SIN and SIS tunneling in cuprates. Ar. Abanov and Andrey V. Chubukov Department of Physics, University of Wisconsin, Madison, WI 53706 (February 1, 2008) 0 We calculate the SIN and SIS tunneling conductances for the spin-fermion model. We argue that 0 at strongspin-fermion coupling,relevanttocuprates,bothconductanceshavedipfeatures nearthe 0 threshold frequencies when a tunneling electron begin emitting propagating spin excitation. We 2 argue that the resonance spin frequency measured in neutron scattering can be inferred from the n tunneling data byanalyzing thederivativesof SINand SISconductances. a J PACS numbers:71.10.Ca,74.20.Fg,74.25.-q 0 The electron tunneling experiments are powerful tools tances display a behavior which is generally expected in 1 to study the spectroscopyof superconductors. These ex- a d wave superconductor: SIN conductance is linear in − ] periments measure the dynamical conductance dI/dV voltage for small voltages, and has a peak at eV = ∆ n through a junction as a function of applied volt- where ∆ is the maximum value of the d wave gap [1], o − age V and temperature [1,2]. For superconductor- while SIS conductance is quadratic in voltage for small c - insulator-normal metal (SIN) junctions, the measured voltages, and has a near discontinuity at eV = 2∆ [2]. r dynamical conductance is proportional to the elec- These features have been explained by a weak-coupling p u tron density of states (DOS) in a superconductor theory, without specifying the nature of the pairing in- s N(ω) = (1/π) dk ImG(k,ω) at ω = eV [3]. teraction [11]. However, above the peaks, both SIN and t. Forsuperco−nductorR-insulator-superconductor(SIS)junc- SIS conductances have extra dip/hump features which a tions, the conductance dI/dV G(ω = eV), where become visible ataroundoptimaldoping,andgrowwith m ω ∝ G(ω) = dΩN(ω Ω) ∂ N(Ω) is proportional to the underdoping [1,2]. We argue that these features are sen- - derivativRe0over volt−age ofΩthe convolution of the two sitive to the type of the pairing interaction and can be d DOS [3]. explained in the spin-fluctuation theory. n o For conventional superconductors, the tunneling ex- Asawarm-upforthestrongcouplinganalysis,consider c periments have long been considered as one of the most first SIN and SIS tunneling in a d wavesuperconductor [ relevant ones for the verification of the phononic mech- in the weak coupling limit. In th−is limit, the fermionic 1 anism of superconductivity [4]. In this communication self-energy is neglected, and the superconducting gap v we discuss to which extent the tunneling experiments on does not depend on frequency. For simplicity, we con- 5 cuprates may provide the information about the pair- sider a circular Fermi surface for which ∆ =∆cos2φ. k 2 ing mechanismin high-T superconductors. More specif- We begin with the SIN tunneling. Integrating c 1 ically, we discuss the implications of the spin-fluctuation G(k,ω)=(ω+ǫ )/(ω2 ǫ2 ∆2) over ǫ =v (k k ) 1 k − k− k k F − F mechanismofhigh-temperaturesuperconductivityonthe we obtain 0 forms of SIN and SIS dynamical conductances. 