Competition between electron pairing and phase coherence in superconducting interfaces
G. Singh, A. Jouan, L. Benfatto, F. Couedo, P. Kumar, A. Dogra, R. Budhani, S. Caprara, M. Grilli, E. Lesne, A. Barthelemy, M. Bibes, C. Feuillet-Palma, J. Lesueur, N. Bergeal
CCompetition between electron pairing and phase coherence in superconductinginterfaces
G. Singh , † , A. Jouan , † , L. Benfatto , , F. Couedo , , P. Kumar , A. Dogra , R. Budhani , S. Caprara , , M.Grilli , , E. Lesne , A. Barth´el´emy , M. Bibes , C. Feuillet-Palma , , J. Lesueur , , N. Bergeal , ∗ Laboratoire de Physique et d’Etude des Mat´eriaux, ESPCI Paris,PSL Research University, CNRS, 10 Rue Vauquelin - 75005 Paris, France. Universit´e Pierre and Marie Curie, Sorbonne-Universit´es,75005 Paris, France. Institute for Complex Systems (ISC-CNR), UOS Sapienza, Piazzale A. Moro 5, 00185 Roma, Italy Dipartimento di Fisica Universit`a di Roma“La Sapienza”, Piazzale A. Moro 5, I-00185 Roma, Italy. National Physical Laboratory, Council of Scientific and IndustrialResearch (CSIR)Dr. K.S. Krishnan Marg, New Delhi-110012, India. Condensed Matter Low Dimensional Systems Laboratory,Department of Physics, Indian Institute of Technology, Kanpur 208016, India. Unit´e Mixte de Physique CNRS-Thales, 1 Av. A. Fresnel, 91767 Palaiseau, France. † : Both authors contributed equally to this work. ∗ : Correspondence and request should be sent to N. B.([email protected]).The large diversity of exotic electronic phases dis-played by two-dimensional superconductors confrontsphysicists with new challenges. These include therecently discovered quantum Griffith singularity inatomic Ga films [1], topological phases in proximizedtopological insulators [2] and unconventional Isingpairing in transition metal dichalcogenide layers [3].In LaAlO /SrTiO heterostructures, a gate tunablesuperconducting electron gas is confined in a quantumwell at the interface between two insulating oxides [4].Remarkably, the gas coexists with both magnetism [5, 6]and strong Rashba spin-orbit coupling [7, 8] and is acandidate system for the creation of Majorana fermions[9]. However, both the origin of superconductivity andthe nature of the transition to the normal state over thewhole doping range remain elusive. Missing such crucialinformation impedes harnessing this outstanding systemfor future superconducting electronics and topologicalquantum computing. Here we show that the supercon-ducting phase diagram of LaAlO /SrTiO is controlledby the competition between electron pairing and phasecoherence. Through resonant microwave experiments,we measure the superfluid stiffness and infer the gapenergy as a function of carrier density. Whereas a goodagreement with the Bardeen-Cooper-Schrieffer (BCS)theory is observed at high carrier doping, we find thatthe suppression of T c at low doping is controlled by theloss of macroscopic phase coherence instead of electronpairing as in standard BCS theory. We find that only avery small fraction of the electrons condenses into thesuperconducting state and propose that this correspondsto the weak filling of a high-energy d xz/yz band, moreapt to host superconductivity.The superconducting phase diagram ofLaAlO /SrTiO interfaces defined by plotting thecritical temperature T c as a function of electrostatic doping has the shape of a dome. It ends into a quantumcritical point, where the T c is reduced to zero, as carriersare removed from the interfacial quantum well [4, 10].Despite a few proposals [11, 12], the origin of thisgate dependence and in particular the non-monotonicsuppression of T c remains unclear. There are two funda-mental energy scales associated with superconductivity.On the one hand, the gap energy ∆ measures the pairingstrength between electrons that form Cooper pairs. Onthe other hand, the superfluid stiffness J s determines thecost of a phase twist in the superconducting condensate.In ordinary BCS superconductors, J s is much higherthan ∆ and the superconducting transition is controlledby the breaking of Cooper pairs. However, when thestiffness is strongly reduced, phase fluctuations play amajor role and the suppression of T c is expected to bedominated by the loss of phase coherence [13]. Tunnelingexperiments in the low doping regime of LaAlO /SrTiO interfaces evidenced the presence of a pseudogap in thedensity of states above T c [14]. This can be interpretedas the signature of pairing surviving above T c whilesuperconducting coherence is destroyed by strong phasefluctuations, enhanced by a low superfluid stiffness [15].Superconductor-to-Insulator quantum phase transitionsdriven