Interplay between Superconductivity and Antiferromagnetism in a Multi-layered System H. T. Quan and Jian-Xin Zhu∗ Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Based on a microscopic model, we study the interplay between superconductivity and antifer- romagnetism in a multi-layered system, where two superconductors are separated by an antiferro- magnetic region. Within a self-consistent mean-field theory, this system is solved numerically. We find that the antiferromagnetism in the middle layers profoundly affects the supercurrent flowing 1 acrossthejunction,whilethephasedifferenceacrossthejunctioninfluencesthedevelopmentofan- 1 0 tiferromagnetism in the middle layers. This study may not only shed new light on material design 2 and material engineering, but also bring important insights to building Josephson-junction-based quantumdevices, such as SQUIDand superconductingqubit. n a PACSnumbers: 74.50.+r,75.70.-i,74.81.-g J 5 I. INTRODUCTION merical diagonalization, we are able to solve the Hamil- ] n tonian, which allows us to study the interplay between o DSC and AFM in detail. Varying some external param- As a remarkable aspect of high-T superconductivity, c c eters, such as the SC phase difference across the system - its unique characteristics may result from the competi- δϕ and Coulomb interaction U in the AF layers leads to r tion between more than one type of order parameter.1–3 p interesting results about the interplay. u Historically,thetypicalantagonisticrelationshipbetween s superconductivity and magnetism has led researchers to t. avoidusingmagnetic elements,suchasiron,aspotential a building blocks of superconducting (SC) material. How- m ever, it is now well accepted that the unconventional - superconductivity emerging upon doping is closely re- d lated to the antiferromagnetism (AFM) in parent com- n o pounds4–7. Recent research theme in high-Tc cuprate c community centers on how to establish the connection [ between antiferromagnetic (AF) and d-wave SC (DSC) 2 orderings8, i.e., whether they compete with each other FIG. 1: Schematic drawing of a multi-layered system. The v or coexist microscopically. This issue is a subject of cur- middle region consists of AF layers (in X-Y plane) and the 1 rent discussions. Depending on material details, some two sides are SC layers with d-wave pairing symmetry. The 9 compounds show the coexistence of DSC and AFM9 phases of the SC order parameter (or the pairing potential) 1 while others exhibit microscopic separation of these two in thefirst (far left) and thelast (far right) layerare fixedat 2 phases10. Engineered heterogeneous systems and multi- ϕL and ϕR, respectively. . 9 layered high-T cuprates offer a unique setting to study c 0 the interplay between DSC and AFM. In these systems, 0 the disorder effect can be minimized significantly with 1 atomically smooth interfaces. Experimentally, no mix- II. MODEL AND SETUP : v ing of DSC and AFM was reported in the heterostruc- Xi ture artificially grown by stacking integer number of Themodelsystemunderconsiderationisschematically La Sr CuO and La CuO layers11. Meanwhilemi- r 1.85 0.15 4 2 4 presented in Fig. 1, for which the Hamiltonian can be a croscopic evidence for the uniform mixed phase of AFM written as: and DSC in outer CuO planes was reported12 on a Hg- 2 basedfive-layeredcuprate. Theoretically,Demleret al.13 H = − t c† c + U (n − 1)(n − 1) havestudiedtheproximityeffectandJosephsoncoupling i,j i,σ j,σ i i,↑ 2 i,↓ 2 iX,j,σ Xi intheSO(5)theoryofhigh-T superconductors. Depend- c 1 ingonthethickness,themiddleantiferromagneticregion − V n n −µ n , (1) ij i j i could behave like a superconductor, metal or insulator. 