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Renormalization of spectral widths by screening interactions
J. Phys. Chew. Sdids Vol. 47, No. 5. pp. 473.476, Printed in Great Britain.
1986
002%3691186 Pergamon
$3.00 + 0.00
Jotmmb Ltd.
RENORMALIZATION OF SPECTRAL WIDTHS BY SCREENING INTERACTIONS PETER S. RISEBOROUGH Department of Physics, Polytechnic Institute of New York, 333 Jay Street, Brooklyn, NY 11201, U.S.A. (Received 21 August 1985; accepted 5 December
1985)
Abstra&--In some of the rare earth compounds the width of the rlfelectron spectral density is dominated by processes which involve the transfer of electrons between the f shell and the conduction band. We examine the effects that screening of the ilfshell’s charge density by the conduction electrons have on these hybridization widths. The screening interaction is treated by low-order perturbation theory. We conclude that the screening of the excess charge on the 4fion may result in a reduction of this hybridization width, in a manner analagous to the enhancement of the effective mass of the small polaron. Keywords: rare earths, 4f spectra1 density, hybridization
widths, screening interaction.
2. THR MODEL SYSTEM
1. ~RODU~ON In many of the lanthanide and actinide compounds, the width of thefelectron spectral density is predominantly produced by hybridization processes [l]. The materials in this catagory have itinerant character introduced into the f states by the overlap between the wavefunctions of the f orbital and the itinerant band states. Among the compounds in this category, one finds the impo~ant classes of mixed valent 121and heavy fermion materials [3]. The widths of the f derived features in the electronic spectral density play an important role in the description of experimental properties [4]. The width of the f spectral density is significantly reduced by many-body screening interactions. The physics behind this reduction is particularly simple. The decay rate of a quasi-particie in the localized f Ievei can be calculated from the matrix elements which express the time evolution from the initial to the final states. In the absence of the screening interaction, the electrons in the screening channels have identical initial and final states, and therefore their overlap matrix elements are unity. The presence of the screening interaction modifies the initial and final states of the conduction electrons, since it gives rise to the appropriately relaxed charge densities. Thus, the screening interaction reduces the overlap between the initial and final states of the electrons in the conduction band screening channels to a factor less than unity. The matrix elements between the total initial and total final states are reduced by this same factor. This gives rise to the reduction of the widths of the f derived features in the spectral density. In this note, we investigate the many-body narrowing outlined above, In Section 2, we present the model system and in Section 3 we outline the calculation of the f spectral density. In Section 4 we shall discuss our results.
We describe the system by the single site Anderson Hamiltonian. The system consists of a localized, highly correlated, f level on an impurity site and the itinerant conduction bands of the host metal. The f level mixes with the itinerant band states through a hybridization interaction. This allows the number of electrons in the f level to change. The conduction band electrons will screen the charge imbalance associated with the state of occupation of the f level. We shall model this by the short-ranged Fahcov-Kimball interaction. The Hamiltonian of the system can be written as the sum of three terms: A=ri,+&.+lii,,.
(I)
The term &r represents the Hamiltonian which govems the f level, fid represents the Hamiltonian of the itinerant band states, and the last term fi,d represents the coupling between the f and d states. The Hamiltonian describing the f level may be written as ri, = 2: .&X+,&k + $r c f +J+~wf~h,s. In* rnSni#
(21
wheref+, and f,, respectively create and destroy an electron of spin u in the state labeled by the orbital angular momentum quantum number m. The first term in eqn (2) represents the binding energy, E,, of a single electron. The second term represents the Coulomb interaction, U,, between pairs of electrons in the f shell. The conduction band states are governed by the H~iltonian fi,,, where
413
Ei, = C &) La
d&&.
