Paramagnon-induced dispersion anomalies in the cuprates

0.23 MB·English

by  R. S. Markiewicz

Paramagnon-induced dispersion anomalies in the cuprates
R.S. Markiewicz, S. Sahrakorpi, and A. Bansil
Physics Department, Northeastern University, Boston MA 02115
(Dated: February 5, 2008)
7 We report the self-energy associated with RPA magnetic susceptibility in the hole-doped
0 Bi2Sr2CuO6 (Bi2201) and the electron-doped Nd2−xCexCuO4 (NCCO) in the overdoped regime
0 within theframework of a one-band Hubbard model. Strong weight is found in the magnetic spec-
2 trum around (π,0) at about 360 meV in Bi2201 and 640 meV in NCCO, which yields dispersion
anomalies in accord with therecently observed ‘waterfall’ effects in the cuprates.
n
a
J PACSnumbers: 79.60.-i,71.38.Cn,74.72.-h,71.45.Gm

2
2
Veryrecentangle-resolvedphotoemission(ARPES)ex- function. The RPA susceptibility is given by

] periments in the cuprates have revealed the presence of
n an intermediate energy scale in the 300-800 meV range χ0(~q,ω)
χ(~q,ω)= , (3)
o where spectral peaks disperse and broaden rapidly with 1−Uχ0(~q,ω)
c
- momentum,givingthisanomalousdispersiontheappear-
r withU denotingtheHubbardparameter. Theself-energy
ance of a ‘waterfall’[1, 2, 3, 4, 5, 6]. Similar self-energies
p canbeobtainedstraightforwardlyfromthesusceptibility
u have also been adduced from optical data[7]. This new
via the expression[16] (at T =0)
s energy scale is to be contrasted with the well-known
.
mat lboewenendeisrcguyss‘ekdinfkrse’quinentthlye i5n0-t7h0emcuepVrartaesngaes,awrihsiinchgfhraovme Σ(~k,ω)= 23Z¯U2 Z ∞ dπω′Imχ(~q,ω′)
X 0
the bosonic coupling ofthe electronic systemwith either q~
-
d phonons[8] and/ormagnetic modes[9]. Although low en- f¯ 1−f¯
on ebrogsyonp,laansmaloynsiss[1in0,di1c1a]teasrethaanttohbevpioluassmchooniscleiefoartttohoehniegwh ×hω−ξ¯~k~k−−q~q~+ω′ + ω−ξ¯~k−~kq~−−q~ω′i. (4)
c an energy of ∼1 eV to constitute a viable candidate[12].
[ Concerning technical details, we note that for the
Herewedemonstratethatparamagnonsprovidenotonly
genericpurposesofthisstudy,allcomputationsinthisar-
1
an explanation of the energy scale but also of the other ticleemployafixedvalueZ¯ =0.5,whichisrepresentative
v
observed characteristics of the waterfall effect in both
4 of the band dispersions observed experimentally in hole
hole and electron doped cuprates.
2 as well as electron doped cuprates.[17] Self-consistency
5 For this purpose, we have evaluated the self-energy is then achieved approximately by determining values of
1 associated with the RPA magnetic susceptibility in the
the chemical potential µ and the Hubbard parameter U
0 hole-dopedBi2Sr2CuO6 (Bi2201)andtheelectron-doped tokeepafixeddopinglevelandtoensurethatthebands
7
0 Nd2−xCexCuO4 (NCCO).[13] In order to keep the com- are indeed renormalized by the average factor Z¯ = 0.5.
putations manageable, the treatment is restricted to the
/ The procedure is relativelysimple, but it should capture
t
a overdoped systems where magnetic instabilities are not the essential physics of the electron-paramagnon inter-
m expected to present a complication. Our analysis pro-
action, although our treatment neglects the energy[18]
- ceeds within the framework of the one-band Hubbard and momentum dependencies of Z¯. Note also that in
d Hamiltonian, where the bare band is fit to the tight-
the overdoped regime considered, the effective U values
n
binding LDA dispersion[14, 15]. We incorporate self-
o in Bi2201 and NCCO are small enough that the system
consistency by calculating the self energy and suscep-
c remainsparamagneticandthe complicationsofthe anti-
v: tibility using an approximate renormalized one-particle ferromagnetic instability are circumvented. Specifically,
Green function
i the presented results on Bi2201 are for x = 0.27 with
X G=Z¯/(ω−ξ¯ +iδ), (1) µ=−0.43eVandU =3.2t,while forNCCO,x=−0.25
r k
with µ=0.18 eV and U =4t.