0 ω 2π dφ 0 The spin-fluctuation mechanism implies that the pair- N(ω)=Re / ing between electrons is mediated by the exchange of 2π Z0 ω2 ∆2cos2(2φ) t − a theircollectivespinexcitationspeakedatornearthean- 2 K(∆/ωp) for ω >∆ m tiferromagnetic momentum Q. This mechanism yields a = , (1) π (cid:26)(ω/∆)K(ω/∆) for ω <∆ - d wave superconductivity [6], and explains [5,7] a num- d b−erofmeasuredfeaturesinsuperconductingcuprates,in- where K(x) is the elliptic integral. We see that n cluding the peak/dip/humpfeatures inthe ARPESdata N(ω) ω for ω ∆ and diverges logarithmically as o c near (0,π) [8], and the resonance peak below 2∆ in the (1/π)l∼n(8∆/∆ ≪ω ) for ω ∆. At larger frequencies, : inelastic neutron scattering data [9]. Moreover, in the N(ω) gradua|lly−dec|reases t≈o a frequency independent, v spin-fluctuation scenario, the ARPES and neutron fea- normal state value of the DOS, which we normalized to i X turesarerelated: thepeak-dipdistanceinARPESequals 1. The plot of N(ω) is presented in Fig 1a. r the resonance frequency in the dynamical spin suscepti- We now turn to the SIS tunneling. Substituting the a bility [7]. This relation has been experimentally verified resultsfortheDOSintoG(ω)andintegratingoverΩ,we in optimally doped and underdoped YBCO and opti- obtaintheresultpresentedinFig1b. Atsmallω,G(ω)is mally doped Bi2212 materials [10]. Here we argue that quadratic in frequency, which is an obvious consequence the resonance spin frequency can also be inferred from ofthefactthatthe DOSislinearinω. Atω =2∆,G(ω) the tunneling data by analyzing the derivatives of SIN undergoes a finite jump. This discontinuity is related to and SIS conductances. the fact that near 2∆, the integral over the two DOS The SIN and SIS tunneling experiments have been includes the region Ω ∆ where both N(Ω) and N(ω performed on YBCO and Bi2212 materials [1,2]. At Ω) are logarithmically≈singular, and ∂ N(Ω) diverges a−s Ω low/moderate frequencies, both SIN and SIS conduc- 1/(Ω ∆). The singular contribution to G(ω) from this − 1 Similarly,expandingΣ2 F2 neareachofthe maxima dI/dV a) dI/dV b) of the gap we obtain Σ2(φ−,ω) F2(φ,ω) (ω ∆)+ − ∝ − B(φ φ )2, where B >0. Then 1 8∆ − max πln |∆−ω| dφ˜ ln Ω ∆ ω ∼ω2 N(ω) Re | − | (4) ∆ 1 ∝ Z Bφ˜2+(Ω ∆) ≈− √B q − 1 1 This result implies that the SIN conductance in an ar- bitrary Fermi liquid still has a logarithmic singularity at eV = ∆, although its residue depends on the strength of the interaction. The logarithmical divergence of the ∆ eV 2∆ eV DOS causes the discontinuity in the SIS conductance by FIG. 1. The behavior of SIN and SIS tunneling conduc- the same reasons as in a Fermi gas. tances, dI/dV, in a d−wave BCS superconductor (figures a In the presence of impurities, the logarithmical singu- and b, respectively) larity is smeared out, and the DOS acquires a nonzero . value at zero frequency (at least, in the self-consistent T matrix approximation [12]). However, for small con- − centration of impurities, this affects the conductances region can be evaluated analytically and yields only in narrowfrequency regions near singularities while ∞ away from these regions the behavior is the same as in 1 dx ln x 1 G(ω)= P | | = sign(ω 2∆) (2) the absence of impurities. −π2 Z−∞ x+ω 2∆ −2 − We now show that a strong spin-fermion interaction − givesriseextrafeaturesintheSISandSINconductances, We see that the amount of jump in the SIS conductance not present in a gas. The qualitative explanation of is a universal number which does not depend on ∆. these features is the following. At strong spin-fermion The results for the SIN and SIS conductances in a coupling, a d-wave superconductor possesses propagat- d wave gas agree with earlier studies [11]. In previous − ing, spin-wave type collective spin excitations near an- studies,however,SISconductancewascomputednumer- tiferromagnetic momentum Q and at frequencies below ically, and the universality of the amount of the jump at 2∆. These excitations give rise to a sharp peak in the 2∆ was not discussed, although it is clearly seen in the dynamical spin susceptibility at a frequency