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Photoemission and Bremsstrahlung isochromat spectra of mixed valence and Kondo lattice materials
Journal of Magnetism and Magnetic Materials 52 (1985) 135-140 North-Holland, Amsterdam
INVITED
135
PAPER
PHOTOEMISSION
AND BREMSSTRAHLUNG ISOCHROMAT LATUICE MATERIALS
SPECTRA OF MIXED
VALENCE AND KONDO J.W. A L L E N
Xerox Palo Alto Research Center, Palo Alto, CA 94304, USA
Through a pedagogical treatment of the Gunnarsson-Sch~nhammer calculation, the relevance of electron spectra to ground state properties in the presence of relaxation and screening is described.
1. Introduction For about a decade there has been a large effort by many electron spectroscopy groups around the world to measure the f-electron photoemission (PES) and bremsstrahlung isochromat (BIS) spectra of rare earth and actinide materials with unusual low temperature and low energy properties. Initially the focus of effort was to determine the ionization energy or the affinity energy of the f-electrons relative to the Fermi energy. This focus came from the general notion of mixed valence, that the unusual low temperature properties were a consequence of having an ionization energy coincident with the Fermi level. However, it soon became apparent that the spectra of some materials, notably cerium materials, displayed screening and relaxation phenomena which took the form of multiple peaks in the spectra. This occurrence has led to two kinds of disputes. First, within the community of theorists and experimentalists studying these effects there has been debate as to the exact mechanism of their origin. Second the f-electron community at large has often taken the skeptical view that these effects intrinsically invalidate the electron spectroscopy results, so far as the low energy physics is concerned. The oral version of this talk dealt speculatively with the first of these questions, focusing on whether the Anderson Hamiltonian will describe all aspects of these spectra for rare earth systems such as cerium, and whether it will describe the spectra of other systems, such as uranium or transition metal compounds. In this written version I address the second question in an attempt to dispel some of the skepticism. The G u n n a r s s o n - S c h 6 n h a m m e r (GS) calculation [1] of the 0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
local orbital spectral weight of the degenerate impurity Anderson Hamiltonian was not only a major step forward in providing a theoretical description of P E S / B I S spectra of cerium materials, but it also serves as a clear and rigorous example of what electron spectroscopy measures in the presence of screening and relaxation effects. As such it has great pedagogical value and so I offer a pedagogical version of the simplest GS calculation, the PES spectrum for infinite f - f Coulomb interaction, infinite degeneracy, and an f-occupation between 0 and 1. This treatment has the merit for the present purposes of being unsophisticated, with the consequence that it is not easily extended quantitatively to more complex situations. But it does enable one to think qualitatively about more complex situations.
2. Pedagogical version of the Gunnarsson-Sch6nhammer calculation 2.1. The response function Theoretical treatments of screening and relaxation effects proceed from the understanding that the response function for electron spectroscopy is, apart of the PES and BIS cross sections, the imaginary part of the trace of the general correlation function known as the single-particle Green's function G, defined in the early portions of many-body text books. There exists an exact representation of G, due to Lehman, which makes its connection to PES and BIS easy to see. In the Lehman representation the diagonal part of G for the ith single particle state is given by G , ( E ) = Y' [~n'N+II~*IO'N)]2 ,, E - [ E , ( N + I)-Eo(N)]+iTq
J. w. Allen / Photoemission and brernsstrahlung isochromat spectra
136 + ~
I(n'N-11~'IO'N>I2
(a)
"DISCRETE"
CONTINUUM STATES
STATES
,, E + [ E , ( N - 1 ) - E o ( N ) ] - i ~ " _
In this expression ~ , ~* are destruction and creation operators for an electron in state i, and i n , N + 1), In,N-1) label exact eigenstates of t h e - s y s t e m with N + I and N - 1 electrons. ]0, N ) is the exact g r o u n d state of the system with N electrons. The poles of the first and second terms occur, respectively, at the exact differences in the energies of the ground state of the N-electron system a n d all the eigenstates of the N + 1 and N-1 electron systems. The n u m e r a t o r gives the probability for reaching the various states of the N + 1 a n d N - 1 electron system as the overlap of these states with the state produced by adding or removing an electron in state " i " to or from the g r o u n d state of the N particle system. Thus the first a n d second terms are the response functions for BIS a n d PES, respectively, with two assumptions. The first is that the BIS and PES cross sections are slowly varying and the second is that the s u d d e n a p p r o x i m a t i o n holds. The latter requires that the outgoing or incoming electron have sufficiently high kinetic energy that it does not interact with the N - 1or N-electron system, respectively. This is opposite to the intuitive idea that high-energy excitation invalidates the technique.
DEGENERA~_f
I
(b)
t~ E 1 = EF
~%=o E1 = 0 ~E2
~, ~
w=~v
E1 = EF
E1 = 0
I --E F + Ef
I 0
I Ef
= EF/2
I Ef + E F
ENERGY
Fig. 1. (a) Reduced basis set of the GS calculation; (b) the resulting sequence of Fano problems. In (b) the straight vertical line is the "discrete" state, and the vertical arrow is the bound state.
2.2. The Hamiltonian
where the first term is the symmetrized projection of the c o n d u c t i o n b a n d states that hybridize to the local orbital, e f is the energy of the local-orbital a n d v runs over its Nf-fold degeneracy, V(¢) is the hybridization matrix element, including the energy d e p e n d e n c e of the cond u c t i o n b a n d density of states, and U is the local-orbital C o u l o m b interaction.
set ~s given. The basis set, shown in fig. la, has two types of states. First, there are " d i s c r e t e " states, so-called here for reasons described below. These have no localorbital electron a n d have excitations in which one electron is p r o m o t e d from energy E 2 up to the Fermi energy E v. The lowest energy state has no excitation, i.e., E 2 = E v. For fixed E 2 , a second group of " c o n t i n u u m " states are o b t a i n e d from the discrete state by transferring an electron from the b a n d at energy E 1 to one of the degenerate local orbitals at energy ~r. ( N o t e E v - ~ f is the local-orbital energy separation from the Fermi level.) The variation of El defines the c o n t i n u u m , which is Nf-fold degenerate. Taking the energy of the E 2 = E v discrete state to be zero, the other discrete states have energy E v - E z, and the c o n t i n u u m states have energies E v - E 2 + ~f - El.
