- Email: [email protected]
A resonant photoemission model for nickel metal
Solid State Communications, Printed in Great Britain.
Vol. 41, No. 5, pp. 439-443,1982.
0038-1098/82/050439-05$02.00/0 Pergamon Press Ltd.
A RESONANT PHOTOEMISSION
MODEL FOR NICKEL METAL
J.C. Parlebas Universite Louis Pasteur, LMSES: LA 306 CNRS, 67070 Strasbourg Cedex, France and A. Kotani and J. Kanamori Department
of Physics, Faculty of Science, Osaka University,
Toyonaka
560, Japan
(Received 25 September 1981 by Y. Tuyozawa) We present a model for the resonance near the 3p threshold in the photoemission associated with a static 3d pair-hole bound state in paramagnetic nickel. We show that, following Auger processes, the strong intraatomic screening of the pair hole by the 4s-4p conduction band electrons is essential to explain experimental photoemission data not only in copper but also in nickel. From the energy relaxation of the conduction band states we estimate the bare Coulomb interaction between two d-holes in nickel. 4s-4~ bands and all the related effects: screening of the two d-holes by 4s-4~ electrons, Auger emission arising from final states where the core electron goes into 4s-4p band. We propose a model for paramagnetic nickel which is complementary to theirs. One of our purposes is to summarize a formal description of the satellite and Auger contributions to the resonant photoemission intensity, using a different formalism as the one used in [ 121; a full account of the derivation has already been given elsewhere [ 131 and applied to copper metal in a somewhat more quantitative way as compared to that by Davis and Feldkamp [ 11, 121. A second purpose is to extend our theory, with little change, to nickel, using as for copper a similar two hybridized band model [ 141 which simulates a kind of realistic density of states [ 131. Instead of considering direct excitations of the 3d band and related Fano effect we will focus attention on 4s-4~ screening effect and Auger electron emission involving both 3d and 4s-4~ bands. We investigate a simple system consisting of(i) a conduction band denoted by s-band and analogous to the 4s-4p bands of nickel; (ii) a narrow d-band treated here to be non-degenerate; (iii) non-degenerate core states, denoted by p-states and representing 3p-states of nickel. Only one spin direction is calculated and spin degeneracy is included at the end. The Idk) and lsk) Bloch states are supposed to be strongly mixed through a k-dependent (interatomic) hybridization /rid. For the whole system the ground state lJ/#) is written by
IN d-BAND photoemission spectra, the satellite at 6 eV below the Fermi level for nickel [l-4] and recently at 15 eV for copper [S] has attracted much attention. Following Mott’s suggestion, Hiiffner and Wertheim [l] had already proposed that the satellite is somewhat associated with a two hole bound state in the 3d-bands. In the case of copper, Iwan et al. [S] suggested that, to some extent similarly to nickel, the resonant enhancement in photoemission arises from Auger transitions called super Coster-Kronig transitions, i.e. a 3p core hole state is decaying into shake-up final states with the core level filled and a 3d photoelectron ejected, but with a two hole bound state in the 3d-bands. When analyzed within the itinerant bound state picture and without taking into account the screening due to the 4sAp electrons explicitly, the effective Couloumb interaction U,, between two d-holes in nickel is found about or below 2 eV [6,7]; also it is difficult to reconcile the satellite position with the observed exchange splitting [8]. The importance of the screening [9] of the d-holes by the 4s-4~ electrons was pointed out by one of the present authors [lo] in order to explain the satellite position in a way consistent with the magnetic properties of nickel. Recently, Davis and Feldkamp [ 11, 121 treated the 4s-4~ electrons of copper within a rectangular single band model neglecting the sd hybridization; they showed that the screening is qualitatively important in the persistence of the satellite above the 3p resonance threshold. Also they presented a model for ferromagnetic nickel [ 121 with (i) 3p to 3d band absorption (ii) super Coster-Kronig decay and (iii) interference (or Fano) effect with the direct excitation of the 3d band. However they disregarded the 439
