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0 0 0 2 n Destruction of long-range antiferromagnetic order a J 0 2 by hole doping ] l e - r F. Carvalho Dias,† I. R. Pimentel and R. Orbach∗ t s . t a Department of Physics and CFMC, University of Lisbon, 1649 Lisboa, Portugal m - ∗ d Department of Physics, University of California, Riverside, CA92521, USA n o c [ Abstract 1 v Westudytherenormalizationofthestaggeredmagnetizationofatwo- 9 8 dimensional antiferromagnet as a function of hole doping, in the frame- 2 1 workofthet-Jmodel. Itisshownthatthemotionofholesgeneratesdecay 0 0 of spin waves into ”particle-hole” pairs, which causes the destruction of 0 / t a thelong-rangemagneticorderatasmallholeconcentration. Thiseffectis m mainly determined by the coherent motion of holes. The value obtained - d n for the critical hole concentration, of a few percent, is consistent with o c experimental data for thedoped copper oxide high-Tc superconductors. : v i PACS: 74.25.Ha, 75.40.Cx, 75.50.Ep X r a 1 One of the interesting features of the copper oxide high-T superconductors c is the dramatic reduction, with doping, of the long-range magnetic order of theirparentcompounds.1 Theundopedmaterials,e.g. La CuO ,areantiferro- 2 4 magnetic (AF) insulators. Doping, e.g., in La Sr CuO , introduces holes in 2−δ δ 4 the spin lattice of the CuO planes, and the long-range AF order is destroyed 2 at a small hole concentration, δ 0.02. The CuO planes are described by c 2 ∼ a spin-1/2 Heisenberg antiferromagnet on a square lattice, with moving holes that strongly interact with the spin array. The motion of holes generates spin fluctuations that tend to disrupt the AF order. It has been shown that hole motionproducesstrongeffectsonthemagneticproperties,leadinginparticular to significant softening and damping of the spin excitations as a function of doping.2−5 The critical concentration δ where long-range magnetic order dis- c appears has often been identified with the concentration where the spin wave velocity vanishes. However important damping effects occur, which have to be taken into account. In particular, all spin waves become overdamped at a concentrationwellbelowthe oneforwhichthe spinwavevelocityvanishes,sug- gestingthatthelong-rangeAFordermaydisappearatasmallerconcentration.5 The critical hole concentration δ is provided by the vanishing of the staggered c magnetization order parameter. In this workwe use the t-J modelto calculatethe doping dependence of the staggered magnetization of a two-dimensional antiferromagnet, and determine thecriticalholeconcentrationδ . Itisshownthatthemotionofholesgenerates c decayofspinwavesinto”particle-hole”pairs,leadingtobroadeningofthespin- 2 wave spectral function. This broadening gives rise to a drastic reduction of the staggered magnetization and the disappearance of the long-range order at low doping, in agreement with experiments. Such a process was suggested some years ago by Ramakrishnan.6 The vanishing of the staggered magnetization as aconsequenceofdoping,hasalreadybeenstudiedinthe t-J modelby Ganand Mila,7 considering the scattering of spins by moving holes, and by Khaliullin and Horsch,8 considering spin disorder introduced by the incoherent motion of holes. We describe the copper oxide planes with the t-J model, 1 H = t c† c +H.c. +J S S n n , (1) t−J − iσ jσ i· j − 4 i j <iX,j>,σ(cid:16) (cid:17) <Xi,j>(cid:18) (cid:19) where, S = 1c† σ c is the electronic spin operator, σ are the Pauli ma- i 2 iα αβ iβ trices, n = n +n and n = c† c . To enforce no double occupancy of i i↑ i↓ iσ iσ iσ sites, we use the slave-fermion Schwinger Boson representation for the elec- tron operators c = f†b , where the slave-fermion operator f† creates a hole iσ i iσ i and the boson operator b accounts for the spin, subject to the local con- iσ straint f†f + b† b = 2S. For zero doping, the model (1) reduces to a i i σ iσ iσ P spin-1/2 Heisenberg antiferromagnet, exhibiting long-range Néel order at zero temperature. The Néel state is represented by a condensate of Bose fields b = √2S and b = √2S, respectively in the up and down sub-lattices, and i↑ j↓ the bosons b = b and b = b are then spin-wave operators on the Néel i↓ i j↑ j background. After Bogoliubov-Valatin transformation on the boson Fourier transform b = v β† + u β , with u = (1 γ2)−1/2+1 /2 1/2, v = k k −k k k k − k k (cid:2)(cid:0) (cid:1) (cid:3) 3 sgn(γ ) (1 γ2)−1/2 1 /2 1/2, and γ = 1(cosk +cosk ), we arrive at − k − k − k 2 x y (cid:2)(cid:0) (cid:1) (cid:3) the effective Hamiltonian 1 H = f f† V(q, k)β +V(q k,k)β† + ω0β†β , (2) −√N q q−k − −k − k k k k Xq,k h i Xk having S = 1/2 and N sites in each sub-lattice. In (2), the first term, with V(q,k)=zt(γ u +γ v ),representstheinteractionbetweenholesandspin q k q+k k waves resulting from the motion of holes with emission and absorption of spin waves,andthesecondtermdescribesspinwavesforapureantiferromagnet,with dispersion ω0 = (zJ/2) 1 γ2 1/2, z being the lattice coordination number k − k (cid:0) (cid:1) (z =4). The staggered magnetization is given by M =<Sz > <Sz >=2<Sz >, (3) ↑ − ↓ ↑ with <Sz >= <Sz >, ↑ i i∈XS(↑) where the sum is over the up sub-lattice. Using the Schwinger boson represen- tationforthe spinoperatorSz = 1(c† c c† c ), andthe bosoncondensation i 2 i↑ i↑− i↓ i↓ associated to the Néel state, one has Sz = (1 h†h )(S b†b ), which, after i − i i − i i Bogoliubov-Valatintransformation, leads to M =(1 δ)[M ∆M], (4) 0 − − 4 where M =2 NS v2 (5) 0 " − k# k X is the staggered magnetization for a pure antiferromagnet, and ∆M = 2 (u2 +v2)<β† β > k k −k −k Xk h +u v (<β β >+<β† β† >) . (6) k k k −k −k k i The prefactor in (4) accounts for the spin dilution due to doping, being negli- gible for small hole concentrations. In (5), the order parameter is considerably reduced by quantum fluctuations, to 0.6 2NS. With zero doping the ex- ≃ × pectation values in (6) vanish and ∆M = 0. However, in a doped system the motion of holes generates spin fluctuations, giving rise to nonzero expectation values in (6), even at zero temperature, and then ∆M =0. 6 In order to calculate the staggered magnetization for the doped system, we needthespin-waveGreen’sfunctions,definedasD−+(k,t t′)= i β (t)β†(t′) , − − T k k D E D+−(k,t t′) = i β† (t)β (t′) , D−−(k,t t′) = i β (t)β (t′) , − − T −k −k − − hT k −k i D E D++(k,t t′) = i β† (t)β†(t′) , where represents an average over − − T −k k h i D E the groundstate. Thespin-waveGreen’sfunctions satisfythe Dysonequations: Dµν(k,ω) = Dµν(k,ω)+ Dµγ(k,ω)Πγδ(k,ω)Dδν(k,ω), where µ,ν = . 