Reconstruction of the Fermi surface in the pseudogap state of cuprates

a r X i v :0806.3826v 1 [c o n d -m a t .s u p r -c o n ] 24 J u n 2008

Pis’ma v ZhETF

Reconstruction of the Fermi surface in the pseudogap state of cuprates

E.Z.Kuchinskii,M.V.Sadovskii 1)

Institute for Electrophysics,RAS Ural Branch,620016Ekaterinburg,Russia

Submitted *

Reconstruction of the Fermi surface of high-temperature superconducting cuprates in the pseudogap state is analyzed within nearly exactly solvable model of the pseudogap state,induced by short-range order ?uctu-ations of antiferromagnetic (AFM,spin density wave (SDW),or similar charge density wave (CDW))order parameter,competing with superconductivity.We explicitly demonstrate the evolution from “Fermi arcs”(on the “large”Fermi surface)observed in ARPES experiments at relatively high temperatures (when both the amplitude and phase of density waves ?uctuate randomly)towards formation of typical “small”electron and hole “pockets”,which are apparently observed in de Haas -van Alfen and Hall resistance oscillation experiments at low temperatures (when only the phase of density waves ?uctuate,and correlation length of the short-range order is large enough).A qualitative criterion for quantum oscillations in high magnetic ?elds to be observable in the pseudogap state is formulated in terms of cyclotron frequency,correlation length of ?uctuations and Fermi velocity.

PACS:71.10.Hf,74.72.-h

Pseudogap state of underdoped copper oxides [1,2,3,4]is probably the main anomaly of the normal state of high temperature superconductors.Especially striking is the observation of “Fermi arcs”in ARPES experi-ments,i.e.parts on the “large”Fermi surface around the diagonal of the Brillouin zone (BZ)with more or less well de?ned quasiparticles,while the parts of the Fermi surface close to BZ boundaries are almost completely “destroyed”[5,6,7].

However,the recent observation of quantum oscilla-tion e?ects in Hall resistance [8],Shubnikov -de Haas [9]and de Haas -van Alfen (dHvA)oscillations [9,10]in the underdoped YBCO cuprates,producing evidence for rather “small”hole or electron [11]pockets of the Fermi surface,seemed to contradict the well established ARPES data on the Fermi surface of cuprates.

Qualitatvive explanation of this apparent contra-diction was given in Ref.[12]within very simpli?ed model of hole-like Fermi surface evolution under the ef-fect of short-range AFM ?uctuations.Here we present an exactly solvable model of such an evolution,which is able to describe continuous transformation of “large”ARPES Fermi surface with typical “Fermi arcs”at high-enough temperatures into a collection of “small”hole-like and electron-like “pockets”,which form due to elec-tron interaction with ?uctuations of SDW (CDW)short-range order at low temperatures (in the absence of any kind of AFM (or charge)long-range order).We also for-mulate a qualitative criterion for observability of quan-tum oscillation e?ects in high-magnetic ?eld in this,rather unusual,situation.

a ,

π

2 E.Z.Kuchinskii,M.V.Sadovskii

a ,

π

π2

κq 2y

+κ2

(3)

where κ=ξ?1

is determined by the inverse correlation

length of short-range order.Phase φis also considred to be random and distributed uniformly on the interval [0,2π].

Factorized form of

(3)

is

not very important physi-cally,but allows for an analytic solution for the Green’function which takes the form [18]:G D (ε,k )=

ε?ε(k +Q )+ivκ

?k x,y .

Spectral density A (ε,k )=?1

2

1?

ε(?)

k

2[ε(k )

±ε(k +Q )],E k =

ε(?)2

k

+|D |2

(6)

which is just the same as dispersion in the case of the

presence of long-range AFM order.Equation E (?)

k =0determines the hole “pocket”of the Fermi surface,around the point (π2a )in the Brillouin zone,while E (+)k =0de?nes the electronic “pockets”,centered around (πa ),as shown in Fig.1(a).