a
where ξ¯ = Z¯(ǫ −µ). Here, ǫ are bare energies and µ Figure 1 summarizes the results for Bi2201. We con-
k k k
is the chemical potential, and the renormalizationfactor sider Figs. 1(a) and (b) first, which give the real and
is Z¯ ∼ (1−∂Σ′/∂ω)−1 < 1. The associated magnetic imaginary parts of the self-energy at several different
susceptibility is momenta as a function of frequency. The theoreti-
cal self-energies, which refer to Bi2201, should be com-
f¯ −f¯
χ0(~q,ω)=−Z¯2X~k ǫ¯~k−¯ǫ~k~k+q~+~k+ωq~+iδ, (2) p(gaorleddsdqiureacrtelsy[3w])i,thaltthhoeucgohrraevsapiolanbdliengexepxepreimrimenetnatlaplodianttas
for Bi2212[4] and La2−xSrxCuO4 (LSCO)[5] are also in-
whereδisapositiveinfinitesimal,f¯ ≡f(ǫ¯ )istheFermi cluded for completeness. The agreement between theory
~k ~k
2
andexperimentis seento be quite goodfor the realpart (a)
0.4 Bi2201
ofthe self-energyin(a),while theoryunderestimatesthe
imaginarypartoftheself-energybyafactorof∼2. That V) 0.2
′′ e
the computedΣ issmallerthanthe experimentalone is Σ' ( (0,0)
to be generally expected since our calculations do not 0.0 (π,0)
Bi2201 (π,π)
account for scattering effects beyond those of the para-
LSCO (π/2,π/2)
magnons. Here, we should keep in mind that there are -0.2 Bi2212
uncertainties inherent in the experimental self-energies 0.8 -1 0 1 2 3
(b)
due to different assumptions invoked by various authors
0.6
concerningthebaredispersionsinanalyzingthedata. In V)
particular, Feng et al.[4] extract the bare dispersion by (e 0.4
assuming that Σ′ is always positive and goes to zero at Σ''
large energies. Other groups[3, 19] compare their results 0.2
′
toLDAcalculationsandarguethatΣ mustbecomeneg-
0.0
′
ative at higher energies. Our computed Σ in Fig. 1(a) -1 0 1 2 3
ω (eV)
becomesnegativeovertherange0.35-0.9eVincertain~k-
0.0
directions. Interestingly, various computed colored lines
in (a) and (b) more or less fall on top of one another, ) (c) 1.0
V
indicating that the self-energy is relatively insensitive to e 0.8
momentum, especially below the Fermi level, consistent ω ( -0.5 0.6
0.4
withexperimentalfindings[5], eventhoughΣ possessesa
0.2
fairly strong frequency dependence. 0.0
Fig. 1(c) gives further insight into the nature of the -1.0Γ (π,0)
0.0 0.5 1.0
spectral intensity obtained from the self energy of Eq. 4.
The spectral intensity shown in the color plot of the fig-
ure is representative of the ARPES spectrum, matrix
FIG. 1: (Color online) (a) and (b): Real and imaginary
element effects[20] notwithstanding. The peak of the parts Σ′ and Σ′′, respectively, of theparamagnon self-energy
spectral density function defined by taking momentum inoverdopedBi2201. Theoreticalresultsatvariousmomenta
density cuts (MDCs), shown by yellow dots, follows the are shown by lines of different colors. The values of Σ′ have
renormalizeddispersion(orangedashedline)up tobind- been shifted by a constant, to produce a zero average value
′
ing energy ofabout200meV. It then disperses to higher of the theoretical Σ at the Fermi level. Experimental points
are from the nodal point for: Bi2201 (gold squares, Ref. 3);
energies rapidly (waterfall effect) as it catches up with
LSCO (circles, Ref. 5); and Bi2212 (diamonds, Ref. 4). Thin
the bare dispersion (red solid line) around Γ. In fact,
linesjoiningexperimentalpointsareguidestotheeye. Arrow
near Γ, the dressed spectral peak lies slightly below the ′′
marksthelocation of peakin χ at (π,0). (c): Spectralden-
bare band. The width of the spectral function is largest sityintheenergy-momentumplaneobtainedfromthedressed
in the intermediate energy range of 200-600meV, where Green function is shown in a color plot along with the bare
its slope also is the largest. This behavior of the spec- (red line) and the renormalized (dashed orange line) disper-
tral function results from the presence of peaks in the sions. Dotsmark thepeak positions of theMDCplots ofthe
dressed spectral density.
