Universality of the single-particle spectra of cuprate superconductors

0.51 MB·English

by  Lijun Zhu

Universality of the single-particle spectra of cuprate superconductors
Lijun Zhu, Vivek Aji, Arkady Shekhter and C. M. Varma
Department of Physics and Astronomy, University of California, Riverside, California 92521
All the available data for the dispersion and linewidth of the single-particle spectra above the
superconductinggap and thepseudogap in metallic cuprates for any dopinghas universalfeatures.
The linewidth is linear in energy below a scale ωc and constant above. The cusp in the linewidth
8 at ωc mandates, due to causality, a ”waterfall”, i.e., a vertical feature in the dispersion. These
0 features are predicted by a recent microscopic theory. We find that all data can be quantitatively
0 fittedbythetheory with a coupling constant λ0 and an uppercutoff at ωc which varyby less than
2 50% amongthedifferentcupratesandfor varyingdopings. The microscopic theoryalso givesthese
valuesto within factors of O(2).
n
a
PACSnumbers: 74.20.Mn,74.25.Jb,74.72.-h
J

9

] Introduction. With increased refinement of technique Above E1, the “velocity” dε(k)/dk sharply increases up
el and imaginative use, angle-resolved photoemission spec- to another cutoff E2 where ε(k) resumes the normal dis-
- troscopy (ARPES) on the high temperature supercon- persion. The nearly vertical dispersion has been pic-
r ductors has revealed that novel physical principles de- turesquely termeda “ waterfall”[3]. In the energyrange,
t
.s termine the single-particle spectra in such compounds E1 .ω .E2, thereis alsoanindicationofmultipleε(k)
at [1, 2]. Recently, the single-particle spectra over an en- for fixed ω [6, 8]. E1 varies systematically being largest
m ergy range from the chemical potential to about 1 eV inthe(π,π)directionandsmallestinthe(π,0)direction
have been deduced for various metallic dopings in dif- [8]. Similarly, the position of the “waterfall” in k-space
-
d ferent cuprates [3, 4, 5, 6, 7, 8, 9, 10]. Also, recently a varies systematically.
n microscopic theory [11] has been formulated which de- All these features follow quantitatively from the
o
rives the fluctuations leading to the phenomenological quantum-critical fluctuations derived recently [11]. We
c
[ marginal Fermi liquid(MFL) [12] and their coupling to find that, given the bare band structure ǫk, all available

