- Email: [email protected]
4f photoemission spectra of the light rare earths
Physica 1SOB ( 1985) 66-68 North-Holland. Amsterdam
4f PHOTOEMISSION
SPECTRA
OF THE LIGHT
RARE
EARTHS
P.S. RISEBOROUGH Polytechnic The
Institute
of N. Y., 433 Jay Street.
Rrooklyn.
NY
Il201,
4f photoemission spectra of Ce, Pr, and Nd compounds
screening
channels;
localized
screening
for the Ce, Pr and Nd compounds, conduction moves along reduced.
electrons.
The
the lanthanide
relative
and difiuse
which weights
reflects
screening.
of eliminating
the occupation
of the
The photoemission spectra of pure Ce metal and many Ce compounds have been the subject of much scientific debate [l-9]. The valence band photoemission spectra of these C‘e systems often show two 4f derived features [ 1, 21; one in the vicinity of the fermi level, and another roughly at 2.5 eV below the fermi level. Some authors [S, 61 have adopted an approach which relates the 4f derived features to the magnetic properties of Ce. They propose that the peak of the fermi level is associated with the Kondo resonance, while the peak below the fermi level is related to the bare 4f level. Liu and Ho [7], as well as Riseborough [X] have adopted an approach where the 4f peaks are associated with two processes through which the final state 4f hole is screened. The coulomb interaction between the final state 4f hole and the conduction electrons is large enough to be able to bind an electron to the site of the hole. This produces two types of d states, the localized state that is the bound state split off from the bottom of the d band, and the itinerant states. These two types of electronic states lead to two screening channels for the 4f. Recent experiments [9, 101 have shown that the bimodal character of the 4f photoemission spectrum is common to many systems containing the light rare earth elements, Ce, Pr, and Nd. We note that the two 4f peaks are separated by a fixed Science Publishers Division)
in term\
separation
of the presence
channel
with the effect band.
of two type5 of
of the 3f peaks is almost constant
IO fold degenerate
the conduction
the locally screened
1. Introduction
037%4363/85/$03.30 @ Elsevier (North-Holland Physics Publishing
are interpreted ‘The relative
of the two peaks are correlated
series, the 4f level drops below
This has the new effect
(ISA
band by only two or three of the hybridization.
and the effect
as is observed
As one
of the hybridization
i\
in the experiments.
energy difference of roughly 3.5 eV. also as one traverses across the series (‘c to Pr to Nd, the intitnsity of the well screened peak decrcascs across the series. We shall show how the data [‘I. IO] can he described by our approach [9].
2. The model results and discussion We use as our model the magnetic impurity [I I]. In order to take into account the screening of the 4f hole in the final state of photoemission, we use a localized screening interaction [ 121. The model can be written as the sum of three terms
kl = l-2‘,+ fif + c/f<, In this expression. I?,, which governs the itinerant can be written as
(2.1) is the Hamiltonian electrons. This term
(2.2) in which d;,,, and dknrrrespectively create and destroy an electron of spin cr in the band state labeled by the crystal field index n and the Bloch wave vector k. The term fi, represents the 14 fold degenerate 4f level. The wave functions associated with the 4f level are assumed to be localized. The 3f electron Hamiltonian is given by B.V.
P.S. Riseborough
mfr
mm’ IT‘,’
/ 4f photoemission spectra of ?he light rare earths
67
L
(2.3)
where f& and f,,,urespectively create and destroy an electron of spin cr in the localized orbital, with the z component of angular momentum given by m. The first term represents the binding energy of the f electrons and the second term represents the coulomb interaction between pairs of f electrons. The last term in the Hamiltonian Cifd describes the coupling between the localized orbitals and the itinerant states. The coupling is composed of two mechanisms; the hybridization and the f hole-d electron attraction. The coupling term is written as
R, =
c
V”,(k){
multiple scattering approximation fig. lb and modeling the quantity
f +madknrr + diF”:,,frnJ
knm
k.I,,#
ufd + JNdkfn’u
,d
Fig. 1. Processes in which the final state 4f hole interacts with the conduction band electrons. a) The lowest order processes which rearranges the d electrons in the final state, by creating an electron hole pair; b) typical multiple scattering processes that lead to the formation of a bound d state.
indicated
in
knr,{f “,,fmc- (fmcf fmd.
mm7’ (2.4) The first term represents processes in which an electron from the band state kna’ into the state k’n’a’. It is this last term that is responsible for screening the 4f hole. The 4f derived valence band photoemission spectrum can be related to the imaginary part of the 4f electron Greens function.