real and imaginary parts of the self-energy in the 200-
500meVenergyrangediscussedinconnectionswithFigs.
1(a) and (b) above. It is also in accord with the water-
ergy in going from Bi2201 to NCCO is in good accord
falleffect observedin ARPESexperiments,althoughthe
with the experimentally reported shift of ∼300 meV[6].
sharpness of the theoretically predicted waterfall in Fig.
The dispersion underlying the dressed Green function,
1(c)islessseverethaninexperiments,whichmaybedue
which may be tracked through the yellow dots, is highly
tolimitationsofourmodel,includingtheapproximations
anomalousandpresentsakink-likefeaturequitereminis-
underlying our treatment of the susceptibility.
centofthemorefamiliarlowenergykinksinthe 50meV
Fig. 2considersthecaseofelectrondoped(overdoped) range around the (π,0)-direction[21], which have been
′
NCCO. The peak in Σ in Fig. 2(a) lies at binding ener- discussedfrequentlyinthecuprates. Thisstrongbosonic
giesof0.5-0.6eV(indifferent~k-directions)withaheight coupling is also reflected in the fact that the band bot-
′′
of 0.55-0.7 eV. Correspondingly, the peak in Σ in Fig. tom in NCCO lies severalhundred meVs below the bare
2(b) lies at a binding energy of 0.7-1.1 eV with a height LDA band in Fig. 2(c). It is interesting to note that
of 1-1.4 eV. Comparing these with the results of Fig. 1, the self-energies of Figs. 1 and 2 display a ‘mirror-like’
′
we see that the self-energy effects in NCCO are much symmetry: The peaks below the Fermi energy in Σ and
largerthan in Bi2201. Our computed shift of ∼300meV Σ′′ for Bi2201in Fig. 1 aresmaller than those abovethe
′
in the position of the peak in Σ to higher binding en- Fermi energy, but the situation reverses itself for NCCO
3
FIG.3: (Coloronline)Mapoftheimaginarypartofthemag-
neticsusceptibilityfor(a)hole-dopedBi2201and(b)electron-
dopedNCCO.Spectralweights areinunitsofeV−1. Arrows
mark the positions of the high spectral weights discussed in
thetext.
an energy 8t¯.
A reference to Figs. 1(a) and 2(a), where the energy
ω1 in Bi2201, and the energies ω2 and ω3 in NCCO are
′
markedbyarrows,indicatesthatthepeaksinΣ arecor-
FIG.2: (Coloronline)SameasthecaptiontoFig. 1,except
thatthisfigureshowsonlythecomputedresultsforoverdoped relatedwiththesefeaturesinthemagneticsusceptibility.
