24 January 2000

Ž .Physics Letters A 265 2000 221–224www.elsevier.nlrlocaterphysleta

Spin-polaron excitations in the two-dimensional Kondo latticewith spin frustration

A.F. Barabanov, A.A. Kovalev, O.V. Urazaev, A.M. BelemoukInstitute for High Pressure Physics, Troitsk, Moscow region, 142092, Russia

Received 1 October 1999; accepted 8 November 1999Communicated by V.M. Agranovich

Abstract

In the framework of the two-dimensional Kondo lattice model with hoppings between first-, second-, and third-nearestneighbors the spin-polaron spectrum is investigated. A complex structure of a spin-polaron and an increase of frustration inspin subsystem leads to the evolution in the excitation band close to the evolution on hole hopping which is given byARPES experiments in HTSC. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 75.50.Ee; 74.20.Mn; 71.38; 75.30.Mb

The dispersion of the hole quasiparticle in thenormal state of high temperature superconductorsŽ .HTSC is of great importance in the microscopic

Ž .theory of the two-dimensional 2D antiferromagnetŽ .AFM . The results of angle resolved photoemission

Ž .spectroscopy ARPES reveal a similar origin of theundoped valence-band and optimally doped conduc-tivity band in the cuprates. The most interesting

w xresults of ARPES are 6,9–14 :1. the spectrum of a single hole in 2D AFM has the

lowest band bottom located close to the pointŽ .pr2,pr2 in k-space and the spectrum close toband bottom is isotropic;

2. as the doping increases, the non-rigid band evolu-tion is observed. Such an evolution of spectrumleads to closing of the so-called pseudogap for theFermi surface when the doping is optimal forHTSC-T ;c

3. as doping x increases the Fermi surface centerŽ .shifts to p ,p point, the Fermi surface takes the

form close to the Fermi surface for tight-bindingband model with 1qx filling, and the experimentmanifests a violation of the Luttinger theorem;

4. ARPES demonstrates a sudden drop in the inten-Ž .sity of ARPES peaks as k goes from pr2,pr2

Ž . Ž .to p ,p or 0,0 ;5. another striking aspects of ARPES experiments is

an existence of very flat bands close to Fermisurface in doped CuO planes of HTSC. The2

existence of such flat regions is often used toexplain a large value of the superconducting tem-perature.In the present work we show that many features

of 2D AFM mentioned above are explained qualita-tively in the framework of the spin-polaron conceptif one takes into account a complex structure of aspin-polaron and the frustration of the spin subsys-tem. It is generally believed that the frustration in the

Žspin subsystem is governed mainly by doping see,w x.for example, Ref. 16 . In our calculation below

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00794-X

( )A.F. BarabanoÕ et al.rPhysics Letters A 265 2000 221–224222

namely, the frustration in the spin subsystem modi-fies the spin-polaron spectrum. That is why when weshall speak about the evolution of the spectrum ondoping we suppose some qualitative equivalence be-tween the doping and frustration in the spin subsys-tem.

The problem is studied on the bases of a Kondolattice Hamiltonian with hopes to first-, second- and

Žthird-nearest neighbors such a Hamiltonian givesw x.some mapping for the Emery model 4,5 .

The Hamiltonian has the form:

H tot sH qH qH ,0 1 2

H s t aq a q t aq aÝ Ý0 g rqg r d rqd rr,g r,d

q t aq a ,Ý 2 g rq2 g rr ,2 g

q q̃H sJ a S a ,Ý1 r r rr

1 1a a a aH s I S S q I S S . 1Ž .Ý Ý2 1 rqg r 2 rqd r2 2r , g r , d

Here gs"g "g are the vectors of the nearestx y

neighbors; d and 2 g are the vectors of the second-and third-nearest neighbors. The Fermi operator aq

r ,s

creates a hole with spin Ss1r2 at the lattice site rand spin projection sr2. We omit the spin indexes

ˆand in the insite Kondo interaction H we use the˜ a a ˆnotation S sÝ S s . The Hamiltonian T de-r s r

scribes the hole hopping between the first-, the sec-ond- and the third-nearest neighbors with the ampli-