G(Z)=
lV(k)12
z-E’-~~_~
-’ (kj-Z(z)] .
d
*wy+w2 - w2)p+1’2, 1) (
we find the approximate state formation is
condition
for
bound
(2.5)
In this expression, E(z) represents the electronic self-energy. Processes which introduce a rearrangement of the distribution of the conduction electrons by the 4f hole, are shown in fig. la. Multiple scattering processes, such as those depicted in fig. lb, can lead to the formation of a bound state by splitting off a d state fromhhe bottom of the conduction band [13]. Since the hybridization by itself, may also lead to the formation of a bound state, we expect that V may strongly influence this occurrence. Using the
where
4P) = 22p+3(P + I)
l’(p + 3)r‘(p+ 4) l’(zp + 3) ’
When the f level is close to the bottom of the d band, the hybridization may enhance the effect of the screening interaction Ufd_ For the light rare earths & is not sufficiently strong enough to produce a bound state, if V = 0 (i.e. &/ W < $).
hX
P.S.
Riseborough
/ 4f photoemission spectra of the light me eurlhs
However, we assume that in Ce the hybridization is large enough such that the approximate criterion (3.2) is satisfied. The final state in the Jf photoemission. involves the 4f hole. l’hcre arc two possible screening channels; one which involvcs the filling of the bound state hv a conduc.tion electron, and the continuum port& of the d band is filled up to a fermi level. ‘l‘his is the local screening channel. In the other case, the bound state remains empty and the continuum portion of the d band is filled up to the fermi level. ‘[‘his i\ the diffuse screening channel. These two scrccliing channels lead to the two peaks in the If photoemission spectra. Energy conservation considerations lead to the separation of the two peaks being given by the energy required to transfer an electron between the bound state and the fermi level. Thus the separation must be slightly greater than the width of the occupied portion of the d band. For the light rare earths this energy is approximately 2 eV. These COIV siderations are consistent with the constant separation between the experimentally observed peaks [‘I, IO]. The relative weights of the two peaks arc strongly determined by the magnitude of the hybridization. For systems with strong hybridi/+ tion. such as Ce, the localized d bound state strongly overlaps with the occupied portion of the initial state conduction band, and the Idealized screening peak has large weight. As the hybridization decreases. the strength of the
localized peak decreases and the ditfuse peak dominates the spectrum. ‘l‘hus wc correlate the shift in the relative weights of the two peaks with the decrease of the hybridization matrix element\ across the Innthanide series.
References R.1)
Parks. N. Martensson
\tahilltica. Holland. J.W.
I’.
Wachter
~4mstcrdam.
Allen.
S.J
and 13. Rcihl.
and
H
IYX2).
Oh,
I.
J.M.
cd\.
11,~
!North
p. ?3Y
l.indau.
.lokmsson and S.H. Hiqwom, IIYXI)
\‘alence
Hopparl. J.M.
I.awrenc~.
Phys.
Re\.
1I
lxtt
Jh
IIOO. Lawrence.
Prop. Phys. 44
P.S. Riseborough
and R.11
Parhs. Rep.
( 1YX I ) 1.
J.W. Allen and R.M.
Martin.
Phv\. Rc\.
Lxtt
-tY
I l’)S17r
I IOh
M. Schluter
and (‘.M.
Wachter
and
Qerdam.
19x2).
S.H.
t1.
P.S. Riscborough. R.1).
\‘alencc
~6.
Ho. Php
H ?h I IYX21
KC\.
51 (lYX?)
Wileczka,
C’.Ci. Olson and D.W.
Lxtt.
52 (lYX4)
21x0
IIll I’.W. Anderson, I’hp Kc\ I 7X 11’1 L.M. Falico\ and .I.(‘. Kimball. , I YhY) YY7. Hewson 77
(1077)
7052
M.I_. Den Boer. L .S. (‘hang
Phys. Rev. Ixtt.
D.M.
A.(‘.
I‘ 4m
to he putdished
Parks. S. Raaen.
"lull.
Instahilitlcx.
(North-Holland
p 250.
Liu and K.M.
C;.P. Williams.
V;Irma
I3oppart.
77Y
I
c iY67)
5V)
Phy\.
and P.S. Riseborough.
;tud
3176.
