Antiferromagnetic phase diagram of the cuprate superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Antiferromagnetic Phase Diagram of the Cuprate Superconductors
L. H. C. M. Nunes b , A. W. Teixeira b , E. C. Marino a a Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brazil b Departamento de Ciˆencias Naturais, Universidade Federal de S˜ao Jo˜ao del Rei, 36301-000 S˜ao Jo˜ao del Rei, MG, Brazil
Abstract
Taking the spin-fermion model as the starting point for describing the cuprate superconductors, we obtain an e ff ectivenonlinear sigma-field hamiltonian, which takes into account the e ff ect of doping in the system. We obtain an expres-sion for the spin-wave velocity as a function of the chemical potential. For appropriate values of the parameters wedetermine the antiferromagnetic phase diagram for the YBa Cu O + x compound as a function of the dopant concen-tration in good agreement with the experimental data. Furthermore, our approach provides a unified description forthe phase diagrams of the hole-doped and the electron doped compounds, which is consistent with the remarkablesimilarity between the phase diagrams of these compounds, since we have obtained the suppression of the antifer-romagnetic phase as the modulus of the chemical potential increases. The aforementioned result then follows byconsidering positive values of the chemical potential related to the addition of holes to the system, while negativevalues correspond to the addition of electrons. Keywords: high-Tc superconductors, phase transitions
PACS:
1. Introduction
Cuprates are puzzling materials; the undoped parentcompounds are Mott insulators presenting an antiferro-magnetic (AF) arrangement at a finite N´eel temperature.As the system is doped, either with electron acceptorsor donors, which, in any case would simply mean anincrease of the amount of charge carriers to the system,it develops high-temperature superconductivity. The su-perconducting (SC) critical temperature presents a char-acteristic dome-shaped dependence on dopant concen-tration, reaching a maximum value at an optimal dop-ing and vanishing as the system is doped even further,becoming a normal metal. So far, there is no consensusregarding the microscopic mechanism which is respon-sible for the appearance of superconductivity in thosesystems. However, it is widely accepted that some sortof AF spin fluctuations are the interaction responsiblefor the Cooper pairs formation.Recently, starting from a spin-fermion model, whichhas been employed previously to describe the cupratesuperconductors [1] we have derived an e ff ective model Email addresses: [email protected] (L. H. C. M.Nunes), [email protected] (E. C. Marino) for the charge carriers [2] and the superconductivity inour model arises from a novel mechanism, that yieldsa high critical temperature which is comparable to theexperimental values. Moreover, by including doping ef-fects a dome-shaped dependence of the critical temper-ature is found as charge carriers are added to the system,in agreement with the experimental phase diagram [3].Presently instead of focusing on the SC phase, we in-vestigate the magnetic order, by calculating the dopingdependence of the N´eel temperature, which is calcu-lated providing the AF phase diagram that can be com-pared with experimental results. As shall be seen below,our results are in good agreement with the data for theYBa Cu O + x (YBCO) system [4, 5]. This calculationis an alternative to the one performed before [6], whichwas based on the fact that a skyrmion topological exci-tation is created in association to a doped charge, beingactually attached to it.Our approach provides a unified description for thephase diagrams of the hole-doped and the electrondoped compounds, which is consistent with the exper-imental results [7]. As the dopant concentration in-creases, a dome-shaped SC phase appears adjacent tothe AF order, both for electron-doped and hole-dopedcuprate compounds as well. Our calculations present Preprint submitted to Solid State Communications September 30, 2020 he suppression of the AF phase as the modulus ofthe chemical potential increases. Positive values of thechemical potential are related to the addition of holesto the system, while negative values correspond to theaddition of electrons.
