a r X i v :c o n d -m a t /0303018v 1 [c o n d -m a t .s t r -e l ] 2 M a r 2003
Quasiparticles in high temperature superconductors:consistency of angle resolved
photoemission and optical conductivity
A J Millis 1,H.D.Drew 2
1Department of Physics,Columbia University 538W.120th St,N.Y.,N.Y.10027
Department of Physics,University of Maryland
College Park,MD 20742
The consistency of angle-resolved photoemission and optical conductivity experiments on high temperature superconductors is examined.In the limit (apparently consistent with angle-resolved photoemission data)of an electron self energy with a weak momentum dependence and a strong frequency dependence formulae are derived which directly related quantities measured in the two experiments.Application of the formuale to optimally and overdoped Bi 2Ca 2SrCu 2O 8+δshows that the total self energy inferred from photoemission measurements cannot be interpreted as a transport scattering rate (in agreement with work of Varma and Abrahams),but that the inelastic part may be so interpreted,if Landau parameter e?ects are non-negligible.
The fundamental concept underlying the modern un-derstanding of the physics of metals is the quasiparticle 1:the idea that the crucial low energy eigenstates of in-teracting electron systems are are su?ciently similar to conventional electrons that they may be used in stan-dard ways to calculate transport and other quantities.The utility of the quasiparticle concept in the case of the high temperature copper oxide superconductors remains the subject of controversy 2.Some authors argue that an intrinsically non-fermi-liquid picture involving unconven-tional excitations or a nontrivial critical point is needed.Others assert that a more or less conventional picture of electrons scattered by some (perhaps somewhat uncon-ventional)scattering mechanism su?ces.Intermediate views exist also .
Recent improvements in angle-resolved photoemission experiments (for reviews see 3,4)have provided new in-sight into this issue.Detailed measurements of the mo-mentum,temperature and energy dependence of the elec-tron spectral function 5,6,7,8,9have demonstrated the ex-istence,near the fermi surface of optimally doped mate-rials,of reasonably well de?ned peaks.It seems natural to interpret the peak position in terms of a quasiparticle dispersion and the peak width in terms of a quasipar-ticle scattering rate.The question of the relation be-tween the scattering rate and dispersion deduced from angle resolved photoemission experiments and those de-duced from the frequency and temperature dependence of the electrical conductivity immediately arises.A similar question concerns the relation between the optical and photoemission data and the predictions of speci?c mod-els,for example various versions of the ’spin fermion’model of carriers interacting with spins 10,11,12or the ’marginal fermi liquid’model of Varma and co-workers 13.In the existing literature,it is generally assumed that knowledge of the (low frequency)self energy measured e.g.via photoemission or calculated via a model deter-mines the frequency dependent conductivity.
In this paper we examine the issue in more detail.We show how to compare,with minimal assumptions,the photoemission and optical data.We show that the scat-tering rates and mass enhancements deduced from pho-toemission do not by themselves describe the high-T c optical data 14in optimally doped cuprates:?rst,some portions of the self energy inferred from data must be dis-carded (as noted earlier by Varma and Abrahams 15)and even after this is done an extra modi?cation (in fermi liq-uid language a Landau parameter)must be introduced.Implications of these results for our physical understand-ing of the cuprates and for attempts at modelling cuprate properties are outlined.
The rest of this paper is organized as follows.Section II summarizes the formalism needed to dscuss the pho-toemission data ,section III presents the theory for the conductivity and section IV relates the theory to avail-able data.Section V is a conclusion.
The propagation of an electron in a solid is described by the electron Green function
G (r,t )=
where ψis the electron annihilation operator.Photoe-mission measurements 3,4allow the determination of the imaginary part of the Fourier transform of G ,i.e.the spectral function
A (p,ω)=ImG (p,ω)
These measurements are typically interpreted in terms of di?erences between the actual G and that corresponding to a reference system in which the electron propagates without scattering according to some reference dispersion εp .One de?nes the self energy Σvia
Σ(p,ω)=ω?εp ?G ?1(p,ω)
The self energy has both real (Σ′
)and imaginary (Σ′′
)parts.The real part depends on the choice of reference energy εp ,and the condition εp +Σ(p,ω=0)=0de-?nes the fermi surface.A common choice for reference dispersion is the dispersion εp,band predicted by the local density approximation to the Kohn-Sham band theory equations;when band theory results are needed we use the tight binding parametrization of the LDA band struc-ture determined by Andersen et.al.16and presented in more detail in the Appendix.
