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Effect of electronic correlations on thequasiparticle dispersion of USb2

To cite this article Xiaodong Yang et al 2010 J Phys Conf Ser 200 012164

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Effect of electronic correlations on the quasiparticle

dispersion of USb2

Xiaodong Yang1 Peter S Riseborough1 Tomasz Durakiewicz2 PMOppeneer3 S Elgazzar31Physics Dept Temple University Philadelphia PA 19122 USA2Los Alamos National Labs Los Alamos New Mexico 87545 USA3Department of Physics Uppsala University Box 530 Sweden

E-mail prisebortempleedu

Abstract Angle resolved photoemission experiments have been performed on USb2 and verynarrow quasiparticle peaks have been observed in a band which LSDA predicts to osculate theFermi-energy The observed band is found to be depressed by 17 meV below the Fermi-energyfurthermore the inferred quasiparticle dispersion relation for this band exhibits a kink at anenergy of about 23 meV below the Fermi-energy The kink is not found in LSDA calculationsand therefore is attributable to a change in the quasiparticle mass renormalization by a factorof approximately 2 The existence of a kink in the quasiparticle dispersion relation of a bandwhich does not cross the Fermi-energy is unprecedented The origin of the observed depressionof the band its quasi-particle mass enhancement and the characteristic energy are discussedon the basis of a theoretical model

1 IntroductionThe dispersion relations of quasiparticles that are subjected to large mass enhancements areexpected to exhibit kinks in the vicinity of the Fermi energy These kinks occur at a characteristicenergy below which the quasi-particles are renormalized by the electronic correlations and abovewhich they are unrenormalized The kinks in the dispersion relations were first observed byARPES experiments [1] almost four decades after they were first predicted [2] Since enhancedquasi-particles are usually described Landau Fermi-liquid theory the kinks are expected to beconfined to the vicinity of the Fermi surface Hence the experimental discovery of kinks inUSb2 in a band that does not cross the surface is unexpected [3] In this paper we presentthe experimental results and explain theoretically the origin of the observed quasiparticle massenhancements

2 Experimental ResultsUSb2 is a tetragonal material that orders antiferromagnetically with a relatively low Neeltemperature of 209K and a relatively large ordered magnetic moments of 188 microB [4 5] Thematerial has a quasi-two-dimensional character presumably due to the presence of uraniumplanes in the structure The high-quality flux grown single crystals used in the experiment werecleaved in situ under ultra-high vacuum conditions Our measurements have determined thatthe bands show minimal dispersion along the c-direction [3] thereby attesting to the quasi-two-dimensional nature of the bands We have observed that there is a large discrepancy between

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

ccopy 2010 IOP Publishing Ltd 1

the observed low-energy bands near the Γ point and the LSDA bands The LSDA calculationsshow a 5f band that osculates the Fermi-energy at the Γ point and disperses quadratically askx is increased The experimentally determined valence band spectra are shown in fig(1) fork values along the Γ-X direction The data were taken with a photon energy of 34 eV Itis seen that at the Γ point the observed quasi-particle band is depressed to about 17 meVbelow the Fermi energy in contrast to the predictions of LSDA The red markers indicate thequasi-particle energy and detailed analysis shows that the curvature of the dispersion relationincreases by a factor of 2 at kxa asymp 018π indicating a change in the quasiparticle mass Close tothe Γ point the intrinsic peak width is estimated to be about 3 meV however the peak widthincreases abruptly for kxa gt 018π The abrupt change in the curvature and the rapid change inthe width of the peak signifies a change in regime from highly renormalized quasi-particles forexcitation energies of hω lt 21meV to a regime of modest renormalizations for hω gt 21meV

X

Figure 1 The electronic spectral density forUSb2 measured along the Γ-X direction

2

3

4

hωD

A(k

ω)

000000250050007501000125015001750 200

0

1

-2 -15 -1 -05 0 05 1h

ωωD

0200

Figure 2 The calculated dimensionless electronicspectral density for various values of kaπ Thespectral density is plotted in units of ωD againstthe dimensionless frequency ωωD where ωD is theDebye frequency

3 TheoryThe electronic structure is modeled by two bands taken from the LSDA One band osculatesthe Fermi energy at the Γ point and the other has a maximum at the Γ point of about 110 meVand cuts the Fermi energy at the wave vector kF a asymp 055π

The spectral density was obtained by treating the electron-phonon interaction as aperturbation via Greenrsquos function techniques The unperturbed single-electron Greenrsquos functionsare denoted by Gστ

