Theory of f-electron photoemission in light rare-earth insulating systems

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Theory of f-electron photoemission in light rare-earth insulating systems

J.C. Parlebas, T. Nakano, A. Kotani

To cite this version:

J.C. Parlebas, T. Nakano, A. Kotani. Theory of f-electron photoemission in light rare-earth insulating

systems. Journal de Physique, 1987, 48 (7), pp.1141-1146. �10.1051/jphys:019870048070114100�. �jpa-

00210537�

Theory ^of f-electron photoemission

in light rare-earth insulating systems

J. C. Parlebas (*), ^{T. Nakano} (+ ) ^{and A. Kotani} (+ )

L.M.S.E.S. (UA ^{CNRS N°} 306) Université Louis Pasteur, 4, ^{rue Blaise} Pascal, 67070 Strasbourg Cedex, ^France

(+ ) Department ^of Physics, Faculty ^of Science, ^Osaka University, Toyonaka 560, Japan (Reçu le 27 octobre 1986, accept6 ^sous forme définitive ^{le 9 mars} 1987)

Résumé.

Nous présentons ^une méthode pour calculer les spectres ^de photoémission directe f dans le cas des

systèmes isolants de Terres Rares légères ; pour ^ce faire nous utilisons le modèle d’impureté d’Anderson avec une bande pleine ^{et un état de base} qui ^{est un} doublet de spin. ^Nous explorons ^l’allure générale ^des spectres ^en fonction des divers paramètres du modèle. Nous proposons d’expliquer ^ces spectres ^à partir ^{de la limite} Uff ^infinie qui ^est entièrement analytique. ^Un exemple typique ^de comparaison ^{de nos résultats avec les} spectres expérimentaux ^de photoémission ^{directe est le cas du} composé Ce2O3.

Abstract.

We present ^a method for calculating f-electron photoemission spectra ^of light ^rare-earth insulating systems within the filled band Anderson impurity ^{model and} starting ^{from a} spin ^doublet ground

state. We explore ^the overall lineshapes ^{of the} spectra ^with respect ^to the various parameters entering ^the

model. An analytical explanation of the calculated spectra ^is proposed ^{in the} infinitely large Uff ^limit.

Typically, ^{contact can} be made with experimental ^direct photoemission spectra ^{in the case of} Ce2O3 compounds, ^for example.

Classification

Physics ^Abstracts

79.60E

1. Introduction.

The splitting ^of 4 f-derived photoemission spectra due to the mixing ^{of the 4 f} configurations ^{in the} ground ^state (i.e. ^{due to the} hybridization ^{in the}

initial state which induces the mixed valence effect)

has been well obtained in some compounds ^of heavy

Rare Earth elements like Sm, Eu, ^{Tm and Yb} [1].

For Sm and Tm, the valence deduced from the

experimental photoemission (XPS) spectra ^{is in} agreement ^with the valence deduced from the macro-

scopic measurements (susceptibility, ^lattice par-

ameter...). ^For light ^{rare earth} systems, especially

for Ce metal and metallic compounds, ^{the situation}

is more complicated, ^{first of all,} because it is then hard to separate the f contribution from the other conduction states contribution in the XPS spec-

trum [1] ^{unless resonant} XPS is used. Nevertheless,

direct f photoemission ^{has often been used to} study

the position ^and occupancy of the f level in Ce

(*) ^{Present address :} c/o Prof. P. Fulde, Max-Planck-

Institut, Heisenbergstrasse 1, ^Postfach 80 06 65, ⁷⁰⁰⁰ Stuttgart 80, ^F.R.G.

^{A von} Humboldt fellow.

metallic compounds [2, 3]. ^{This kind of} experiments

has been rather well understood theoretically ^within

the impurity model in the limit of large degeneracy (Nf - oo ) [4].