2. The spin-fermion model
So far, there is no consensus regarding the minimalmodel which entails the vast phenomenology presentedby the high- T c superconductors, however, since it iswell established that superconductivity emerges fromthe CuO planes of the cuprates, the paradigmatic three-band Hubbard model proposed by Emery [8] is a goodcandidate to describe the physics in these planes. How-ever, due to its several parameters this model is a com-plicated starting model for the study of the electronicproperties in the CuO planes. Therefore, many theo-retical studies have considered instead a one-band Hub-bard model or a single band t - J model [9] which rep-resents the lower “Zhang-Rice singlet” [10] band of theoriginal three-band Hubbard model. However, the ap-proach that a simpler one-band Hubbard model or t - J models are capable of describing correctly the dopedCuO planes has been challenged recently [11] and weagree with this particular point of view that a stronglycorrelated single-band model cannot provide the appro-priate description for the cuprates. Indeed, our startingpoint for the description of the CuO planes is the spin-fermion model, which has been also extensively used todescribe the high- T c superconductors [1]Henceforth, consider a single CuO plane containinglocalized spins located at the sites of a square lattice,which is the appropriate topology for the cuprates. Thespin degrees of freedom are modelled by the spin 1 / H H = J P < i j > S i · S j , where S i is thelocalized spin operator. As the system is doped, chargecarriers are bumped into the planes and the localizedspins interact with the spin degrees freedom of the itin-erant fermionic charge carriers via a Kondo coupling, H K = J K P i S i · s i , where s i = P α,β c † i α ~σ αβ c i β denotesthe spin operator of an itinerant charge carrier, which iswritten in terms of the Pauli matrices ~σ = ( σ x , σ y , σ z )and c † i α denotes the creation operator for a charge carrierat site i with spin α = ↑ , ↓ . Combining the Heisenbergmodel, the Kondo coupling and the kinetic term associ-ated to the itinerant charge carries, one obtain the spin-fermion model.We formulate this model in the continuum limit, byemploying the spin coherent states. This amounts to re-placing the localized spin operators by S N ( x ), where S is the spin quantum number and N ( x ) is a classical vec-tor such that | N ( x ) | = N is then decomposed intotwo perpendicular components, L and n ( L · n = N ( x ) isdecomposed as N ( x ) = a L ( x ) + ( − | x | n ( x ) + O ( a ),where a denotes the lattice parameter and we also have | n ( x ) | = H H = (cid:16) ρ s |∇ n | + χ ⊥ S | L | (cid:17) + iS L · ( n × ∂ τ n ) , (1)where ρ s = JS is the spin sti ff ness, χ ⊥ = Ja is thetransverse susceptibility and the last term in the rhs ofthe above expression describes the Berry phase.On the same token, the hamiltonian density of theKondo interaction becomes H K = J K S L · X α,β ψ † α (cid:0) ~σ (cid:1) αβ ψ β , (2)where the continuum fermion field ψ α ( x ) correspondsto c i α . Also notice that the oscillating contribution fromthe antiferromagnetic fluctuations cancels out as we in-tegrate it over space [13].For the cuprates, it is well known that Dirac pointsappear in the intersection of the nodes of the d -wavesuperconducting gap and the two-dimensional (2D)Fermi surface. In that case, the quasiparticles disper-sion exhibit a Dirac-like linear energy dispersion [15].Presently, we assume that the dispersion of the chargecarriers can be linearized close to the Fermi surface andtherefore the carrier kinematics in the continuum limitis described by the Dirac-Weyl density hamiltonian, aspreviously seen in [16], H = ψ † (cid:16) i ~ v F ~σ · ~ ∇ − µ (cid:17) ψ , (3)where ψ † σ has spinorial components ψ † σ = ( ψ † σ , ψ † σ )and v F is the Fermi velocity. The indices 1 and 2 denoteodd and even lattice sites respectively. Notice that thechemical potential µ controls the total number of chargecarriers that are added to the itinerant band as the systemis doped.Hence, in the continuum limit we may express thepartition function of the spin-fermion model as the fol-lowing functional integral in the complex time represen-2ation Z = Z D ψ D ψ † D L D n δ (cid:16) | n | − (cid:17) × exp " − Z β d τ Z d x (cid:16) H − ψ † i ∂ τ ψ (cid:17) , (4)where β = / k B T , with k B denoting the Boltzmann’sconstant and T the system temperature, and H = H H + H K + H , is given by (1), (2) and (3) respectively.