ˆtudes t ,t ,t . The exchange Hamiltonian I corre-g d 2 g

sponds to AFM frustrated interaction between theŽ .spins, p 0FpF1 – the frustration parameter,

Ž .I s 1yp I and I spI are the exchange constants1 2

for the first- and second-nearest neighbors.Treating this model in the two sublattice spin

w xstructure Schrieffer pointed out 8 that the takinginto account of certain coherence factors is of crucialimportance. We shall treat the system in the spheri-

w xcally symmetric approach 3 . In order to take intoaccount the mentioned coherent factor at finite tem-perature one should treat the spin-polaron operator asthe superposition of local spin polaron states A ,r ,1

A and A , A – two local polarons dressed in ar ,2 r ,3 r ,4

spin waves with momenta q which are close toŽ .antiferromagnetic vector Qs p ,p , where the spin

w xsusceptibility is sharply peaked. 1,15 These opera-tors have the form:

˜ ˜Ž1.A sa ; A sS a ; A sQ a ;r ,1 r r ,2 r r r ,3 r r

˜Ž1. ˜ ˜Ž1. y1 i qŽ rqr . ˜A sQ S a ; Q sN e S ,Ýr ,4 r r r r rqr

r ,qgV

< <Vs q , "pyq -L . 2Ž .� 4x , y

w xAs was shown in Refs. 1,15 , two operators A ,r ,3Ž .A spin-polarons of intermediate radius are impor-r ,4

tant for description of the splitting of the localpolaron bands. We take the region V as a squareregion around Q describing by q which satisfy

Žinequality linear dimension L four squares VsL=L related to corners of the first Brillouin zoneŽ ..BZ . Below we take Ls0.5p .

Ž .To obtain the one-hole spectrum ´ k we use theŽ .two-time retarded matrix Green’s function G v,ki j

for the Fourier transformation of the operators A .r , i

We use the standard Mori–Zwanzig projectionw xmethod 7 to solve the Green function equations and

Žtreat the problem in the limit of low doping we will.consider doping x-0.25 . The projection method

Žmatrix elements have a cumbersome form they will.be reproduced elsewhere and they are expressed

over a static spin–spin correlation function Thisspin–spin correlation function was calculated in aspherically symmetric approach taking into account a

w xfrustration 3 . As a result the Green’s functions havea form:

4 Ž l .Z kŽ .i jG v ,k s . 3Ž . Ž .Ýi j

vy´ kŽ .lls1

Ž . Ž1.Ž .In particular, the value of Z k sZ k corre-h 1,1

sponds to the number of bare oxygen holes with thefixed spin s and the momentum k in the state< : Ž .k,s of the lowest quasi-particle band ´ k which1

will be represented below. Let us mind that spectralŽ . Ž .weight residue Z k satisfy the sum rulei, j

Ž s.Ž .Ý Z k s1. This means that in this model thes 11

Luttinger theorem is not fulfilled and the maximumnumber of holes per cell is equal to two despite thepresence of four bands.

The qualitative reduction of CuO plane effec-2Ž w x.tive three band model see Ref. 2 to the model

under discussion leads to the following choice ofHamiltonians parameters: t s 0.5t ; t s 0.25t ;g d

( )A.F. BarabanoÕ et al.rPhysics Letters A 265 2000 221–224 223

t s0.2t ; Js3t ; Is0.4t , where ts t 2r´ . Be-2 g p d

low we put ts1. We present the results of thespin-polaron spectrum calculation for temperatureTs0.3 I. Let us mention that for the frustrationvalue parameter p)0.1 the spectrum has a weaktemperature dependence up to the temperature T;

0.5I.Ž .In Fig. 1 the spectrum ´ k is presented by the1

Ž .equal-energy lines ´ k sconst for the case of the1

frustration parameter ps0.1. We suppose that thisvalue of frustration correspond to a small doping

Ž .case. As it is seen from Fig. 1 the minimum of ´ k1Ž .is close to pr2,pr2 and the spectrum is rather

isotropic near the band bottom. The dispersion alongŽ . Ž . Ž .the directions G 0;0 yM p ,p , and GyX p ,0 y