In this paper ^we focus attention on the f-XPS spectra ⁱⁿ light rare-earth insulating systems, ^es- pecially ⁱⁿ Ce203 compounds. ^{One of our motiva-}

tions was the recent experimental ^XPS spectra ^{for a} Ce02 ^{film converted} progressively ^to Ce203 [5]. ^In

the case of Ce02, ^a cluster model [6] ^{has first been} proposed ^{to calculate core level} spectra, ^{then the} impurity ^{model has been used} by ^{several authors} [7- 10] ; ⁱⁿ particular Wuilloud et al. [7] calculated and discussed the f-spectra ^of Ce02 compounds ^{but the}

case of Ce203 ^{has never been} investigated theoretically. In section 2 of the present paper, using

the filled band impurity ^{model we} develop ^{an exact}

formulation of the f-photoemission spectrum ^{in a}

Ce203 type compound ^{with a} spin ^doublet ground

state. Then (Sect. 3) ^we present ^an analytical ^calcu-

lation of the f-XPS spectrum ^{in the} limiting ^{case of} large electron-electron interactions (Uff -+ oo). ^In

section 4 we present ^our main numerical results on f-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048070114100

1142

XPS spectra and section 5 is devoted to some

discussion and concluding ^remarks.

2. Formulation.

We consider ^a system consisting ^{of a filled valence}

band (VB) and 4 f levels of energy sf with spin (but

without orbital) degeneracy ^{and located on the} photoexcited ^site. During ^the f-photoemission ^pro-

cess, the Rare-Earth compound ^{will be described} by

the following general Hamiltonian (valid ^{for both}

initial and final states) :

where

Here atu ^and aku ^are, respectively, the creation and annihilation operators of the VB electron with energy Ek (k =1, ... N ) ^and spin a ; afQ ^and

af. ^are those of the 4 f electron with _energy Ef and spin a (i.e. Nf = 2) ; Uff ^is the Coulomb interaction between 4 f electrons and V kf ^{is the} hybridization between the 4 f and VB states.

For simplicity ^we disregard ^{the f orbital} degener-

acy in equations (2.1) ^and (2.2) because it is not so

essential, ^{in a first} step, ^for insulating systems ⁱⁿ

contrast to metallic systems (see ^Ref. [10] ^{as well as}

the present ^Sect. 3). Considering ^a system ^{state with}

one electron in the 4 f level (4 ^fl configuration), ^the physical meaning of the energy E f appears ^{in the} difference _{6f - Ek} which expresses the total _energy

required for the transition 4 f - 4 f2 corresponding

to the excitation of a valence electron of _energy E k ^to the 4 f level and in the special ^{case of} V kf

^{0 and} Uff

0. In order to diagonalize H, ^{it is} convenient to rewrite H in the form :

where Ho ^{is the same as} Ho except that sf ⁱⁿ (2.2) ^is replaced by s f ⁺ Uff. ^{We can} easily diagonalize

Ho ^{as follows :}

by ^the transformation :

where En ^is given by the solution of

and

In the case of a filled VB it is very convenient ^to introduce a

Slater determinant » state 0 ^{which is} defined by all VB and f states being occupied :

and the corresponding energy E, ^{is given by :}

Equation (2.10) ^{means that in the} special ^{case of our}

filled system ^{there is no} energy variation which would result from the presence of hybridization.

In the initial state of the photoemission ^we suppose there is essentially ^one hole, Ef’ ^{Thus, the} ground

state is a spin ^{doublet state deduced from the} previously considered I t/1 > ^state by :

and the corresponding ground ^state energy is :

where £g is here the highest energy level among

{£n} ^of equation (2.6) ^{and which} automatically

includes Uff. By absorbing ^a photon of energy v, ^an f electron is emitted with energy ^E. So in the final state the two-hole ( I m) ) eigenstates ^{of H are} given by :

From Him>

Em 1m> , Em ^and B£, ^{are determined} by ^the following equations :

where :