3. The e ff ective nonlinear sigma model We start by Fourier transforming our model Hamil-tonian assuming that the ferromagnetic component ofthe vector spin L is small and approximately constantin space, since we investigate the system in the longrange AF state ordering. Therefore, the Kondo inter-action from (2) is approximated by H K = J K S Z d k Z d k ′ (5) × X α,β ψ † α ( k ) "Z d x e − i ( k − k ′ ) · x L · (cid:0) ~σ (cid:1) αβ ψ β ( k ′ ) ≈ J K S L · Z d k X α,β ψ † α ( k ) (cid:0) ~σ (cid:1) αβ ψ β ( k ) . (6)In this approximation, we may introduce the Nambufield Φ † = (cid:16) ψ † , ↑ ψ † , ↑ ψ † , ↓ ψ † , ↓ (cid:17) in order to expressthe fermionic part of our model Hamiltonian H ψ ≡H K + H − ψ † i ∂ τ ψ in momentum space, p = ~ k , as H ψ = Φ † ( k ) A Φ ( k ), where the matrix A above is givenby A = ˜ µ + k − L − k + ˜ µ + L − L + µ − k − L + k + ˜ µ − , (7)with the following definitions in the above expression, k ± = − ~ v F ( k x ± ik y ), ˜ µ ± = − i ω n + µ ± J K S L z and L ± = J K S (cid:16) L x ± iL y (cid:17) .We may now integrate exactly the fermionic contri-bution of the partition function in (4), which is a sim-ple Gaussian path integral and, hence, proportional todet A . Therefore,ln Z ψ = ln Y p , n det A = Z d k X n ln h ( ~ v F k ) − ( µ − − i ω n ) i + Z d k X n ln h ( ~ v F k ) − ( µ + − i ω n ) i , (8) where µ ± = µ ±| J K L | and ω n = (2 n + πβ − are the Mat-subara frequencies for fermions. Performing the sumover ω n and after some algebra we getln Z ψ = Z d k X s = ± n β ~ v F k + ln h + e − β ( ~ v F k + µ s ) i + ln h + e − β ( ~ v F k − µ s ) io , (9)which is the same result obtained for a noninteractingrelativistic system with a Zeeman term applied to it(e.g. [17]), but presently with | J K L | corresponding to anexternal “magnetic field”. Furthermore, notice that (9)yields to the partition function of a free fermion systemwhen L →
0, as should be expected.Now, we can integrate the above expression over thefirst Brillouin zone in momentum space. Also, noticethat the Fermi surface of the YBCO system has rota-tional symmetry around the point ( π, π ). Using trans-lational invariance, we shift the momentum around thispoint, thereby simplifying the integration over the firstBrillouin zone. Introducing the change of variable y = a D k /π , where a D is the lattice spacing between dopants,we getln Z ψ = π γ + π X s = ± Z dy y (cid:8) ln (cid:0) + e − γ y z s (cid:1) + ln (cid:16) + e − γ y z − s (cid:17)o , (10)where we have introduced the dimensionless parameters γ = β ~ v F π/ a D and z s = e − βµ s , with s = ±
1. Moreover,since we have assumed that | L | is small, we can expand Z ψ as a Taylor series, Z ψ = exp h A ( T , µ ) + B ( T , µ ) | L | + O (cid:16) | L | (cid:17) i , (11)where the first factor in the rhs of the above expres-sion, exp (cid:2) A ( T , µ, γ ) (cid:3) , does not contribute to the e ff ec-tive NLSM that will be obtained at the end of this sec-tion or to the calculation of the N´eel temperature in thenext section and hence shall be neglected. On the otherhand, B ( T , µ ) seen in (11) is given by B ( T , µ ) = π J K S γ ! βγ " + sinh ( βγ )cosh ( βγ ) + cosh ( βµ ) + X s = ± n ln (cid:16) + e s βµ (cid:17) + ln h + e β ( γ + s µ ) io . (12)Combining (12) and (1), the NLSM becomes˜ H H = h ρ s |∇ n | + ˜ χ ⊥ ( T , µ ) | L | i + iS L · ( n × ∂ τ n ) , (13)3here the above new transverse susceptibility˜ χ ⊥ ( T , µ ) = χ ⊥ S − β B ( µ, T ) includes the e ff ectof doping, since it depends on the chemical potential.Inserting the expression for ˜ χ ⊥ in (4) and integratingover L we finally get the partition function, except for amultiplicative factor, Z = Z D n δ (cid:16) | n | − (cid:17) exp − Z β d τ Z d x ˜ H e ff ! , (14)where the e ff ective NLSM is˜ H e ff = ρ s " |∇ n | + c ( T , µ ) | ∂ τ n | , (15)with the new spin wave velocity given by c ( T , µ ) = s ρ s χ ⊥ " − B ( µ, T ) βχ ⊥ . (16)Notice that c in the above expression reduces to c = √ ρ s χ ⊥ in the absence of the Kondo coupling, as shouldbe expected.In the present approach the spin wave velocity is fi-nite for µ =
0, which is related to the parent compoundsof the cuprates and it is an even function with respect tothe chemical potential. Indeed, positive values of µ arerelated to hole-doped cuprate superconductors, while µ < ff erence between holes movingon an otherwise inert O-lattice, in the presence of an AFbackground on the Cu sites and electrons moving on theCu sites in the presence of the same AF background andan inert O-lattice. The only di ff erence perhaps would beon the value of the exchange coupling.