ŽM reproduces the ARPES results compare for ex-ample Fig. 1 with the dispersion in Fig. 3 in Ref.w x.14 . The band width is also close to ARPES resultsif we take a realistic value of parameter ts0.4 eV.The spin-polaron spectrum in Fig. 1 has a symmetryclose to the symmetry of magnetic BZ but the

Ž1.Ž .residues Z k have the symmetry of the initial BZ1,1Ž Ž1.Ž .. Ž1.Ž Ž ..for example Z ks0,0 s0.1/Z ks p ,p1,1 1,1

. Ž .s0.21 . The residues Z k are close to 0.3 near1,1

band bottom. If we suppose that the frustration valueparameter ps0.1 is related to the doping xs0.032then the Fermi surface corresponds to the equi-en-

Fig. 1. Spectrum of the lowest band for ps0.1, T s0.3 I. Equal-Ž .energy lines ´ k sconst, line ´ sy4.44 correspond to the1 f

Fermi surface for doping xs0.032.

Ž .Fig. 2. Spectrum of the lowest band for ps0.3, T s0.3 I. aŽ .Equal-energy lines ´ k sconst, line ´ sy4.437 correspond to1 f

Ž .the Fermi surface for doping xs0.25. b Residues contour linesŽ1.Ž . Ž .Z k sconst for ´ k ps0.3.1,1 1

ergetical line ´ sy4.44. If we determine a pseudo-fŽ . Ž w x.gap d as ds´ p ,0 y´ see Ref. 12 then the1 f

value of the pseudogap is equal 0.19.In Fig. 2 we give the spin-polaron spectrum and

Ž1.Ž .residues Z k for the frustration parameter value1,1

ps0.3 which we relate to the doping xs0.25. Theevolution of the spectrum on frustration leads to theFermi surface ´ sy4.437. which crosses X–Mf

w xboundary 12 . Such a crossing is given by ARPESŽ .experiments for optimally doped or overdoped

Ž w x.HTSC see, for example, Refs. 13,12,14 . Thepseudogap in this case is equal ds0.04. Let us

Ž1.Ž .mention that the residues Z k have a sudden drop1,1Ž . Ž .on the pr2,pr2 to 0,0 cut. We don’t reproduce

the second band, but the residues in the second bandŽ2.Ž .Z k are not small in the region close to G and we1,1

( )A.F. BarabanoÕ et al.rPhysics Letters A 265 2000 221–224224

Fig. 3. The electronic spectrum along symmetrical lines G – M –Ž . Ž . Ž . Ž .XG ; G s 0,0 , X s p ,0 , 0,p , Ms p ,p . The energy unit is

Ž .eV. Zero energy corresponds to the Fermi level. a The first andsecond bands for ps0.3, T s0.3 I. The part of the both bandswhich has considerable residues is denoted by the solid line. The

Ž .part which has small residues is denoted by the dotted line. bŽThe first band for ps0.1, T s0.3 I the cut from Fig. 1 along

.symmetrical lines .

think that namely this band is given by ARPES forthe mentioned area. Another important feature of thecalculated spectrum is the existence of flat bandregion close to point X.

In Fig. 3 we give the electronic spectrum alongŽsymmetric directions of BZ this is turned over hole

. Ž .spectrum . The energy unit is eV. In Fig. 3 a forps0.3, Ts0.3 I we take into account the second

Ž .band the lower band on the figure . The part of boththe bands that has considerable residues is denotedby the solid line. So in the energetical region fromy0.15 eV to 0.10 eV we obtain the effective elec-tronic band. The part that has small residues is

denoted by the dotted line. Fig. 3b represents thelower electronic band for ps0.1, Ts0.3 I. Thecomparison of Fig. 3a,b with the ARPES experimentŽ w x.see Fig. 3 from Ref. 14 shows that our spin-polaron approach qualitatively reproduce the evolu-tion of the hole spectrum from undoped to optimallydoped regime.

Acknowledgements

ŽThis work was supported by the INTAS project. ŽNo.97-11066 and by RFFI project No. 98-02-

.17187 .

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