Interpretation of 3p-core-excitation spectra in Cr, Mn, Fe, Co, and Ni

Solid State Communications,Vol. 19, PP. 413—416, 1976.

Pergamon Press.

Printed in Great Britain

INTERPRETATION OF 3p-CORE-EXCITATION SPECTRA IN Cr, Mn, Fe, Co, AND Ni L.C. Davis and L.A. Feldkamp Engineering and Research Staff, Ford Motor Company, Dearborn, MI 48121, U.S.A. (Received 26 January 1976 by R.H. Silsbee) Recent measurements of the 3p-core.excitation spectra of Cr, Mn, Fe, Co, and Ni are interpreted with an atomic model. The dispersionlike line shape 63d”T observed in these metals is attributed to the interference of 3p 3ps3dN~,which decays to 3p63dN_l ef via a super Coster—Kronig transition, with the direct excitation of 3p63dN 3p63dN_I ef. The overall width of the line and some weaker features associated with it are related to the multiplet splittings of 3ps3dN~ The more symmetric line thape found for Cr, which is thought to be due to the absence of a large local moment, is explained in terms of the greater number of multiplets that contribute for a small moment (S = 1/2) as compared to a large moment .~

-~

.

(S=5/2).

THE EXCITATION of an electron from the 3p core level in a 3d transition metal has been of interest lately.14 In both absorption of synchrotron radiation and inelastic scattering of electrons, the transition from 3p to empty 3d levels has a dispersionlike or interference line shape in Mn, Fe, Co, and Ni, but a more symmetric shape for Ti, V and Cr. Dietz et al.2 have suggested that the line shape is caused by an interference between -÷ 3d and the process whereby a 3d electron is excited to the continuum (well above the vacuum level), e.g., 3d -÷ ef. They have analyzed the case of Ni using the formalism of Fano for the interference of a discrete (excited) state with a continuum.5 The analysis for Ni is simple since the ground state is taken to be 3p63d9 and the excited state is 3p53d’°,thereby involving only 1 hole in an otherwise closed shell. Since high-resolution measurements of the inelastic scattering of electrons exist for the transition metals Cr to Ni,3 it is worthwhile to analyze these spectra also. The additional complications introduced for a metal whose ground state is 3p63dN, N < 9, arise from the splittings of the multiplets associated with the excited state configuration 53d”~ (see reference 6). The effects of finite life3p time (due to coupling with the continuum) and solidstate bandwidths obscure the actual splittings, the overall width of the transition is clearly relatedbut to these splittings, and some weaker features may be due to multiplets. 63dN ~ Here we present model calculations3p63d~~ for 3p ef. 3ps3dN~ interfering with 3p63d’~ Interference is possible since 3p53d’~ decays via a super Coster—Kronig transition7 into 3p63d”’ ef. Other continuum states are possible, namely, ep and c/i, but the relevant Coulomb matrix elements are small.2’8 -~

The theoretical formalism is simpler than that given by Starace9 for a related problem in the rare-earth spectra, and it involves somewhat different decay channels. (In the present work the core hole is filled by the super Coster—Kronig transition, whereas in references 9 and 6 the core hole is not filled.) We extend the formalism of Fano5 to the case of many discrete states mixing with many continua. Our formalism differs from that of Mies’°in that we use stationary states rather than scattering states. An atomic model i~often used with some success for calculations of this type involving solids2’6’7”’5 in spite of obvious conceptual difficulties. We attempt to alleviate some of the difficulties by including a cubic potential in the ground state and by heuristically ineluding bandwidths. For simplicity we omit spin—orbit coupling. The success of an atomic model appears to be due to the strong perturbation caused by the core hole, the small widths of the 3d bands, and the rapidity of the Auger effect. Within the context of the Fano formalism, the discrete states IØ~)are of the form Pmsdm ~ dm 2s2 dm~DIO),where p,,~ 8is an annihilation operator for a 3p orbital with l~= m and s~= s, dms is the correspond63d10, and ing= operator for continuum a 3d orbital,states 0) represents D 9 N. The I ~E) are 3p of the form f~dmsdms ~ wheref,~ 5is a creation operatore of forthe theefeforbital orbitalisand C =by11subtracting N. The kinetic energy found the energy of the ion 3p63dN_l from the total energy E. The mixing of the discrete states with the continuum states is given by matrix elements —

. . .