The high T c superconductors have a fundamentally two dimensional dispersion and a topologically simple fermi surface,so it is convenient to parametrize momentum by the reference energy εp and an angular coordinate θdescribing position on a surface of constant reference en-ergy.The observed photoemission spectra involve mainly energies ω 0.2eV in which range the calculated band dispersion is (especially in high symmetry directions)lin-ear in momentum (for the direction perpendicular to the fermi surface).For these energies the observed spectral functions display a reasonably well de?ned,reasonably symmetrical peak of approximately Lorentzian form,if A is measured as a function of p at constant ω,θ,(’MDC’)but a rather broad,asymmetric structure if A is mea-sured as a function of ωat constant θ,εp (EDC).The sharpness of the observed MDC curves implies that in the ω<0.2eV region where angle resolved photoemission data are available,the imaginary part of the self energy is small compared to the range over which εp varies.This will be important in the theory of the optical conductiv-ity to be discussed below.Further,the data suggest that in the range ω<ωc 0.2eV ,Σ′′depends reasonably strongly on ωand θand reasonably weakly on εp .How-ever,as we shall see it is likely that at higher energies,beyond the measurement range,Σdepends more strongly on εp .
It is convenient to introduce a frequency ωc separating high and low energy scales and to write
Σ(p,ω)=Σlow (ω,θ)+Σhigh (ω,p )
with Im Σlow (ω)=Im Σ(ω)for |ω|<ωc ,Im Σlow =
Im Σ(ωc )for |ω|>ωc and Re Σlow the Kramers-Kronig transform of Im Σlow .For energies well below ωc and mo-menta not too far from p F we linearize Σhigh (ω,p )in ωand p ?p F .(This procedure is unfortunately clumsy–it leads at intermediate stages in calculations to non-analytic behavior at ω=ωc ,which of course cancels from physical quantities but we shall not need to con-sider ω=ωc in this paper).
In the low energy region one may therefore write
?v θ(p ?p F )?Σlow (ω,θ))
|p F (7)
a (possibly angle-dependent)velocity which may di?er from the band velocity if Σhigh has signi?cant momentum dependence.
Eq 5implies that if measured as a function of p,A has a peak centered at a momentum p ωset by
εp ω+Σhigh (ω<<ωc ,p ω)=
v θZ (θ)
.If Σ′′is not too large then one may linearize εp near p ω,and if ωis not too large then p is close to p F so that we may linearize everything about the fermi surface,obtaining (εp +Σhigh (ω<<ωc ,p ))=
Similarly,the slope v ?
ω=?ω/?p ωof the dispersion curve ω=εp ωat p =p ωis given by
where χGI jj is the gauge invariant current-current correla-tion function,which may be expressed in the usual way in terms of electron Green functions and a vertex opera-tor,T .The photoemission data discussed above indicate that at least in the ω<0.2eV energy range,the self energy has negligible dependence on the magnitude of the momentum and is small compared to the range over
whichεp varies.In this case,the usual arguments of fermi liquid theory17may be applied and in particular by integrating over the magnitude of the energy?rst one ?nds(ωn+=ωn+?n)
χGI jj(i?n,T)=πT ωn p F(θ)dθ
Here we note that the momentum-independence of the self-energy(in the energy range of interest)implies that the’bare’current operator is the’bare’velocity vθde?ned above in Eq7(’bare’in quotes because vθdoes depend onΣhigh and the choice of reference dispersion).T is the vertex operator,de?ned by
T?x(θ,ω)=v x(θ)+T ω′ dθ′
with I the generalization to nonzero frequencies of the usual Landau interaction function(into which we have absorbed the factors of v and p F).