A RESONANT PHOTOEMISSION
440
where ai and b& are creation operators respectively for a lp) atomic orbital centred at the irradiated site 0 with energy ep and for a hybridized Bloch state I&$ with energy eEi; the label L means the total number of s and d states up to the Fermi level eF and 10) is the vacuum which includes all the p-states but the one at site 0. The corresponding ground state energy is:
Vol. 41, No. 5
MODEL FOR NICKEL METAL
In order to calculate the satellite and Auger contributions to the photoelectron spectrum we use second order perturbation processes [ 131 involving a radiation interaction HW between I$I,> and I$,) states and an Auger interaction Hpa,(e) between I$,-Jand I tip(e)) states. Let e’; be a creation operator for a photon of energy, vs, the perturbed Hamiltonian for the whole system is:
Following absorption of a photon, creation of a p-hole and photoexcitation of the corresponding pelectron into an empty d or s conduction state, the system state becomes:
+ c &a(~)I1L,d@Wal+
kc. . 1
%P where cGi is a creation operator for an “impurity” state 117i)of energy wni [ 13, 141 associated with the p-hole potential U,; we assume that perturbation U, only acts in the unit cell 0 containing the core hole. In equation (3) the subscript OLstands for a particular set of {ni) picked up arbitrarily out of 2N available lni) states, N being the number of atoms. The corresponding energy E, is:
The photoemission intensity is expressed by the transition rate from the initial state eilJ/,> to the final state l$p(e)); a detailed formulation has been derived elsewhere [ 13 1. From now on, we will define all the electronic energies (e, f&, tici . . .) with eF as origin. It is then possible to express the result in terms of the binding energy, EB = uq - E, of the photoelectron: 2 WB>
vq>
c
=
a
g
& 4
L+l
E,
ITi) = F IEj)CEjlrli)*
c+;
The states I$,> decay by Auger transitions into final states l,$p(ey with (i) the core level flied; (ii) creation of a d-hole with a given spin bound to another d hole with opposite spin; the pair hole is treated, like the previously considered p-hole, as a static impurity at site 0; (iii) emission of a photoelectron with energy e, the orbital symmetry of which is disregarded:
+H$f)F+ g
ir 0
a L+l
(4)
- E(d’)
j=1
i=l
(7)
-
c i=l
wsi + ifl qi 5
,
(8)
where I’, = trl~~[~ is the decay rate of any I$,> state into all final states lJ/p(e)) and v. is the corresponding super Coster-Kronig matrix element, assumed here to be independent of e. In equation (8) the decay term Hpar(e) is expressed by: Ho&e) =
(~pl~Jvo; vo- (P; 4e2/4d; d),
(9)
L+l II//~(E))
a;led*@&
=
I@&= n c;JO), i=l
where cii is now a creation operator for the “impurity” state l{i> of energy wri [13] and associated with the double d-hole potential Ud supposed to act on the s-electrons only and at site 0; due to this attractive potential there is an increase of the local density of s-symmetry in the region below the d-band giving rise to a strong s-screening effect. Now the subscript /3 stands for a particular set of Isi> states. The energy associated with 19,(e)>is: L+l Ep(e)
=
ep
+
E +
E(d8) +
1
osi;
i=l P
=
krl,
s‘2,.
. . c-L+*),
where E(d*) takes explicitly energy of the two d-holes.