fermions. MFL had previously been tested only for low datacanbefittedwiththetwoparametersofthistheory,
3
v energies and near optimal doping in Bi2212 [13, 14, 15] a sharp cutoff ωc and a coupling constant λ0 calculable
7 with adjustable couplings. Here we test crucial new fea- to factors of O(2).
8 turesofthemicroscopictheoryincluding itsuniversality, Single-particle spectral function. The single-particle
1 its cut-offand coupling functions, by comparingwith re- spectral function deduced by ARPES is given
2 cent ARPES data in 4 different cuprate families and at
0
different dopings. −ImΣ(ω,k)/π
7 A(k,ω)= , (1)
0 The most important results of these recent ARPES [ω−ReΣ(ω,k)−ǫk]2+[ImΣ(ω,k)]2
/ experiments may be summarized as follows:
t
a (i) The spectra for energies ω in the range of inter- where Σ(ω,k) is the self-energy function. The band
m
est (above the superconducting gap and the pseudogap structure ǫ is in general fitted by the tight-binding
k
- energy scales) are universal; they have the same func- dispersion[16].
d
tional form for all cuprates and for all metallic dopings. Microscopic Theory. A microscopic theory for the
n
o Moreover, even the parameters in the functional form cuprates [11, 17, 18] is basedon the realization,that the
c vary less than by a factor of 2 over the entire range central organizing feature in the physics of the metallic
v: of cuprates for which data is available, irrespective of phase of the cuprates are quantum critical fluctuations
i whethertheyareunderdoped(UD),optimallydoped(OP) of loop currents. In this theory the ordering of these
X
or overdoped(OD). loop currents below T (PG in inset of Fig. 1) leads to
g
r
a (ii)Themomentumdistributioncurves(MDC)atcon- a phase which breaks time-reversal symmetry but pre-
stant energy ω is a Lorentzianwith width wk(ω). In the serves translationalsymmetry. Direct evidences for such
energyrangeofinterestwk varieslinearlywithω uptoa anorderedstatehavebeenobtainedbypolarizedneutron
cutoff above which it is approximately a constant. This scatteringinYBa2Cu3O6+x [19]andbydichroicARPES
is modified if the bare velocity v(k) varies within wk, experiments in Bi2Sr2CaCu2O8+x [20], for various x in
whichhappens asthe bottomofthe bandis approached. the pseudo-gap phase. The properties in the entire fun-
See Fig. 3 below for representative experimental data. nel shaped region (I) in inset of Fig. 1 are determined
(iii) The peak of the MDC as a function of ω moves bythequantumcriticalfluctuationsoftheloopcurrents.
with k defining the renormalized dispersion ε(k). The Therefore universal properties are predicted for ω larger
observed dispersion ε(k) follows the band structure ǫ than the superconducting gap or the pseudogapfor all x
k
with a smooth renormalization factor up to ω ≈ E1. in the metallic phases on either side of xc.
2
4
Themicroscopictheoryofthe quantum-criticalfluctu-
T
ations [11] gives their absorptive part to be Tg StMreatnagle Tf
3 I
Imχ(q,ν) = −0,χ0tanh2νT, ||νν||><ωωc;. (2) AFM PGSC Tc FL w-ReS(w)
c x
2 QCP
ωc is a cutoff and χ0 gives the integrated weight of -ImS(w)
the fluctuations. In the microscopic theory [11], ωc2 ≈ 1
2EVR2, where E is the local repulsion or charging en-
-ReS(w)
ergy parameter, V the nearest neighbor Cu-O interac-
tion, and R is the dimensionless loop-current order pa- 00 w1 w/1w w2 2
rameter. χ0 ≈ R/ωc. For E ≃ 5eV, V ≃ 1−2eV and c
R ≃ 0.1 which is consistent with the neutron measure-
FIG. 1: The self-energy, and ω−ReΣ(ω) as functions of ω
ments, we expect ω ≈0.3−0.5eV.
c