0 γδ 0 ± The free Green’s functionsPare D−+(k,ω) = 1/(ω ω0 + iη), D+−(k,ω) = 0 − k 0 1/( ω ω0 +iη), D−−(k,ω) = D++(k,ω) = 0, (η 0+), and Πγδ(k,ω) are − − k 0 0 → theself-energiesgeneratedbytheinteractionbetweenholesandspinwaves. We calculate the spin-wave self-energies in the self-consistent Born approximation (SCBA), which corresponds to consider only ”bubble” diagrams with dressed 5 hole propagators, as illustrated in Figure 1. These diagrams describe decay of spin waves into ”particle-hole” pairs. The spin-wave self-energies can then be written in terms of the hole spectral function, ρ(q,ω), as 1 Πγδ(k,ω)= Uγδ(k,q)[Y(q, k;ω)+Y(q k,k; ω)], (7) N − − − q X with +∞ 0 ρ(q,ω′)ρ(q k,ω′′) Y(q, k;ω)= dω′ dω′′ − , − ω+ω′′ ω′+iη Z0 Z−∞ − and, U−−(k,q) = U++(k,q) = V(q, k)V(q k,k), U+−(k,q) = V(q − − − k,k)2, U−+(k,q) = V(q, k)2. The relations Π−+(k,ω) = Π+−( k, ω) and − − − Π−−(k,ω)=Π++(k,ω)areverified,thelastimplyingD−−(k,ω)=D++(k,ω). The SCBAprovidesa spectralfunction forthe holes,9−15 thatis composedofa coherent quasi-particle peak and an incoherent continuum, taking the approximate forms, respectively, ρcoh(q,ω) = a δ(ω ε ) with a (J/t)2/3, and 0 q 0 − ≃ ρincoh(q,ω) = hθ(ω zJ/2)θ(2zt+zJ/2 ω ) with h (1 a )/2zt, the 0 | |− −| | ≃ − energies are measured with respect to the Fermi level, and the quasi-holes fill up a Fermi surface consisting of pockets, of approximate radius q = √πδ , F locatedatmomentaq =( π/2, π/2)inthe Brillouinzone,the quasi-particle i ± ± dispersion being, near q , written as ε = ε +(q q )2/2m, with an ef- i q min i − fective mass m 1/J. The self-energies will then present three contributions, ≃ Πγδ(k,ω) = Πγδ(k,ω)+Πγδ (k,ω)+Πγδ (k,ω), corresponding, respectively, c,c c,ic ic,ic totransitionsofholeswithinthecoherentband,betweenthecoherentandinco- herentbands,andwithinthe incoherentband. We havecalculatedthe different contributions to lowest order in the hole concentration δ. 6 The change in the staggered magnetization induced by the interaction between holes and spin waves (6), is written in terms of the spin-wave Green´s functions, as 2 +∞ dω ∆M = 2ImD+−(k,ω) − (1 γ2)1/2 2π k − k Z0 X (cid:2) γ Im D++(k,ω)+D−−(k,ω) . (8) k − (cid:0) (cid:1)(cid:3) To lowest order in the hole concentration δ, (8) gives 2 γ ∆M = k ReΠ−−(k,ω0) − (1 γ2)1/2 −2ω0 k k − k (cid:20) k X +∞ dω ImΠ−+(k,ω) ImΠ−−(k,ω) + +γ . (9) π (ω+ω0)2 k ω2 (ω0)2 Z0 (cid:18) k − k (cid:19)(cid:21) Evaluating (9), one finds that the behavior of the staggered magnetization is essentiallydeterminedby the coherentmotionofholes,andmoreover,that itis governedby the imaginary part of the self-energies, i.e., the contributions 2 t 1 ImΠ−±(k,ω) = zJ√δa2 F−±(k,ω) c,c 0 J √πk(1 γ2)1/2 (cid:18) (cid:19) − k 1 s2(g)θ(1 s(g)) 1 s2( g)θ(1 s( g)) , × − −| | − − − −| − | hp p i with F−−(k,ω) = [cos(k g)+cos(k g)] γ [cosk cos(k g)+cosk cos(k g)], x y k x x y y − F−+(k,ω) = [(cosk cosk )(cos(gk ) cos(gk ))/2 x y x y − − (1 γ2)1/2(sink sin(gk )+sink sin(gk )) 2(1 γ2) , − − k x x y y − − k i 7 where s(g)=(1 g)k/2q and g =2ω/Jk2, while F − t 2 γ k2 (sin2k +sin2k ) ReΠ−−(k,ω0)= zJδa2 k x y . c,c k − 0 J 8(1 γ2 (k/2)4) (1 γ2)1/2 (cid:18) (cid:19) − k− − k Regarding the incoherent contributions, π ImΠ−±(k,ω)+ImΠ−± (k,ω)= zJ√δ(1 a )2 c,ic ic,ic ∓ − 0 32 ω t ω t a 0 1 I (ω)+ 4 +1 I (ω)+4 I (ω) 1 2 3 × 4J − J − 4J J (1 a ) (cid:20)(cid:16) (cid:17) (cid:18) (cid:19) − 0 (cid:21) 1 k3 (sin2k +sin2k ) θ(2q k)+√δG±(k) x y θ(k 2q ) , × 2√π(1 γ2)1/2 F − (1 γ2)1/2 − F (cid:20) − k − k (cid:21) with G−(k)=γ , G+(k)=1+(1 γ2)1/2, k − k I (ω)=θ(ω/4J 1)θ(2t/J +1 ω/4J), 1 − − I (ω)=θ(ω/4J 1 2t/J)θ(4t/J+1 ω/4J), 2 − − − I (ω)=θ(2t/J +1/2 ω/4J)θ(ω/4J 1/2), 3 − − is one to two orders of magnitude smaller than ImΠ−±(k,ω), while c,c t 1 a t ReΠ−−(k,ω0)+ReΠ−− (k,ω0)=zJ√δ(1 a )2 ln2+ 0 ln 1+4 c,ic k ic,ic k − 0 J 4 1 a J (cid:20) − 0 (cid:18) (cid:19)(cid:21) 1 k3 (sin2k +sin2k ) θ(2q k)+√δγ x y θ(k 2q ) , × 2√π(1 γ2)1/2 F − k (1 γ2)1/2 − F (cid:20) − k − k (cid:21) is of the same order of