(3)
in which d+, and & respectively create and destroy
474
F%m S.
an electron of spin u, in the conduction band state labeled by the Bloch wave vector k. The conduction electrons are assumed to be non-interacting. The f level and the conduction band states are coupled through Ijld, where
k,k’a’
The first two terms in this expression represent the hyb~d~tion. The first term represents a process in which an electron is removed from the conduction band and is transferred into theforbital. The second term represents the reverse process. This interaction conserves the electron spin B. The last term represents the screening of the charge of the f electrons by the conduction electrons. The interaction causes an electron of spin Q to be scattered by the f charge density, from the conduction band state k into the state k’. The interaction is assumed to be short ranged. This Hamiltonian is exactly soluble [5] in the limit V,,,(k) = 0, and it has been shown to be relevant to the description of the photoemission spectra of Ce and other light rare earth materials [6,7-lo]. In the following treatment we shall make the mathematical simplification of considering a non-degenerate forbita and we shall assume the electrons to be spinless. In the next section, we shall calculate the effect that the Coulomb screening interaction U,, has on the widths of the f spectral density.
RSESEBOROIJGH
The
first-order
correction to the self-energy, both frequency-independent and frequency-dependent terms. The impurity breaks the translational invariance of the system, through the hybridization, and thus causes the Hartree self-energy to become f~uency~e~ndent. The frequencyindependent contribution to the Hartree term can be absorbed into a renormalization of the f electron binding energy Ef. To lowest order in V*, the frequency-dependent part of the self-energy Zj”(o) can be written as
Xjn(w), contains
z:‘)(w) =
r, u,,v(k)v*(k’)/[W
-t-
The lowest order contribution to the self-energy, in powers of urd, is given by the expression
This is the zeroth-order term self-energy. We note that the a broadening of thefspectral energy and renormalizes the
6d(pIll*
i
W
v@)v*(k’) [f(Er) -for’)]
-6dw)
+
o-E,(k)
c
kk'
ufd
I
- E,- cdw) l +
W
-+tLd(k)-Edw)
v@)v*or’) _fW&--f(k)
Er- Q(k) 1 W -E,+Ed(k)--t&‘)
1
+
ufd.
-
u,d
1 x
(7) cd(kl)-Ed(k)
in which f(w)is the Fermi-Dirac dist~bution function. This lowest-order correction, due to the screening interaction U,,, is in agreement with the calculations of Schlottman, [ll], in that it suggests that the hybridization matrix elements are renormalized. This can best be seen by evaluating i
I$?(@) = ;IV(k)~/@
c
kk
1
’
3. THE / SPECTRAL DENSITY
We shall assume that the self-energy C&J) can be calculated perturbatively in the Coulomb interaction
11-fWl
f f@‘f-f@ll -Ef+@+6,(k’)--~
’ In this section, we shall calculate the spectral density associated with the f electrons. We shall follow the usual procedure of calculating the spectral density from the one electron Green’s function, and we express the later function in terms of the proper f self-energy I+) defined by Dyson’s equation. Thus we write thefspectral density, I(U), in the form
-cd(k’)]
kk
= Im (X$?)(w) t X~)(o))lo+.
(8)
The quantity (8) is related to the lifetime of a hole in the f level, l/r. On evaluating eqn (8) at zero temwe recover the second-order Bornperature, approximation for the decay rate:
(6)
in the expansion of the hybridization produces density, shifts the peak total f spectra1 weight.
in which p(w) is the itinerant band density of states.
Renormalization of spectral widths by screening interactions The effect of the Coulomb screening interaction U,d is to introduce the second term in the parenthesis. Thus one can conclude that the effective hybridization matrix elements are enhanced. This does not imply that the hyb~dization matrix elements are more etIective in broadening the f spectral density, since the widths also depend on the properties of the real part of the self-energy. The Coulomb interaction produces a second-order contribution to the self-energy of Pi(W) I
= - W$
dw”p(w’)p(o”)
This actually reduces be simply understood eqn (9). We interpret the overlap between
475
the width at E,, This result can in a manner similar to that of this as being the width due to the bare f wavefunction
and the wave-function corresponding to an extra conduction electron and an f hole. The effect of the Coulomb interaction is evaluated by perturbation theory, we find the particle could be scattered or electron hole pairs could be excited from the Fermisea.