The vertex function has two purposes:it converts the ’single-particle’scattering rate and mass enhancements described byΣto a transport rate and mass enhance-ment described by a new functionΣtr by suppressing the contribution from’forward scattering’and it expresses the’back?ow’arising because in an interacting system motion of one electron a?ects the motion of others,so the current is not given accurately by the single particle velocity.Note that if?Σ/?p≡0at all frequencies and momenta then the current operator is given by the deriva-tive of the reference dispersionεp and the back?ow part,Λ,of the vertex correction vanishes identically.However, aΛwhich is nonnegligible in the frequency range of in-terest may arise from a?Σ/?p which is non-negligible only at frequencies beyond the frequency range of inter-est.For an explicit example see Ref18.
It is perhaps instructive to restate this conclusion in the language of the quantum Boltzmann equation.There are two nontrivial terms in this equation:one gives the interaction induced’feedback’of the excitation of one particle-hole pair on the behavior of others(i.e.accounts for back?ow);the other is the collision term represent-ing scattering of a quasiparticle from one state to an-other.The collision term involves a probability W(p,p′) for scattering an electron from state p to state p′and the resulting conductivity depends on the structure of W(p,p′):for example,if the scattering is mostly forward (W(p,p′)appreciable only for p near p′)then the scat-tering will have little e?ect on the conducitivity.The self energy is proportional to (dp′)W(p,p′)ζ(p′)withζa factor relating to the probability that state p′is avail-able as a?nal state,that any other excitation needed in the scattering process can be created,etc.Measure-ment of the self energy by itself thus does not contain enough information to reconstruct W(p,p′);and in dia-grammatic language this information is contained in the vertex function.
Eq14a cannot be analysed without further assump-tions.We shall assume that the self energy has two con-tributions:one coming from low energies and essentially observable by present-day angle-resolved photoemission experiments(this is the’quasiparticle part’of the elec-tron Green function)and one coming from high energies, not directly observable in present-day angle-resolved pho-toemission experiments but contributing indirectly to low energy physics via the Landau parameter and the velocity renormalization.We shall further assume,following15, that the low energy contribution toΣconsists of two parts:an inelastic part with a negligible momentum de-pendence but a sign?cant frequency dependence and one arising from a quasistatic scattering highly peaked in the forward direction.Thus
(15) We shall now write an expression for the low fre-quency conductivity which separates the e?ects of the ’quasiparticle’and high energy contributions.Eq15im-plies that the Landau interaction function consists of two parts:one from the forward scattering contribution,of the form2πI forward(θ;?)φ(θ?θ′)(independent ofω,ω′because the scattering is taken to be quasistatic,and with peakedness in the forward direction speci?ed byφ)and one,coming from high energies,which is independent of ?,ω,ω′in the frequency range of interest.We de?ne
(19) while the conductivity per CuO2plane becomes
We think of the function B′as the function B with the forward scattering contributions removed.
We now consider the low frequency expansion ofχjj. We expect
4 For example,if the self energy is momentum-independent
?? Σ′+?Σ′? ?i Σ′′++Σ′′? (22)
T′(θ;?)=v x(θ)+i? p F(θ′)dθ′Γtr(θ′)(25)
Combining Eqs21,and25yields a low frequency ex-
pansion for the conductivity of the form
whereσqp is the contribution obtained from the quasipar-ticle scattering and dispersion andσLP arises from the Landau or back?ow renormalization.Observe that the Landau renormalization a?ects the?rst frequency cor-rection to the conductivity,but not the dc value.
It is very convenient to write this expression in terms of an inverse’transport’mean free path W and a transport velocity de?ned analogously to Eq10:
W tr(θ)=Γ(θ)/v F(θ)(29)
v?tr(θ)=v(θ)/Λ(θ)(30) Then one has
I high(θ,θ′)v x(θ′)
W tr(θ)W tr(θ′)(32)
In particular,the dc limit of the conductivity is
and is given entirely in terms of fermi surface geometry
and the transport mean free path.