Ω < 2∆ numerical data. res [9], and also contribute to the damping of fermions near We now turn to the main subject of the paper and hot spots (points at the Fermi surface separated by Q), discusstheformsofSINandSISconductancesforstrong where the spin-mediated d wave superconducting gap spin-fermion interaction. − is at maximum. If the voltage for SIN tunneling is such Wefirstshowthatthefeaturesobservedinagasarein thateV =Ω +∆,thenanelectronwhichtunnelsfrom fact quite general and are present in an arbitrary Fermi res the normal metal, can emit a spin excitation and fall to liquidaslongastheimpurityscatteringisweak. Indeed, the bottom of the band (see Fig. 2a) loosing its group in an arbitrary d wave superconductor, − velocity. Thisobviouslyleadstoasharpreductionofthe current and produce a drop in dI/dV. Σ(φ,ω) N(ω) Im dφ , (3) Similar effect holds for SIS tunneling. Here however ∝ Z (F2(φ,ω) Σ2(φ,ω))1/2 − one has to first break an electron pair, which costs the energy 2∆. After a pair is broken, one of the electrons whereφistheanglealongtheFermisurface,andF(φ,ω) becomesaquasiparticleinasuperconductorandtakesan and Σ(φ,ω) are the retarded anomalous pairing vertex energy ∆, while the other tunnels. If eV = 2∆+Ω , and retarded fermionic self-energy at the Fermi surface res the electron which tunnels through a barrier has energy (the latter includes a bare ω term in the fermionic prop- ∆+Ω , and can emit a spin excitation and fall to the agator). The measured superconducting gap ∆(φ) is a res bottom of the band. This again produces a sharp drop solution of F(φ,∆(φ))=Σ(φ,∆(φ)). in dI/dV (see Fig. 2b). Intheabsenceofimpurityscattering,ImΣandImF in Inthe restof the paper we consider this effect in more asuperconductorbothvanishatT =0uptoafrequency detail and make quantitative predictions for the experi- which for arbitrary strong interaction exceeds ∆. The ments. Our goal is to compute dI/dV for SIN and SIS Kramers-Kronig relation then yields at low frequencies tunneling for strong spin-fermion interaction. ReΣ(φ,ω) ω, ReF(φ,ω) (φ φ ) where φ is node node ∝ ∝ − Thepointofdepartureforouranalysisisthesetoftwo a position of the node of the d wave gap. Substituting − Eliashberg-type equations for the fermionic self-energy these forms into (3) and integrating over φ we obtain Σ , and the spin polarization operator Π . The later is N(ω) ω although the prefactor is different from that ω Ω ∝ related to the dynamical spin susceptibility at the anti- in a gas. The linear behavior of the DOS in turn gives ferromagnetic momentum by χ−1(Q,Ω) 1 Π . The rise to the quadratic behavior of the SIS conductance. Ω ∝ − 2 normalstate are overdampeddue to strong spin-fermion normal superconducting superconducting superconducting interaction. In a superconducting state, the form of the spinpropagatorismodifiedatlowfrequenciesbecauseof the gap opening, and this gives rise to a strong feedback from superconductivity on the electron DOS. More specifically, we argued in [7] that in a super- eV Ωres ∆ cΩo2n/d∆u,ctio.re,.,ΠcΩollaetctliovwe fsrpeiqnueenxcciietsatΩion≪s a2r∆e ubnedhaamvepseads, Ω ∆ eV res propagating spin waves. This behavior is peculiar to a superconductor – in the normal state, the spin ex- ∆ citations are completely overdamped. The propagat- a) b) ing excitations give rise to the resonance in χ(Q,Ω) at Ω (∆ω )1/2 ω where ReΠ(Ω )=1 [7]. This res sf sf res FIG. 2. The schematic diagram for the dip features in reson∼anceaccounts≪forthepeakinneutronscattering[9]. SINandSIStunnelingconductances(figuresaandb,respec- The presence of a new magnetic propagating