2.3. The limited basis set
2.4. The Fano eigenoalue problem
The simplest G S calculation evaluates G after solving exactly for the eigenstates of H using a limited basis set which becomes fully adequate in the limit of infinite Nf. Here we shall take the validity of the limited basis
The "discrete" states are not actually discrete, but only states with the same E 2 are coupled by the hybridization matrix element V of the Hamiltonian. Further, it can be shown that in the limit of infinite Nf the " d i s -
G S evaluated G ( E ) for the single-impurity degenerate A n d e r s o n Hamiltonian,
;-;=2 [f
+fd~[V(~)q'Jq',~+h.c.l+u~n~n,1,
137
J. W. Allen / Photoemission and bremsstrahlung isochromat spectra crete" states with different E 2 are not coupled to one another by perturbation loops through the continuum states. Thus diagonalizing H separates into a series of problems, identical except for an overall energy shift, in which one "discrete" state is coupled to its degenerate continuum, as shown in fig. lb. The coupling matrix element for given E 2 is, using the expression above for the continuum energies, VE2( E ) = V( c = E F - E 2 + c f E). The solution of such problems has been given by F a n o [2]. There is a linear transformation of the N r degenerate continua such that N f - 1 of them decouple from the discrete state, to be denoted I E2), leaving it coupled to one new continuum, to be denoted [E2, E,), with new matrix element WE2(E) = V E 2 ( E ) ~ ff . We shall see below that the decoupled continua do not enter further. The eigenstates of the Hamiltonian are given by
IE,E2)=aEE~I2)+fdE,bEE~(E,)[E2,E,).
(1)
For each continuum energy there is an eigenvalue E and Fano has solved for the coefficients a and b of the corresponding eigenstates. Substituting (1) in the Schr6dinger equation and equating coefficients of like terms yields two coupled equations for aEE 2 and bEE2(E1). The first of these has the formal solution which relates aEE 2 and bEE2(Ej),
bE<(E,) =
[ P1/(E-
E')
+ zE2( E ) 6 ( E - E')] WE~( E ' ) a e E 2,
(2)
where P means principal value and E ' = E F - E 2 + c r -E~, i.e., the continuum energies. An equation determining zE,(E ) is obtained by combining (2) with the second equation, the result being
[E + F~:(E)+ZE2(E)IW&(E)I2-E]aE&=O,
(3)
where E = E v - E 2 , the discrete state energy, and FE2(E) = Re 4e2(E) with qe2(E) = fd E 1 I W E 2 ( E ' ) I 2 / ( E - E'). Using the identity 1 / ( E E') = P 1 / ( E - E') + i~rS( E - E'), ~ separates into real and imaginary parts as
O<(E) = PfdE,
[W~(E') 12/(E - E')
+ i'nl WE2( E ) [ 2
(4)
The quantity zE2(E ) is chosen to make the bracket of (3) be zero, and the solution is completed by choosing aEE 2 tO normalize the wavefunction (1). Fano's result
for I aGE_, ] 2 can be written
[aEE 2 [ 2 = (1/nr) Im ~ , & ( E ) ,
(5)
where gE2 ( E ) = [ E - E ~ 0E2(E)] 1. For energies E within the continuum, using eq. (4), this has the Lorentzian form given by Fano, la~:~l 2 = [ W ( E ) ] 2 / { [ E -
E¢p- F ( E ) ] 2
+ ~ r 2 1 W ( E ) [4}.
(6)
2.5. The bound state/ground state It is very important that there also exists a discrete state solution at energy 8 below the continuum, shown in fig. l b as a vertical arrow. From eq. (2) this requires that an E outside the continuum be found such that the bracket of eq. (3) is zero with the z ( E ) term absent. The resulting trancendental equation is, for E 2 = E v, eq. (3.12) of GS. It turns out that a solution exists whenever the continuum has a step function cutoff, as is the case here where the Fermi energy defines the lowest energy edge. There may also be a discrete state pushed out of the upper end of the continuum, depending on the details of the band edge shape and the values of e f and W. For simplicity we assume that this does not occur. The coefficients " a " and " b " of the bound-state wavefunction are given by eq. (2) with the z-term absent and by eq. (5), which has a pole at the bound state energy. The weight of the pole is [ a [ 2, (Note that for E outside the continuum Im ~ = 0, so that the denominator of g is just the equation for the bound state, and also that the P operator in (2) is not needed.) The ground state of the system is the E 2 = E v bound state, and 8 is the (Kondo) binding energy. Appendix C of GS solves for 8 in the case of W constant within the band, for which the various integrals above can easily be done. Since a~& is the coefficient of the portion of the state having no local-orbital electron, IaG [ 2 = 1 n f, where n r is the ground state local orbital occupation. Note that [dEE212 is the same, 1 - n f , for all b o u n d states, independent of E 2. Appendix C of GS evaluates l a o l 2 for the case of constant W, with the very useful result n f / ( 1 - - n f ) = N r A / ' n & where A = ~V 2.
2.6. The photoemission spectrum Fig. 2 shows the local (0-orbital photoemission process. The f-orbital is occupied in the continuum parts of the N-electron ground state, as in (a), which can drawn
J. W. Allen / Photoemission and bremsstrahlung isochromat spectra
138
"
EF(N)
El(N) . _ ~ (a)
tl
o- ~---
--BIS SPECTRALWEIGHT
h It
El(N)
~
(b)
I I
I
//-PES SPECTRALWEIGHT
I I
EF(NI--_~~E
-= El(N)/ 3 El(N)
)
E2(N- l-)~t__~//_[_ __EF(N-1}
(c)
I
I EF EF+8
J-~
E2(N -- 1) = Et(N)
(d)
Fig. 2. Sequence of configurations showing that f-photoemission from the continuum part of the N-electron ground state results in ( N - 1)-electron "discrete" states.