Calculation of the Self-energy. The loop current fluc- for λ0 = 1. All quantities are dimensionless in units of ωc.
tuationsscatterfermionsfromktok′ withtheamplitude TEqh.e(o6b)siesrveqabulievadleisnptetrosiothneεi(nkt)erfsoerctaiognivoefnωb−arReeǫΣk,(ωso)lvweidthbya
γ(k,k′). From the microscopic model, we find [21] horizontallineatǫk. Asω−ReΣ(ω)hasawidereentrantre-
gionforω1 ≤ω≤ω2,theobserveddispersionfallsfromω1 to
γ(k,k′)=±iV (s (k+k′)s (k−k′)−x↔y)S (k,k′), ω2 for averysmall variation in ǫk producingthe“waterfall”.
2 x y xy Insert shows thephase-diagram of thecuprates.
(3)
where s (k) ≡ sin(k a/2), s (k) ≡ s2(k)+s2(k),
x,y x,y xy x y
produce no singular corrections. At low energies com-
S (k,k′) ≡ (s−1(k)+s−1(k′)). The leaqding self-energy
xy xy xy pared to ω , Eq. (5) reduces to the MFL form deduced
c
contribution is
earlier [12], except for the weak momentum dependence.
Σ(iω ,k)=T |γ(k,k+q)|2G(iω +iν ,k+q)χ(q,iν ), Given the sharp change of the slope in the imaginary
n n n n
part near ω , the real part has a logarithmic divergence
c
qX,iνn in its slope at ω before changing from its low energy
(4) c
form ∝ ωlog|ω| to 1/ω for ω ≫ ω . This sharp varia-
where ω , ν are Matsubara frequencies of the quasi- c
n n
tion of ReΣ(ω) near ω is responsible for the observed
particle and the fluctuating mode, respectively. Given a c
q-independent χ and γ(k,k′) of the form of Eq. (3), the “waterfall” feature as we now proceed to show.
The Waterfall. The dispersion of the quasi-particles,
self-energy variation with k on a Fermi surface comes
only from the separable s-wave part of |γ(k,k′)|2 which ε(k) given by
′
is ∝ (1−cosk acosk a)(k → k ). This gives λ ∝ (1−
x y k ε(k)−ReΣ(ε(k))−ǫ =0. (6)
cosk acosk a) which varies by about a factor of 2 from k
x y
the (π,π) to the (π,0)directionsfor the Fermisurfaceof
As shownin Fig. 1, ω−ReΣ(ω) has a wide reentrantre-
Bi2212 near optimal doping.
gionfromω1 ≤ω ≤ω2. ThesolutionofEq.(6)therefore
At T =0 the self-energy is easily evaluated to be
produces a “waterfall” in the dispersion ε(k) because it
varies over the large energy range ω1 to ω2 for a very
π |ω|, |ω|<ω
ImΣ(ω,k)=−λ(k) c small variation in k. The multiple solutions obtained
2 (ωc, |ω|>ωc in this region are within ImΣ(ω) for λ of O(1). Above
ReΣ(ω,k)=− λ(k) ωln ωc +(ω−ω )ln|ω−ωc| ω ≃ω2,thedispersionbecomesjustarenormalizedband
c structure. The calculated “waterfall” is shown in Fig. 2.
2 |ω| ω
c
h ThespectralintensitymapsinFig.2(d-f)shouldbecom-
−(ω →−ω) , (5) pared with Fig. 1(a-c) of Ref. [8].
i Comparison with Experiments: The calculated self-
whereλ(k)=λ0hγ2ik′;λ0 =N(0)(V2/4)χ0 andhγ2ik′ is energy at λ0 = 1 (suitable to fit the experimental data
theaverageof|γ(k,k′)|2overk′ontheFermisurface. For [8] for La1.83Sr0.17CuO4) is shown in Fig. 1. The experi-
thedensityofstatesperonespinspeciesN(0)≈1(eV)−1 mentalMDCwidthforthiscompoundandthecalculated
and other parameters used above, we expect λ0 ≈1. widths for three different cuts are compared with exper-
Givensuchaweaklymomentumdependentself-energy, iment in Fig.3. In Fig.4(a)-(c), we compare the experi-
the vertex corrections [22] to the self-energy are only of ments [3] for the dispersion of three Bi2212 samples at
O(λωc/W), where W is the bare bandwidth of the con- differentdopingswithcalculationswithωc =0.5,λ0 ∼1.
duction band. Using the ω and λ fitted to the experi- In Fig.4(d), we compare the measured linewidth for an
c
ments, this ratio is of O(0.1). The remaining processes, UD-LSCO sample, an OP-Bi2201 sample and an OP-
repeatedscattering(self-consistentBornapproximation) Bi2212 sample with calculations with parameters given
3