2Xi,j Xi Ontheotherhand,itwasshownintheperturbationthe- ory14 that the spin exchange coupling in the insulating wheret denotesthehoppingintegralbetweenthenear- i,j AF layer can allow the tunneling of Cooper pairs. est neighbor sites. For in-plane (X-Y plane) hopping, t = t, and for inter-layer (Z direction) hopping, t = Inthisarticle,westudymicroscopicallyamulti-layered i,j i,j system with two superconductors separated by AF lay- t⊥. ci,σ (c†i,σ) is the annihilation (creation) operator of ers. Through self-consistent mean-field theory and nu- electrons on the ith lattice site with spin σ (σ =↑,↓). 2 The quantity n = c† c is the number operator on a genuine cuprate compound, there exists the inter-layer i,σ i,σ i,σ theithlatticesite. BothU andV arepositive. U indi- coupling, which is usually weakerthan the in-plane hop- i ij i cates the on-site repulsive Coulombinteraction,which is ping. Because of the inter-layer coupling, there arises nonzero only in the middle layers, and V describes the interesting interplay between SC layers and AF layers ij in-planenearestneighborattractiveinteraction,whichis (see Fig. 1). nonzero only in the SC layers. We are going to fix the ThestrategyofsolvingtheHamiltonianisbrieflysum- phase of the SC order parameter in the first (far left) marized as follows: Due to the translational symmetry and the last (far right) layer at ϕ and ϕ , respectively, L R in the 2D X-Y plane, the 3D problem can be decom- which can be introduced by a gauge flux. posed into a 2D (X-Y) plus 1D (Z) problem. In the X-Y When there is no inter-layer coupling, t = 0, the ⊥ plane, ~k = (k ,k ) is a good quantum number, where system is decomposed into individual two-dimensional x y (2D) systems. It is known that in a 2D SC layer, the kx = 12(Nnxx −Nnyy)π, ky = 21(Nnxx +Nnyy)π, andnx =−21Nx, in-plane nearestneighborattractiveinteractionVij leads −21Nx +1, ···, 21Nx −1, ny = −12Ny, −21Ny +1, ···, toanonzeroenergygap∆withd-wavesymmetry,andin 1N −1. Inorderto simplify the Hamiltonian,we adopt 2 y a 2D AF layer the on-site repulsive interaction U leads the mean-field approximation,which leads to the follow- i to a nonzero (π,π) spin density wave (SDW) order. In ing Bogoliubov-de Gennes equation15: ξ~knδnm+tnm −U2nMnδnm ∆~k,nδnm 0 u~αk,m,↑ u~αk,n,↑ Xm  −∆U2n~∗k,Mn0δnnδmnm ξ~k+∆Q~~∗k,n+δQ~n0,mnδ+nmtnm −ξ−−~kU2nnδMn0mnδ−nmt∗nm −ξ−(−∆~k+U~k2Q~+n)QM~n,δnnnδδmnnmm−t∗nm uv~k~ααkv++~kα,QQ~~m,,mm,↓,,↓↑ =Eα uv~k~ααkv++~kαQ,Q~~n,,,nn↓,,↓↑  , (2) wQ~h=ere(πm,πa)nddennoltaebtehlethwealvaeyevrenctuomrsbeinr;t~kheanfidrs~kt+BrQ~illwouitihn uis~αk,tnh,↑ev~kαF,,en∗r,↓m−iud~αki+stQ~r,inb,u↑tvi~kαo+,n∗Q~.,nT,↓h,raonudgfh(Ethαis)=sim1/p(l1ifi+caetβiEoαn), zone of the X-Y plane, tmn = t⊥(δn,m+1 +δn,m−1) de- the multi-layered system is then solved numerically. We scribes the nearest neighbor inter-plane hopping, ξ~kn = areespeciallyinterestedintheinterplaybetweentheDSC −2t(coskx + cosky) − µ is the normal metal disper- and the AFM, which is characterized by the SDW Mn sion relation, µ is the chemical potential, the variables and the SC pairing wavefunction Ψ . We will focus n Un is equal to U in the AF layers and zero otherwise on zero temperature T = 0. The size of X-Y plane is while Vn is equal to V in the SC layers and zero oth- Nx =Ny =40. The total layer number is Nz =22. The erwise. The superconducting pairing potential ∆~k,n = two middle layers are AF layers (Un = U, Vn = 0, for V Ψ (cosk −cosk )/2, where Ψ is the superconduct- n=11,12),andthe restareSC layers(U =0, V =V, n n x y n n n ing pairing wavefunction. Obviously, the superconduct- forn=1,2,···,10,andn=13,···,22). Forsimplicity, ing pairing potential, which is the SC order parameter we choose t = 