N-11 N
{
t I ~--f-~,
as in (b) since, for the ground state, E 2 ( N ) = E F ( N ) . Removal of the f-electron gives the state shown in (c), which is a state of the " d i s c r e t e " type for the N - 1 electron system, as shown in (d), with the link E 1 ( N ) = E 2 ( N - 1). N o t e that E v in (d) is for N - 1 electrons. The eigenvalue p r o b l e m for N - 1 is, of course, identical to that just described for N except that all energies are reduced by E p N o t e that the N r - 1 decoupled c o n t i n u a m e n t i o n e d in subsection 2.4. above do not c o n t r i b u t e because only states that couple to the "discrete" states are ever involved. We can now write an explicit expression for the imaginary part of the PES part of G ( E ) , to be denoted p f ( E ) , as Ib2(E1)I 2 a& N _ E1 . & 2 .
(7)
The first squared coefficient is the a m p l i t u d e of an E 1 c o n t i n u u m state in the N-electron ground state a n d the second squared coefficient is the distribution of the " d i s c r e t e " state, p r o d u c e d by electron-removal, a m o n g the ( N - 1)-electron eigenstates. T h e second (E2) index on I a 12 is set equal to E t to fulfill the link between the N and (N-1) electron states. T h e first (eigenvalue) index is set equal to E G - E because taking the imaginary part of G yields v 6 ( E + E ( N - 1 ) - E G ( N ) ) , which defines the response to occur at the poles of G. Note from fig. 1 or 3 that E a = - E v + (f 8. Fig. 3 shows the above ingredients and the construction of the PES spectrum, which is the sum of the [ a [ 2 spectrum for each E 2 ( N - 1), weighted by the value of [ b I 2 for E l ( N ) = E 2 ( N - 1). The I a [ 2 spectrum is given by eq. (6) plus a delta-function of weight (1 - n f) at the b o u n d state, as sketched, a n d the I bl 2 spectrum is sketched at the right, following eq. (2). T r a n s i t i o n s
EF
~1 L--8 "
-
ibG(E1)I2
t'
)
p~(EI=~fdE,
3EF+8
)
EG(N-1)
EG(N)
I --2E F +
Ef
L - E F + Ef
I
I
0
Ef
ENERGY Fig. 3. Superposition of distributions of "discrete" states in the ( N - D - e l e c t r o n eigenstates, weighted by distribution of the continuum states in the N-electron ground state, sketched at right, produces the photoemission spectral weight. Apart from a shift of E F, the ( N - l ) - e l e c t r o n eigenvalue spectrum is identical to that for N-electrons, which is not shown. The BIS spectrum is shown dashed.
occur from the g r o u n d state of the N-electron system to all states of the ( N - 1)-electron system, starting at E v a n d extending away from E F a total of 2 E F + & The eigenvalue spectrum of the N - 1-electron system is the same as for the N-electron system, but shifted by E v. (Excited states of the N-electron system are not shown in fig. (3).) The lowest energy transition at E F occurs between the g r o u n d states of the two systems, a n d the PES spectrum over the first 8 of energy is entirely from transitions to the sequence of b o u n d states. Each b o u n d state has the same value of ] a l 2 = l - n r as does the N-electron g r o u n d state, so using eq. (2) for I b~Jl 2 this part of the spectrum is simply
- -
p f ( E ) = (1 - n f ) 2 N f i V ( E ) [ 2 / ( E - E f + E o . )
2
(8)
which is G S eq. (6.23). This part of the spectrum is the K o n d o resonance, a n d it m a p s out the c o n t i n u u m state c o n t r i b u t i o n to the g r o u n d state. F o r constant V and at the Fermi energy Of ( E F ) = (1 -- n f)2NfA/'n'8 2, and using the relation given at the end of section 2.5 above, this
J. W. Allen / Photoemission and bremsstrahlung isochromat spectra
becomes p f ( E v ) = 'rrnf2/Nf A. It is a remarkable consequence [3] of the Friedel sum rule that this is the same value that one obtains for a system with U = 0 and the same value of n f, i.e., that given by a Lorentzian of degeneracy Nf and width A centered above E v. Integrating down from the Fermi level, the total area under p t ( E ) is nf, and the area under the resonance portion is roughly the product of 8 and p f ( E v ) , which is ( , n ~ n f / N f A ) n f = (l - n f ) n r. Because the local orbital occupation of the N - 1 electron (bound) states reached in the resonance is the same as in the N-electron ground state, the resonance can be regarded as the relaxation of the system back to its ground state after creation of the f-hole. Thus this part of the spectrum is closely akin to the result of a density-functional calculation, which obtains its potential from the ground state charge distribution. For energies greater than 6, there are contributions from transitions to the continuum states, for which n f = 1 - ] a 12 is smaller than for the ground state, so this is the "ionization" part of the spectrum. This part is then conceptually related to the results of a "frozen-f-hole" density functional calculation [4] or a renormalized-atom [5] calculation of f-ionization energies.
139
practice, as discussed by GS, the basis set must be enlarged to obtain a realistic result. The qualitative shape of the BIS spectrum is sketched as a dashed line in fig. 3. Note that the width is - 6 / N t. That this vanishes as Nf goes to infinity is a signal of the need for the larger basis set, giving corrections to order 1 / N f . As with the PES spectrum, the lowest energy transition, i.e. the one just at E v is between the ground states of the N and N + 1 electron systems, and the spectrum then extends away from E v to various excited states of the N + 1 electron system. Integrating away from E F, the total area under the BIS spectrum is the number of f-holes, which is Nf - n r, so that the total area under the P E S / B I S spectrum is just N t, the total capacity of the degenerate local orbital. The BIS area is divided into a portion (1 - nf)Nf in the peak just above E v, and into a portion n f ( N f - 1) in a peak pushed off to higher energies by U. The first is the f0 ~ fl weight, which is the product of the f0 probability and the number of holes for f0, and the second is the fl _, f2 weight, which is the product of the probability of f~ and the number of holes for fl. When U is small enough, weight is transferred from the peak at U to the E v peak due to relaxation of electrons out of the f2 state into the band.