mode tively). ForSINtunneling,theelectron whichtunnelsfrom a changes the electronic self-energy for electrons near hot normalmetalcanemitapropagatingspinwaveifthevoltage spots. Intheabsenceofapropagatingmode,anelectron eV =∆+Ωres whereΩres istheminimumfrequencyforspin can decay only if its energy exceeds 3∆. Due to reso- excitations. After emitting a spin-wave, the electron falls to nance, an electron at a hot spot can emit a spin wave the bottom of the band which leads to a sharp reduction of already when its energy exceeds ∆+Ω . It is essential thecurrentandproducesadropindI/dV. ForSIStunneling, res thatcontrarytoaconventionalelectron-electronscatter- the physics is similar, but one first has to break an electron ing, this process gives rise to a discontinuity in ImΣ(ω) pair, which costs energy 2∆. at the threshold. Indeed, using the spectral represen- tation to transform from Matsubara to real frequencies same set was used in our earlier analysis of the relation in the first equation in (5), integrating over momentum between ARPES and neutron data [7]. In Matsubara and neglecting for simplicity unessential q2 in the spin x frequencies these equations read (Σ˜ωm =iΣ(ωm)) susceptibility, we obtain for ω ≥ωth =∆+Ωres Σ˜ωm =ωm+ 83πR2 Z qx2+Σ˜Σ˜2ωωmm++ΩΩmm+F2pqx2d+Ω1m−ΠΩ ImΣ(ω)∝ZΩωre−s(∆ω−dωΩth√)1ω/2−1Ω−∆ √1Ω−1Ωres π ΠΩ = 1 dωm  Σ˜Ωm+ωm Σ˜ωm +F2 1. (5) ∝Z0 dx√ω−ωth−x2 = 2, (6) 2Z ωsf Σ˜2 +F2 Σ˜2 +F2 − WeseethatImΣ(ω)jumpstoafinitevalueatthethresh- q Ωm+ωm q ωm  old. Thisdiscontinuityispeculiartotwodimensions. By This set is a simplification of the full set of Eliashberg Kramers-Kronigrelation,the discontinuityin ImΣ gives equations that includes also the equationfor the anoma- rise to a logarithmical divergence of ReΣ at ω = ω . th lous vertex F(ω) [13]. As in [7] we assume that near op- This in turn gives rise to a vanishing spectral function timal doping, the frequency dependence of F(ω) is weak near hot spots, and accounts for a sharp dip in the atω ∆relevanttoouranalysis,andreplaceF(ω)bya ARPES data [8]. ∼ frequency independent input parameter F. Other input We now show that the singularity in ReΣ(ω) causes parametersin(5)arethedimensionlesscouplingconstant the singularity in the derivatives over voltages of both R = g¯/(vFξ−1) and a typical spin fluctuation frequency SIN and SIS conductances d2I/dV2. Indeed, near a hot ωsf = (π/4)(vFξ−1)2/g¯. They are expressed in terms of spot,F(φ)=F(1 λφ˜2)whereφ˜=φ φmax,andλ>0. the effective spin-fermion coupling constantg¯, the Fermi Then, quite gener−ally,ReΣ(φ,ω) ln−ω ω (φ) where th velocity at a hot spot vF, and the magnetic correlation ω (φ) = ω +Cφ˜2, and C > 0∝. Su|bst−ituting t|his ex- th th length ξ. By all accounts, at and below optimal doping, pressionintotheDOSanddifferentiatingoverfrequency, R 1 [14], i.e., the system behavior falls into the strong we obtain after a simple algebra ≥ coupling regime. Strictly speaking, the set (5) is valid near hot spots ∂N(ω) F2(φ) ∂ Σ(φ,ω)dφ where φ φ . Away from hot spots F(φ) is re- ∂ω ∼−Z Σ3(φ,ω) ω max ≈ duced compared to F. We, however, will demonstrate 1 Θ(ω ω) th thatthe newfeaturesdue tospin-fermioninteractionare − , (7) produced solely by fermions from hot regions. ∼ ln3|ω−ωth| √ωth−ω Asin [7],weconsiderthesolutionof(5)fortheexper- whereΘ(x)isastepfunction. Weseethat∂N(ω)/∂ωhas imentally relevant case F Rω when the measured a one-sided, square-root singularity at ω = ω . Physi- sf th superconducting gap ∆ ≫F2/(R2ω ) ω . In this cally,thisimpliesthattheconductancedropswhenprop- sf sf ∼ ≫ situation,atfrequencies ∆,fermionicexcitationsinthe