1, t = 0.1, V = 1.5, and µ = 0. We will ⊥ has the same phase as that of the superconducting pair- varythecontrolparameters,suchasU andδϕ=ϕ −ϕ , L R ing wavefunction. M = hn i −hn i describes the tostudythe competitionbetweenDSC andAFM.When n n,↑ n,↓ SDW in the n-th layer. The average electron number the phase difference δϕ is introduced, there is a current hn i, hn i and the SC pairing wavefunction Ψ can flowing across the system and its expression is given by: n,↑ n,↓ n bedeterminedself-consistentlythroughiterationoverthe t e following relation: I = 2 ⊥ Im uα,∗ uα + ~k ↔~k+Q~ f 2 (cosk −cosk ) βE N X~k,α (cid:26)h ~k,n,↑ ~k,n+1,↑ (cid:16) (cid:17)i + x y α Ψ = g(u,v)tanh , (3a) n N X~k,α 2 2 +hv~kα,n,↓v~kα,,n∗+1,↓+v~kα+Q~,n,↓v~kα+,∗Q~,n+1,↓if−(cid:27), (4) 1 2 hnn,↑i= N X~k,α(cid:12)(cid:12)u~αk,n,↑+u~αk+Q~,n,↑(cid:12)(cid:12) f(Eα), (3b) where f± =f(±Eα). (cid:12) (cid:12) 1 2 hn i= vα +vα f(−E ), (3c) III. INFLUENCE OF ANTIFERROMAGNETISM n,↓ N X~k,α(cid:12)(cid:12) ~k,n,↓ ~k+Q~,n,↓(cid:12)(cid:12) α ON SUPERCONDUCTIVITY (cid:12) (cid:12) where N = Nx×Ny is the sites number in a 2D plane. In the model introduced above, the change of ~k samples half of the first Brillouin zone. g(u,v) = the Coulomb interaction U in the middle layers 3 can drive a metal-Mott insulator phase transition at I(cid:144)10-3 a certain value U . Correspondingly, the multi- c layered system changes from a superconductor/normal 4 metal/superconductor (SNS) weak link16 to a su- perconductor/insulator/superconductor(SIS) Josephson 3 junction17. The studies of SNS and SIS junctions with static potential barrier have been well documented16–18. 2 Here we start from a microscopic model and drive the middlelayerstochangefromoneelectronicstateintoan- 1 other by tuning the on-site Coulomb interaction U. We ∆j are interested in how the SC pairing wavefunction and 0.2 0.4 0.6 0.8 1.0 thecurrentacrossthejunctionchangewiththeCoulomb Π interaction U. We plot the current as a function of the I(cid:144)10-5 phase difference δϕ in Fig. 2. It can be seen that below 6 a threshold value of U ≈ 0.8, the current is a piecewise c periodic function. In each periodic region, it varies lin- 4 early with δϕ. This agrees with the result obtained in 2 Ref. 16 for a junction with a normalmetal. We also find ∆j tchuarrtewntheanstahefuCnocutiloonmbofinδtϕercahctainogneissglarargdeuratllhyanwUithc,tthhee -2 0.5 1.0 1.5 2.0 Π increase of U. When U is in the range 0.8 < U < 1.5, -4 the currentshows a shape intermediate between a piece- -6 wise linear function and a sinusoidal function. Similar resultshavebeenobservedinRefs.7,13,16 whenonevaries FIG.2: Currentacrossthejunctionasafunctionofthephase the temperature or the thickness of the middle layersin- difference δϕ of two superconductors for different U. Here stead of the the Coulomb interaction U. When U ≈2.0, the current is well approximated by a sinusoidal func- VU == 10.,5,0.t1⊥, 0=.2,0.·1·,·,N1x.5=, aNndy δ=ϕ4=0, 0Nz∼=π.22D. oUwnp ppaanneell:: tion of δϕ (see Fig. 2), which is a typical feature of the U =2.0 and δϕ=0∼2π. dc Josephsonjunction (JJ) current17. The magnitude of current decreases rapidly with the increase of U. Inordertohaveabetterunderstandingoftheinfluence becomes more opaque and the superconducting pairing ofAFMonDSC,weplottheabsolutevalueandthephase wavefunctionis completely suppressedin this region. As of the SC pairing wavefunction in Fig. 3. It can be seen a result, the tunneling of Cooer pairs will be completely that when U is small, there is a nonzero paring poten- quenched, and the two comprising superconductors be- tial in the middle layers due to the proximity effect, and come decoupled. the two superconductors