(6)
account for the creation
with +#~~lrJ~)representing the overlap between system states (@P)and I J/J [ 151; it is written in terms of the determinant of ~jl~i> mixing coefficients. Toyozawa and one of the present authors have studied the numerical evaluation of finite determinant systems of this type [ 161. Also the coupling term HW in equation (8) is given by a dipole transition determinant [ 141. Although obtained through a different derivation, our result on the the photoemission intensity [equation (S)] completely agrees with [ 12 1. In oure previous work [ 131 we already mentioned that the result of [l l] contained a spurious approximation. Numerical calculations are performed within the following assumptions: (i) Eigensolutions (I~i>, eti) for pure nickel are obtained from two one-dimensional energy bands with linear dispersions [ 141 and k-sinusoidal hybridization
Vol.41,No.5
A RESONANT PHOTOEMISSION MODEL FOR NICKEL METAL J
441
(v-v,)
Fig. 1. n”(e) and rid(e))hybridized densities of states (per spin) for nickel. [13]; In Fig. 1 we plot the partial d(e) and rid(e)) hybridized DOS (densities of states) which mimic the realistic bands of nickel, in particular, the strong depletion in the s-DOS and the d-band tail due to sd hybridization. (ii) Only the effects arising from final-state interactions, i.e. u+ = 0, & # 0 are taken into account; then the core hole threshold u. equals (eF - 9) and the dipole transition determinant Hag reduces to a single matrix element: (tL+r lHlP) = ((‘&+rlsk)& + (&+I 1dk)M,),
(10)
where &f, and jkfd are electron dipole matrix elements representing p to s and p to d photoexcited transitions: equation (10) clearly yields a combination of different shake up final states [S]. (iii) Equation (8) is computed for a finite system with N = 40 and with the delta function replaced by a gaussian function [13, 141. We calculate the overall line shape of F(EB, v) for various photon energies (u. + 6~) with (- 4 < 6~ < 8) in eV. According to previous estimations we assume @#!&=4[17],F,,= lSeV[18]andU~rU,~ - 12.5 eV [ 131. Our result (Fig. 2) reproduces the satellite and Auger parts of the experimental spectrum of Ni [3,4] in a good semi-quantitative way. As u is swept through v,, a strong resonance occurs in the vicinity of the lowest binding energy E,f: L+l
E,B = E(d’)+R;
R =
c osi I=1
i et i=l
9
(11)
where R represents the relaxation energy of the conduction electrons around the two holes. Above resonance, the peak persists because .!& # 0 [l 1, 12) and another peak, the Auger peak, develops. This type of spectra has been already discussed for copper by Davis
-2
(eV1 (EB-E:)
Fig. 2. Photoelectron spectra F(EB, v) vs. binding energy EB for various photon energies v near v. in nickel and Feldkamp [ 11, 121 and more extensively by the present authors [ 131; especially the influence of ud and hid on the photoelectron line shape has been tested [13]. Let us just recall here that for v = vo, the resonance height essentially varies with n’(&) in copper: for example it decreases if hybridization increases because of a consequent deeper depletion in the s-DOS. This effect is related to the M, contribution of equation (10) which is dominant in copper but smaller in nickel than in copper since in our N&model we take a somewhat larger hybridization [3 eV in Ni instead of 2.25 eV in Cu for (hS,d),,] and also the Fermi level in nickel lies closer to the maximum caused by the depletion. Nevertheless in nickel we have additional large i& contributions which compete with the previously considered effect because of a large value of nd(+). As far as we know, we perform the first calculation of the total cross section p(v), i.e. Auger + satellite contribution, in nickel metal; it is given by integrating F(EB, u) over all the binding energies. For v + u,, the satellite and Auger spectra coalesce into one, peaked at Et, so that we are unable to separate them; however for v > u. we obtain both contributions by deconvolution of the total spectra (Fig. 3); we also extrapolate our results to the vicinity of v. by a dotted line on Fig. 3. As compared with copper, we have additional absorption associated with extra shake up levels due to (more) dband holes and large p to d transitions (j&&l’&> 1): the
442
A RESONANT PHOTOEMISSION MODEL FOR NICKEL METAL
(eV)
(v-v,)
Fig. 3. Auger (A) and satellite (S) contributions to the total (T) cross-section p(v) in nickel. empty portion of the d-DOS which is quickly decreasing into the d-band tail gives rise to an absorption peak over a range of about 5 eV in the Auger cross-section of nickel in contrast with the copper case [5,13]. The asymmetry of this Auger peak yields a somewhat asymmetric shape for the total cross-section. Now, as far as the resonance part is concerned our result is in qualitative accord with the experimental data exhibited by Iwan et al. [5] : the shape for the satellite cross-section in nickel is quite symmetric and similar to that obtained for copper. Davis and Feldkamp [ 121 discussed the asymmetry of the satellite peak in nickel in terms of the interference effect with the direct excitations of the d-band, we have no intention to deny the validity of this interpretation (see also [ 141) because our interpretation is based on a complementary model involving s-electrons effect, as we already mentioned. There is a basic question, however, as to the balance of both effects in their contributions to the satellite peak of nickel. Finally we study the relaxation energy R (with respect to U,) in nickel as compared with copper [ 131 (Fig. 4). We clearly see a deviation from a linear law which is given by an atomic model; this deviation comes from all the band effects. It is larger for nickel than for copper, in particular because of a larger hybridization. Keeping the same pair-hole potential lJp 5 U$” = - 12.5 eV we fmd 2RNi z - 18 eV for both spin directions. From experiment we know that E,f = 6 eV which corresponds to the position of the largest singlet state peak ‘G in Ni. Thus from equation (11) we find E(d8) = 24 eV. This energy for the creation of a hole pair is equal to twice the energy for the creation of a single d-hole (% 0.7 eV in Ni [S]) plus the bare Coulomb energy Vi,, between the two d-holes. Therefore it comes: Uz ‘v 22.6 eV as compared with U- 2: 27 eV.