by gate voltage [4] or magnetic field [16] alsohighlighted the predominant role of phase fluctuationsin the suppression of T c .The low superfluid stiffness corresponds to a lowsuperfluid density n s = m ¯ h J s which has to be analyzedwithin the context of the peculiar LaAlO /SrTiO bandstructure. Under strong quantum confinement, thedegeneracy of the t g bands of SrTiO ( d xy , d xz and d yz orbitals) is lifted, generating a rich and complex bandstructure [17]. Experiments performed on interfaceswith different crystallographic orientations ([110] vsconventional [001] orientation) revealed the crucial roleof orbitals hierarchy in the quantum well, and alsosuggested that only some specific bands could hostsuperconductivity [18, 19]. Here, we use a resonantmicrowave experiment to measure the kinetic inductance L k of the superconducting LaAlO /SrTiO interface. a r X i v : . [ c ond - m a t . s up r- c on ] A p r Z L port 1 port 2I dc Z D -10dB-10dB-10dB-10dB 40dB -30dB Z C STO C p C p L R LAO/STOSMD Z L bias-teedirectionalcoupler HEMTAmp. V G =20 nH2R =140 Ω C p =2 µ F LAO/STO c.b. a. A in A out FIG. 1: The LaAlO /SrTiO sample and its microwave mea-surement set-up. a) LaAlO /SrTiO sample inserted betweenthe central strip and the ground of a CPW transmission line,in parallel with SMD inductors L and resistors R . C p areprotective capacitors that avoid dc current to flow through L and R without influencing ω . b) Sample circuit ofimpedance Z L in its microwave measurement that includesan attenuated input line and an amplified output line sepa-rated by a directional coupler. A bias-tee allows dc biasingof the sample. c) Equivalent electrical circuit of the samplecircuit including the SMDs and the LaAlO /SrTiO hetero-structure modeled by a capacitor C STO and an impedance Z . This allows us to determine the evolution of the super-fluid stiffness J s = ¯ h e L k and corresponding superfluiddensity n s in the phase diagram.Figure 1 gives a schematic description of ourexperimental set-up, largely inspired by recent devel-opments in the field of quantum circuits [20, 21]. TheLaAlO /SrTiO sample is mounted on a microwavecircuit board which is anchored to the 18 mK coldstage of a dilution refrigerator. It is embedded into aRLC resonant circuit whose inductor L and resistorR are Surface Mounted microwave Devices, and whosecapacitor C STO is the SrTiO substrate in parallel withthe two-dimensional electron gas (2-DEG) (Fig. 1aand 1c.). After calibration, the measurement of thecomplex reflection coefficient Γ( ω ) = A out A in at the inputof the resonant sample circuit allows to determine thecomplex conductance G ( ω ) = G ( ω ) − iG ( ω ) of the2-DEG in a frequency band centered on the resonancefrequency ω = √ L C STO (see Methods). In the normalstate (
T > T c ), C STO is deduced from ω for eachgate value (Fig. 2a,b). In the superconducting state,the 2-DEG conductance acquires an imaginary part G ( ω ) = L k ω that modifies ω , as the total induc- tance is then given by L in parallel with L k . Thesuperconducting transition observed in dc resistance( R dc =0 Ω) for positive gate voltages V G , coincides witha continuous shift of ω towards high-frequency (Fig.2d,e,f). In absence of superconductivity (for V G < /SrTiO interfaces, and compare them with the BCS theory pre-dictions. In Figure 3a, we show the experimental super-fluid stiffness J exp s = ¯ h e L k as a function of V G at thelowest temperature T = 20 mK ( (cid:39) l -mean free path- < ξ -coherencelength-) and for ω (cid:28) ∆ / ¯ h , J s can be expressed as afunction of the gap energy [22] : J s ( T (cid:39)
0) = π ¯ h e k B R n · ∆( T (cid:39)
0) (1)where R n = R ( T > ∼ T c ) is the normal state resistance(inset Fig. 3b). A remarkable agreement is obtainedbetween experimental data ( J exp s ) and BCS prediction( J BCS ) in the overdoped (OD) regime defined by V G > V optG (cid:39)
27 V, assuming in Eq. (1) a gap energy∆ = ∆