Rev.
Sold
Ixtt.
.‘_T
State t‘om
4f PHOTOEMISSION
SPECTRA
OF THE LIGHT
RARE
EARTHS
P.S. RISEBOROUGH Polytechnic The
Institute
of N. Y., 433 Jay Street.
Rrooklyn.
NY
Il201,
4f photoemission spectra of Ce, Pr, and Nd compounds
screening
channels;
localized
screening
for the Ce, Pr and Nd compounds, conduction moves along reduced.
electrons.
The
the lanthanide
relative
and difiuse
which weights
reflects
screening.
of eliminating
the occupation
of the
The photoemission spectra of pure Ce metal and many Ce compounds have been the subject of much scientific debate [l-9]. The valence band photoemission spectra of these C‘e systems often show two 4f derived features [ 1, 21; one in the vicinity of the fermi level, and another roughly at 2.5 eV below the fermi level. Some authors [S, 61 have adopted an approach which relates the 4f derived features to the magnetic properties of Ce. They propose that the peak of the fermi level is associated with the Kondo resonance, while the peak below the fermi level is related to the bare 4f level. Liu and Ho [7], as well as Riseborough [X] have adopted an approach where the 4f peaks are associated with two processes through which the final state 4f hole is screened. The coulomb interaction between the final state 4f hole and the conduction electrons is large enough to be able to bind an electron to the site of the hole. This produces two types of d states, the localized state that is the bound state split off from the bottom of the d band, and the itinerant states. These two types of electronic states lead to two screening channels for the 4f. Recent experiments [9, 101 have shown that the bimodal character of the 4f photoemission spectrum is common to many systems containing the light rare earth elements, Ce, Pr, and Nd. We note that the two 4f peaks are separated by a fixed Science Publishers Division)
in term\
separation
of the presence
channel
with the effect band.
of two type5 of
of the 3f peaks is almost constant
IO fold degenerate
the conduction
the locally screened
1. Introduction
037%4363/85/$03.30 @ Elsevier (North-Holland Physics Publishing
are interpreted ‘The relative
of the two peaks are correlated
series, the 4f level drops below
This has the new effect
(ISA
band by only two or three of the hybridization.
and the effect
as is observed
As one
of the hybridization
i\
in the experiments.
energy difference of roughly 3.5 eV. also as one traverses across the series (‘c to Pr to Nd, the intitnsity of the well screened peak decrcascs across the series. We shall show how the data [‘I. IO] can he described by our approach [9].
2. The model results and discussion We use as our model the magnetic impurity [I I]. In order to take into account the screening of the 4f hole in the final state of photoemission, we use a localized screening interaction [ 121. The model can be written as the sum of three terms
kl = l-2‘,+ fif + c/f<, In this expression. I?,, which governs the itinerant can be written as
(2.1) is the Hamiltonian electrons. This term
(2.2) in which d;,,, and dknrrrespectively create and destroy an electron of spin cr in the band state labeled by the crystal field index n and the Bloch wave vector k. The term fi, represents the 14 fold degenerate 4f level. The wave functions associated with the 4f level are assumed to be localized. The 3f electron Hamiltonian is given by B.V.
P.S. Riseborough
mfr
mm’ IT‘,’
/ 4f photoemission spectra of ?he light rare earths
67
L
(2.3)
where f& and f,,,urespectively create and destroy an electron of spin cr in the localized orbital, with the z component of angular momentum given by m. The first term represents the binding energy of the f electrons and the second term represents the coulomb interaction between pairs of f electrons. The last term in the Hamiltonian Cifd describes the coupling between the localized orbitals and the itinerant states. The coupling is composed of two mechanisms; the hybridization and the f hole-d electron attraction. The coupling term is written as
R, =
c
V”,(k){
multiple scattering approximation fig. lb and modeling the quantity
f +madknrr + diF”:,,frnJ
knm
k.I,,#
ufd + JNdkfn’u
,d
Fig. 1. Processes in which the final state 4f hole interacts with the conduction band electrons. a) The lowest order processes which rearranges the d electrons in the final state, by creating an electron hole pair; b) typical multiple scattering processes that lead to the formation of a bound d state.
indicated
in
knr,{f “,,fmc- (fmcf fmd.
mm7’ (2.4) The first term represents processes in which an electron from the band state kna’ into the state k’n’a’. It is this last term that is responsible for screening the 4f hole. The 4f derived valence band photoemission spectrum can be related to the imaginary part of the 4f electron Greens function.