′
NCCO. Experimental self-energies are not available and are Inthis spirit,the shift inthe peakinΣ tohigherenergy
therefore not shown (see text). Solid (dashed) arrow in (a) ingoingfromBi2201to NCCO reflectsthe factthat fea-
shows location of peak in χ′′ at (π,0) ((π/2,π/2)). ture ω3 in χ′′ at (π/2,π/2) in NCCO (dashed arrow in
Fig.2(a))liesatahigherenergythanthe(π,0)featureω1
inBi2201(arrowinFig.1(a)). Notably,whentheStoner
in Fig. 2 in that now the peaks below the Fermi energy factorS =1/(1−Uχ0) is large,a peak in χ′′ arises from
become larger than those above the Fermi energy. apeak inχ′(ω), whichinturnisassociatedwithnesting
0
′
The aforementioned shift of the peak in Σ to higher offeaturesseparatedbyω inenergy. Inthepresentcase,
energyinNCCOcanbeunderstoodintermsofthechar- the nesting is fromunoccupied states nearthe Van Hove
acteristics of the magnetic susceptibility. Figure 3 com- singularity (VHS) at (π,0) to the vicinity of the band
paresinBi2201andNCCOtheimaginarypartχ′′,which bottomatΓ,soω1 ∼2(t+2t′)∼0.32eVinBi2201. The
is seen from Eq. 4 to be related directly to the real as largervalueofω2 inNCCOreflectstheshiftoftheFermi
well as imaginary part of the self-energy. χ′′ is seen to energy to higher energies in an electron-doped material.
be quite similar in shape along the Γ to (π,0) line in A notable difference between electron and hole dop-
′′
Bi2201 and NCCO, except that in NCCO the band of ing is the low-ω behavior of Σ , which is quadratic in
high intensity (the yellowish trace) extends to a signifi- ω for electron-doping in Fig. 2(b), but nearly linear for
′′
cantlyhigherenergyscale. Incontrast,χ inthetwosys- hole-doping in Fig. 1(b). The linearity for hole-doping,
tems differs sharply around (π,π). These differences re- reminiscent of marginal Fermi liquid physics, is associ-
flectthoseinthelow-energymagneticresponseofthetwo atedherewiththe proximityofthechemicalpotentialto
cuprates. NCCO with strong magnetic response around theVHS.ThispointisconsideredfurtherinFig.4where
′′
(π,π) exhibits anearlycommensurateAFMorder,while Σ is showninBi2201atthe (π,0)pointforthreediffer-
Bi2201 is very incommensurate, with peaks shifted to- ent values of the chemical potential. When the chemical
′′
ward (π,0). In fact, the high energy peaks in the self- potential lies at the VHS (red line), Σ varies linearly,
energy in Figs. 1 and 2 are tied to the flat-tops near but when it is shifted by 75 meV above or below the
(π,0) at ω1 ∼0.36 eV in Bi2201 (solid arrow), and near VHS, the behavior changesrapidly to become parabolic.
both (π,0) at ω2 =0.62 eV (solid arrow) and (π/2,π/2)
at ω3 = 0.9 eV (dashed arrow) in NCCO. Above these Thestrongmagneticscatteringdiscussedinthisstudy
′′
energiesthe weightin χ falls rapidly,goingtozeronear in the case of overdopedcuprates should persist into the
4
30x10-3 tron and hole-doped cuprates, providing an explanation
of such effects observed recently in ARPES. Although
our analysis is limited to the overdoped regime, we ex-
) 20
V
pectstrongmagneticscatteringtopersistintotheunder-
e
Σ'' ( 10 doTpehdisrwegoirmkei.sTsuhpispporotiendt hbyowtehveerUbSeDarespfaurrttmheerntstoufdEy.n-
ergy contract DE-AC03-76SF00098 and benefited from
0 the allocation of supercomputer time at NERSC and
-40 -20 0 20 40 NortheasternUniversity’sAdvancedScientificComputa-
ω (meV) tion Center (ASCC).
FIG.4: (Coloronline)Low-energybehaviorofΣ′′(ω)at(π,0)
in Bi2201 for three different values of the chemical potential
µ in relation to the position of the energy EVHS of the van [1] F. Ronninget al.,Phys. Rev.B71, 094518 (2005).
Hove singularity. EVHS −µ=0 (red solid line); EVHS −µ= [2] J. Graf et al.,to be published,Phys. Rev.Lett.
+75meV(greendottedline);andEVHS−µ=–75meV(blue [3] W. Meevasana et al., unpublished.
dashed line). [4] B.P. Xie, et al., cond-mat/0607450.
[5] T. Valla et al., cond-mat/0610271.
[6] Z.-H. Panet al.,cond-mat/0610442.