BCS = 1.76 k B T c (Fig. 3a). In this regime, thesuperfluid stiffness J exp s takes a value much higher than +50 V0.1 0.2 0.3T (K) 0.40.1 0.2 0.3T (K) 0.4 ω / π ( G H z ) ω / π ( G H z ) R d c ( k Ω / ) -34 V +14 V0.1 0.2 0.3T (K) 0.4 0.1 0.2 0.3T (K) 0.4 0.2 ω / π ( G H z ) ω / π ( G H z ) c. d.e. f. +24 V 0.20.30.40.50.60.10.20.30.40.50.60.1 0.20.30.40.50.60.1T=450mK +24 V ω/2π (GHz)0.2 0.3 0.4 0.50.10-10-20 Γ ( d B ) pha s e (r ad ) π - π G (V) + V ω / π ( G H z ) a. b. R d c ( k Ω / ) R d c ( k Ω / ) R d c ( k Ω / ) Γ (dB)-100-20-30 FIG. 2: Resonance shift in the superconducting state. a)Γ( ω ) in dB (color scale) as a function of frequency and V G at T=450mK. b) Amplitude (left axis) and phase (right axis)of Γ( ω ) showing the resonance frequency for V G = +24V atT=450mK. c,d,e,f) Γ( ω ) in dB (color scale) as a function offrequency and temperature for the selected gate values, V G =-34V (a), V G =+14V (b), V G =+24V (c), V G =+50V (d). Thecorresponding dc resistance as a function of temperature isshown on the right axis. T c in agreement with the BCS paradigm. However, inthe underdoped (UD) regime, corresponding to V G 27 V)is ∆ exp s ≈ µeV . By using tunneling spectroscopyon planar Au/LaAlO /SrTiO junctions, Richter etal. have reported an energy gap in the density ofstates of ∼ µ eV for optimally doped LaAlO /SrTiO interfaces [14], which corresponds to ∆ BCS (cid:39) . k B T c in agreement with our result. However, the tunnelinggap was found to increase in the UD regime, which isdifferent from the behavior of ∆ exp s reported here. Inaddition, a pseudogap has been observed above T c inthis regime, as also reported in High- T c superconductingcuprates [25] or in strongly disordered films of conven-tional superconductors [22, 26]. The results obtainedby the two experimental approaches can be reconciledby considering carefully the measured quantities. Inour case, the superconducting gap ∆ exp s probed bymicrowaves is directly converted from the stiffness ofthe superconducting condensate and is therefore onlyreflective of the presence of a true phase-coherent state. ∆∆ BCS -20 -10 0 10 20 30 40 5005101520 ∆ ( µ e V ) 2D Superconductivity V G (V) homogenousJJ array -30 J s T c J BCS J ( K ) 10 20 30 400 V G (V) 5010100 L k ( n H / ) expexp -20 0 20 40-40 V G (V) R ( k Ω / ) a.b. V G opt ODUD FIG. 3: Superfluid stiffness and phase diagram. a) Experi-mental superfluid stiffness J exp s as a function of V G comparedwith T c (taken at R=0 Ω) and with the BCS theoretical stiff-ness J BCS . Inset) L k as a function of V G . b) Superfluid stiff-ness converted into a gap energy ∆ exp s as a function of V G compared with the BCS gap energy ∆ BCS . Inset) Normalsheet resistance as a function of V G . On the other hand, tunneling experiments probe thesingle particle density of states and can evidence pairingeven without phase coherence. The two experimentalmethods provide complementary informations whichindicate that in the UD region of the phase diagram,the superconducting transition is dominated by theloss of phase coherence rather than the pairing. In theregion V G < 0, some non-connected superconductingislands could already exist without contributing to themacroscopic stiffness of the 2-DEG.A simplified scheme of the band structure in theinterfacial quantum well is presented in Figures 4a and4b. The degeneracy of the three t g bands is lifted byconfinement in the z direction, leading to a splitting thatis inversely proportional to the effective masses m z alongthis direction. d xy subbands are isotropic in the interfaceplane with an effective mass m xy =0.7 m whereas the d xz / d yz bands are anisotropic with a correspondingaverage mass m xz/yz = √ m x m y (cid:39) m . At lowcarrier densities, we expect several d xy subbands tobe populated, whereas at higher density ( V G > d xz / d yz bands,leading to multiband transport. Recent measurementsof quantum oscillations showed that, in addition to amajority of low-mobility carriers (LMC), a small amountof high-mobility carriers (HMC) is also present, with an B(T) R H a ll ( Ω ) -10 0 10 20 30 40 500.00.51.01.52.0 n s (d xz /d yz ) n HM n ( e - / c m ) V G (V)-40 -20 0 20 400.81.01.21.41.61.8 n Hall n tot= n LM+ n HM n LM V G (V) n ( e - / c m ) c. d. e. 2D supercond. 2D supercond. d yz d xz k x d xy z E E d xz/yz d xy a. SrTiO b. multiband transport FIG. 4: Superfluid density and Hall effect analysis. a) Schemeof the interfacial quantum well showing the splitting of the t g bands. b) Simplified scheme of the band structure takinginto account only the last filled d xy subband, the d xz bandand the d yz band. c) Hall resistance as a function of mag-netic field for different gate voltages fitted by at two-bandmodel (see Supplementary Information). d) Hall carrier den-sity n Hall = BeR Hall extracted in the limit B → n LM extracted from the two-band analysis. The to-tal carrier density n tot is obtained by matching the charg-ing curves of the gate capacitance with n Hall at negative V G .The unscaled T c dome in the background indicates the regionwhere superconductivity is observed. e) Superfluid density n s calculated from J exp s using a mass m xz/yz , compared withthe HMC density n HM . effective mass close to the m xz/yz one [27]. Despite aband