Vol. 41, No. 5
Fig. 4. Relaxation energy R (per spin) of the conduction electrons with respect to U,: full line for Ni; dashed line for Cu; dotted line in an atomic model. Moreover, the effective on-site Coulomb interaction U..* is obtained like (I,_ but with E(d8) replaced by E,f which yields UI$$‘v- 4.6 eV. In this estimate the Coulomb interaction is defined between two d-holes screened fully by s-electrons. The value of U,, is strongly reduced from U,, by: Uen = U,,
-21RI
(12)
through extra-atomic relaxation process which cannot be described in a simple atomic model (since 2lRl < 21Udl). One of the essential differences between the photoelectron spectrum in the metal and that in the free atom is precisely the corresponding shift of the spectrum due to the extra-atomic relaxation. For a more quantitative description of the photoelectron spectrum we need to extend our theory to ferromagnetic nickel with more realistic 3p, 3d and 4s-4~ states and a more careful treatment of the sd hybridization, it is also possible to generalize equation (8) to include the direct excitations. Nevertheless we have shown that our simple s-screening model can explain general trends of the photoemission line shape and the corresponding cross-section as far as the satellite and Auger contributions are concerned near the 3p threshold of Cu and Ni. Our theory is of course extensible to similar materials involving other core levels as 4p, 4d, 4f. But compared with other 3d transition metals, Cu and Ni are simpler test cases for studying screening effects (associated with super Coster-Kronig mechanism) because of their filled or almost filled dshell ground state [lo]. - The authors thank Mrs A. Egusa for preparing the manuscript. One of the authors (JCP) is grateful to the Japan Society for the Promotion of Science for a research Fellowship at Osaka University. This work was supported partially by a Grant-in-Aid for Scientific Research given by the Ministry of Education, Science and Culture of Japan.
Acknowledgements
Vol. 41, No. 5
A RESONANT PHOTOEMISSION MODEL FOR NICKEL METAL in Narrow-Band Systems (Edited by T. Moriya),
REFERENCES 1. 2. 3. 4. 5. 6. 7. ;10:
S. Htifner & G.K. Wertheim,Phys. Left. SlA, 299 (1975). P.C. Kemeney & N.J. Shevchik, Solid State Commun. 17,255 (1975). C. Guillot, Y. Ballu, T. Paigne, J. Lecante, KS. Gain, P. Thiry, R. Pinchaux, Y. Petroff & L.M. Falicov, Phys. Rev. Letf. 39,1632 (1977). J. Barth, G. Kaekoffen & C. Kunz, Phys. Lett.
74A, 360 (1979). M. Iwan, F.J. Himpsel & D.E. Eastman, Phys. Rev. Lett. 43,1829 (1979). D.R. Penn, Phys. Rev. Lett. 42,921 (1979). G. Treglia, F. Ducastelle & D. Spanjaard, J. Phys. 41,281 (1980). A. Liebsch, Phys. Rev. Lett. 43,143l (1979). J. Friedel,Nuovo Cim. Suppl. 7,287 (1958). J. Kanamori, Electron Correlation and Magnetism
443
11.
p. 102. Springer-Verlag, Berlin (1981). L.C. Davis & L.A. Feldkamp, Phys. Rev. Lett. 44, 673 (1980).