Neglecting here emission from the valence band (see

Ref. [12]), ^the Hamiltonian describing ^{the direct}

transition from the initial state to the final state of

the f-electron photoemission process is the follow-

ing :

where MD ^denotes the direct f transition matrix element. For singlet ^final states {I m) }, ^{the f-}

spectrum is written as :

where w is the binding energy of the photoemitted

electron :

and r represents ^the spectral broadening ^{due to the}

finite lifetime of the f-hole, ^{as well as to the} experimental resolution width. The matrix element

Before testing equation (2.18) numerically ^{let us}

calculate the f-XPS spectrum ^{in the} analytical ^limit of Uff -+> 00.

3. Analytical ^limit.

/

In the case of Uff -+> ^oo it is convenient to write the

ground ^{state as :}

where the state 1°) ^{of energy} Eo is defined by ^{all VB}

states being occupied. ^The ground ^state energy of

1 .0 0) ^is then just given by Eo ⁺ Ef. Furthermore the f-XPS spectrum ^is expressed ^{as :}

with z being ^{here the} opposite ^{of the} binding energy defined in equation (2.19) :

Taking ^{account of our} simplified ground ^state, equation (3.2) ^can be rewritten immediately ^{as :}

Because of Uff -+ 00, hybridization V kf ^of equation (2.2) ^cannot play any role in the initial state of the photoemission process. However V kf ^can

again ^{act in the final state of the} photoemission

process when the f-electron has been photoemitted

and this final ^state effect can be exactly represented by ^the self-energy part -V(z) :

with

Equation (3.6) ^{can also} straightforwardly ^{be written}

as :

moreover, if we ^{assume a constant} hybridization, Vkf

V / BIN _{(cf. [6]} ^and [71), X(z) ^is essentially given by ^{the local Green function} associated with the valence band :

In equations (3.7) ^and (3.8) the factor of 2 is in fact the degeneracy N f. ^{It is} interesting ^to notice that the

degeneracy ^effect only appears associated ^with

V 2, ^at least in the present analytical ^limit (Uff -+ ^oo ),

so that the spectrum ^{is not} changed ^with N f, provided ^that N V2iS fixed. For finite Uff ^{it has been}

shown (for ^core photoemission) ^that the essential features of the spectrum ^{are not much} changed ^with N f ^if Nf V2 ^{is fixed,} although ^{some minor} changes happen ^{to show} up [10]. ^{Our f-XPS} spectrum (3.5) ^is formally given by ^{the same} expression ^{as that used}

for the f-local density ^{of states in} (an ^insulator

version of) the Anderson f impurity

^{or f extra}

orbital-model (see ^for example [11]) : ^there might ^be

two bound states (b.s.), ^{one b.s. or no b.s. at all} depending ^{on the model} (V, ^{Ef, centre,} shape ^and

width of the VB) ^and according ^{to the} following equation

where GR ^{is the} principal part ^{of G.}

4. Numerical analysis and discussion.

In this short paper ^we would like to analyse ^our

numerical results for direct f-photoemission ^with respect ^{to the main} parameters ^{which enter our} model. Complementary ^{results as well as a detailed}

fit to available experimental ^{data will be} presented

in a further publication [12]. ^{In our} calculation, ^we

treat the filled VB as a finite system consisting ^{of N}

discrete levels with equal energy spacing

1144

where W is the VB width and we put ^W

^{2 eV} except ^{when we} study ^{the VB width on the} spectrum.

The cluster model and the band model correspond ^to

the limits of small and large N, respectively. ^The hybridization Vkf ^{is assumed to be} Vkf

V/ J N

with a constant V as already assumed in section 3.

In figure ^{1 we show the} dependence ^{of the calcu-}

lated photoemission spectra F (w ) ^{on the number} N, by using V =1 eV ; Uff = 6 eV ; F = 0.5 eV and Ef

0.5 eV where E f ^{and also the} binding energy w

are measured with the centre of the VB as origin.