+ I1-f~~~~lfw# -fW)l This wave-function the value
w--EI-+o’-w”
This does not contribute to the imaginary part of the self-energy at Er, since the imaginary contribution is finite only in the region E,>w
= 1 -i-cIr’$J-pfbJ
> Er-2W.
By contrast, the real part is fmite and rapidly varying in the vicinity of EP The derivative of the self-energy is do”
cklkr
is not normalized to unity but to
PW)P(4f(~‘){1 -fw’H~ (E,+ cu’-a “)*
The normalized
hyb~di~tion
has matrix elements
(I1) (14)
This term strongly renormalizes the spectral weight in the vicinity of Ef, and reduces the width of the spectral density near E,, as can be seen by rewriting
The similarity between eqns (13) and (14) is immediately obvious, except for the second term in the denominator of eqn (14). This difference is not significant, since in the Fermi golden rule expression derived from eqn (14) one would evaluate the matrix elements near fL.N Ep Under these conditions, the second term is negligible compared to the third. Thus the correspondent between eqns (13) and (14) is actually quite close.
Im[w-E,-ReZ/(o)-iImZ,(o)]-’ as 1
4. RESULTS AND DISCUSSION This suggests that the hybridization nated by
~[
w-i
VOf2: k
1
width is domi-
Ufd(1-A 1 %‘-%i S(&-E,).
UjiAJl -A) l +cLY (es-e k)* 1
4
(13)
We have examined the effect that the screening interaction U,, has on the hybridization width of a hole in the f level. We find that the hyb~d~tion matrix elements can be thought of as being enhanced by U,,. This enhancement becomes quite pronounced at E,, especially when the f level lies close to the Fermi level. Nevertheless, the hybridization contribution to the width of the spectral density at Ef is reduced
416
PETERS. RISEBOROUGH
by the Coulomb correlations. This is due to the renormalization of the spectral density, by the creation of many electron-hole pairs in the conduction band. The effect is similar to that of orthogonality catastrophe in the X-ray or Kondo problems. A similar reduction of the width has been previously found [lo] in the strong screening limit, where the ratio U,,jW is large. Since the form of the spectral density found in ref. [lo] is quite different from that of weak Coulomb correlation limit, there is no immediate relation between the resulting reduction of the hybridization widths. The polaronic type of narrowing of the hybridization widths which we find, is seen to be most important in the integral valence regime p >>E, and less important in the mixed valence regime. We do not find any region in which the Coulomb interaction enhances the hybridization widths to the extent of becoming equal to the width of the underlying conduction band, as has been previously suggested [ 121.
Acknowledgements-This work was supported by the United States Department of Energy under the auspices of grant No. DE-FGO2-84 ER 45127. REFERENCES 1. Smith J. L. and Kmetko E. A., .I. Less-Common Metals 90, 83 (1983). 2. Lawrence J. M., Riseborough P. S. and Parks R. D., Rep. Prog. Phys. 44, 1 (1981). 3. Stewart G. R., Rev. Mod. Phys. 56, 755 (19&Q). 4. Newns D. M. and Hewson A. C., J. Phys. F 10, 2429 (1980). 5. Hewson A. C. and Riseborough P. S., Solid State Commun. U, 379 (1977). 6. Parks R. D., Raaen S., den Boer M. L., Chang Y-S. and Williams G. P., Phys. Reu. Lett. 52, 2176 (1984). 7. Wieliozka D. M., Olson C. G. and Lynch D. W., Phys. Reu. Lett. 52, 2180 (1984). 8. Liu S. H. and Ho K. M., Phys. Rev. B 28,422O (1983). 9. Riseborough P. S., PhySica B 130, 66 (1984). IO. Liu S. H. and Ho K. M.. Phvs. Rev. B 30,3039 (1984). 11. Schlottmann P., Phys. I&. *E 22, 613 (1980). 12. Khomskii D., Quantum Theory of Solids (Edited by I. M. Lifshits). M.I.R., Moscow (1982).