0 (k ω) and are given by

Gστ0 (k ω) =

δστ

hω + micro minus εσk + i ηk

(1)

where σ and τ are band indices The Greenrsquos functions Gστ (k ω) for the interacting electronsare given in terms of the bare Greenrsquos function and the self-energies Σσρ(k ω) via the matrix

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

2

Dysonrsquos equations

Gστ (k ω) = Gστ0 (k ω) +

sumρλ

Gσρ0 (k ω) Σρλ(k ω) Gλτ (k ω) (2)

Using the structure of the unperturbed Greenrsquos functions Dysonrsquos equations can be re-writtenas a pair of matrix equations The matrix of electronic self-energies is calculated from thephonon-emission and absorption processes [2] and is given by

Σσρ(k ω) =1N

sumτqα

λστqα λτρ

qαlowast

[ 1minus f τkminusq + Nq

hω + microminus ετkminusq minus hωqα

+f τ

kminusq + Nq

hω + microminus ετkminusq + hωqα

](3)

where λqα are the electron-phonon coupling constants The vertex corrections have beenneglected in accordance with Migdalrsquos theorem [6] For the intraband scattering processesthe electron-phonon coupling constant will be assumed to be independent of the band indices σand τ

The self-energy which is off-diagonal in the band indices is quite unremarkable but does havethe effect of repelling the two branches of the electronic dispersion relation and contributes tothe depression of the observed band below the Fermi energy The intra-band processes for theosculating band does has an imaginary part which gradually falls to zero as the Fermi energyis approached From a Kramers-Kronig analysis one finds that this has the consequence thatthe real part of the intra-band self-energy has a maximum near this energy and therefore doesnot result in a renormalization of the quasi-particle mass On the other hand the inter-bandself-energy describes the scattering of an electron involving a momentum change of qa asymp 054πwith either the emission or absorption of a virtual phonon This produces a narrow gap in theimaginary part of the inter-band self-energy of the order of 2 hωq where the imaginary part fallsrapidly to zero The abrupt drop of the imaginary part of the inter-band self-energy results ina rapidly varying real part which produces the quasi-particle mass enhancement The spectraldensity calculated from the imaginary parts of the Greenrsquos function is shown in fig(2) It isseen that in addition to the quasi-particle peak the spectra shows the presence of an incoherentstructure at energies near ωD2

4 DiscussionFor small values of k the calculated spectral density shows a narrow quasiparticle widthwhich disperses quadratically however the inferred effective mass is increased by a factor ofapproximately two over the band mass For larger values of the momentum the width ofthe quasi-particle peak rapidly broadens and the peak disperses quadratically but at a ratedetermined by the unrenormalized band mass as shown in fig(3) It is seen that the cross-overfrom the renormalized quasiparticle peak to the unrenormalized peak occurs at an energy givenby approximately qaπ ωD in contrast to the standard description of electron-phonon massenhancements where the cross-over occurs at ωD This difference occurs since the interbandscattering process is responsible for the quasi-particle mass srenormalization and this involveslow-energy excitations of electrons between the bands near which require a finite momentumtransfer This finding is in accordance with the experimental results since the observed cross-over occurs at an energy which is significantly lower than the reported Debye frequency [5]

41 AcknowledgmentsThe work at Temple University was supported by the US Department of Energy Office of BasicEnergy Sciences through grant FG02-01E=R45872 The work at LANL was performed underthe auspices of the US DOE and by LANL through the LDRD program The SRC is supported

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

3

0

04

08

-02 -01 0 01 02(kaπ)

-16

-12

-08

-04

ωω

D

Figure 3 The dispersion for the various features in the calculated electronic spectrum (solidlines) The dashed redline represents the rigidly shifted band dispersion relation found from theLSDA calculations and the dashed blue line represents the shifted dispersion relation with therenormalized quasiparticle mass

by the NSF The work was also supported by the Swedish Research Council (VR) SNIC andthe European Commission (JRC-ITU) The authors of this work would like to commemoratethis paper to the memory of Cliff Olson

References[1] Lanzara A Bogdanov PV Zhou XJ Kellar SA Feng DL Lu ED Yoshida T Eisaki H Fujimori A

Kishio K Shimoyama JI Noda T Uchida S Hussain Z and Shen Z-X (2001) Nature 412 510[2] Engelsberg S and Schrieffer JR (1963) Phys Rev 131 993[3] Durakiewicz T Riseborough PS Olson CG Joyce JJ Bauer ED Sarrao JL Elgazzar S Oppeneer