First of all we recover that F (w ) depends ^{on N for}

N , 3 but almost _converges for N ;:= 4 [7, 8]. ^{In the} following calculations we choose N = 4 and F

0.5 eV.

In figure 2, ^{we show} the behaviour of photoemis-

sion spectra F (to) for various values of the hy-

bridization V. According ^{to the}

extraorbital

pic-

ture (see ^Sect. 3), ^{in order to} reproduce ^a two-peak

structure, ^{V must} be sufficiently large (V

0.5 eV :

see Fig. 2b). Actually ^{in our} rectangular (discrete)

band model we always get ^{two b.s. on each side of} the VB but in order to obtain sizeable weights ^of

these b.s., ^{V has to be} sufficiently large. ^{For small}

values of V (Fig. 2a), ^{the spectra} essentially ^show one-peak structures corresponding ^{to a resonance}

within the VB and near Ef. The role of ^r is to broaden the structures especially ^{in the case of b.s.}

outside the VB.

Fig. ^1.

f-photoemission intensity ^versus binding

energy w for various values of the number N of the electronic levels in VB. Other parameters ^{are fixed as}

ef = 0.5 eV, W = 2 eV, V =1 eV, Uft = 6 eV; Ef ^{and w}

are measured with the centre of VB as origin ; ^{a Lorentzian} broadening ^r

0.5 eV has been used (see Eq. (2.18)).

Fig. ^2.

f-photoemission intensity ^versus binding

energy w for various values of the hybridization ^V = 0.1, 0.3, ^0.5 (eV) (Fig. 2a) ^and V = 0.6, 0.8, ^1.0 (eV) (Fig. 2b). ^Other parameters ^are fixed as sf

0.5 eV ;

W = 2 eV and Ua = 6 eV.

In figure ^{3 we test the} sensitivity ^{of the.} two-peak

structure upon the position E f ^{of the f level for a} given large hybridization (V ^{= 1} eV). Our results

can be understood by ^the analytic expression (3.5) ^of F (Cù ) ^{in the limit of} U ff -+ ^oo. By defining -V (w ) = + i y (co ), ^{we obtain} the relations 5 (to ) =

-

8 (- w ) ^{and y} (w )

^y (- w ), ^from equations

(3.8) ^and (4.1), ^{so that} F ((o ) satisfies the following

symmetric ^relation

Fig. ^3.

f-photoemission intensity ^versus binding

energy

for various values of the f level position

Bf = - 1.5, - 1.0, - 0.5, 0.0, 0.5, 1.0, ^1.5 (eV) ^{and for} Uff ^{= 12 eV} (Fig. 3a) ^or Ua

^{3 eV} (Fig. 3b). Other par- ameters are fixed as W

2 eV and V = 1 eV.

For U = 12 eV i.e. strong electron-electron interac- tion (Fig. 3a) ^we already begin ^{to recover this} symmetry ^{in the} spectra as Ef is varied gradually

from - 1.5 eV to 1.5 eV. It is not yet ^{the case for}

U

3 eV (Fig. 3b).

In figure ^{4 we show the} photoemission spectrum

as a function of the VB width W, ^{still in the case of a} two-peak ^structure (V =1 eV ). ^{As we double the}

band width from W = 1.25 eV to W

2.50 eV, keeping ^{the centre of the band at} the energy origin

we remark that both peaks stay ^almost unchanged ⁱⁿ position ^{and that} only ^the shape between the peaks

is affected by ^{this VB} change.

The Uff convergence ^of the two-peak ^structure

towards the Uff -+ ^oo limit is shown in figure ^{5. Let}

us just ^{recall here that our} calculation is strictly ^exact

Fig. ^4.

f-photoemission spectra ^{for two} values of the VB width W = 1.25 eV and 2.5 eV with Ef

^0.5 eV, V = 1 eV, U = 6 eVe

Fig. ^5.