2. 7. Relation to the T-finear specific heat 2.9. Two basic points
The T-linear specific heat coefficient "t is related to p f ( E v ) , the spectral weight at E v, as follows. For a many-body system the single-electron result y = ~ r 2 k ~ N ( E F ) / 3 continues to hold if N ( E F ) is taken to be the quasi-particle density of states. N ( E F ) is generally related to p f ( E v ) by a multiplicative factor Z -1, i.e., N ( E F ) = Z -1 p f ( E v ) . Z -1 has a general definition [6] in terms of the derivative of the self-energy part of G, and for the present problem Z turns out to be the weight of the pole part of eq. (5), which is just [ a G 12 = ( l - n t). For the case of constant W, using results given in the text above, one has p f ( E v ) = v n Z f / N f A and Z -1 = NfA/,rrnf~, leading to an enhanced y = ~ r 2 k 2 n f / 3 6 , as obtained by others [7]. Note that p f ( E r ) is not enhanced - indeed, as pointed out above, it is pinned by the Friedel sum rule to have the same value as for U=0.
Two basic points are elegantly illustrated by the GS calculation. (1) The local orbital tries to relax back to its ground state after an electron is added or removed by exchanging electrons with the band. This is a kind of screening, in the broad sense of energy-lowering, and has been called [8] hybridization-screening in connection with the spectrum of NiO. Since the spectral weight for adding and removing f-electrons meets at E v in the K o n d o resonance, the U-value is screened down essentially to zero (or more likely to 8), for electrons in the resonance. (2) All energy scales of the system, including the small energy scale 6 characteristic of the ground state, are observed in the spectrum because all the states of the N and N - 1 particle systems are identical so far as the local orbital is concerned.
2.8. The B I S spectrum
3. F u r t h e r theoretical results
In principle the BIS spectrum is calculated in exactly the same way - by constructing the eigenstates of the N + 1 electron system and then finding their overlap w i t h the state produced by adding a local orbital electron to the ground state of the N-electron system. In
There are two more repent results that should be mentioned, because, in my opinion, they go far toward accounting for the presence of more spectral weight near E v than is expected from the Kondo resonance itself.
140
J. W. Allen / Photoemission and bremsstrahlung isochromat spectra
One is that the K o n d o resonance has spin orbit sidebands. The possibility for this follows from the discussion above. If the local orbital degeneracy is split, as by the s p i n - o r b i t interaction, then the ground state is largely made from the lower energy component, a n d there are excited states involving the higher energy component. In the P E S / B I S spectrum there is some probability that the system will relax back to these states, rather than the g r o u n d state. The a m o u n t of weight in these sidebands can be m u c h larger than for the resonance itself. If the relevant valence states have more complex q u a n t u m structure then a spin orbit splitting this will add further sidebands a r o u n d E v. This is p r o b a b l y very i m p o r t a n t for u r a n i u m systems [9] because they have larger n u m b e r s of f-electrons. G S treated the sidebands in a n a p p e n d i x and they have been explored further by others [10,11]. The second [12] is that for finite U, the basis set must be enlarged to consider states with two local-orbital electrons. This leads to there being in the ground state some f2 configurations, a n d there is added spectral weight due to f 2 ~ fl transitions near E v in the PES spectrum.
4. Concluding remarks In the G S calculation one sees explicitly that the relaxation a n d screening effects in the P E S / B I S spect r u m are intimately a n d naturally connected to all the other interesting properties of the system, such as its low energy scale 8. T h e serious limitation of the meas u r e m e n t is its resolution because the larger is the specific heat y, the smaller is 8. On the other hand, the P E S / B I S spectrum also reveals the existence of other, larger energy scales, such as the ionization energy ~t,
a n d the C o u l o m b interaction U, so it can show simple model having a single energy scale, like p r o m o t i o n a l model for Ce, must be replaced multiple-energy-scale model, like the K o n d o collapse [13] picture for Ce.
when a the old with a volume
It is a pleasure to t h a n k S.-J. Oh for his help and e n c o u r a g e m e n t with this exercise, and O. G u n n a r s s o n for his remarkable patience in n u m e r o u s discussions.
References [1] O. Gunnarsson and K. SchOnhammer, Phys. Rev. B28 (1983) 4315. [2] U. Fano, Phys. Rev. 124 (1961) 1866. [3] D.C. Langreth, Phys. Rev. 150 (1966) 516. [4] M.R. Normann, D.D. Koelling, A.J. Freeman, H.J.F. Jansen, B.J. Min, T. Oguchi and Ling Ye, Phys. Rev. Lett. 53 (1984) 1673. [5] J.F. Herbst, R.E. Watson and J.W. Wilkens, Phys. Rev. BI7 (1978) 3089. [6] J.M. Luttinger, Phys. Rev. 119 (1960) 1153. [7] J.W. Rasul and A.C. Hewson, J. Phys. C17 (1984) 2555. [8] G.A. Sawatzky and J.W. Allen, Phys. Rev. Lett. 53 (1984) 2339. J.W. Allen, Magn. Magn. Mat. 47&48 (1985) 168. [9] J.W. Allen, S.-J. Oh, L.E. Cox, W.P. Ellis, M.S. Wire, Z. Fisk~ J.L. Smith, B.B. Pate, I. Lindau and A.J. Arko, Phys. Rev. Lett. 54 (1985) 2635. [10] N.E. Bickers, D.L. Cox and J.W. Wilkins, Phys. Rev. lett. 54 (1985) 230. [11] O. Sakai, M. Takeshige and T. Kasuya, J. Phys. Soc. Japan 53 (1984) 3657. [12] O. Gunnarsson and K. SchOnhammer, J. Magn. Magn. Mat. 47&48 (1985) 266; Phys. Rev. B 31 (1985) 4815. [13] J.W. Allen and R.M. Martin, Phys. Rev. Lett. 49 (1982) 1106.
