Dispersion anomalies induced by the low-energy plasmon in the cuprates

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by  R. S. Markiewicz

Dispersion anomalies induced by the low-energy plasmon in the cuprates
R.S. Markiewicz and A. Bansil
Physics Department, Northeastern University, Boston MA 02115
(Dated: February 6, 2008)
7 We discuss the characteristic effects of the electron plasmon interaction resulting from the ∼
0 1 eV plasmon, which is a universal feature in the cuprates. Using the framework of a one-band
0 tightbindingmodel,weidentifysignaturesofthislowenergyplasmonintheelectronicstructureof
2 metallic overdoped Bi2212 as well as half-filled insulating SCOC. Theelectron-plasmon interaction
is found to yield renormalizations near the Fermi energy in reasonable accord with experimental
n
a observations, and toproduce dispersion anomalies at higher energies.
J
PACSnumbers: 79.60.-i,71.38.Cn,74.72.-h,71.45.Gm
8
1

Quasiparticledispersions nearthe Fermi energyin the as follows[1]:
]
on 0cu.3p−ra0te.6s[1a]reinrceonmorpmaarilsizoendto(rethdeucbeadn)dbtyheaorfaycptorredoifctZion∼s ǫk =−2t[cx(a)+cy(a)]−4t′cx(a)cy(a)−2t′′[cx(2a)
-c based on the conventional LDA picture, interrupted by +cy(2a)]−4t′′′[cx(2a)cy(a)+cy(2a)cx(a)], (1)
r theappearanceoflowenergy‘kinks’ordispersionanoma-
p where cα(na) = cos(nkαa) for α = x or y, n is an in-
lies in the 50-70meV energy range,which arisefrom the
u teger, and a is the in-plane lattice constant. Hopping
s couplingoftheelectronicsystemwithphonons[2]and/or parameters appropriate for Bi2212 (neglecting bilayer
.
t magnetic modes[3]. Interestingly, the ‘spaghetti’ of vari- splitting)[1] are: t=360 meV, t′ = −100 meV, t′′=35
a
m ous hybridized and unhybridized Cu and O bands start- meV, t′′′ = 10meV. We model the dynamic dielectric
ingaround1eVinthe cupratesseemstoremainmoreor
function, ǫ(q,ω), in the form[6, 11]
-
d lessunrenormalized.[4]Recallingthatoptical[5]andelec-
n tron energy loss spectroscopy (EELS) experiments[6, 7] ǫ(q,ω)=ǫ 1− ωp2(q) , (2)
o have shown the presence of dispersive plasmons in the 0 ω(ω−iγ(q))
c cuprates lying at ∼1 eV, we explore the effects of such a h i
[ where ǫ0 = 6 and the dispersion and damping of the
lowenergyplasmoninthisLetter. Theplasmonisfound
plasmon in Bi2212 obtained from EELS data are given
1 to induce renormalizations in the low energy regime in
by[6]: ω = 1.1eV[1 + 4S ], γ = 0.7eV[1 + 16S ],
v p xy xy
both the metallic and the insulating cuprates, which are
5 Sxy = sin2(qxa/2)+sin2(qya/2). Notably, due to the
in substantialaccordwith the corresponding experimen-
6 large broadening, the experimental plasmon peak lies at
4 talresults. We emphasizethatplasmonsin3Dmaterials a lower energy than the bare value ω . Using the bare
1 typicallylieat∼10eVandthereforehavelittle influence p
dispersion of Eq. 1 and the dielectric function of Eq. 2,
0 on the electronic states in the 0-1 eV range.
it is straightforward[12] to obtain the first order correc-
7
0 We approach the problem within the framework of tion to the electronic self energy Σ due to the screened
/ Coulomb interaction[13]. The formulas can also be de-
t a one-band tight-binding model and consider the lim-
a rived from those for the insulator (Eqs. 5-11 below) by
iting cases of a metal with the example of overdoped
m
letting the magnetic gap ∆ →0.