magnitude as ReΠ−−(k,ω), though smaller. c,c Asaresult,wefindthatthestaggeredmagnetization(4),calculatedwith(9), is strongly reduced with doping, vanishing at a small hole concentration, as il- lustratedinFigure2. Thereductionofthestaggeredmagnetizationisgenerated by the imaginary part of the self-energies, ImΠ−±, which imply broadening of 8 the spin-wave spectral function. The real part of the self-energy, ReΠ−−, gives rise to an increase of the staggered magnetization, which however is one order of magnitude smaller than the decrease due to the imaginary part of the self- energies. Theincreaseofthe staggeredmagnetizationarisingfromtherealpart of the self-energy results from the coherent motion of holes, while the incoherent motion leads to a decrease, though with a smaller amplitude. We find a critical hole concentration that for t/J =3 is δ 0.07, whereas for t/J =4 is c ≃ δ 0.05. The value for δ , of a few percent, is consistent with experimental c c ≃ data for the copper oxide high-T superconductors. The critical hole concen- c tration δ is smaller than the hole concentration leading to the vanishing of c the spin wave velocity (e.g., δ 0.23 for t/J = 3), or the concentration at sw ≃ which all spin waves become overdamped (δ∗ 0.17 also for t/J = 3), in the ≃ same approach.5 This is because the staggeredmagnetization is specially influ- enced by the strong damping effects induced by hole motion. Khaliullin and Horsch8 did not consider damping effects, and concluded that the long-range order disappears as a result of the incoherent motion of holes, however having estimated a decrease of the staggered magnetization due to the incoherent motion of holes that is over one order of magnitude larger than the one calculated by us. Gan and Mila7 studied the effects of damping on the staggered magnetization, though considering the scattering of spins by holes, i.e. a four-particle interaction with ”uncondensed” bosons. Our results, giving the vanishing of the magnetization for a hole concentration where the spin wave velocity is still finite,suggestthat,evenwhenlong-rangeorderhasdisappeared,strongAFcor- 9 relationspersist,whichallowspinwaveexcitationstoexist,forlengthscalesless than the magnetic correlation length. This is in fact experimentally observed.1 In conclusion, we have shown that the staggered magnetization of a two- dimensional antiferromagnet is significantly reduced as a function of doping due to the strong interaction between holes and spin waves. The motion of holes generates decay of spin waves into ”particle-hole” pairs, leading to the destructionofthelong-rangemagneticorderatasmallholeconcentration. This effect is mainly determined by the coherent motion of holes. The calculated criticalholeconcentrationisinagreementwithexperimentaldataforthedoped copper oxide high-T superconductors. c WealsonotethatNMRmeasurements,reportedinRef. 16,showdampingof thelow-energyspinexcitationsinthedopedCuO planesdueto“particle-hole” 2 excitations, which supports the mechanism for destruction of the long-range order presented in this work. We thank T. Imai for bringing Ref. 16 to our attention. References: †[email protected] 1R.J. Birgeneau and G. Shirane, in Physical Properties of High Temperature Superconductors, ed. D.M. Ginzberg (World Scientific, New Jersey, 1990). 2J.Gan,N.AndreiandP.Coleman,J.Phys.: Condens. Matter3,3537(1991). 3I. R. Pimentel and R.Orbach, Phys. Rev. B 46, 2920 (1992). 4K.W. Becker and U. Muschelknautz, Phys. Rev. 48, 13826 (1993). 10