INVITED
135
PAPER
PHOTOEMISSION
AND BREMSSTRAHLUNG ISOCHROMAT LATUICE MATERIALS
SPECTRA OF MIXED
VALENCE AND KONDO J.W. A L L E N
Xerox Palo Alto Research Center, Palo Alto, CA 94304, USA
Through a pedagogical treatment of the Gunnarsson-Sch~nhammer calculation, the relevance of electron spectra to ground state properties in the presence of relaxation and screening is described.
1. Introduction For about a decade there has been a large effort by many electron spectroscopy groups around the world to measure the f-electron photoemission (PES) and bremsstrahlung isochromat (BIS) spectra of rare earth and actinide materials with unusual low temperature and low energy properties. Initially the focus of effort was to determine the ionization energy or the affinity energy of the f-electrons relative to the Fermi energy. This focus came from the general notion of mixed valence, that the unusual low temperature properties were a consequence of having an ionization energy coincident with the Fermi level. However, it soon became apparent that the spectra of some materials, notably cerium materials, displayed screening and relaxation phenomena which took the form of multiple peaks in the spectra. This occurrence has led to two kinds of disputes. First, within the community of theorists and experimentalists studying these effects there has been debate as to the exact mechanism of their origin. Second the f-electron community at large has often taken the skeptical view that these effects intrinsically invalidate the electron spectroscopy results, so far as the low energy physics is concerned. The oral version of this talk dealt speculatively with the first of these questions, focusing on whether the Anderson Hamiltonian will describe all aspects of these spectra for rare earth systems such as cerium, and whether it will describe the spectra of other systems, such as uranium or transition metal compounds. In this written version I address the second question in an attempt to dispel some of the skepticism. The G u n n a r s s o n - S c h 6 n h a m m e r (GS) calculation [1] of the 0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
local orbital spectral weight of the degenerate impurity Anderson Hamiltonian was not only a major step forward in providing a theoretical description of P E S / B I S spectra of cerium materials, but it also serves as a clear and rigorous example of what electron spectroscopy measures in the presence of screening and relaxation effects. As such it has great pedagogical value and so I offer a pedagogical version of the simplest GS calculation, the PES spectrum for infinite f - f Coulomb interaction, infinite degeneracy, and an f-occupation between 0 and 1. This treatment has the merit for the present purposes of being unsophisticated, with the consequence that it is not easily extended quantitatively to more complex situations. But it does enable one to think qualitatively about more complex situations.
2. Pedagogical version of the Gunnarsson-Sch6nhammer calculation 2.1. The response function Theoretical treatments of screening and relaxation effects proceed from the understanding that the response function for electron spectroscopy is, apart of the PES and BIS cross sections, the imaginary part of the trace of the general correlation function known as the single-particle Green's function G, defined in the early portions of many-body text books. There exists an exact representation of G, due to Lehman, which makes its connection to PES and BIS easy to see. In the Lehman representation the diagonal part of G for the ith single particle state is given by G , ( E ) = Y' [~n'N+II~*IO'N)]2 ,, E - [ E , ( N + I)-Eo(N)]+iTq
J. w. Allen / Photoemission and brernsstrahlung isochromat spectra
136 + ~
I(n'N-11~'IO'N>I2
(a)
"DISCRETE"
CONTINUUM STATES
STATES
,, E + [ E , ( N - 1 ) - E o ( N ) ] - i ~ " _
In this expression ~ , ~* are destruction and creation operators for an electron in state i, and i n , N + 1), In,N-1) label exact eigenstates of t h e - s y s t e m with N + I and N - 1 electrons. ]0, N ) is the exact g r o u n d state of the system with N electrons. The poles of the first and second terms occur, respectively, at the exact differences in the energies of the ground state of the N-electron system a n d all the eigenstates of the N + 1 and N-1 electron systems. The n u m e r a t o r gives the probability for reaching the various states of the N + 1 a n d N - 1 electron system as the overlap of these states with the state produced by adding or removing an electron in state " i " to or from the g r o u n d state of the N particle system. Thus the first a n d second terms are the response functions for BIS a n d PES, respectively, with two assumptions. The first is that the BIS and PES cross sections are slowly varying and the second is that the s u d d e n a p p r o x i m a t i o n holds. The latter requires that the outgoing or incoming electron have sufficiently high kinetic energy that it does not interact with the N - 1or N-electron system, respectively. This is opposite to the intuitive idea that high-energy excitation invalidates the technique.
DEGENERA~_f
I
(b)
t~ E 1 = EF
~%=o E1 = 0 ~E2
~, ~
w=~v
E1 = EF
E1 = 0
I --E F + Ef
I 0
I Ef
= EF/2
I Ef + E F
ENERGY
Fig. 1. (a) Reduced basis set of the GS calculation; (b) the resulting sequence of Fano problems. In (b) the straight vertical line is the "discrete" state, and the vertical arrow is the bound state.
2.2. The Hamiltonian
where the first term is the symmetrized projection of the c o n d u c t i o n b a n d states that hybridize to the local orbital, e f is the energy of the local-orbital a n d v runs over its Nf-fold degeneracy, V(¢) is the hybridization matrix element, including the energy d e p e n d e n c e of the cond u c t i o n b a n d density of states, and U is the local-orbital C o u l o m b interaction.
set ~s given. The basis set, shown in fig. la, has two types of states. First, there are " d i s c r e t e " states, so-called here for reasons described below. These have no localorbital electron a n d have excitations in which one electron is p r o m o t e d from energy E 2 up to the Fermi energy E v. The lowest energy state has no excitation, i.e., E 2 = E v. For fixed E 2 , a second group of " c o n t i n u u m " states are o b t a i n e d from the discrete state by transferring an electron from the b a n d at energy E 1 to one of the degenerate local orbitals at energy ~r. ( N o t e E v - ~ f is the local-orbital energy separation from the Fermi level.) The variation of El defines the c o n t i n u u m , which is Nf-fold degenerate. Taking the energy of the E 2 = E v discrete state to be zero, the other discrete states have energy E v - E z, and the c o n t i n u u m states have energies E v - E 2 + ~f - El.