12. 13.
L.C. Davis 8c L.A. Feldkamp, Phys. Rev. 23,
6239 (1981).
:::
J.C. Parlebas, A. Kotani & J. Kanamori, Submitted to J. Phys. Sot. Japan. A. Kotani, J. Phys. Sot. Japan 46,488 (1979). P. Nozieres & C.T. de Dominicis, Phys. Rev. 178,
16.
A. Kotani & Y. Toyozawa, J. Phys. Sot. Japan
17.
18.
1097 (1969).
35,1082
(1973).
M.B. Stearns & L.A. Feldkamp, Magnetism and Magnetic Materials (Edited by J.J. Becker, G.H. Lander and J.J. Rhyne), p, 286. Am. Inst. of Phys. (1975). R.E. Dietz, E.G. McRae, Y. Yafet & C.W. Caldwell, Phys. Rev. Lett. 33, 1372 (1974).
Vol. 41, No. 5, pp. 439-443,1982.
0038-1098/82/050439-05$02.00/0 Pergamon Press Ltd.
A RESONANT PHOTOEMISSION
MODEL FOR NICKEL METAL
J.C. Parlebas Universite Louis Pasteur, LMSES: LA 306 CNRS, 67070 Strasbourg Cedex, France and A. Kotani and J. Kanamori Department
of Physics, Faculty of Science, Osaka University,
Toyonaka
560, Japan
(Received 25 September 1981 by Y. Tuyozawa) We present a model for the resonance near the 3p threshold in the photoemission associated with a static 3d pair-hole bound state in paramagnetic nickel. We show that, following Auger processes, the strong intraatomic screening of the pair hole by the 4s-4p conduction band electrons is essential to explain experimental photoemission data not only in copper but also in nickel. From the energy relaxation of the conduction band states we estimate the bare Coulomb interaction between two d-holes in nickel. 4s-4~ bands and all the related effects: screening of the two d-holes by 4s-4~ electrons, Auger emission arising from final states where the core electron goes into 4s-4p band. We propose a model for paramagnetic nickel which is complementary to theirs. One of our purposes is to summarize a formal description of the satellite and Auger contributions to the resonant photoemission intensity, using a different formalism as the one used in [ 121; a full account of the derivation has already been given elsewhere [ 131 and applied to copper metal in a somewhat more quantitative way as compared to that by Davis and Feldkamp [ 11, 121. A second purpose is to extend our theory, with little change, to nickel, using as for copper a similar two hybridized band model [ 141 which simulates a kind of realistic density of states [ 131. Instead of considering direct excitations of the 3d band and related Fano effect we will focus attention on 4s-4~ screening effect and Auger electron emission involving both 3d and 4s-4~ bands. We investigate a simple system consisting of(i) a conduction band denoted by s-band and analogous to the 4s-4p bands of nickel; (ii) a narrow d-band treated here to be non-degenerate; (iii) non-degenerate core states, denoted by p-states and representing 3p-states of nickel. Only one spin direction is calculated and spin degeneracy is included at the end. The Idk) and lsk) Bloch states are supposed to be strongly mixed through a k-dependent (interatomic) hybridization /rid. For the whole system the ground state lJ/#) is written by
IN d-BAND photoemission spectra, the satellite at 6 eV below the Fermi level for nickel [l-4] and recently at 15 eV for copper [S] has attracted much attention. Following Mott’s suggestion, Hiiffner and Wertheim [l] had already proposed that the satellite is somewhat associated with a two hole bound state in the 3d-bands. In the case of copper, Iwan et al. [S] suggested that, to some extent similarly to nickel, the resonant enhancement in photoemission arises from Auger transitions called super Coster-Kronig transitions, i.e. a 3p core hole state is decaying into shake-up final states with the core level filled and a 3d photoelectron ejected, but with a two hole bound state in the 3d-bands. When analyzed within the itinerant bound state picture and without taking into account the screening due to the 4sAp electrons explicitly, the effective Couloumb interaction U,, between two d-holes in nickel is found about or below 2 eV [6,7]; also it is difficult to