G(Z)=
lV(k)12
z-E’-~~_~
-’ (kj-Z(z)] .
d
*wy+w2 - w2)p+1’2, 1) (
we find the approximate state formation is
condition
for
bound
(2.5)
In this expression, E(z) represents the electronic self-energy. Processes which introduce a rearrangement of the distribution of the conduction electrons by the 4f hole, are shown in fig. la. Multiple scattering processes, such as those depicted in fig. lb, can lead to the formation of a bound state by splitting off a d state fromhhe bottom of the conduction band [13]. Since the hybridization by itself, may also lead to the formation of a bound state, we expect that V may strongly influence this occurrence. Using the
where
4P) = 22p+3(P + I)
l’(p + 3)r‘(p+ 4) l’(zp + 3) ’
When the f level is close to the bottom of the d band, the hybridization may enhance the effect of the screening interaction Ufd_ For the light rare earths & is not sufficiently strong enough to produce a bound state, if V = 0 (i.e. &/ W < $).
hX
P.S.
Riseborough
/ 4f photoemission spectra of the light me eurlhs
However, we assume that in Ce the hybridization is large enough such that the approximate criterion (3.2) is satisfied. The final state in the Jf photoemission. involves the 4f hole. l’hcre arc two possible screening channels; one which involvcs the filling of the bound state hv a conduc.tion electron, and the continuum port& of the d band is filled up to a fermi level. ‘l‘his is the local screening channel. In the other case, the bound state remains empty and the continuum portion of the d band is filled up to the fermi level. ‘[‘his i\ the diffuse screening channel. These two scrccliing channels lead to the two peaks in the If photoemission spectra. Energy conservation considerations lead to the separation of the two peaks being given by the energy required to transfer an electron between the bound state and the fermi level. Thus the separation must be slightly greater than the width of the occupied portion of the d band. For the light rare earths this energy is approximately 2 eV. These COIV siderations are consistent with the constant separation between the experimentally observed peaks [‘I, IO]. The relative weights of the two peaks arc strongly determined by the magnitude of the hybridization. For systems with strong hybridi/+ tion. such as Ce, the localized d bound state strongly overlaps with the occupied portion of the initial state conduction band, and the Idealized screening peak has large weight. As the hybridization decreases. the strength of the
localized peak decreases and the ditfuse peak dominates the spectrum. ‘l‘hus wc correlate the shift in the relative weights of the two peaks with the decrease of the hybridization matrix element\ across the Innthanide series.
References R.1)
Parks. N. Martensson
\tahilltica. Holland. J.W.
I’.
Wachter
~4mstcrdam.
Allen.
S.J
and 13. Rcihl.
and
H
IYX2).
Oh,
I.
J.M.
cd\.
11,~
!North
p. ?3Y
l.indau.
.lokmsson and S.H. Hiqwom, IIYXI)
\‘alence
Hopparl. J.M.
I.awrenc~.
Phys.
Re\.
1I
lxtt
Jh
IIOO. Lawrence.
Prop. Phys. 44
P.S. Riseborough
and R.11
Parhs. Rep.
( 1YX I ) 1.
J.W. Allen and R.M.
Martin.
Phv\. Rc\.
Lxtt
-tY
I l’)S17r
I IOh
M. Schluter
and (‘.M.
Wachter
and
Qerdam.
19x2).
S.H.
t1.
P.S. Riscborough. R.1).
\‘alencc
~6.
Ho. Php
H ?h I IYX21
KC\.
51 (lYX?)
Wileczka,
C’.Ci. Olson and D.W.
Lxtt.
52 (lYX4)
21x0
IIll I’.W. Anderson, I’hp Kc\ I 7X 11’1 L.M. Falico\ and .I.(‘. Kimball. , I YhY) YY7. Hewson 77
(1077)
7052
M.I_. Den Boer. L .S. (‘hang
Phys. Rev. Ixtt.
D.M.
A.(‘.
I‘ 4m
to he putdished
Parks. S. Raaen.
"lull.
Instahilitlcx.
(North-Holland
p 250.
Liu and K.M.
C;.P. Williams.
V;Irma
I3oppart.
77Y
I
c iY67)
5V)
Phy\.
and P.S. Riseborough.
;tud
3176.
Rev.
Sold
Ixtt.
.‘_T
State t‘om