2.3. The limited basis set
2.4. The Fano eigenoalue problem
The simplest G S calculation evaluates G after solving exactly for the eigenstates of H using a limited basis set which becomes fully adequate in the limit of infinite Nf. Here we shall take the validity of the limited basis
The "discrete" states are not actually discrete, but only states with the same E 2 are coupled by the hybridization matrix element V of the Hamiltonian. Further, it can be shown that in the limit of infinite Nf the " d i s -
G S evaluated G ( E ) for the single-impurity degenerate A n d e r s o n Hamiltonian,
;-;=2 [f
+fd~[V(~)q'Jq',~+h.c.l+u~n~n,1,
137
J. W. Allen / Photoemission and bremsstrahlung isochromat spectra crete" states with different E 2 are not coupled to one another by perturbation loops through the continuum states. Thus diagonalizing H separates into a series of problems, identical except for an overall energy shift, in which one "discrete" state is coupled to its degenerate continuum, as shown in fig. lb. The coupling matrix element for given E 2 is, using the expression above for the continuum energies, VE2( E ) = V( c = E F - E 2 + c f E). The solution of such problems has been given by F a n o [2]. There is a linear transformation of the N r degenerate continua such that N f - 1 of them decouple from the discrete state, to be denoted I E2), leaving it coupled to one new continuum, to be denoted [E2, E,), with new matrix element WE2(E) = V E 2 ( E ) ~ ff . We shall see below that the decoupled continua do not enter further. The eigenstates of the Hamiltonian are given by
IE,E2)=aEE~I2)+fdE,bEE~(E,)[E2,E,).
(1)
For each continuum energy there is an eigenvalue E and Fano has solved for the coefficients a and b of the corresponding eigenstates. Substituting (1) in the Schr6dinger equation and equating coefficients of like terms yields two coupled equations for aEE 2 and bEE2(E1). The first of these has the formal solution which relates aEE 2 and bEE2(Ej),
bE<(E,) =
[ P1/(E-
E')
+ zE2( E ) 6 ( E - E')] WE~( E ' ) a e E 2,
(2)
where P means principal value and E ' = E F - E 2 + c r -E~, i.e., the continuum energies. An equation determining zE,(E ) is obtained by combining (2) with the second equation, the result being
[E + F~:(E)+ZE2(E)IW&(E)I2-E]aE&=O,
(3)
where E = E v - E 2 , the discrete state energy, and FE2(E) = Re 4e2(E) with qe2(E) = fd E 1 I W E 2 ( E ' ) I 2 / ( E - E'). Using the identity 1 / ( E E') = P 1 / ( E - E') + i~rS( E - E'), ~ separates into real and imaginary parts as
O<(E) = PfdE,
[W~(E') 12/(E - E')
+ i'nl WE2( E ) [ 2
(4)
The quantity zE2(E ) is chosen to make the bracket of (3) be zero, and the solution is completed by choosing aEE 2 tO normalize the wavefunction (1). Fano's result
for I aGE_, ] 2 can be written
[aEE 2 [ 2 = (1/nr) Im ~ , & ( E ) ,
(5)
where gE2 ( E ) = [ E - E ~ 0E2(E)] 1. For energies E within the continuum, using eq. (4), this has the Lorentzian form given by Fano, la~:~l 2 = [ W ( E ) ] 2 / { [ E -
E¢p- F ( E ) ] 2
+ ~ r 2 1 W ( E ) [4}.
(6)
2.5. The bound state/ground state It is very important that there also exists a discrete state solution at energy 8 below the continuum, shown in fig. l b as a vertical arrow. From eq. (2) this requires that an E outside the continuum be found such that the bracket of eq. (3) is zero with the z ( E ) term absent. The resulting trancendental equation is, for E 2 = E v, eq. (3.12) of GS. It turns out that a solution exists whenever the continuum has a step function cutoff, as is the case here where the Fermi energy defines the lowest energy edge. There may also be a discrete state pushed out of the upper end of the continuum, depending on the details of the band edge shape and the values of e f and W. For simplicity we assume that this does not occur. The coefficients " a " and " b " of the bound-state wavefunction are given by eq. (2) with the z-term absent and by eq. (5), which has a pole at the bound state energy. The weight of the pole is [ a [ 2, (Note that for E outside the continuum Im ~ = 0, so that the denominator of g is just the equation for the bound state, and also that the P operator in (2) is not needed.) The ground state of the system is the E 2 = E v bound state, and 8 is the (Kondo) binding energy. Appendix C of GS solves for 8 in the case of W constant within the band, for which the various integrals above can easily be done. Since a~& is the coefficient of the portion of the state having no local-orbital electron, IaG [ 2 = 1 n f, where n r is the ground state local orbital occupation. Note that [dEE212 is the same, 1 - n f , for all b o u n d states, independent of E 2. Appendix C of GS evaluates l a o l 2 for the case of constant W, with the very useful result n f / ( 1 - - n f ) = N r A / ' n & where A = ~V 2.
2.6. The photoemission spectrum Fig. 2 shows the local (0-orbital photoemission process. The f-orbital is occupied in the continuum parts of the N-electron ground state, as in (a), which can drawn
J. W. Allen / Photoemission and bremsstrahlung isochromat spectra
138
"
EF(N)
El(N) . _ ~ (a)
tl
o- ~---
--BIS SPECTRALWEIGHT
h It
El(N)
~
(b)
I I
I
//-PES SPECTRALWEIGHT
I I
EF(NI--_~~E
-= El(N)/ 3 El(N)
)
E2(N- l-)~t__~//_[_ __EF(N-1}
(c)
I
I EF EF+8
J-~
E2(N -- 1) = Et(N)
(d)
Fig. 2. Sequence of configurations showing that f-photoemission from the continuum part of the N-electron ground state results in ( N - 1)-electron "discrete" states.