(cau)t 2 8 12 (cbu)t 4 (ccu)t 7 0eV).0 (a) UD64K (b) OP91K (c) OD65K
) (
, FS y
kA( erg
0.1eV 0.1eV 0.05eV En
0.5
0.8eV 0.7eV 0.4eV
0.-00.6 -0.4 -0 .2 0.0 0.2 -0.4 -0 .2 0.0 0.2 -0.4 -0 .2 0.0 0.2
(d) (e) (f) -0.3 0.0 -0.3 0.0 -0.3 0.0
cut 2 cut 4 cut 7 1 k-kF k-kF k-kF
00 (eV)..42 k (k)1 E1k (k) k(k) 0M (/a).4 (d)
H
0.6 FW

C
D
0.-80.6 -0.4 -0.2 0.0 0.2 -0.4 -0.2 0.0 0.2 -0.4 -0.2 0.0 0.2 M
ky(/a) ky(/a) ky(/a)
0.0
FIG. 2: (a),(b) and (c): calculated MDCs for three of the 0.0 0.5
momentum cuts 2, 4 and 7 shown in theinset to Fig. (a) for Energy(eV)
whichdataisavailablefromRef.8. TheMDC’sareshownat FIG.4: Comparisonbetweenexperimentalandtheoryresults
variousenergies ω labeled inthefigures. Figs. (d),(e)and(f) (represented by symbols and lines, respectively) for various
arespectralintensitymaps,forthesamecutscorrespondingly. cuprate samples. (a)-(c) are calculated dispersions for three
From the intensity maps, we can identify thedispersion ε(k) Pb-dopedBi2212samplesalongthenodalcuts: UDwithTc =
markedbycircles;thebaredispersionǫk areshownindashed 64K, OP with Tc = 91K and OD with Tc = 65K. The
lines. TheinsetofFig.(a)showstheFermisurfaceandeight experimentaldatashownareextractedfromFig. 1ofRef.[3].
momentum cuts done in experiments [8]. It also shows the The tight-binding fitting parameters of the band structure
positions where the “waterfall” are expected in k-space for are taken from Ref. [23]. All these samples are fitted by the
radialcutswiththeadditionalcontourdrawninsidetheFermi parametersωc =0.5eV(forall)andλ0=0.98, 1.01,and1.05,
surface. respectively. (d) shows the MDC linewidths (full width at
half maximum) for different cuprate samples. ◦, ×, • and (cid:3)
represent OP-Bi2201 (nodal cut, Ref.[9]), OP-Bi2212 (nodal
in the figure caption. cut, Ref. [24]), LSCO 0.17 (cut 2 in Fig. 2 of Ref. [8]) and
LSCO 0.145 (cut 1 in Fig. 3 of Ref. [25]), respectively. The
correspondingtheory fittingparameters are: λ0=0.99, ωc =
0.4 0.5eV; λ0 =1.01, ωc=0.5eV; λ0 =1.09, ωc=0.41eV and λ0=
1.64, ωc=0.41eV.
)
a
0/.3 5
(