reconcile the satellite position with the observed exchange splitting [8]. The importance of the screening [9] of the d-holes by the 4s-4~ electrons was pointed out by one of the present authors [lo] in order to explain the satellite position in a way consistent with the magnetic properties of nickel. Recently, Davis and Feldkamp [ 11, 121 treated the 4s-4~ electrons of copper within a rectangular single band model neglecting the sd hybridization; they showed that the screening is qualitatively important in the persistence of the satellite above the 3p resonance threshold. Also they presented a model for ferromagnetic nickel [ 121 with (i) 3p to 3d band absorption (ii) super Coster-Kronig decay and (iii) interference (or Fano) effect with the direct excitation of the 3d band. However they disregarded the 439
A RESONANT PHOTOEMISSION
440
where ai and b& are creation operators respectively for a lp) atomic orbital centred at the irradiated site 0 with energy ep and for a hybridized Bloch state I&$ with energy eEi; the label L means the total number of s and d states up to the Fermi level eF and 10) is the vacuum which includes all the p-states but the one at site 0. The corresponding ground state energy is:
Vol. 41, No. 5
MODEL FOR NICKEL METAL
In order to calculate the satellite and Auger contributions to the photoelectron spectrum we use second order perturbation processes [ 131 involving a radiation interaction HW between I$I,> and I$,) states and an Auger interaction Hpa,(e) between I$,-Jand I tip(e)) states. Let e’; be a creation operator for a photon of energy, vs, the perturbed Hamiltonian for the whole system is:
Following absorption of a photon, creation of a p-hole and photoexcitation of the corresponding pelectron into an empty d or s conduction state, the system state becomes:
+ c &a(~)I1L,d@Wal+
kc. . 1
%P where cGi is a creation operator for an “impurity” state 117i)of energy wni [ 13, 141 associated with the p-hole potential U,; we assume that perturbation U, only acts in the unit cell 0 containing the core hole. In equation (3) the subscript OLstands for a particular set of {ni) picked up arbitrarily out of 2N available lni) states, N being the number of atoms. The corresponding energy E, is:
The photoemission intensity is expressed by the transition rate from the initial state eilJ/,> to the final state l$p(e)); a detailed formulation has been derived elsewhere [ 13 1. From now on, we will define all the electronic energies (e, f&, tici . . .) with eF as origin. It is then possible to express the result in terms of the binding energy, EB = uq - E, of the photoelectron: 2 WB>
vq>
c
=
a
g
& 4
L+l
E,
ITi) = F IEj)CEjlrli)*
c+;
The states I$,> decay by Auger transitions into final states l,$p(ey with (i) the core level flied; (ii) creation of a d-hole with a given spin bound to another d hole with opposite spin; the pair hole is treated, like the previously considered p-hole, as a static impurity at site 0; (iii) emission of a photoelectron with energy e, the orbital symmetry of which is disregarded:
+H$f)F+ g
ir 0
a L+l
(4)
- E(d’)
j=1
i=l
(7)
-
c i=l
wsi + ifl qi 5
,
(8)
where I’, = trl~~[~ is the decay rate of any I$,> state into all final states lJ/p(e)) and v. is the corresponding super Coster-Kronig matrix element, assumed here to be independent of e. In equation (8) the decay term Hpar(e) is expressed by: Ho&e) =
(~pl~Jvo; vo- (P; 4e2/4d; d),
(9)
L+l II//~(E))
a;led*@&
=
I@&= n c;JO), i=l
where cii is now a creation operator for the “impurity” state l{i> of energy wri [13] and associated with the double d-hole potential Ud supposed to act on the s-electrons only and at site 0; due to this attractive potential there is an increase of the local density of s-symmetry in the region below the d-band giving rise to a strong s-screening effect. Now the subscript /3 stands for a particular set of Isi> states. The energy associated with 19,(e)>is: L+l Ep(e)
=
ep
+
E +
E(d8) +
1
osi;
i=l P
=
krl,
s‘2,.