N-11 N
{
t I ~--f-~,
as in (b) since, for the ground state, E 2 ( N ) = E F ( N ) . Removal of the f-electron gives the state shown in (c), which is a state of the " d i s c r e t e " type for the N - 1 electron system, as shown in (d), with the link E 1 ( N ) = E 2 ( N - 1). N o t e that E v in (d) is for N - 1 electrons. The eigenvalue p r o b l e m for N - 1 is, of course, identical to that just described for N except that all energies are reduced by E p N o t e that the N r - 1 decoupled c o n t i n u a m e n t i o n e d in subsection 2.4. above do not c o n t r i b u t e because only states that couple to the "discrete" states are ever involved. We can now write an explicit expression for the imaginary part of the PES part of G ( E ) , to be denoted p f ( E ) , as Ib2(E1)I 2 a& N _ E1 . & 2 .
(7)
The first squared coefficient is the a m p l i t u d e of an E 1 c o n t i n u u m state in the N-electron ground state a n d the second squared coefficient is the distribution of the " d i s c r e t e " state, p r o d u c e d by electron-removal, a m o n g the ( N - 1)-electron eigenstates. T h e second (E2) index on I a 12 is set equal to E t to fulfill the link between the N and (N-1) electron states. T h e first (eigenvalue) index is set equal to E G - E because taking the imaginary part of G yields v 6 ( E + E ( N - 1 ) - E G ( N ) ) , which defines the response to occur at the poles of G. Note from fig. 1 or 3 that E a = - E v + (f 8. Fig. 3 shows the above ingredients and the construction of the PES spectrum, which is the sum of the [ a [ 2 spectrum for each E 2 ( N - 1), weighted by the value of [ b I 2 for E l ( N ) = E 2 ( N - 1). The I a [ 2 spectrum is given by eq. (6) plus a delta-function of weight (1 - n f) at the b o u n d state, as sketched, a n d the I bl 2 spectrum is sketched at the right, following eq. (2). T r a n s i t i o n s
EF
~1 L--8 "
-
ibG(E1)I2
t'
)
p~(EI=~fdE,
3EF+8
)
EG(N-1)
EG(N)
I --2E F +
Ef
L - E F + Ef
I
I
0
Ef
ENERGY Fig. 3. Superposition of distributions of "discrete" states in the ( N - D - e l e c t r o n eigenstates, weighted by distribution of the continuum states in the N-electron ground state, sketched at right, produces the photoemission spectral weight. Apart from a shift of E F, the ( N - l ) - e l e c t r o n eigenvalue spectrum is identical to that for N-electrons, which is not shown. The BIS spectrum is shown dashed.
occur from the g r o u n d state of the N-electron system to all states of the ( N - 1)-electron system, starting at E v a n d extending away from E F a total of 2 E F + & The eigenvalue spectrum of the N - 1-electron system is the same as for the N-electron system, but shifted by E v. (Excited states of the N-electron system are not shown in fig. (3).) The lowest energy transition at E F occurs between the g r o u n d states of the two systems, a n d the PES spectrum over the first 8 of energy is entirely from transitions to the sequence of b o u n d states. Each b o u n d state has the same value of ] a l 2 = l - n r as does the N-electron g r o u n d state, so using eq. (2) for I b~Jl 2 this part of the spectrum is simply
- -
p f ( E ) = (1 - n f ) 2 N f i V ( E ) [ 2 / ( E - E f + E o . )
2
(8)
which is G S eq. (6.23). This part of the spectrum is the K o n d o resonance, a n d it m a p s out the c o n t i n u u m state c o n t r i b u t i o n to the g r o u n d state. F o r constant V and at the Fermi energy Of ( E F ) = (1 -- n f)2NfA/'n'8 2, and using the relation given at the end of section 2.5 above, this
J. W. Allen / Photoemission and bremsstrahlung isochromat spectra
becomes p f ( E v ) = 'rrnf2/Nf A. It is a remarkable consequence [3] of the Friedel sum rule that this is the same value that one obtains for a system with U = 0 and the same value of n f, i.e., that given by a Lorentzian of degeneracy Nf and width A centered above E v. Integrating down from the Fermi level, the total area under p t ( E ) is nf, and the area under the resonance portion is roughly the product of 8 and p f ( E v ) , which is ( , n ~ n f / N f A ) n f = (l - n f ) n r. Because the local orbital occupation of the N - 1 electron (bound) states reached in the resonance is the same as in the N-electron ground state, the resonance can be regarded as the relaxation of the system back to its ground state after creation of the f-hole. Thus this part of the spectrum is closely akin to the result of a density-functional calculation, which obtains its potential from the ground state charge distribution. For energies greater than 6, there are contributions from transitions to the continuum states, for which n f = 1 - ] a 12 is smaller than for the ground state, so this is the "ionization" part of the spectrum. This part is then conceptually related to the results of a "frozen-f-hole" density functional calculation [4] or a renormalized-atom [5] calculation of f-ionization energies.
139
practice, as discussed by GS, the basis set must be enlarged to obtain a realistic result. The qualitative shape of the BIS spectrum is sketched as a dashed line in fig. 3. Note that the width is - 6 / N t. That this vanishes as Nf goes to infinity is a signal of the need for the larger basis set, giving corrections to order 1 / N f . As with the PES spectrum, the lowest energy transition, i.e. the one just at E v is between the ground states of the N and N + 1 electron systems, and the spectrum then extends away from E v to various excited states of the N + 1 electron system. Integrating away from E F, the total area under the BIS spectrum is the number of f-holes, which is Nf - n r, so that the total area under the P E S / B I S spectrum is just N t, the total capacity of the degenerate local orbital. The BIS area is divided into a portion (1 - nf)Nf in the peak just above E v, and into a portion n f ( N f - 1) in a peak pushed off to higher energies by U. The first is the f0 ~ fl weight, which is the product of the f0 probability and the number of holes for f0, and the second is the fl _, f2 weight, which is the product of the probability of f~ and the number of holes for fl. When U is small enough, weight is transferred from the peak at U to the E v peak due to relaxation of electrons out of the f2 state into the band.