4. N´eel temperature calculation and comparisonwith experimental data
We start our analysis pointing out that the Coleman-Mermin-Wagner-Hohenberg theorem prevents the ap-pearance of a long-range magnetic order at finite tem-peratures for any 2D system [18]. Therefore, we addan small out-of-plane interlayer coupling J ⊥ , so that thepartition function for a stack of CuO planes labeled bythe subscript i becomes Z = Y i Z D n i exp ( − Z ~ β d τ Z d x × δ (cid:16) | n i | − (cid:17) " ˜ H e ff ~ + ρ s α ~ ( n i + − n i ) , (17) where we have introduced the parameter α = (cid:16) / a (cid:17) J ⊥ / J .Assuming that there is an external magnetic field ap-plied to the system, one may calculate T N with sev-eral approaches, among them spin-wave theories (SWT)and field-theoretical calculations, which takes into ac-count the contribution of the spin-fluctuation excitations(neglected in the SWT). Both the standard and self-consistent SWT are shown to be insu ffi cient to quan-titatively describe the experimental data for the parentcompounds of the cuprates superconductors, while theresults calculated for a large N expansion are in goodagreement with the experiments [19]. Hence, we takethe expression for T N obtained to order 1 / N in a large N expansion from the e ff ective model in (17), which isgiven by [20] T N = πρ s ln T N α ( ~ c ′ ) + πρ s T N ! − . − , (18)where c ′ = ( a / ~ ) c . Moreover, we have set k B = c in (16) is ex-pressed in terms of µ .In the remaining of this section we compare the re-sults of the N´eel temperature as a function of dopingwith the available experimental data for YBCO. In-deed, this compound has an almost circular shape forthe Fermi surface centered at ( π, π ) in the reciprocalspace [21] and the low energy dispersion can be approx-imated to a linear relation in the vicinity of the Fermilevel, which is consistent with the kinetic term given by(3).In terms of the three-band Hubbard model parame-ters, the exchange coupling between the Cu magneticmoments is given by [22, 23] J = t pd (cid:16) ∆ E + U pd (cid:17) U d + ∆ E + U p ! (19)and the Kondo coupling of an itinerant oxygen hole spinand the nearest local Cu spin is [22, 23] J k = t pd ∆ E + U d − ∆ E ! . (20)Estimates for the microscopic parameters of the three-band model Hamiltonian have been obtained from local-density functional techniques [24]: t pd = . U d = . U p = U pd = .
87 and ∆ E = .
5, all given in units of4V. Inserting the above parameters values in (18) yields T N ≈
420 K for the undoped system ( µ = a c [ e VÅ ] -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 µ [eV] hole-dopedelectron-doped Figure 1: Spin wave velocity multiplied by the lattice parameter ac asa function of the chemical potential µ for T = K . Fom (16), one can calculate the spin wave velocityas a function of the chemical potential numerically fora particular temperature, provided the values for ρ s , χ ⊥ and γ are given. In our case, these are calculated from J K and J in (19) and (20), with the parameters valuesgiven above, and also ~ v f = .
15 eV Å, a D = .