1986
002%3691186 Pergamon
$3.00 + 0.00
Jotmmb Ltd.
RENORMALIZATION OF SPECTRAL WIDTHS BY SCREENING INTERACTIONS PETER S. RISEBOROUGH Department of Physics, Polytechnic Institute of New York, 333 Jay Street, Brooklyn, NY 11201, U.S.A. (Received 21 August 1985; accepted 5 December
1985)
Abstra&--In some of the rare earth compounds the width of the rlfelectron spectral density is dominated by processes which involve the transfer of electrons between the f shell and the conduction band. We examine the effects that screening of the ilfshell’s charge density by the conduction electrons have on these hybridization widths. The screening interaction is treated by low-order perturbation theory. We conclude that the screening of the excess charge on the 4fion may result in a reduction of this hybridization width, in a manner analagous to the enhancement of the effective mass of the small polaron. Keywords: rare earths, 4f spectra1 density, hybridization
widths, screening interaction.
2. THR MODEL SYSTEM
1. ~RODU~ON In many of the lanthanide and actinide compounds, the width of thefelectron spectral density is predominantly produced by hybridization processes [l]. The materials in this catagory have itinerant character introduced into the f states by the overlap between the wavefunctions of the f orbital and the itinerant band states. Among the compounds in this category, one finds the impo~ant classes of mixed valent 121and heavy fermion materials [3]. The widths of the f derived features in the electronic spectral density play an important role in the description of experimental properties [4]. The width of the f spectral density is significantly reduced by many-body screening interactions. The physics behind this reduction is particularly simple. The decay rate of a quasi-particie in the localized f Ievei can be calculated from the matrix elements which express the time evolution from the initial to the final states. In the absence of the screening interaction, the electrons in the screening channels have identical initial and final states, and therefore their overlap matrix elements are unity. The presence of the screening interaction modifies the initial and final states of the conduction electrons, since it gives rise to the appropriately relaxed charge densities. Thus, the screening interaction reduces the overlap between the initial and final states of the electrons in the conduction band screening channels to a factor less than unity. The matrix elements between the total initial and total final states are reduced by this same factor. This gives rise to the reduction of the widths of the f derived features in the spectral density. In this note, we investigate the many-body narrowing outlined above, In Section 2, we present the model system and in Section 3 we outline the calculation of the f spectral density. In Section 4 we shall discuss our results.
We describe the system by the single site Anderson Hamiltonian. The system consists of a localized, highly correlated, f level on an impurity site and the itinerant conduction bands of the host metal. The f level mixes with the itinerant band states through a hybridization interaction. This allows the number of electrons in the f level to change. The conduction band electrons will screen the charge imbalance associated with the state of occupation of the f level. We shall model this by the short-ranged Fahcov-Kimball interaction. The Hamiltonian of the system can be written as the sum of three terms: A=ri,+&.+lii,,.
(I)
The term &r represents the Hamiltonian which govems the f level, fid represents the Hamiltonian of the itinerant band states, and the last term fi,d represents the coupling between the f and d states. The Hamiltonian describing the f level may be written as ri, = 2: .&X+,&k + $r c f +J+~wf~h,s. In* rnSni#
(21
wheref+, and f,, respectively create and destroy an electron of spin u in the state labeled by the orbital angular momentum quantum number m. The first term in eqn (2) represents the binding energy, E,, of a single electron. The second term represents the Coulomb interaction, U,, between pairs of electrons in the f shell. The conduction band states are governed by the H~iltonian fi,,, where
413
Ei, = C &) La
d&&.