Evolution of the f-photoemission spectra ^with increasing values of Uff = 1, 3, 5, 7, 9,

1000 (eV).

Other parameters ^are fixed as -ef = 0 eV, ^{V == 1} eV, W

2 eV. As Ef

^{0 and} Ua - ^{oo the} spectral shape

appears ^to be symmetric ^as expected ^{from our} simple

analytic calculation.

1146

for any value of Uff ; ^for Uff ^{= 100 eV} convergence towards the infinite limit begins ^{to be achieved i.e.}

the dependence ^of Uff upon the overall lineshape

vanishes with increasing Uff ^values.

Finally ^{we show how our f-XPS in} Ce203 type system ^{with the} spin ^doublet ground ^{state is different}

from that in a Ce02 type system ^{with the} spin singlet ground ^state. By using ^{the same} parameter ^{values as} those of figure ^3a and ef ^{= 1.5} eV, ^we calculate the f-XPS for the spin singlet ground ^state. The result is shown in figure 6 with the solid curve and compared

with that for the spin ^doublet ground ^state (the

dashed curve). ^The spectral ^features of the solid

curve are similar to that of Ce02 calculated by

Wuilloud et al. [7]. ^{It is} clearly ^seen that the spec-

Fig. ^6.

f-photoemission spectra ^{for a} spin singlet ground ^{state i.e.} Ce02 (solid curve) ^{and for a} spin ^doublet ground ^{state i.e.} Ce2o3 (dashed curve). ^The parameter values are the same as used in figure ^3a with Ef ^{= 1.5 eV}

so that the dashed curve is equivalent ^{to the} corresponding

curve in figure ^3a.

trum in ^a Ce203 type system ^is quite ^{different from}

that in ^a Ce02 type system ^{in its} intensity, spectral shape ^and peak positions. ^This difference comes

mainly from the difference in the f electron number n f in the ground ^{state :} nf -- 1.0 for Ce203 ^and

nf -- 0.5 for Ce02 (nf - 0.32 with the present par- ameters for Ce02 type model). ^{When we also take}

the valence band (0 2p band) photoemission ^into

account, ^a strong ^{and broad} spectrum ^{will occur} around w

0 and it will be superposed ^{on the} spectra ^of figure 6. Then it is expected ^{that for a} Ce203 type system ^the spectrum consists of a sharp ^f-

derived peak ^{and a broad 0} 2p-derived peak ^which

is superposed ^{on the weak f-derived} component. ^For

a Ce02 type system, ^{on the other} hand, ^the spectrum will only ^{have one} broad feature corresponding ^to

the 0 2p-derived spectrum superposed ^{on the f-}

derived one. In this _{way, the} present calculation may be consistent with the experimental ^data [5, 7], although ^{a more detailed} analysis taking ^{the 0} 2p-

derived photoemission ^{into account} [12] is necessary.

In the present paper ^we restricted ourselves to the

spin singlet ^{final states, but in the case of finite} Uff, spin triplet ^{final states can also occur. In the}

limit of Uff -+ ^{oo, as shown in} paragraph 3, ^we only

have the spin singlet ^{final state which} couples ^with

the singlet state 0) through Vkf. ^{We can show that}

for realistic values of U ff (6 -- ¹¹ eV ) the contribution of the triplet ^{final state is} much smaller than that of the singlet ^{final state} [12]. ^{A more detailed} study,

which includes the orbital degeneracy ^{of the f state} (Nf =14), the contribution of the photoexcitation

of valence band electrons and a more detailed

comparison ^with experimental data, ^{is now in} pro- gress and will be published ^{in the near future} [12].

Acknowledgments.

The present paper ^was prepared ^{when one of the}

authors (A. K.) stayed ⁱⁿ L.M.S.E.S., ^Universite Louis Pasteur, and he would like to thank Prof.

F. Gautier and L.M.S.E.S. for their hospitality ^and support.

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