PM Guziewicz E Moore DP Butterfield MT and Graham KS (2008) Europhysics Letters 84 37003[4] Leciejewicz J Troc R Murasik A and Zygmunt A (1967) Phys Stat Sol 22 517[5] Wawryk R (2006) Phil Mag 86 1775[6] Migdal AB (1957) Sov Phys JETP 5 333

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

4

the observed low-energy bands near the Γ point and the LSDA bands The LSDA calculationsshow a 5f band that osculates the Fermi-energy at the Γ point and disperses quadratically askx is increased The experimentally determined valence band spectra are shown in fig(1) fork values along the Γ-X direction The data were taken with a photon energy of 34 eV Itis seen that at the Γ point the observed quasi-particle band is depressed to about 17 meVbelow the Fermi energy in contrast to the predictions of LSDA The red markers indicate thequasi-particle energy and detailed analysis shows that the curvature of the dispersion relationincreases by a factor of 2 at kxa asymp 018π indicating a change in the quasiparticle mass Close tothe Γ point the intrinsic peak width is estimated to be about 3 meV however the peak widthincreases abruptly for kxa gt 018π The abrupt change in the curvature and the rapid change inthe width of the peak signifies a change in regime from highly renormalized quasi-particles forexcitation energies of hω lt 21meV to a regime of modest renormalizations for hω gt 21meV

X

Figure 1 The electronic spectral density forUSb2 measured along the Γ-X direction

2

3

4

hωD

A(k

ω)

000000250050007501000125015001750 200

0

1

-2 -15 -1 -05 0 05 1h

ωωD

0200

Figure 2 The calculated dimensionless electronicspectral density for various values of kaπ Thespectral density is plotted in units of ωD againstthe dimensionless frequency ωωD where ωD is theDebye frequency

3 TheoryThe electronic structure is modeled by two bands taken from the LSDA One band osculatesthe Fermi energy at the Γ point and the other has a maximum at the Γ point of about 110 meVand cuts the Fermi energy at the wave vector kF a asymp 055π

The spectral density was obtained by treating the electron-phonon interaction as aperturbation via Greenrsquos function techniques The unperturbed single-electron Greenrsquos functionsare denoted by Gστ

0 (k ω) and are given by

Gστ0 (k ω) =

δστ

hω + micro minus εσk + i ηk

(1)

where σ and τ are band indices The Greenrsquos functions Gστ (k ω) for the interacting electronsare given in terms of the bare Greenrsquos function and the self-energies Σσρ(k ω) via the matrix

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

2

Dysonrsquos equations

Gστ (k ω) = Gστ0 (k ω) +

sumρλ

Gσρ0 (k ω) Σρλ(k ω) Gλτ (k ω) (2)

Using the structure of the unperturbed Greenrsquos functions Dysonrsquos equations can be re-writtenas a pair of matrix equations The matrix of electronic self-energies is calculated from thephonon-emission and absorption processes [2] and is given by

Σσρ(k ω) =1N

sumτqα

λστqα λτρ

qαlowast

[ 1minus f τkminusq + Nq

hω + microminus ετkminusq minus hωqα

+f τ

kminusq + Nq

hω + microminus ετkminusq + hωqα

](3)

where λqα are the electron-phonon coupling constants The vertex corrections have beenneglected in accordance with Migdalrsquos theorem [6] For the intraband scattering processesthe electron-phonon coupling constant will be assumed to be independent of the band indices σand τ

The self-energy which is off-diagonal in the band indices is quite unremarkable but does havethe effect of repelling the two branches of the electronic dispersion relation and contributes tothe depression of the observed band below the Fermi energy The intra-band processes for theosculating band does has an imaginary part which gradually falls to zero as the Fermi energyis approached From a Kramers-Kronig analysis one finds that this has the consequence thatthe real part of the intra-band self-energy has a maximum near this energy and therefore doesnot result in a renormalization of the quasi-particle mass On the other hand the inter-bandself-energy describes the scattering of an electron involving a momentum change of qa asymp 054πwith either the emission or absorption of a virtual phonon This produces a narrow gap in theimaginary part of the inter-band self-energy of the order of 2 hωq where the imaginary part fallsrapidly to zero The abrupt drop of the imaginary part of the inter-band self-energy results ina rapidly varying real part which produces the quasi-particle mass enhancement The spectraldensity calculated from the imaginary parts of the Greenrsquos function is shown in fig(2) It isseen that in addition to the quasi-particle peak the spectra shows the presence of an incoherentstructure at energies near ωD2