—

Vkfl(E) 413

=

<~kEI11Iøn),

(1)

414

3p-CORE-EXCITATION SPECTRA IN Cr, Mn, Fe, Co AND Ni

where H is the Hamiltonian. Only the Coulomb interaction contributes to (1), and the relevant matrix elements are the form (Pmsfm’s’ 1e2/r 12 Idm1sj dms) which can RK(3d3d; be expressed in terms theThese Slaterare integrals2’8”6 3pef), K = of 1,3. the same matrix elements which give rise to the super Coster— Kronig transitions.’7 For any stationary solution of energy E we can write

=

Vol. 19, No. 5

~ I(iITI’I’~I2

(7a)

2 =

h x

~.2

+it 4(E) 21

IT’ —

I~ ~

~

~1

k

Vhfl(EYIITI’I’kE)I

z~(E)I2

Iqv +

I

~ I (7b)

where =

~ a,~(E)IØ,~)+ ~

(2)

JdE’bk(E,E’)qIkE’).

k

q~ =

Following Fano,5 it can be thown that (assuming the number of discrete states does not exceed the number of continua) bk(E, E’)

=

F[E —E’ + z(E>5(E E’)lJ ~°

—

‘

(3) where P is the principal value and z(E) is an eigenvalue (for each value of the parameter E) satisfying —

El

a + FEE) a +

FEE)a iT

fmn(E)

=

71

=

0, (4a)

—

~ Vm(E)Vkn(E),

(4b)

1 Hmn

dE’

= =

Fmn(E’), (~‘tmIHIø~).

it! [1r2+

(4d)

M

z2(E)].

(5)

For any two orthogonal solutions, ‘i4’~and a(V)t F(E)a0’~ = 0.

Equation (4) represents a well-defmed eigenvalue problemtransformation whose solution be found by making a congruent as can discussed by Goldstein.”7 For the vth solution, the result is a(v)

=

CVA~,A(P)tA(V)

C~I2=

=

l~

(6a)

[ir2 + 4(E)}1,

(6b)

(E — En),

(6c)

E~ = A(~(H+ F)A~~,

(6d)

z~(E)=

=

it —

A~t rA(v).

In our numerical calculations we neglect the E dependence of E~,F~,and qp. These quantities are calculated only once for E equal to some mean energy of the transition (for each metal or N value). We also neglect F and the second term of (7d), both of which are presumed small. For N = 9 (1 hole) our formulas reduce to those of Dietz et al.2 To calculate the multiplet splittings we have used the direct (F’~)and exchange (G”)

(4c)

The matrix H is not diagonal in the chosen basis due to the Coulomb interaction. The normalization is such that a~f(E)a =

(7d)

Slater integrals’6 given by Watson’8 for atoms with configuration d’1. For the super Coster—Kronig transitions we have used the values of RK(3d3d; 3pef) given by McGuire.8 It is thought that the original values were too large,’4”9 so we have scaled them to give the

k

Fmn(E)

(7c) dE’ I~~) = Iøn)+~PfEE,Vkn(E’)IlPkE’).

~ Vkfl(E’~fl(E) ,~

~

(6e)

For any transition represented by an operator T the transition rate from the initial state 0 to I~44)is