Bi Sr CaCu O (Bi2212)[8,9]andtheinsulatorwiththe m
2 2 2 8
- Figure1displaysthespectraldensityintheoverdoped
d example of half-filled Sr2CuO2Cl2 (SCOC)[10]. The di-
metallic case where effects ofthe electron-plasmoninter-
n electric functions computed via the random phase ap-
o proximation (RPA) yield loss functions and the asso- action are included. We consider the low energy region
c along Γ → (π,0) shown in (b) first. A region of high
ciated plasmon dispersions in both Bi2212 and SCOC,
v: which are in reasonable accord with the corresponding spectralintensity depicted by the thin white trace in (b)
Xi EELS data. The generic effects of the electron-plasmon can be seen clearly (note logarithmic scale), indicating
that the LDA band continues to persist in the presence
interaction on the electronic structure and how the low
r
of the plasmon, albeit with some renormalization. An
a energy plasmon in the metal and the insulator induces
averagefactor of Z =0.5 is seen to more or less describe
characteristicdispersion anomalies and renormalizations
therenormalizationoftheLDAbandasseenbycompar-
of the spectrum are then delineated via a first order cal-
ing the dashed blue line with the whitish trace of high
culation of the electronic self energy.
intensity. TheZ =0.5valuesodeterminedisinsubstan-
We discussfirstthe metalliccaseofoverdopedBi2212. tial accordwith the correspondingexperimentalvalue of
Here the magnetic gap has collapsed fully and we take Z = 0.28,[1] indicating that the plasmon constitutes a
the bare electronic dispersion to be given by a one-band substantial part of the bosonic dressing of the electron.
tight-binding modelfitted tothe LDA-basedbandstruc- Moreover,thespectralweightspreadsoutfromtherenor-
ture of Bi2212 in the vicinity of the Fermi energy (E ) malizedbandtowardsthebareband(redline),givingthe
F
2
40
30
q
V 20
q
10
0
Γ (π,0) (π,π) Γ
1 2 3
FIG. 2: (color online) Different calculations of the Coulomb
FIG. 1: (Color online) (a) Dressed spectral density function potential(seetext)intheformofqVq arecompared. Full3D
layered lattice (solid line); A single 2D layer (dashed); and
in overdoped (metallic) Bi2212. Bare dispersion (red line) is
2D layer without theon-site U term (dotted).
compared with dispersion renormalized with Z = 0.5 (blue
dashed line). Dotted lines indicate plasmon dispersion. (b)
Blowup of region enclosed by white box in (a). Logarithmic
color scale is used to reveal weaker features. malized by a factor of Z=0.28 to account for the renor-
malization observed in ARPES experiments on Bi2212.
The treatment of the long-range Coulomb interaction
resulting yellowish trace the visual appearance of a ’wa-
V(r) for the correlated electronic system in a layered
terfall’ terminating around the Γ point. These results
structure requires some care and we model V(r) as an
bear some resemblance to ’waterfall’-like anomalies re-
on-site Hubbard U =2 eV term[16] plus a 1/r contribu-
portedrecentlyin the ARPESspectrafromthe cuprates
tion screened by the background dielectric constant ǫ
0
overthe0.4-0.8eVenergies[8,9]. However,thewaterfall-
at other lattice sites. V is then obtained via a lattice
q
like features induced by the plasmon and those reported
Fourier transform by summing the contributions of all
in ARPES differ significantly insofar as their location in in-plane Cu terms for r ≤ R ≡ 4a (a being the in-plane
energy and momentum is concerned, and therefore, on
lattice constant) and approximating the remaining in-
the face of it, the plasmonwould notseem to provide an
plane as well as interplane contributions for r > R by a
explanation of the recent ARPES results. We return to continuum in order to recover the correct q →0 limit of
comment further on this point below.
V . Fig. 2 shows qV along high symmetry lines in the
q q
We now turn to Fig. 1(a) which exposes effects of the
Brillouinzone(BZ).V isseentodisplayacrossoverfrom
q
plasmon over a wider energy scale. Here we see that 3D behavior ∼ 1/q2 to 2D behavior ∼ 1/q for qc > 1,
the plasmon induces ’plasmaron’ features[14] – a new
where c is the distance between successive CuO planes.