2. 7. Relation to the T-finear specific heat 2.9. Two basic points
The T-linear specific heat coefficient "t is related to p f ( E v ) , the spectral weight at E v, as follows. For a many-body system the single-electron result y = ~ r 2 k ~ N ( E F ) / 3 continues to hold if N ( E F ) is taken to be the quasi-particle density of states. N ( E F ) is generally related to p f ( E v ) by a multiplicative factor Z -1, i.e., N ( E F ) = Z -1 p f ( E v ) . Z -1 has a general definition [6] in terms of the derivative of the self-energy part of G, and for the present problem Z turns out to be the weight of the pole part of eq. (5), which is just [ a G 12 = ( l - n t). For the case of constant W, using results given in the text above, one has p f ( E v ) = v n Z f / N f A and Z -1 = NfA/,rrnf~, leading to an enhanced y = ~ r 2 k 2 n f / 3 6 , as obtained by others [7]. Note that p f ( E r ) is not enhanced - indeed, as pointed out above, it is pinned by the Friedel sum rule to have the same value as for U=0.
Two basic points are elegantly illustrated by the GS calculation. (1) The local orbital tries to relax back to its ground state after an electron is added or removed by exchanging electrons with the band. This is a kind of screening, in the broad sense of energy-lowering, and has been called [8] hybridization-screening in connection with the spectrum of NiO. Since the spectral weight for adding and removing f-electrons meets at E v in the K o n d o resonance, the U-value is screened down essentially to zero (or more likely to 8), for electrons in the resonance. (2) All energy scales of the system, including the small energy scale 6 characteristic of the ground state, are observed in the spectrum because all the states of the N and N - 1 particle systems are identical so far as the local orbital is concerned.
2.8. The B I S spectrum
3. F u r t h e r theoretical results
In principle the BIS spectrum is calculated in exactly the same way - by constructing the eigenstates of the N + 1 electron system and then finding their overlap w i t h the state produced by adding a local orbital electron to the ground state of the N-electron system. In
There are two more repent results that should be mentioned, because, in my opinion, they go far toward accounting for the presence of more spectral weight near E v than is expected from the Kondo resonance itself.
140
J. W. Allen / Photoemission and bremsstrahlung isochromat spectra
One is that the K o n d o resonance has spin orbit sidebands. The possibility for this follows from the discussion above. If the local orbital degeneracy is split, as by the s p i n - o r b i t interaction, then the ground state is largely made from the lower energy component, a n d there are excited states involving the higher energy component. In the P E S / B I S spectrum there is some probability that the system will relax back to these states, rather than the g r o u n d state. The a m o u n t of weight in these sidebands can be m u c h larger than for the resonance itself. If the relevant valence states have more complex q u a n t u m structure then a spin orbit splitting this will add further sidebands a r o u n d E v. This is p r o b a b l y very i m p o r t a n t for u r a n i u m systems [9] because they have larger n u m b e r s of f-electrons. G S treated the sidebands in a n a p p e n d i x and they have been explored further by others [10,11]. The second [12] is that for finite U, the basis set must be enlarged to consider states with two local-orbital electrons. This leads to there being in the ground state some f2 configurations, a n d there is added spectral weight due to f 2 ~ fl transitions near E v in the PES spectrum.
4. Concluding remarks In the G S calculation one sees explicitly that the relaxation a n d screening effects in the P E S / B I S spect r u m are intimately a n d naturally connected to all the other interesting properties of the system, such as its low energy scale 8. T h e serious limitation of the meas u r e m e n t is its resolution because the larger is the specific heat y, the smaller is 8. On the other hand, the P E S / B I S spectrum also reveals the existence of other, larger energy scales, such as the ionization energy ~t,
a n d the C o u l o m b interaction U, so it can show simple model having a single energy scale, like p r o m o t i o n a l model for Ce, must be replaced multiple-energy-scale model, like the K o n d o collapse [13] picture for Ce.
when a the old with a volume
It is a pleasure to t h a n k S.-J. Oh for his help and e n c o u r a g e m e n t with this exercise, and O. G u n n a r s s o n for his remarkable patience in n u m e r o u s discussions.
References [1] O. Gunnarsson and K. SchOnhammer, Phys. Rev. B28 (1983) 4315. [2] U. Fano, Phys. Rev. 124 (1961) 1866. [3] D.C. Langreth, Phys. Rev. 150 (1966) 516. [4] M.R. Normann, D.D. Koelling, A.J. Freeman, H.J.F. Jansen, B.J. Min, T. Oguchi and Ling Ye, Phys. Rev. Lett. 53 (1984) 1673. [5] J.F. Herbst, R.E. Watson and J.W. Wilkens, Phys. Rev. BI7 (1978) 3089. [6] J.M. Luttinger, Phys. Rev. 119 (1960) 1153. [7] J.W. Rasul and A.C. Hewson, J. Phys. C17 (1984) 2555. [8] G.A. Sawatzky and J.W. Allen, Phys. Rev. Lett. 53 (1984) 2339. J.W. Allen, Magn. Magn. Mat. 47&48 (1985) 168. [9] J.W. Allen, S.-J. Oh, L.E. Cox, W.P. Ellis, M.S. Wire, Z. Fisk~ J.L. Smith, B.B. Pate, I. Lindau and A.J. Arko, Phys. Rev. Lett. 54 (1985) 2635. [10] N.E. Bickers, D.L. Cox and J.W. Wilkins, Phys. Rev. lett. 54 (1985) 230. [11] O. Sakai, M. Takeshige and T. Kasuya, J. Phys. Soc. Japan 53 (1984) 3657. [12] O. Gunnarsson and K. SchOnhammer, J. Magn. Magn. Mat. 47&48 (1985) 266; Phys. Rev. B 31 (1985) 4815. [13] J.W. Allen and R.M. Martin, Phys. Rev. Lett. 49 (1982) 1106.