4 (who compared 2, 3—M4 5M4 to ~ rates of Yin et Auger widths experiment foral.’ Cu and other solids with higher Z). For the ratio of the dipole matrix elements (3pIrj3d)/(3dlrlef), we have used the same value for all N: 0.4 of McGuire’s value for Ni,2 as discussed below. Additional broadening of the amount 0.35, 0.5, 0.8, 1.1, and 1.0 eV for 1—5 holes, respectively, has been added to each I’~,.This represents 0.2 eV for M 2, 3—M4 5N1 ~ and 20 the remaining is the half-width d bands. In Fig.of1, the the empty line shapes for 1—5 holes in the 3d subshell (N = 9 to 5) are shown. These correspond with Ni to Cr only roughly, since the d occupancy is noninteger.21 The dip-to-peak heights have been normalized and the transition energies have been measured from the mean energy. For the 5 hole case, the cubic potential in the ground state was chosen large enough to make = 1/2 (1 ODq = 5 eV), whereas for 1—4 holes Hund’s Rule was allowed to prevail and S = (10 —N)/2 (lODq = 1 eV). This corresponds roughly with the known magnetic character of the metallic ground states. For 1, 2, and 4 holes in the ground state, a cubic potential as small as lODq = 1 eV has no effect on the line shape as compared to lODq = 0.22 For three holes, it has a small effect. A plot (similar to Fig. 1) of the

S

Vol. 19, No. 5

3p-CORE-EXCITATION SPECTRA IN Cr, Mn, Fe, Co AND Ni

415

I-I (p

z uJ

w

i-I

z

FE

w >

‘-4

‘-4 4-

a: -J

a: -J w

—20

-10 0 10 20 RELATIVE ENERGY LEVI

Fig. 1. The calculated 3p-core-excitation spectra for 1—5 holes in the 3d subshell with dipole selection rules. S = 1/2 for 5 holes whereas S equals half the number of holes for 1—4 holes. Dip-to-peak heights are normalized and the energy is measured relative to the mean energy. energy-loss measurements of reference 3 for Ni to Cr is shown in Fig. 2. Background scattering has not been subtracted from the experimental data, In comparing the two figures, the following features are to be noted. (i) In Ni, the spin—orbit splitting of the line is observed but is not included in our calculation. The spin—orbit splitting is not observed in the remaining metals. For the individual lines the calculated and experimental widths (HWHM) are in good agreement (for Ni, 1.3 and 1.2 eV respectively). Using McGuire’s value2 for the ratio (3pIrI3d)/(3dIrIeJ), we find a value of q which is about 2.5 times larger than the experimental value of 1.2. Hence we have adjusted this ratio to give the correct q for Ni. This could represent modification of the wave functions due to solid-state effects or to inadequacies in the model, (ii) In Co and Mn a small peak is observed preceding the main peak. An additional peak is seen in the calculated curves for 2—4 holes and is due to low-lying multiplets of the excited-state configuration (3ps3dN+1) This extra peak is absent in Fe although present in our

-20 -10 0 10 20 REL~TIVFENERGY LEVI

Fig. 2. Energy-loss measurements of reference 3 for Ni to Cr. Background scattering has not been subtracted. Spin—orbit splitting is observed only in Ni. calculation for 3 holes. No multiplet splitting occurs for I hole.2 (iii) The width of the lines increases from Ni to Fe as they do in the calculation for 1—3 holes. This is due to the multiplet splittings. (iv) Some narrowing of the line occurs for Mn as compared to Fe. There is also evidence of this in the calculation when the width for 4 holes is compared to that for 3 holes. (v) Experimentally the line shape for Cr is broader than all the rest and is not dispersionlike. In the calculations for 5 holes, since S = 1/2 in our ground state, many more multiplets contribute than ifS = 5/2. This broadens the structure and obscures most of the dispersion-like character. There appears to be some correlation between the small bumps near the top of the main peak occurring in both the experimental spectra and the calculations. The most striking feature, however, is the apparently abrupt change in the lineshape between Cr, which does not possess a large local moment in the ground state, and the metals Mn to Ni, which do have large local moments. In the atomic picture this corresponds to S being (10 —N)/2 for 1—4 holes and S being