In the absence of Landau renormalization,the imagi-
nary partσ′′→σ′′qp given by
In experimental analyses of optical conductivity it is
conventional(see,e.g.19)to de?ne an optical mass and
scattering rate via
where K is a constant related to the optical spectral
weight in the frequency range of interest.The values
ofΓand m?/m depend on the value used for K,leading
to ambiguity in the values ofΓopt and m?/m opt similar
to the ambiguity in the single-particle self energy arising
from uncertainty as to the correct choice of reference ve-
locity.One quantity which is independent of the choice
of K is the ratio
5 We have selected this material because extensive photoe-
mission and optical data are available.In other high-T c
materials insu?cient information exists to perform the
analysis at present.
Experiments show that in optimally doped BSCCO the
fermi surface is to reasonable approximation a circle of
radius p F=0.71?A?1centered at the(π,π)point.The
quasiparticle velocity v?=1.8eV-?A with negligible varia-
tion around the fermi surface(in the normal state),and
the’MDC full width’2W(θ,T,ε)is reasonably well rep-
withΓ0=8×10?5 ?A?1K?1 ,Γ1=0.05?A?1Γ2=
0.01?A?1andθ=0at the antinodal point(0,π).Com-
prehensive data from other groups are not available as of
this writing but we note that the zone-diagonalω=0
MDC widths reported in Ref7are very close to the zone
diagonal numbers obtained from the formula above.
Let us?rst make the assumption that vertex correc-
tions are negligible.Then from Eqs33and34we obtain
(c is the mean interplane distance)
The two integrals are
(41) I2= dθW(θ,T,ε)2=
TABLE I:ρ[μ?cm]calculated for optimally doped Bi2Ca2SrCu2O8+δat temperatures indicated,using photoe-mission data6assuming(A)directly measured MDC width (B)Only T andω-linear parts,and compared to data14.
TABLE III:ρ[μ?cm]calculated for overdoped Bi2Ca2SrCu2O8+δat temperatures indicated,using di-rectly measured MDC width9and compared to resistivity data from the same paper.
TABLE IV:E?ective scattering rateΓ?[meV]calculated for overdoped Bi2Sr2CaCu2O8+δfrom measured MDC widths9 and compared to data20
For other doping levels the comparison is more di?cult to undertake at this stage,because the photoemission data are less extensive.A recent paper9presents evidence that in an overdoped sample of Bi2Sr2CaCu2O8+δ, W(θ,ω=0)is only weakly dependent on angle,but varies more rapidly than linearly with temperature,being about W=0.02?A?1at70K rising to0.04?A?1at120K and.057?A?1at160K.The fermi surface radius(mea-sured from the(π,π)point)is slightly larger(0.78?A?1 vs0.72?A?1in the optimally doped material studied in6. The frequency dependence of W is still found to be linear (at least at very low T).Converting from the units of9to the conventions of this paper yields,in convenient units
W OD(ω,T→0) ?A?1 =1.2×10?5ω[K](43)
The crossover betweenω-dominated and T?dominated regimes is not discussed,however as can be seen from Eq43,for a frequency corresponding to600K the frequency dependent contribution is only ?W OD=7.2×10?3?A?1negligible compared to the dc scattering rate.Therefore we may to reasonable accu-racy simply neglect theω-dependence of the scattering rate,and use a Drude model.Our results are given in Tables III and IV.
Here,as noted by the authors of Ref9the photoemis-sion and resistivity data appear to have an inconsistent temperature dependence.Also,the optical scattering rate is again underpredicted,suggesting the importance of a Landau parameter.
For underdoped materials su?cient photoemission data does not yet exist to make the comparison feasi-ble.Determining the behavior of the Landau parameter with doping would be very important.