4 DiscussionFor small values of k the calculated spectral density shows a narrow quasiparticle widthwhich disperses quadratically however the inferred effective mass is increased by a factor ofapproximately two over the band mass For larger values of the momentum the width ofthe quasi-particle peak rapidly broadens and the peak disperses quadratically but at a ratedetermined by the unrenormalized band mass as shown in fig(3) It is seen that the cross-overfrom the renormalized quasiparticle peak to the unrenormalized peak occurs at an energy givenby approximately qaπ ωD in contrast to the standard description of electron-phonon massenhancements where the cross-over occurs at ωD This difference occurs since the interbandscattering process is responsible for the quasi-particle mass srenormalization and this involveslow-energy excitations of electrons between the bands near which require a finite momentumtransfer This finding is in accordance with the experimental results since the observed cross-over occurs at an energy which is significantly lower than the reported Debye frequency [5]

41 AcknowledgmentsThe work at Temple University was supported by the US Department of Energy Office of BasicEnergy Sciences through grant FG02-01E=R45872 The work at LANL was performed underthe auspices of the US DOE and by LANL through the LDRD program The SRC is supported

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

3

0

04

08

-02 -01 0 01 02(kaπ)

-16

-12

-08

-04

ωω

D

Figure 3 The dispersion for the various features in the calculated electronic spectrum (solidlines) The dashed redline represents the rigidly shifted band dispersion relation found from theLSDA calculations and the dashed blue line represents the shifted dispersion relation with therenormalized quasiparticle mass

by the NSF The work was also supported by the Swedish Research Council (VR) SNIC andthe European Commission (JRC-ITU) The authors of this work would like to commemoratethis paper to the memory of Cliff Olson

References[1] Lanzara A Bogdanov PV Zhou XJ Kellar SA Feng DL Lu ED Yoshida T Eisaki H Fujimori A

Kishio K Shimoyama JI Noda T Uchida S Hussain Z and Shen Z-X (2001) Nature 412 510[2] Engelsberg S and Schrieffer JR (1963) Phys Rev 131 993[3] Durakiewicz T Riseborough PS Olson CG Joyce JJ Bauer ED Sarrao JL Elgazzar S Oppeneer

PM Guziewicz E Moore DP Butterfield MT and Graham KS (2008) Europhysics Letters 84 37003[4] Leciejewicz J Troc R Murasik A and Zygmunt A (1967) Phys Stat Sol 22 517[5] Wawryk R (2006) Phil Mag 86 1775[6] Migdal AB (1957) Sov Phys JETP 5 333

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

4

Dysonrsquos equations

Gστ (k ω) = Gστ0 (k ω) +

sumρλ

Gσρ0 (k ω) Σρλ(k ω) Gλτ (k ω) (2)

Using the structure of the unperturbed Greenrsquos functions Dysonrsquos equations can be re-writtenas a pair of matrix equations The matrix of electronic self-energies is calculated from thephonon-emission and absorption processes [2] and is given by

Σσρ(k ω) =1N

sumτqα

λστqα λτρ

qαlowast

[ 1minus f τkminusq + Nq

hω + microminus ετkminusq minus hωqα

+f τ

kminusq + Nq

hω + microminus ετkminusq + hωqα

](3)

where λqα are the electron-phonon coupling constants The vertex corrections have beenneglected in accordance with Migdalrsquos theorem [6] For the intraband scattering processesthe electron-phonon coupling constant will be assumed to be independent of the band indices σand τ

The self-energy which is off-diagonal in the band indices is quite unremarkable but does havethe effect of repelling the two branches of the electronic dispersion relation and contributes tothe depression of the observed band below the Fermi energy The intra-band processes for theosculating band does has an imaginary part which gradually falls to zero as the Fermi energyis approached From a Kramers-Kronig analysis one finds that this has the consequence thatthe real part of the intra-band self-energy has a maximum near this energy and therefore doesnot result in a renormalization of the quasi-particle mass On the other hand the inter-bandself-energy describes the scattering of an electron involving a momentum change of qa asymp 054πwith either the emission or absorption of a virtual phonon This produces a narrow gap in theimaginary part of the inter-band self-energy of the order of 2 hωq where the imaginary part fallsrapidly to zero The abrupt drop of the imaginary part of the inter-band self-energy results ina rapidly varying real part which produces the quasi-particle mass enhancement The spectraldensity calculated from the imaginary parts of the Greenrsquos function is shown in fig(2) It isseen that in addition to the quasi-particle peak the spectra shows the presence of an incoherentstructure at energies near ωD2