2
’shadow’bandlying1.5eVbelowthe baredispersionvia
TheuseofV andχ inEq. 3yieldsthedielectricfunc-
q 0
the plasmon emission process along the filled portion of
tion based directly on the band structure of Eq. 1. The
the LDA bands along the Γ to (π,0) and the Γ to (π,π)
plasmon dispersion is found from the peak of the EELS
lines. A similar shadow band associated with plasmon
spectrum, which is proportional to the loss function Im
absorption appears 1.5 eV above the unfilled part of the
[−1/ǫ], plotted in Fig. 3. The theoretically predicted
LDAbandalongthe (π,0)to(π,π)andΓto(π,π)lines.
plasmon energy given by the maximum of the loss func-
At low energies, the plasmondresses the electrons, mak-
tion (white trace in the figure) is compared to the avail-
ing them heavier and renormalizing their dispersion. In
able experimental results in Bi2212 (red dots). Given
contrast,athighenergies,the plasmonbecomes progres-
that the calculation is not self-consistent, the agreement
sively less effective in dressing the electrons which move
is seen to be reasonable. Notably, our calculations as-
with ever increasing speeds, even though the electronic
sume purely in-plane excitations (q =0).[17]
z
states are still broadened by plasmon emission effects.
We consider nextthe morechallengingcaseof insulat-
Wenowdemonstratethattheexperimentallyobserved
ing SCOC, where a magnetic gap is present and it has
plasmondispersioninBi2212is infactdescribedreason-
notbeenclearhowtheEELSspectra[7]arerelatedtothe
ablywellby the bandstructureofEq. 1, consistentwith
plasmon excitation. As in the metallic case, the starting
the results of Ref. [15]. For this purpose, we directly
point is the dispersion of Eq. 1, except that the hop-
evaluatethe dielectricfunctionfromthebarechargesus-
pingparametersarechosentofittheLDA-basedbandsin
ceptibility χ0 and the Coulomb interaction Vq via SCOC:t=420meV,t′ =−57 meV,t′′=52meV, t′′′ =26
meV.Thecorrelatedbandstructureismodeledbyintro-
ǫ(q,ω)=ǫ [1+V χ (q,ω)]. (3)
0 q 0
ducing an on-site Hubbard term U=2 eV and treating
χ is calculated straightforwardly within the RPA ap- the system as a uniform half-filled antiferromagnet[18]
0
proximation where the band structure of Eq. 1 is renor- withsaturated(staggered)momentm =0.5. Theband
Q
3
FIG.3: (Color online)LossfunctionIm[−1/ǫ]asafunction FIG. 4: (Color online) Dressed spectral density function in
ofωandqinBi2212plottedonalogarithmiccolorscale. Red SCOC at half-filling. Bare dispersion (red line) is compared
dots give theexperimental plasmon peak positions [6]. withdispersionrenormalizedwithZ =0.3(bluedashedline).
Dotted lines indicate plasmon dispersion. Logarithmic color
scale is thesame as in Fig. 1.
structure of Eq. 1 then splits into upper and lower mag-
netic bands with gap ∆ = Um . The dielectric re-
m Q
sponse given by Eq. 2 in the metallic case is modified
strongly at low frequencies and our analysis indicates
that the dielectric function in the insulator can be mod-
eled as
ω2
ǫ=ǫ 1− p , (4)
0 ω(ω−iγ)−ω2
0
h i
where ω ≃ U.[19] Taking ω =2 eV and the same γ as
0 0
in Eq. 2, the EELS data[7] on SCOC can be fitted to
Eq. 4 with ω =1.3eV[1+3S ]. Again, the broadening
p xy
shifts the peak in Im [ǫ−1] from its expected position
ω˜p =[ωp2+ω02]1/2; the experimental peak can be fit to a FIG.5: (Color online)LossfunctionIm[−1/ǫ]asafunction
similar q-dependence, but with ωpe =2.4eV[1+0.33Sxy]. ofω andqinSCOCathalf-filling. Solidlinegivestheexper-
Note that ωp is not very different in metallic Bi2212and imental plasmon dispersion (ω˜p) for SCOC [7]. Logarithmic
insulatingSCOC,sothatmostoftheshiftintheposition scale as in Fig. 3.
ω˜ of the plasmonpeak in the loss function is due to the
p
gap ω .