Quasiparticle damping as given by the imaginary part of (5)is,in fact,changing rather drastically as par-ticle moves around the “pocket”of the Fermi surface.Being practically zero in the nearest to point Γ=(0,0)nodal (i.e.on the diagonal of the Brillouin zone)point of this trajectory on the hole “pocket”,it becomes of the

order of ≈v n

F κin the far (from Γ)nodal point.Here

we have introduced v n

F =|v x (k |)+|v y (k )||ε(k )=0,k x =k y —particle velocity at the nodal point of the “bare”

Reconstruction of the Fermi surface 3

a

is the velocity in the antinodal point of “bare”Fermi surface.

Of course,the complete theory of quantum (Shub-nikov -de Haas or de Haas -van Alfen)oscillations for such peculiar situation can be rather complicated.How-ever,a rough qualitative criterion for the observability of quantum oscillations in our model can be easily for-mulated as follows.E?ective width of spectral densities in our model,which determines smearing of the Fermi surfaces,can be roughly compared to impurity scatter-ing contribution to Dingle temperature and estimated as τ?1～

～

ωH

a

?1(7)

where ωH is the usual cyclotron frequency.

As the most unfavourable estimate (overestimating the e?ective damping)we take:

=

v n F for hole “pocket”

v a

F for electronic “pocket”(8)Experimentally oscillations become observable in mag-netic ?elds larger than 50T [8,9,10,11].Taking

the large correlation length ξ=100a and magnetic ?eld H =50T we get ωH τ≈0.8for hole “pocket”and ωH τ≈1.3for electronic “pockets”in our model.Thus we need rather large values of correlation length ξ～50?100a for oscillations to be observable.How-ever,this value may be smaller in the case of cyclotron mass larger than the mass of the free electron used in the above estimates.

From Luttinger theorem it follows that the number

of electrons per cell is given by n =2a 2S

fs

π2

calculating the

area S sh of the “shadow”Fermi surface (ε(k +Q )=0)around the point M (πa ).Obviously S sh =S fs .Then,in the limit of |D |→0,for hole doping we get [19,20]:

p =1?n =a

2S h

?S ′e

π2

(9)

where S h is the area of hole “pocket”and S ′

e is the area

of the parts of electronic pocket inside the quarter of

a

S , p

D/t

Fig.2.The area of hole (a 2S h

π2

)“pockets”in the quarter of Brillouin zone and “doping”

p =a 2(S h ?S ′e

)

Fig.3.Formation of the Fermi“arcs”in the high-temperature regime of pseudogap?uctuations(n=0.9, t′/t=?0.4,κa=0.01).Shown are intensity plots of spectral density forε=0.(a)–?=0.2t;(b)–?=0.4t;(c)–?=0.7t;(d)–?=1.5t;Dashed line denotes“bare”Fermi surface.

distribution of amplitude?uctuations given by Rayleigh distribution[18]:

P D(|D|)=2|D|

?2 (10)

Then the averaged Green’s function takes the form: G?(ε,k)= ∞0d|D|P D(|D|)G D(ε,k)(11) Pro?les of the spectral density at the Fermi level(ε=0), corresponding to(11)and di?erent values of the pseu-dogap width?are shown in Fig. 3.The growth of the pseudogap width leads to the“destruction”of the Fermi surface close to Brillouin zone boundaries and for-mation of typical Fermi“arcs”,qualitatively(and quan-titatively)similar to that obtained in our previous work [13,14]and in accordance with the results of ARPES experiments,which are typically done at much higher temperatures,than experiments on quantum?uctua-tions.

This work is supported by RFBR grant08-02-00021and RAS programs“Quantum macrophysics”and “Strongly correlated electrons in semiconductors,met-als,superconductors and magnetic materials”.MVS is gratefully acknowledges a discussion with L.Taillefer at GRC’07Conference on Superconductivity,which stim-ulated his interest in this problem.