M
0HWH.2 3 perconducting gap or the pseudogapthe self-energyis of
C MFLformandgivenintermsofonlythetwoparameters
MD 2
0 .1 ωc, λ0 for each compound for all x. Weak dependencies
intheseparametersfromvariationinmicroscopicparam-
etersduetovaryingxorT mayoccurofcourse. We find
0.0
0.0 0.2 0.4 0.6
however that for a given compound, a single value of
(eV)
these parameters is adequate to fit all the available data
FIG. 3: The MDC half-width at half-maximum wk(ω) is
for different x and for all momentum directions.
shown for the cuts 2 ,3 and 5 of the inset of Fig. 2(a). The
experimental data for the same cuts from Fig. 2 of Ref. 8 is It is worth noting that the spectra for energies below
also shown. Note that the experiments quote are done with the pseudogap energy and T ≤ T is also scale-invariant
g
an energy resolution of 30 meV, which accountsfor thedevi- with a new scale ∝T (x) [26, 27].
g
ationfrom thetheoryatlowenergies. Higherresolution data
(ii) Suppose at certain energy ω, Eq. (6) is satis-
[15] confined to lower energies is consistent with thetheory.
fied for k = k0. Since the self-energy does not de-
pend significantly on k, we can expand the spectral
UniversalityoftheData: Thedataandthecomparison function in (k − k0). The MDC is then a Lorentzian
withexperimentsinFig.3andFig. 4(a)-(d)attesttothe with width wk given by ImΣ(ω)/v(k0) where v(k0) =
universality of the single-particle spectra of the cuprates vy(k0)+vx(k0)(kx −kx0)/(ky −ky0), is the bare veloc-
and of the quantitative success of the theory. Now we ity in the momentum-cut direction. This expansion also
consider in detail each of the points (i) to (iii) of the requires that within (k−k0) ≈ wk, the velocity vk is
experimental data and explain them successively. nearly a constant.
(i) The physical properties in any quantum critical As discussed above ImΣ(ω) increases linearly in ω for
regime are universal, controlled by the scale-invariant ω .ωc andisconstantbeyond. Thereforeifv0(k)varies
criticalfluctuations. Specifically,forωlargerthanthesu- slowly with k as in cut 2 in Fig. 2, MDC linewidths also
4
vary linearly in ω, i.e., w ∝ ω. Away from the nodal experimental evidence has also been adduced.
k
momentumdirections,v0(k)variesconsiderablyasincut Acknowledgments. CMV is especially grateful to J.
4 and higher of Fig. 2. As a result, MDCs’ linewidth Mesot for interesting him in the problem and to him, J.
deviates from the linear-ω dependence. This accounts Chang, S. Paillh`es and C. Mudry for a detailed discus-
for the MDC width of cut 5 shown as an example in sion of the data. Thanks are also due to J. Graf and A.
Fig. 3 and the higher cuts. If the MDC linewidth is Lanzara for communications regarding their data.
multipliedbythebarevelocityateachkinanydirection,
alineardependenceofthewidthwithω isobtainedboth
in theory and the experiments.
(iii) Comparing Figs. 2(d-f), we can see that there are
two distinct reasons for the “waterfalls”. If ǫ reaches [1] J. C. Campuzano, M. R. Norman, and M. Randeria, in
k
The Physics of Superconductors Vol. II, ed. K. H. Ben-
ω1−ReΣ(ω1) atk≈k0 as kis variedalongthe momen-
nemann and J. B. Ketterson (Springer, Berlin, 2004), p.
tumcut,e.g.,cut2inFig.2,ε(k)followsthe“waterfall”
167.
between ω1 and ω2, which correspond to E1 and E2 de- [2] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod.
fined in experiments. Phys. 75, 473 (2003).
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nodal cut such that the bottom of the band is very shal- [4] B. P. Xie et al., Phys.Rev.Lett. 98, 147001 (2007).
[5] T. Valla et al., cond-mat/0610249.
low, ǫk never reaches ω1 − ReΣ(ω1); e.g., cuts 5-8 in
[6] J. Graf, G.-H. Gweon, A. Lanzara, cond-mat/0610313.
Fig.2. Theobserveddispersionε(k)thenfollowsEq.(6)
[7] Z.-H. Pan et al.,cond-mat/0610442.
toitsmaximumvalueatthebottomofthebandk . For
m [8] J. Chang et al.,Phys. Rev.B 75, 224508 (2007).
higher energies,there are no solutions to Eq.(6). In this [9] W. Meevasana et al.,Phys. Rev.B 75, 174506 (2007).
casethe MDC curvesstaycenteredatkm whichleads to [10] In D.S. Inosov et al., cond-mat/0703223, it is pointed
another type of “waterfall”. E1 in this case is nearly the out that the variation of the photoemission matrix el-
energyofthe bottom ofthe renormalizedband, andgets ements with momentum can lead to misleading conclu-
continuously smaller as the bottom of the band (where sions about thespectralfunction andproducefaux“wa-
terfalls”. To avoid this one should only take seriously
the velocity is zero) becomes continuously more shallow
thespectralfunctiondeducedfromtheexperimentwhich
from the (π,π) to the (π,0) direction. The variation of
obey Kramers-Kronig relations.
the position of the “waterfall”s, Fig. 3 of Ref. [8] and [11] V. Aji and C. M. Varma, Phys. Rev. Lett. 99, 067003
Fig. 3 of Ref. [6] is thereby explained. In addition, the (2007).
linewidth is no longer given by ImΣ/v0, leading to an [12] C. M. Varma et al.,Phys. Rev.Lett. 63, 1996 (1989).
additional cusp in linewidth at E1 (e.g., cut 5 in Fig. 3). [13] T. Valla et al., Science285, 210 (1999).
However, if radial cuts are taken to avoid the shallow [14] E. Abrahams and C. M. Varma, Proc. Natl. Acad. Sci.
USA 97, 5714 (2000).
band,thepositionofthe“waterfall”inmomentumspace
[15] A. Kaminski et al.,Phys. Rev.B 71, 014517 (2005).
is always the locus of k where ε(k) ≈ ω1. This locus is [16] The band structure ǫk = −2t(coskxa + coskya) −
shown in Fig. 2(a) for radial cuts and is to be compared 4t′coskxacoskya−2t′′(cos2kxa+cos2kya)−µwasused
with data in Fig. 4 of Ref. [8] and Fig. 4 of Ref. [6]. in Ref. [8] to fit the Fermi surface of LSCO, with t′/t=
Concluding remarks. The experimental results dis- −0.144,t′′/t=0.072,andµ/t=−0.84.Weusethesame
cussedplace strongconstraintsona theory applicable to valueswitht=0.48eV consistentwithfitstoLDAcalcu-
the cuprates. Specifically, the experiments give a scat- lations, see,e.g., R.J.RadtkeandM.R.Norman,Phys.
Rev. B 50, 9554 (1994).
tering rate linear in ω up to a sharp cut-off at ω and
c [17] C. M. Varma, Phys. Rev.Lett. 83, 3538 (1999).
constant above with a coefficient which is a weak func-
[18] C. M. Varma, Phys. Rev.B. 73, 155113 (2006).
tion of k. This behavior is found in the entire ’strange [19] B. Fauqu´eet al.,Phys. Rev.Lett.96, 197001 (2006).
metal’regionofthephasediagram. Wedonotknowany [20] A. Kaminski et al.,Nature (London) 416, 610 (2002).
ideas proposed for cuprates besides those discussed here [21] Thisisastraightforward extensiontoq6=0oftheq=0
which give these properties. result of Eq. (44) in Ref. [18].
Inthispaper,wehavepointedouttheuniversalaspects [22] A. B. Migdal, Zh. Eksp. Teor. Fiz. 34 1438 (1958) [Sov.
Phys. JETP, 7 996 (1958)].
of the measured single-particle self-energy in cuprates
[23] A. A.Kordyuk et al.,Phys. Rev.B 67, 064504(2003).
and shown that its functional form and even its mag-
[24] J. Graf and A.Lanzara, private communication.
nitude is consistent with the recent microscopic theory [25] J. Chang et al.,arXiv:0708.2782.
of quantum critical fluctuations [11]. These fluctuations [26] A. Kanigel et al.,NaturePhysics 2, 447 (2006).
are predicated on the existence of an unusual symmetry [27] C. M. Varma and L. Zhu, Phys. Rev. Lett. 98, 177004
breaking in underdoped cuprates for which considerable (2007).