0
The spectral density for the dressed Greenfunction in where the prime means summation over the magnetic
the insulator is shown in Fig. 4. The results are similar Brillouin zone only,
to those of Fig. 1 for Bi2212 in that the dressed lower
[1−f(ǫ )]f(ǫ )
magnetic band in Fig. 4 displays the waterfalleffect and g2 = i,k j,k+q , (6)
ij ǫ −ǫ +ω+iδ
that it is renormalized in comparison to the bare band i,k j,k+q
(compare red line with dashed blue line). The renormal-
f is the Fermi function,
izationofthelowenergyspectrumbyafactorofZ ∼0.3
in SCOC due to the plasmonin Fig. 4 is againcompara-
ǫ =ǫ(+)∓E , (7)
ble tothe correspondingexperimentalvalue ofZ =0.64. 1,2k k k
Finally, we consider briefly the calculation of the loss with the 1(2) going with the -(+) sign, ǫ(±) = (ǫ ±
function and the self-energy in the insulating system. k k
ǫ )/2,
With two bands involved in this case, the formalism is k+q
less familiar than the one-band metallic case, and there-
fore, an overview of the relvant results is appropriate. u2 = 1 1± ǫk(−)ǫk(−+)q+∆2m , (8)
±
[The metallic case can be recovered by taking the limit 2 EkEk+q
h i
∆ → 0.] The RPA expression for the charge suscepti-
m
bility χ in the insulator is andE =[ǫ(−)2+∆2 ]1/2. Theprecedingequationshave
0 k k m
been applied previously to the Hubbard model[20, 21],
1
χ (q,ω)= ′[u2 g +g +u2 g +g ], (5) but we have extended their usage to include the long-
0 N − 12 21 + 11 22 range Coulomb interaction.
k
X (cid:0) (cid:1) (cid:0) (cid:1)
4
In the presence of a gap, the electronic self energy Σ Inconclusion,wehaveshownthatthelow-energyplas-
becomes a tensor with indices 1(2) corresponding to the mon, which is a universal feature of the cuprates, pro-
lower (upper)magnetic band[12, 22]: duces renormalizations and dispersion anomalies in the
electronic structure. Our study indicates that the plas-
d2q
Σ (p,ω,σ)= [v2 Σˆ +u2 Σˆ ], (9) mon is a very significant player for understanding the
11 ZBZ (2π)2 m 1 m 2 physics of the cuprates.
Acknowledgments: We thank B. Barbiellini for im-
d2q portant discussions. This work is supported by the US
Σ =Σ = u v σ[−Σˆ +Σˆ ], (10)
12 21 (2π)2 m m 1 2 DOE contract DE-AC03-76SF00098 and benefited from
ZBZ the allocation of supercomputer time at NERSC and
where NortheasternUniversity’sAdvancedScientificComputa-
tion Center (ASCC).
V2χ (q,ω−ǫ )
Σˆ =[Θ(ω−ǫ )−Θ(−ǫ )] q 0 i,p+q ;
i i,p+q i,p+q ǫ(q,ω−ǫ )/ǫ
i,p+q 0
1 ǫ(−) 1 ǫ(−)
u = 1+ p+q ; v = 1− p+q , (11)
m m
s2 Ep+q s2 Ep+q [1] R.S. Markiewicz et al.,Phys. Rev.B72, 054519 (2005).
h i h i
[2] A.Lanzaraetal.,Nature(London)412,510(2001);X.J.
and Σ has the same form as Σ , but with u and v
22 11 m m Zhou, et al., Phys.Rev.Lett. 95, 117001 (2005).
interchanged. [3] A. Kaminski et al., Phys. Rev. Lett. 86, 1070 (2001);
The lossfunctioninSCOCcomputedfromEqs. 5 and P.D.Johnson et al.,Phys.Rev.Lett.87, 177007 (2001);
3 is shown in Fig. 5. The experimentally observed plas- S.V.Borisenkoetal.,Phys.Rev.Lett.90,207001(2003);
mon dispersion (solid line) is seen to be in good accord A.D. Gromko et al., Phys.Rev.B68, 174520 (2003).
[4] K. McElroy et al.,Bull. A.P.S. 51, 141 (2006).
with the theoretical prediction. A comparison of Figs.
[5] I. Bozovic, Phys. Rev.B42, 1969 (1990).
3 and 5 reveals how the gap in the underlying spectrum
[6] N. Nu¨cker et al., Phys. Rev. B39, 12379 (1989); M.R.
of the insulator shifts the plasmon energy in SCOC to
Norman et al., Phys. Rev. B59, 11191 (1999); L. Hedin
higher values with little weightatenergies below≈ 2 eV and J.D. Lee, Phys. Rev.B64, 115109 (2001).
in contrast to the metallic case of Bi2212. [7] Y.Y. Wanget al.,Phys. Rev.Lett. 77, 1809 (1996).