We have presented a precise and reasonably model-independent method for comparing the photoemission and optical scattering rates and mass enhancements,and have applied the method to optimally doped and over-doped Bi2Ca2SrCu2O8+δ.Our method provides re-lations between directly measured quantities and there-fore provides an unambiguous test of whether the’MDC width’measured in angular resolved photoemission ex-periments corresponds to a transport mean free path.In agreement with previous authors,we?nd that it does not.The discrepancy is particularly severe in the case of optimally doped Bi2Ca2SrCu2O8+δwhere use of the full MDC width grossly overpredicts the resistivity.We conclude,in agreement with previous authors15that the broadening of the photomission spectra in the vicinity of the(0,π)point of the fermi surface should not be re-garded as a contribution to the transport part of the self energy.The authors of Ref15argued that the large broad-ening in this part of the zone arises from elastic scattering by out of plane impurities.In our view the ubiquity of the zone-corner broadening in cuprate materials argues instead in favor of an intrinsic,probably many-body ori-gin to the phenomenon;understanding why it does not remains a very challenging theoretical problem.How-ever,assuming that this broadening enters transport in the usual way is inconsistent with data.In our view the apparent irrelevance of the large zone-corner self-energy to the low frequency transport casts doubt on the at-tempts to describe transport and optical propeties with a’spin-fermion’model11,12,because in these models it is precisely the zone-corner scattering rate which is taken to be crucial for the conductivity.HOwever,it is possi-ble to?nd parameter regimes in spin-fermion models for which the scattering is not so strongly angle-dependent and reasonable(modulo Landau-parameter e?ects)?ts to the conductivity may be achieved12.
Our main new?nding is that even if one is selective in the part of the photoemission data one interprets as giving rise to a transport rate(for example by selecting only the’inelastic’part,agreement between calculation and experiment cannot be obtained unless a’Landau pa-rameter’(corresponding to an interaction-induced vertex correction to the conductivity)is introduced.The im-portance of this vertex correction casts doubt on most of the existing calculations of the frequency and tem-perature dependent conductivity,which neglect vertex corrections.This conclusion may be stated in a di?er-ent way.If(as,for example,was very elegantly done in Ref14)a model self energy is constructed which re-produces(without vertex corrections)the conductivity spectrum,this self energy will necessarily fail to?t the photoemission spectrum.Determining the doping de-pendence of the vertex correction factor is an important topic for future research.We suspect that this must be large,because the low frequency optical spectral weight (which is closely related to theΓ?discussed above,dis-
plays a strong doping dependence whereas the observed low energy photoemission velocity does not.We also note that information on the temperature and frequency de-pendence of the photoemission-determined quaisparticle velocity would considerably help in making this compar-ison precise.
Acknowledgements:H.D.D.was supported in part by NSF-DMR-0070959. A.J.M was supported in part by NSF DMR00081075,in part by the Institute for Theo-retical Physics in part by the Department of Energy at Brookhaven National Laboratory and in part by the ES-PCI(France),and thanks the ESPCI for hospitality.We thank Eilhu Abrahams,Nicole Bontemps,J.C.Cam-puzano,L.B.Io?e and Andres Santander for helpful discussions and Nicole Bontemps and A.Santanders for provision of unpublished data.
The natural choice for the reference dispersion is that given by band theory.There is general agreement that the dispersion in a single CuO2plane of a high T c super-conductor is given to a good approximation by
εLDA(p)=?2t(cos(p x a)+cos(p y a))
+4t′cos(p x a)cos(p y a)(44)
?2t′′(cos(2p x a)+cos(2p y a))The best choice of the parameters is a subtle issue16. Especially,the behavior in the vicinity of the zone cor-ners(0,π),(π,0)depends sensitively on details,but the zone-diagonal velocity is reasonably robust,varying be-tween about3.8?4.1eV?A depending on calcula-tion and precise doping.We adopt here t=0.38eV, t′=0.32t and t′′=0.5t16implying a zone-diagonal ve-locity?εLDA,p/?p=3.9?4.1eV?A with the variation arising mainly from nonlinearities in the dispersion.The fermi line parameter p F S≈0.7A?1.It is also sometime convenient to de?ne the kinetic energy K via