretical results for the spin difference ∆I+ obtained by model calculations. Left: the strength x of the spin–orbit coupling varied between 0 and 1. Right: only the spin diagonal (zz) and only the spin mixing (xy) parts of the full spin–orbit coupling have been kept for the calculations

0 -10

-8 -6 -4 -2 Binding energy (eV)

0

-10

-8 -6 -4 -2 Binding energy (eV)

0

Fig. 3. As Fig. 1 but for fcc-Ag

ference spectrum amounts for Ag to about 14%. However, it should be noted that for heavier elements higher-order spin–orbit coupling contributions to ∆Iλ might become important. In the right panel of Fig. 2 results are shown for a second set of calculations. Within these the spin–orbit coupling was split into its spin diagonal (l z σz ) and a part that gives rise to a hybridization of the spin systems (l x σx + l y σ y ) [18]. As can be seen, the spin diagonal part l z σz reproduced more or less the experimental spin difference ∆I + . This also demonstrates the fact that the Fano-effect is indeed a first-order effect with respect to the spin–orbit coupling. The rather simple expressions (5) and (6) allow us to discuss the Fano-effect in the photoemission from paramagnetic solids in a very transparent way. From (6) one can see that the spin difference from d → p and d → f transitions have reversed sign. In addition, if the helicity of the radiation is reversed, the spin difference simply changes sign. That means, in particular, that if unpolarized radiation is used for excitation, i.e., an average with respect to left and right circularly polarized radiation is taken, the spin difference vanishes exactly. Of course, this is what one would expect. 3 Summary Results of fully relativistic calculations for the VB XPS spectra of fcc-Cu and fcc-Ag for circularly polarized radiation were presented. Recent experimental data for both metals were reproduced very satisfactorily. Based on model calculations, we presented a simple non-relativistic description of the Fano-effect in photoemission based on perturbation theory. It was demonstrated that the Fano-effect in the case of Cu and Ag is a first-order effect with respect to the spin– orbit coupling. In addition, it turned out that the spin difference induced by spin–orbit coupling essentially reflects the energy derivative of the density of states, n(E). These results imply that the spin-resolved angle-integrated VB XPS can be used as a simple and sensitive probe to determine the average spin–orbit coupling strength for the valence-band electrons. Acknowledgements. This work was supported by the German Ministry for Education and Research (BMBF) under contract 05 SC8WMA 7 within the program Zirkular polarisierte Synchrotronstrahlung: Dichroismus, Magnetismus und Spinorientierung.

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