agating electrons start emitting spin excitations. Note ∼ 3 and ω Ω ∆, and both ∂ N(ω Ω) and ∂ N(ω) are ω ω dI/dV dI/dV -d 2I/dV2 singula−r. ≈ − -d 2I/dV2 Again, it is very plausible that the singularity of the Ω derivative causes a dip at a frequency ω ω∗ , and ∆ Ω~ eV ∼ω2 2∆ eV a hump at even larger frequency. We stres≥s, htohwever, ∗ ∼ω that at exactly ω , the SIS conductance has an infinite th derivative, while the dip occurs at a frequency which is 1 1 somewhat larger than ω∗ . The behavior of the SIS con- th ductance is presented in Fig 3. Qualitatively, the forms of conductances presented in Fig 3 agree with the SIN and SIS data for YBCO and ∆ eV 2∆ eV Bi2212 materials [1,2]. Moreover, recent SIS tunneling a) b) data for Bi2212 [2] indicate that the relative distance FIG. 3. The schematic forms of SIN (a) and SIS (b) tun- between the peak and the dip (Ω /(2∆) in our theory) res neling conductances for strong spin-fermion interaction. We decreases with underdoping. More data analysis is how- addedsmallimpurityscatteringtosoftensingularfeaturesre- ever necessary to quantitatively compare tunneling and latedtothesharpnessoftheFermisurface(seethetext). The neutron data. dip/hump features above the peaks are the strong coupling To summarize, in this paper we considered the forms effects not present in a gas. The insets show the derivatives of SIN and SIS conductances both for noninteracting of conductances above ∆ for SIN tunneling and 2∆ for SIS fermions, and for fermions which strongly interact with tunneling. We argue in the text that these derivatives have their own collective spin degrees of freedom. We ar- maxima at voltages eV = Ω˜ = ∆+Ωres for SIN tunneling gue that for strong spin-fermion interaction, the reso- andeV =Ω¯ =2∆+Ωres forSIStunneling,whereΩres isthe nance spin frequency Ω measured in neutron scatter- resonance spin frequency measured in neutron experiments. res ingcanbe inferredfromthe tunneling databy analyzing the derivatives of SIN and SIS conductances. We found thatthe typicalφwhichcontributetothesingularityare that the derivative of the SIN conductance diverges at small(oforder ωth ω 1/2), whichjustifiesourassertion eV = ∆+Ωres while the derivative of the SIS conduc- | − | that the singularity is confined to hot spots. tance diverges at eV =2∆+Ωres, where ∆ is the maxi- The singularity in ∂N(ω)/∂ω is likely to give rise to a mum value of the d wave gap. − dip in N(ω) at ω ω . The argument here is based ItisourpleasuretothankG.Blumberg,A.Finkel’stein th ≥ on the fact that if the angular dependence of ω (φ) and particularly J. Zasadzinski for useful conversations. th is weak (i.e., C is small), then Σ(ω ) F(ω ), and The research was supported by NSF DMR-9979749. th th ≫ N(ω ) reaches its normal state value with infinite neg- th ative derivative. Obviously then, at ω >ω , N(ω) goes th below its value in the normal state and should there- fore have a minimum at some ω ω . Furthermore, at th ≥ larger frequencies, we solved (5) perturbatively in F(ω) and found that N(ω) approaches a normal state value from above. Thisimpliesthatbesidesadip,N(ω)should [1] Ch. Renner et al, Phys. Rev. Lett. 80, 149 (1998); Y. alsodisplayahumpsomewhereaboveω . Thebehavior DeWilde et al, ibid 80, 153 (1998). th [2] N. Miyakawa et al, Phys. Rev. Lett. 83, 1018 (1999); oftheSINconductanceisschematicallyshowninFig.3a. L. Ozyurer, J.F. Zasadzinski, and N. Miyakawa, cond- mat/9906205; J.F. Zasadzinski, privatecommunication. Similar results hold for SIS tunneling. The derivative [3] G.D. 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