68 Åand a = √ a D [6]. Our numerical results are shownin Fig. 1 for the particular value of T =
420 K. Noticethat for the undoped parent compound, we assume thatthe Dirac point is at µ =
0, which is exactly the casefor the cuprates. As the chemical potential increasesfor positive values, which means that holes are addedto the CuO planes, c vanishes indicating the destruc-tion of the long range AF order, what is in agreementwith the experimental results. On the other hand, fornegative values of µ , we also have that the spin wavevelocity vanishes as µ increases in modulus, which cor-responds to the doping of electrons, instead of holes, tothe system. Therefore our approach provides a unifieddescription for the phase diagrams of the p -type and n -type cuprate superconductors, where µ > µ < T N as a function of the chem-ical potential from (18). It is well known that the oc-cupancy of charge carriers increases as µ > T N [ K ] Oxygen content
Figure 2: N´eel temperature T N as a function of doping x for YBCO.Squares indicate experimental data from Ref. [5]. logical approach relating the chemical potential and thedoping as µ − µ ∝ ( x − x ) β , where µ and x are thechemical potential and the doping for which T N reacheszero respectively. Since µ is numerically calculatedand x is given by experiments, the proportionality con-stant is uniquely defined and β is the single parameterwhich has been adjusted to the available data . The re-sults of the AF phase diagram for the YBCO parametersand β ≈
5. Conclusions
Starting from the spin-fermion model we have ob-tained an e ff ective nonlinear sigma model which takesinto account the e ff ects of doping in the system. Takingthe appropriate values of the parameters for YBCO, wehave calculated the AF phase diagram as a function ofthe dopant concentration and the results presented hereare in good agreement with the experimental data [4, 5].Notice that several studies indicate that there is a quan-tum phase transition as the ratio J / J K increases, startingfrom a Fermi liquid state, the system becomes a spinliquid [25]. Presently, on the other hand, the interac-tion couplings are not model parameters, but providedby the experimental data available for the YBCO com-pound and henceforth it is remarkable that the systempresents an AF arrangement in the absence of dopingfor the given values of J and J K , which is in agreementwith the phenomenology of the cuprates. Moreover, the5alculated N´eel temperature is consistent with the ex-perimental data for T N , since the results were obtainedwithout resorting to any kind of parameter adjustment.Recently, also starting from the spin-fermion model,we have obtained an e ff ective interaction among thecharge carriers of the system, which produces a dome-shaped SC high critical temperature versus doping [2]plot that qualitatively reproduces the SC phase diagramexperimentally observed. Hence, combining our results,the following picture emerges: for the e ff ective modelof the localized spins presented here, where the itiner-ant fermions have been integrated out, we get the sup-pression of the magnetic order as charge carriers areadded to the system; for the e ff ective model of the itiner-ant fermionic fields, where the localized magnetic mo-ments have been integrated out, we have the appearanceof a dome-shape SC critical temperature with the addi-tion of charge carriers [2]. Therefore, we have a the-ory where the AF order is suppressed and the SC phasearises as charge carriers added to the system, which isthe phenomenology observed for several strongly cor-related electronic systems [26]. However, the completephenomenology of the cuprates, including its strangemetal behaviour in the underdoped regime remain un-explained [27].Furthermore, our results for positive values of thechemical potential are related to the addition of holes,while negative values correspond to the addition of elec-trons. Therefore, our results provide a unified descrip-tion for the phase diagrams of the hole-doped and theelectron doped compounds, which is consistent with thedata provided for the p -type and n -type cuprate super-conductors [7]. Notice that further studies are requiredin order to address the quantitative di ff erences betweenthe values for the ordering temperatures of the com-pounds, since both T c and T N depend on the couplings J and J K of the spin-fermion model, which are givenin terms of the microscopic parameters of the specificsystem under investigation. Nevertheless, taking a sin-gle set of parameter values and assuming that dopinge ff ects produce only a weak dependence on J and J K ,the results presented naturally lead to a qualitative sym-metry between hole and electron doped cuprates.Also, it is worthy to mention that the approach em-ployed here is not just restrained to the cuprates, butmight be applied to several other compounds, since wehave only assumed 2D spatial dimensionality and thepresence of a relativistic dispersion relation for the itin-erant fermionic fields. Remarkably, the presence ofDirac electrons have also been observed for the cupratesand other d -wave superconductors [15], what mightsuggest that Dirac electrons might play a relevant role in some of those condensed matter systems.
6. Acknowledgements
E. C. Marino has been supported in part by CNPq andFAPERJ.
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