are weakly linked by the nor- We also would like to mention that when the middle mal metallic layers. In this weak link regime, the phase layersareinthenormalmetallicstate,thecurrentacross of the SC pairing wavefunction is almost linear in layer the junction is a mesoscopiceffect20. The currentis pro- number n. This means that when the middle layers are portional to the phase gradient of the SC pairing wave- inthe normalmetallic state,it haslittle influence onthe function. That is, when one increases the number of SC SC layers on two sides. However, when U is larger than layers,the currentwilldecreaseandfinallyvanish. How- Uc ≈ 0.8, SDW develops, and the middle layers become ever,whenthe middle layerisinthe AFstate,the whole Mottinsulator. Quantummechanically,Cooperpairscan system becomes a JJ. The current of a JJ is no longer still tunnel through the SDW region when U is not very a mesoscopic effect. It is completely determined by the large. We call this intermediate U regime as the tun- phase difference of the SC pairing wavefunction on both neling regime. From the weak link regime to tunneling ends of the system. Therefore, it will not decrease with regime,thephaseoftheSCparingwavefunctionchanges theincreaseofthenumberoftheSClayers. Inthissense gradually from a (nearly) linear function to a step func- we say that AFM insulator layers enhance supercurrent. tion of the layer number (see Fig. 3). This corresponds to the observation that the current shape changes from beingpiecewiselineartosinusoidal. Therefore,whenU is IV. INFLUENCE OF SUPERCONDUCTIVITY very large, the current-phase relation becomes identical ON ANTIFERROMAGNETISM to the famous Josephson relation17,19. Here for the first time, we establish the connection between the profile of Intheabovediscussion,wefindtheprofoundinfluence the SC phase distribution (Fig.3)andthe current-phase of the Coulomb interaction U (and hence AFM) of the dependence (Fig. 2), which we believe is rather intrigu- middlelayersontheDSC.Wenowstudytheback-action ing. of DSC on AFM. We will fix the phase difference δϕ at When one continues to increase U, the middle layers different values, and see if the SDW in the middle layers 4 0.3 !! ! δϕ= π n δϕ=3π/4 0.4 δϕ=0 0.2 0.3 !!n!"10"2 W D 6 S 0.1 0.2 3 0.1 n 0 10 11 12 13 0 0.3 0.6 0.9 1.2 1.5 U n 0 5 10 15 20 FIG. 4: (Color online) SDW in the middle layer (n = 11) as a function of the Coulomb interaction for different fixed Arg! δϕ. Here U = 0 ∼ 1.5, V = 1.5, t⊥ = 0.1, Nx = Ny = 40, n Nz =22, δϕ=0 (solid), 3π/4 (dashed),and π (dot dashed). 2.0 inwhichthe SDW willdevelopforaninfinitesimal U. In the present model (t 6= 0), when δϕ = 0, the SDW in ⊥ 1.5 themiddlelayersissuppressedduetotheinter-layercou- pling, which weakens the perfect nesting at Q~ = (π,π). 1.0 TheSDWwillnotdevelopuntilU islargerthanathresh- old value U (see Fig. 4). Hence both t and δϕ will in- c ⊥ 0.5 fluence the AFM. The results in Fig. 4 indicate that the AFM shows a crossoverbehavior for δϕ=π rather than n a criticality as obtained for δϕ=0. For δϕ∈(0,π), the 5 10 15 20 SDW critical point U is less than that for δϕ=0. This c result is in agreement with that in Ref. 13. Inordertohaveabetterunderstandingoftheinfluence FIG. 3: Absolute value (Up panel) and argument (Down of the DSC on the AFM in the middle layers,we plot in panel)ofsuperconductingpairingwavefunctionΨn asafunc- Fig. 5 the local density of state (LDOS), as given by tionoflayernumber. ThephaseofSCpairingwavefunctionis the same as that of the pairing wavefunction. Here V =1.5, 1 2 ∂f(ω) t⊥ =0.1, Nx =Ny =40, Nz =22, δϕ=3π/4. U is fixed at ρ (E) = uα +uα − (,5a) different values: U =0, 0.1, 0.2, ···, 1.5. n,↑ N