4 DiscussionFor small values of k the calculated spectral density shows a narrow quasiparticle widthwhich disperses quadratically however the inferred effective mass is increased by a factor ofapproximately two over the band mass For larger values of the momentum the width ofthe quasi-particle peak rapidly broadens and the peak disperses quadratically but at a ratedetermined by the unrenormalized band mass as shown in fig(3) It is seen that the cross-overfrom the renormalized quasiparticle peak to the unrenormalized peak occurs at an energy givenby approximately qaπ ωD in contrast to the standard description of electron-phonon massenhancements where the cross-over occurs at ωD This difference occurs since the interbandscattering process is responsible for the quasi-particle mass srenormalization and this involveslow-energy excitations of electrons between the bands near which require a finite momentumtransfer This finding is in accordance with the experimental results since the observed cross-over occurs at an energy which is significantly lower than the reported Debye frequency [5]

41 AcknowledgmentsThe work at Temple University was supported by the US Department of Energy Office of BasicEnergy Sciences through grant FG02-01E=R45872 The work at LANL was performed underthe auspices of the US DOE and by LANL through the LDRD program The SRC is supported

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

3

0

04

08

-02 -01 0 01 02(kaπ)

-16

-12

-08

-04

ωω

D

Figure 3 The dispersion for the various features in the calculated electronic spectrum (solidlines) The dashed redline represents the rigidly shifted band dispersion relation found from theLSDA calculations and the dashed blue line represents the shifted dispersion relation with therenormalized quasiparticle mass

by the NSF The work was also supported by the Swedish Research Council (VR) SNIC andthe European Commission (JRC-ITU) The authors of this work would like to commemoratethis paper to the memory of Cliff Olson

References[1] Lanzara A Bogdanov PV Zhou XJ Kellar SA Feng DL Lu ED Yoshida T Eisaki H Fujimori A

Kishio K Shimoyama JI Noda T Uchida S Hussain Z and Shen Z-X (2001) Nature 412 510[2] Engelsberg S and Schrieffer JR (1963) Phys Rev 131 993[3] Durakiewicz T Riseborough PS Olson CG Joyce JJ Bauer ED Sarrao JL Elgazzar S Oppeneer

PM Guziewicz E Moore DP Butterfield MT and Graham KS (2008) Europhysics Letters 84 37003[4] Leciejewicz J Troc R Murasik A and Zygmunt A (1967) Phys Stat Sol 22 517[5] Wawryk R (2006) Phil Mag 86 1775[6] Migdal AB (1957) Sov Phys JETP 5 333

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

4

0

04

08

-02 -01 0 01 02(kaπ)

-16

-12

-08

-04

ωω

D

Figure 3 The dispersion for the various features in the calculated electronic spectrum (solidlines) The dashed redline represents the rigidly shifted band dispersion relation found from theLSDA calculations and the dashed blue line represents the shifted dispersion relation with therenormalized quasiparticle mass

by the NSF The work was also supported by the Swedish Research Council (VR) SNIC andthe European Commission (JRC-ITU) The authors of this work would like to commemoratethis paper to the memory of Cliff Olson

References[1] Lanzara A Bogdanov PV Zhou XJ Kellar SA Feng DL Lu ED Yoshida T Eisaki H Fujimori A

Kishio K Shimoyama JI Noda T Uchida S Hussain Z and Shen Z-X (2001) Nature 412 510[2] Engelsberg S and Schrieffer JR (1963) Phys Rev 131 993[3] Durakiewicz T Riseborough PS Olson CG Joyce JJ Bauer ED Sarrao JL Elgazzar S Oppeneer

PM Guziewicz E Moore DP Butterfield MT and Graham KS (2008) Europhysics Letters 84 37003[4] Leciejewicz J Troc R Murasik A and Zygmunt A (1967) Phys Stat Sol 22 517[5] Wawryk R (2006) Phil Mag 86 1775[6] Migdal AB (1957) Sov Phys JETP 5 333

International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics Conference Series 200 (2010) 012164 doi1010881742-65962001012164

4