[7] J. Hwang et al.,cond-mat/0610488.
underdoped regime, where the Stoner factor is expected
[8] A.Lanzaraetal.,Nature(London)412,510(2001);X.J.
to become larger. In fact, this scattering is a precursor
Zhou, et al., Phys.Rev.Lett. 95, 117001 (2005).
to the magnetically ordered state near half-filling and [9] A. Kaminski et al., Phys. Rev. Lett. 86, 1070 (2001);
it is responsible for opening the magnetic gap. In con- P.D.Johnson et al.,Phys.Rev.Lett.87, 177007 (2001);
trast, a number of authors have related the presence of S.V.Borisenkoetal.,Phys.Rev.Lett.90,207001(2003);
waterfall-likeeffectsnearhalf-fillingto‘Mott’physicsas- A.D. Gromko et al., Phys.Rev.B68, 174520 (2003).
sociated with (π,π) AFM fluctuations[22, 23, 24], but [10] N. Nu¨cker et al.,Phys. Rev.B39, 12379 (1989).
[11] Y.Y. Wanget al.,Phys. Rev.Lett. 77, 1809 (1996).
have difficulty explaining why these effects persist into
[12] R.S. Markiewicz and A. Bansil, to be published, Phys.
the overdoped regime.
Rev. B.
The possible doping dependence of U has been an im-
[13] Although not discussed here, we find similar results in
portantissueinconnectionwithelectron-dopedcuprates. LSCO.
Adoping-dependentU issuggestedbyanumberofstud- [14] R.S. Markiewicz et al.,Phys. Rev.B72, 054519 (2005).
iesinthehole-dopedcupratesaswell. Theseinclude: Op- [15] For Bi2201, we use the parameters of Bi2Sr2CaCu2O8
ticalevidenceofMottgapdecrease[25];ARPESobserva- (Bi2212), but neglect the bilayer splitting. Following
′ ′′ ′′′
Ref.14,thehoppingparametersare(t,t,t ,t )=(360,–
tion of very LDA-like bands in optimally and overdoped
100,35,10)meVforBi2201and(420,–100,65,7.5) meVfor
materials; models of the magnetic resonance peak[26];
NCCO.
and, a strongly doping-dependent gap derived from Hall [16] W.F. Brinkman andS. Engelsberg, Phys.Rev. 169, 417
effect studies[27]. The Z¯2-renormalizationof χ0 in Eq. 2 (1968).
bears on this question and gives insight into how the [17] The specific values of Z are 0.28 in Bi2212 and 0.55 in
value of U enters into the magnetic response of the sys- NCCO.[14]
tem. Recall that the susceptibility is often evaluated in [18] Specifically,Z¯istherenormalizationofthecoherentpart
of thedispersion.
the literature via Eq. 3 using experimental band param-
eters, but without the Z¯2 factor of Eq. 2 in χ0, which [[1290]] AS..SLaahnrzaakroar,ppieertsoanl.a,lPchoyms.mRuenvic.aLteiottn..95,157601 (2005).
yields a χ scaling ∼ Z¯−1 rather than the correct scaling [21] A. Kaminski et al.,Phys. Rev.Lett. 86, 1070 (2001).
of χ ∼ Z¯. This can be corrected by replacing the U in [22] Y.KakehashiandP.Fulde,J.Phys.Soc.Japan74,2397
the Stoner factor by (2005).
[23] E. Manousakis, cond-mat/0608467.
U =Z¯2U. (5) [24] Q.-H. Wang, F. Tan, and Y.Wan, cond-mat/0610491.
eff
[25] S. Uchidaet al., Phys.Rev.B43, 7942 (1991).
Indeed, our Hubbard parameter for NCCO of U = 4t [26] H. Woo et al., Nature Physics, 2, 600 (2006), in supple-
is closer to the value at half-filling than is generally mentary materials.
[27] S. Ono, S. Komiya, and Y. Ando, cond-mat/0610361.
found.[28]
Thestrongerdopingdependencemaybeduetothehigh
In conclusion, we have shown that the higher energy
measurement temperatures.
magnetic susceptibility in the cuprates has considerable
[28] The smaller U for Bi2201 may be due to neglect of kz-
weight near (π,0) and that this leads to a high energy dispersion, which broadens features in χ near the VHS.
kink or waterfall-like effect in dispersion in both elec-