. . c-L+*),
where E(d*) takes explicitly energy of the two d-holes.
(6)
account for the creation
with +#~~lrJ~)representing the overlap between system states (@P)and I J/J [ 151; it is written in terms of the determinant of ~jl~i> mixing coefficients. Toyozawa and one of the present authors have studied the numerical evaluation of finite determinant systems of this type [ 161. Also the coupling term HW in equation (8) is given by a dipole transition determinant [ 141. Although obtained through a different derivation, our result on the the photoemission intensity [equation (S)] completely agrees with [ 12 1. In oure previous work [ 131 we already mentioned that the result of [l l] contained a spurious approximation. Numerical calculations are performed within the following assumptions: (i) Eigensolutions (I~i>, eti) for pure nickel are obtained from two one-dimensional energy bands with linear dispersions [ 141 and k-sinusoidal hybridization
Vol.41,No.5
A RESONANT PHOTOEMISSION MODEL FOR NICKEL METAL J
441
(v-v,)
Fig. 1. n”(e) and rid(e))hybridized densities of states (per spin) for nickel. [13]; In Fig. 1 we plot the partial d(e) and rid(e)) hybridized DOS (densities of states) which mimic the realistic bands of nickel, in particular, the strong depletion in the s-DOS and the d-band tail due to sd hybridization. (ii) Only the effects arising from final-state interactions, i.e. u+ = 0, & # 0 are taken into account; then the core hole threshold u. equals (eF - 9) and the dipole transition determinant Hag reduces to a single matrix element: (tL+r lHlP) = ((‘&+rlsk)& + (&+I 1dk)M,),
(10)
where &f, and jkfd are electron dipole matrix elements representing p to s and p to d photoexcited transitions: equation (10) clearly yields a combination of different shake up final states [S]. (iii) Equation (8) is computed for a finite system with N = 40 and with the delta function replaced by a gaussian function [13, 141. We calculate the overall line shape of F(EB, v) for various photon energies (u. + 6~) with (- 4 < 6~ < 8) in eV. According to previous estimations we assume @#!&=4[17],F,,= lSeV[18]andU~rU,~ - 12.5 eV [ 131. Our result (Fig. 2) reproduces the satellite and Auger parts of the experimental spectrum of Ni [3,4] in a good semi-quantitative way. As u is swept through v,, a strong resonance occurs in the vicinity of the lowest binding energy E,f: L+l
E,B = E(d’)+R;
R =
c osi I=1
i et i=l
9
(11)
where R represents the relaxation energy of the conduction electrons around the two holes. Above resonance, the peak persists because .!& # 0 [l 1, 12) and another peak, the Auger peak, develops. This type of spectra has been already discussed for copper by Davis
-2
(eV1 (EB-E:)
Fig. 2. Photoelectron spectra F(EB, v) vs. binding energy EB for various photon energies v near v. in nickel and Feldkamp [ 11, 121 and more extensively by the present authors [ 131; especially the influence of ud and hid on the photoelectron line shape has been tested [13]. Let us just recall here that for v = vo, the resonance height essentially varies with n’(&) in copper: for example it decreases if hybridization increases because of a consequent deeper depletion in the s-DOS. This effect is related to the M, contribution of equation (10) which is dominant in copper but smaller in nickel than in copper since in our N&model we take a somewhat larger hybridization [3 eV in Ni instead of 2.25 eV in Cu for (hS,d),,] and also the Fermi level in nickel lies closer to the maximum caused by the depletion. Nevertheless in nickel we have additional large i& contributions which compete with the previously considered effect because of a large value of nd(+). As far as we know, we perform the first calculation of the total cross section p(v), i.e. Auger + satellite contribution, in nickel metal; it is given by integrating F(EB, u) over all the binding energies. For v + u,, the satellite and Auger spectra coalesce into one, peaked at Et, so that we are unable to separate them; however for v > u. we obtain both contributions by deconvolution of the total spectra (Fig. 3); we also extrapolate our results to the vicinity of v. by a dotted line on Fig. 3. As compared with copper, we have additional absorption associated with extra shake up levels due to (more) dband holes and large p to d transitions (j&&l’&> 1): the