(3)
in which d+, and & respectively create and destroy
474
F%m S.
an electron of spin u, in the conduction band state labeled by the Bloch wave vector k. The conduction electrons are assumed to be non-interacting. The f level and the conduction band states are coupled through Ijld, where
k,k’a’
The first two terms in this expression represent the hyb~d~tion. The first term represents a process in which an electron is removed from the conduction band and is transferred into theforbital. The second term represents the reverse process. This interaction conserves the electron spin B. The last term represents the screening of the charge of the f electrons by the conduction electrons. The interaction causes an electron of spin Q to be scattered by the f charge density, from the conduction band state k into the state k’. The interaction is assumed to be short ranged. This Hamiltonian is exactly soluble [5] in the limit V,,,(k) = 0, and it has been shown to be relevant to the description of the photoemission spectra of Ce and other light rare earth materials [6,7-lo]. In the following treatment we shall make the mathematical simplification of considering a non-degenerate forbita and we shall assume the electrons to be spinless. In the next section, we shall calculate the effect that the Coulomb screening interaction U,, has on the widths of the f spectral density.
RSESEBOROIJGH
The
first-order
correction to the self-energy, both frequency-independent and frequency-dependent terms. The impurity breaks the translational invariance of the system, through the hybridization, and thus causes the Hartree self-energy to become f~uency~e~ndent. The frequencyindependent contribution to the Hartree term can be absorbed into a renormalization of the f electron binding energy Ef. To lowest order in V*, the frequency-dependent part of the self-energy Zj”(o) can be written as
Xjn(w), contains
z:‘)(w) =
r, u,,v(k)v*(k’)/[W
-t-
The lowest order contribution to the self-energy, in powers of urd, is given by the expression
This is the zeroth-order term self-energy. We note that the a broadening of thefspectral energy and renormalizes the
6d(pIll*
i
W
v@)v*(k’) [f(Er) -for’)]
-6dw)
+
o-E,(k)
c
kk'
ufd
I
- E,- cdw) l +
W
-+tLd(k)-Edw)
v@)v*or’) _fW&--f(k)
Er- Q(k) 1 W -E,+Ed(k)--t&‘)
1
+
ufd.
-
u,d
1 x
(7) cd(kl)-Ed(k)
in which f(w)is the Fermi-Dirac dist~bution function. This lowest-order correction, due to the screening interaction U,,, is in agreement with the calculations of Schlottman, [ll], in that it suggests that the hybridization matrix elements are renormalized. This can best be seen by evaluating i
I$?(@) = ;IV(k)~/@
c
kk
1
’
3. THE / SPECTRAL DENSITY
We shall assume that the self-energy C&J) can be calculated perturbatively in the Coulomb interaction
11-fWl
f f@‘f-f@ll -Ef+@+6,(k’)--~
’ In this section, we shall calculate the spectral density associated with the f electrons. We shall follow the usual procedure of calculating the spectral density from the one electron Green’s function, and we express the later function in terms of the proper f self-energy I+) defined by Dyson’s equation. Thus we write thefspectral density, I(U), in the form
-cd(k’)]
kk
= Im (X$?)(w) t X~)(o))lo+.
(8)
The quantity (8) is related to the lifetime of a hole in the f level, l/r. On evaluating eqn (8) at zero temwe recover the second-order Bornperature, approximation for the decay rate:
(6)
in the expansion of the hybridization produces density, shifts the peak total f spectra1 weight.
in which p(w) is the itinerant band density of states.
Renormalization of spectral widths by screening interactions The effect of the Coulomb screening interaction U,d is to introduce the second term in the parenthesis. Thus one can conclude that the effective hybridization matrix elements are enhanced. This does not imply that the hyb~dization matrix elements are more etIective in broadening the f spectral density, since the widths also depend on the properties of the real part of the self-energy. The Coulomb interaction produces a second-order contribution to the self-energy of Pi(W) I
= - W$
dw”p(w’)p(o”)
This actually reduces be simply understood eqn (9). We interpret the overlap between
475
the width at E,, This result can in a manner similar to that of this as being the width due to the bare f wavefunction
and the wave-function corresponding to an extra conduction electron and an f hole. The effect of the Coulomb interaction is evaluated by perturbation theory, we find the particle could be scattered or electron hole pairs could be excited from the Fermisea.