416

3p-CORE-EXCITATION SPECTRA IN Cr, Mn, Fe, Co AND Ni

Vol. 19, No. 5

1/2 (forced by the cubic potential in our model) for 5 in agreement with that observed. A model similar to holes. The calculations clearly reflect this abrupt change. that of Kotani and Toyozawa has been proposed by IfS = 5/2 had been allowed in the ground state, a very Stearns,3 emphasizing the interference between localized narrow line (similar to Ni) would have resulted. d levels and itinerant s and d bands. This model, too, In conclusion, we have presented an atomic model appears incompatible with the data, because it produces which reproduces the principal trends of the 3p corea Fano line shape cut off on the low-energy side by an level excitation spectra for Cr to Ni and have indicated edge at the Fermi energy, whereas the data show only certain features which are direct evidence of multiplets. a Fano line shape. Furthermore, the interference extends The role which the presence or absence of a local well below the Fermi level, which rules out any such moment plays in these spectra has also been discussed. one-electron explanation. (The experimental correlation between the existence of a local moment and a strong Fano line shape in these materials was noted in reference 3.) Other mechanisms Acknowledgements The authors would like to have been proposed but do not appear viable. The calcu- acknowledge useful conversations with Drs. M.B. Stearns lations of Kotani and Toyozawa23 do not yield structure and E.J. McGuire, and R.E. Dietz. —

REFERENCES 1.

SONNTAG B., HAENSEL R. & KUNZ C., Solid State Commun. 7,597(1969).

2.

DIETZ R.E., MCRAE E.G., YAFET Y. & CALDWELL C.W.,Phys. Rev. Lett. 33, 1372 (1974).

3.

5.

STEARNS M.B., FELDKAMP L.A. & SHINOZAKJ S.S., Bull. Amer. Phys. Soc. 20, 358 (1975); STEARNS M.B. & FELDKAMP L.A., Proc. AlP Conf on Magnetism and Magnetic Materials (1975) (to be published). BROWN F.C., in Solid State Physics (Edited by SEITZ F., TURNBULL D. & EHRENREICH H.), Vol. 29. Academic Press, New York (1974). FANO U.,Phys. Rev. 124, 1866 (1961).

6.

DEHMER J.L., STARACE A.F., FANO U., SUGAR J. & COOPER J.W.,Phys. Rev. Lett. 26, 1521 (1971).

7.

MCGUIREE.J.,J.Phys. Chem. Solids 33, 577 (1972).

8.

MC GUIRE E.J., Sandia Laboratories Research Report No. SC-RR-71 0835 (1972).

9.

STARACE A.F.,Phys. Rev. B5, 1773 (1972).

4.

10.

MIES F.H., Phys. Rev. 175, 164 (1968).

11.

FADLEY C.S., SHIRLEY D.A., FREEMAN A.J., BAGUS P.S. & MALLOW J.V.,Phys. Rev. Lett. 23, 1397 (1969); FREEMAN A.J., BAGUS P.S. & MALLOW J.V., mt. J. Magnetism 4, 35 (1973); BAGUS P.S., FREEMAN A.J. & SASAKI F., Phys. Rev. Lett. 30, 850 (1973).

12. 13.

BROWN F.C., GAHWILLER C. & KUNZ A.B., Solid State Commun. 9,487 (1971). YIN L., ADLER I., TSANG T., CHEN M.H. & CRASEMANN B.,Phys. Lett. 46A, 113 (1973).

14.

YIN L.I., ADLER I., TSANG T., CHEN M.H., RINGERS D.A. & CRASEMANN B.,Phys. Rev. A9, 1070 (1974).

15.

ASADA S., SATOKO C. &SUGANO S.,J. Phys. Soc. Japan 38, 855 (1975).

16.

SLATER J.C., Quantum Theory ofAtomic Structure, Vols. I and II. McGraw-Hill, New York (1960).

17.

GOLDSTEIN H., Classical Mechanics, p. 326. Addison-Wesley, Reading, MA (1950).

18. 19.

WATSON R.E., MIT Technical Report No. 12 (1959). MC GUIRE E.J. (private communication).

20. 21.

PARK R.L. & HOUSTON J.E.,Phys. Rev. B6, 1073 (1972). SNOW E.C. & WABER J.T., Acta Met. 17, 623 (1969).

22.

MCCLURE D.S., in Solid State Physics (Edited by SEITZ F., TURNBULL D. & EHRENREICH H.), Vol. 9. Academic Press, New York (1959).

23.

KOTANI A. & TOYOZAWAY.,J. Phys. Soc. Japan 35, 1073, 1083 (1973).

-