X~k,α(cid:12)(cid:12) ~k,n,↑ ~k+Q~,n,↑(cid:12)(cid:12) (cid:20) ∂ω (cid:21) (cid:12) (cid:12) 1 2 ∂f(ω′) ρ (E) = vα +vα − (,5b) changeswithδϕ. WeplotinFig.4theSDWasafunction n,↓ N X~k,α(cid:12)(cid:12) ~k,n,↓ ~k+Q~,n,↓(cid:12)(cid:12) (cid:20) ∂ω′ (cid:21) of U for various values of δϕ. Clearly, the SDW in the (cid:12) (cid:12) middle layers is influenced by δϕ in certain range of U. whereω =E−E andω′ =E+E . FromFig.5,wecan α α Whenδϕ=π,thecriticalvalueUc,afterwhichtheSDW seeasharpintensitydecayofLDOSatEF whenδϕ=0. develops, is much smaller than in the case of δϕ = 0, Whenδϕ=π,theLDOSexhibitsapeakstructureatE . F though the current is zero in both cases. This is very Sincethe developmentofthe SDWis alsodeterminedby similartothe resultinRef. 13thatthe middle AFlayers the normal-state DOS at low energies, it explains the will be influenced by the current fed into the junction. different SDW critical behavior between the cases with When the Coulomb interaction is very weak, the SDW δϕ=π and δϕ=0. doesnotdevelopforδϕ6=π,andthemiddlelayersarein thenormalmetallicstate. Incontrast,whentheCoulomb interaction is very strong, e.g. U > 1.2, the AFM is V. CONCLUSION AND DISCUSSION robust against the phase difference across the junction and strongly suppresses the pairing wavefunction in the We have studied the interplay between the DSC and middle region (see Fig. 3). Only when U is small, is the AFM in a multi-layered system. It is revealed that the emergence of the SDW sensitive to the phase difference. AFlayershaveprofoundinfluence onthe DSC.A metal- When switching off the interlayer coupling (t = 0), Mott insulator phase transition makes the multi-layer ⊥ the system is decoupled into 2D systems. The DSC on system change from a SNS to a SIS junction. We have twosideswillnotinfluence theAFMinthe middlelayer. alsoestablishedtheconnectionbetweentheprofileofthe The middle layer is described by a 2D Hubbard model, SCphasedistributionandthecurrent-phasedependence. 5 of the phase difference. Meanwhile, the phase difference 0.3 δϕ=π across the system will dramatically influence the devel- δϕ=0 opment of SDW in the middle layers. The results from 0.25 oursimulationsmayshednewlightonmaterialengineer- S ing. TheDSCandAFMcancoexistinasinglelayer,but O D 0.2 they also compete with each other. Another important L insight from the present study is the microscopic model of JJ. JJ has been widely used in a lot of quantum de- 0.15 vices,suchasinSQUIDforultra-sensitivemagneticfield measurement and superconducting qubit21 for quantum 0.1 -0.9 -0.6 -0.3 0 0.3 0.6 0.9 computing. Our study provides a microscopic theory for E-E JJ, hence has possible applications in building adaptive F JJ based devices by engineering the middle layers of the FIG. 5: (Color online) LDOS of the middle layer (n = 11) JJ. Finally, with the development of experimental tech- ρ11,↑(E). Here U = 0, V = 1.5, t⊥ = 0.1, Nx = Ny = 40, niques, it is now possible to synthesize heterogeneous Nz =22, δϕ=0, and π. systems with layer by layer at atomic scale22. We ex- pect that our results can be experimentally verified, as some experiments on engineered material have been re- We findthat whenthe barrierof the middle layersis not ported11,12,22. high enough (metal or semiconductor regime), the mid- dle layers cannot maintain a step-function-like phase of Acknowledgments: We thank A. V. Balatsky, two superconductors on two sides. Accordingly, the cur- Quanxi Jia, A. J. Taylor, and S. Trugman for useful rent across the junction will deviate from a sinusoidal discussions. This work was carried out under the aus- function of the phase difference. 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