+ I1-f~~~~lfw# -fW)l This wave-function the value
w--EI-+o’-w”
This does not contribute to the imaginary part of the self-energy at Er, since the imaginary contribution is finite only in the region E,>w
= 1 -i-cIr’$J-pfbJ
> Er-2W.
By contrast, the real part is fmite and rapidly varying in the vicinity of EP The derivative of the self-energy is do”
cklkr
is not normalized to unity but to
PW)P(4f(~‘){1 -fw’H~ (E,+ cu’-a “)*
The normalized
hyb~di~tion
has matrix elements
(I1) (14)
This term strongly renormalizes the spectral weight in the vicinity of Ef, and reduces the width of the spectral density near E,, as can be seen by rewriting
The similarity between eqns (13) and (14) is immediately obvious, except for the second term in the denominator of eqn (14). This difference is not significant, since in the Fermi golden rule expression derived from eqn (14) one would evaluate the matrix elements near fL.N Ep Under these conditions, the second term is negligible compared to the third. Thus the correspondent between eqns (13) and (14) is actually quite close.
Im[w-E,-ReZ/(o)-iImZ,(o)]-’ as 1
4. RESULTS AND DISCUSSION This suggests that the hybridization nated by
~[
w-i
VOf2: k
1
width is domi-
Ufd(1-A 1 %‘-%i S(&-E,).
UjiAJl -A) l +cLY (es-e k)* 1
4
(13)
We have examined the effect that the screening interaction U,, has on the hybridization width of a hole in the f level. We find that the hyb~d~tion matrix elements can be thought of as being enhanced by U,,. This enhancement becomes quite pronounced at E,, especially when the f level lies close to the Fermi level. Nevertheless, the hybridization contribution to the width of the spectral density at Ef is reduced
416
PETERS. RISEBOROUGH
by the Coulomb correlations. This is due to the renormalization of the spectral density, by the creation of many electron-hole pairs in the conduction band. The effect is similar to that of orthogonality catastrophe in the X-ray or Kondo problems. A similar reduction of the width has been previously found [lo] in the strong screening limit, where the ratio U,,jW is large. Since the form of the spectral density found in ref. [lo] is quite different from that of weak Coulomb correlation limit, there is no immediate relation between the resulting reduction of the hybridization widths. The polaronic type of narrowing of the hybridization widths which we find, is seen to be most important in the integral valence regime p >>E, and less important in the mixed valence regime. We do not find any region in which the Coulomb interaction enhances the hybridization widths to the extent of becoming equal to the width of the underlying conduction band, as has been previously suggested [ 121.
Acknowledgements-This work was supported by the United States Department of Energy under the auspices of grant No. DE-FGO2-84 ER 45127. REFERENCES 1. Smith J. L. and Kmetko E. A., .I. Less-Common Metals 90, 83 (1983). 2. Lawrence J. M., Riseborough P. S. and Parks R. D., Rep. Prog. Phys. 44, 1 (1981). 3. Stewart G. R., Rev. Mod. Phys. 56, 755 (19&Q). 4. Newns D. M. and Hewson A. C., J. Phys. F 10, 2429 (1980). 5. Hewson A. C. and Riseborough P. S., Solid State Commun. U, 379 (1977). 6. Parks R. D., Raaen S., den Boer M. L., Chang Y-S. and Williams G. P., Phys. Reu. Lett. 52, 2176 (1984). 7. Wieliozka D. M., Olson C. G. and Lynch D. W., Phys. Reu. Lett. 52, 2180 (1984). 8. Liu S. H. and Ho K. M., Phys. Rev. B 28,422O (1983). 9. Riseborough P. S., PhySica B 130, 66 (1984). IO. Liu S. H. and Ho K. M.. Phvs. Rev. B 30,3039 (1984). 11. Schlottmann P., Phys. I&. *E 22, 613 (1980). 12. Khomskii D., Quantum Theory of Solids (Edited by I. M. Lifshits). M.I.R., Moscow (1982).