Thepresentresultshavesignificantimplicationsforthe [8] J. Graf et al., cond-mat/0607319, Phys. Rev. Lett. (to
low energy physics of the cuprates and provide a route be published).
[9] W. Meevasana et al., cond-mat/0612541.
for connectingthis physicsto the spectrumathigher en-
[10] F. Ronninget al.,Phys. Rev.B71, 094518 (2005).
ergyscalesasoneencountersthe venerable‘spaghetti’of
[11] The static ǫ(q,0), calculated in RPA, screens the ex-
Cu and O related bands spread typically over ∼1-8 eV
changepotential,leadingtoareal,nearlyk-independent
binding energies. We have shown that the plasmon in- self-energy which is presumably already incorporated in
ducesbandrenormalizationsatlowenergiesinthemetal- the LDAcalculation.
liccaseofBi2212aswellasthe insulatingcaseofSCOC, [12] J.D.Mahan,“Many-ParticlePhysics”(2dEd.)(Plenum,
and that these renormalization effects are in substantial New York, 1990). The present Eq. 11 describes only the
accordwith the corresponding experimental values. Our component Σ(res); a second, smaller component, Σ(line)
(Eq. 5.1.25 of Mahan) has also been included in the nu-
results clearly establish the importance of the plasmon
merical calculations.
in the bosonic dressings of the low energy quasiparticles
[13] This is similar totheGW approximation scheme. Going
in the cuprates.
beyond the simplest GW toward full self-consistency is
There have been several recent calculations of complicatedbytheneedtoincludevertexcorrections.See
electronically-inducedkinks,butthesearegenerallyvery F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys.
weak[23]orconfinedtoarestricteddopingrange[24]. The 61, 237 (1998) for a review.
plasmonby contrastis robustand produces strong kink- [14] L. Hedin and S. Lundqvist, in Solid State Physics, Vol.
23,ed.byF.Seitz,D.Turnbull,andH.Ehrenreich,(Aca-
like features which extend over several eVs, due to the
demic, N.Y.,1969), p.1.
large broadening of the plasmon[25]. While these char-
[15] A.C. Sharma and I. Kulshrestha, Phys. Rev. B46, 6472
acteristicfeaturesoftheelectron-plasmoninteractionare
(1992).
reminiscent of those reported in very recent ARPES ex- [16] In theoverdoped metal, possible renormalization of U is
periments on several families of cuprates[8, 9, 10], they neglected in the charge response, and it is assumed that
fall at too high an energy scale. Possible candidates there is no magnetic order.
for the intermediate waterfall scale include the acoustic [17] The role of acoustic plasmons is thus underestimated.
Our preliminary calculations suggest that this is not a
branch of the plasmons and high-energy magnetic exci-
seriouslimitation,butthispointbearsfurtherattention.
tations. The extent to which low energy plasmons are
[18] While applicability of the RPA to hole doped cuprates
involved in an electronic mechanism of superconductiv-
remains a matter of debate, we note that U =8t is con-
ityremainsunclear,althoughsuchsuggestionshavebeen siderably below the criterion for a Brinkman-Rice insta-
made in the literature[26]. bility [W.F. Brinkman and T.M. Rice, Phys. Rev. B2,
5
4302 (1970)]. [23] I.A. Nekrasov et al.,Phys. Rev.B73, 155112 (2006).
[19] The valueof ω is estimated from the ω=0 limit of the [24] Y.KakehashiandP.Fulde,J.Phys.Soc.Japan74,2397
0
calculated ǫ(q,ω).Forsimplicity,theq-dependenceofω (2005).
0
is neglected. [25] The broadening is associated with the plasmon entering
[20] A.V. Chubukov and D.M. Frenkel, Phys. Rev. B46, intotheparticle-hole continuum,andcan bereproduced
′′
11884 (1992). by adding a small Σ to the bareGreen’s function.
[21] A.P.Kampf, Phys.Rev.B53, 747 (1996). [26] J. Ruvalds, Phys. Rev. B35, 8869 (1987); V.Z. Kresin
[22] G. Vignale and M.R. Hedayati, Phys. Rev. B42, 786 and H.Morawitz, Phys. Rev.B37, 7854 (1988).
(1990).