442
A RESONANT PHOTOEMISSION MODEL FOR NICKEL METAL
(eV)
(v-v,)
Fig. 3. Auger (A) and satellite (S) contributions to the total (T) cross-section p(v) in nickel. empty portion of the d-DOS which is quickly decreasing into the d-band tail gives rise to an absorption peak over a range of about 5 eV in the Auger cross-section of nickel in contrast with the copper case [5,13]. The asymmetry of this Auger peak yields a somewhat asymmetric shape for the total cross-section. Now, as far as the resonance part is concerned our result is in qualitative accord with the experimental data exhibited by Iwan et al. [5] : the shape for the satellite cross-section in nickel is quite symmetric and similar to that obtained for copper. Davis and Feldkamp [ 121 discussed the asymmetry of the satellite peak in nickel in terms of the interference effect with the direct excitations of the d-band, we have no intention to deny the validity of this interpretation (see also [ 141) because our interpretation is based on a complementary model involving s-electrons effect, as we already mentioned. There is a basic question, however, as to the balance of both effects in their contributions to the satellite peak of nickel. Finally we study the relaxation energy R (with respect to U,) in nickel as compared with copper [ 131 (Fig. 4). We clearly see a deviation from a linear law which is given by an atomic model; this deviation comes from all the band effects. It is larger for nickel than for copper, in particular because of a larger hybridization. Keeping the same pair-hole potential lJp 5 U$” = - 12.5 eV we fmd 2RNi z - 18 eV for both spin directions. From experiment we know that E,f = 6 eV which corresponds to the position of the largest singlet state peak ‘G in Ni. Thus from equation (11) we find E(d8) = 24 eV. This energy for the creation of a hole pair is equal to twice the energy for the creation of a single d-hole (% 0.7 eV in Ni [S]) plus the bare Coulomb energy Vi,, between the two d-holes. Therefore it comes: Uz ‘v 22.6 eV as compared with U- 2: 27 eV.
Vol. 41, No. 5
Fig. 4. Relaxation energy R (per spin) of the conduction electrons with respect to U,: full line for Ni; dashed line for Cu; dotted line in an atomic model. Moreover, the effective on-site Coulomb interaction U..* is obtained like (I,_ but with E(d8) replaced by E,f which yields UI$$‘v- 4.6 eV. In this estimate the Coulomb interaction is defined between two d-holes screened fully by s-electrons. The value of U,, is strongly reduced from U,, by: Uen = U,,
-21RI
(12)
through extra-atomic relaxation process which cannot be described in a simple atomic model (since 2lRl < 21Udl). One of the essential differences between the photoelectron spectrum in the metal and that in the free atom is precisely the corresponding shift of the spectrum due to the extra-atomic relaxation. For a more quantitative description of the photoelectron spectrum we need to extend our theory to ferromagnetic nickel with more realistic 3p, 3d and 4s-4~ states and a more careful treatment of the sd hybridization, it is also possible to generalize equation (8) to include the direct excitations. Nevertheless we have shown that our simple s-screening model can explain general trends of the photoemission line shape and the corresponding cross-section as far as the satellite and Auger contributions are concerned near the 3p threshold of Cu and Ni. Our theory is of course extensible to similar materials involving other core levels as 4p, 4d, 4f. But compared with other 3d transition metals, Cu and Ni are simpler test cases for studying screening effects (associated with super Coster-Kronig mechanism) because of their filled or almost filled dshell ground state [lo]. - The authors thank Mrs A. Egusa for preparing the manuscript. One of the authors (JCP) is grateful to the Japan Society for the Promotion of Science for a research Fellowship at Osaka University. This work was supported partially by a Grant-in-Aid for Scientific Research given by the Ministry of Education, Science and Culture of Japan.
Acknowledgements
Vol. 41, No. 5
A RESONANT PHOTOEMISSION MODEL FOR NICKEL METAL in Narrow-Band Systems (Edited by T. Moriya),
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