mass substantially higher than the m xy one, thesecarriers acquire a high-mobility as d xz/yz orbitals extenddeeper in SrTiO where they recover bulk-like properties,including reduced scattering, higher dielectric constantand better screening. Multiband transport was alsoevidenced in Hall effect measurements [10, 28]. Whereasthe Hall voltage is linear in magnetic field B in the UDregime corresponding to one-band transport, this is notthe case in the OD regime because of the contributionof a new type of carriers (the HMC). We performeda two-band analysis of the Hall effect data combinedwith gate capacitance measurement to determine thecontribution of the two populations of carriers to thetotal density n tot (Fig. 4c) [10]. The first clear signatureof multiband transport is seen when the Hall carrier density n Hall , measured in the limit B → 0, starts todecrease with V G instead of following the charging curveof the capacitance ( n tot in Fig. 4d). Figures 4d and4e show that LMC of density n LM are always present,whereas a few HMC of density n HM are injected inthe 2-DEG for positive V G , which corresponds to theregion of the phase diagram where superconductivityis observed. In consistency with quantum oscillationsmeasurements, we identify the LMC and the HMC ascoming from the d xy and d xz/yz subbands respectivelyand we emphasize that the addition of HMC in thequantum well triggers superconductivity.To further outline the relation between HMC andsuperconductivity, we extract the superfluid density n s from J exp s assuming a mass m xz/yz for the electrons, andplot it as a function of the gate voltage (Fig. 4e). It in-creases continuously to reach n s (cid:39) × e − · cm − at maximum doping, which is approximately 1% of thetotal carrier density. This behavior is similar to theone observed for the superfluid density measured byscanning SQUID experiments [15]. The comparison of n s with n HM in Fig. 4e shows that, unexpectedly,both quantities have a very similar dependence withthe gate voltage and almost coincide numerically. Thissuggests that the emergence of the superconductingphase is related to the filling of d xz/yz bands, whose highdensity of states is favorable to superconductivity. Thisis consistent with the observation of a gate-independentsuperconductivity in [110] oriented LaAlO /SrTiO interfaces for which the d xz/yz bands have a lower energythan the d xy subbands and are therefore always filled[19]. The fact that n s (cid:39) n HM is somewhat intriguingas the dirty limit that we used in Eq. (1) impliesthat n s should correspond to a fraction of the totalnormal carrier density (approximately 2∆ τ / ¯ h , where τ is the scattering time) and not to n HM . 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MethodsSample growth and gate deposition. In this study, we used 8 uc thick LaAlO epitaxial layersgrown on 3 × TiO -terminated [001] SrTiO single crystals by Pulsed Laser Deposition. The substrates weretreated with buffered HF to expose TiO terminatedsurface. Before deposition, the substrate was heated to830 ◦ C for one hour in an oxygen pressure of 7.4 × − mbar. The thin film was deposited at 800 ◦ C in anoxygen partial pressure of 1 × − mbar. The LaAlO target was ablated with a KrF excimer laser at a rateof 1Hz with an energy density of 0.56-0.65 Jcm − . Thefilm growth mode and thickness were monitored usingRHEED (STAIB, 35 keV) during deposition. Afterthe growth, a weakly conducting metallic back-gate ofresistance ∼ 100 kΩ (to avoid microwave short cut ofthe 2-DEG) is deposited on the backside of the 100 µm thick SrTiO substrate. Complex conductivity and kinetic inductance ofa superconductor. In superconducting thin films, J s is usually assessedeither from penetration depth measurements [1] or fromdynamic transport measurements [2, 3]. This lattermethod was adapted in this work for the specific caseof LaAlO /SrTiO samples which requires the use of alow-temperature dilution refrigerator. While supercon-ductors have an infinite dc conductivity, they exhibita finite complex conductivity σ ( ω ) at non-zero fre-quency, which in 2D translates into a sheet conductance G ( ω ) = G ( ω ) − iG ( ω ). The real part G ( ω ) accountsfor the transport of unpaired electrons existing at T (cid:54) = 0and ω (cid:54) = 0, and the imaginary part G ( ω ) accounts forthe transport of Cooper pairs. The expression of G ( ω )and G ( ω ) have been derived by Mattis and Bardeen ina seminal paper which gives a complete description ofthe electrodynamic response of superconductors basedon the BCS theory [4]. In the limit ¯ hω (cid:28) ∆, whichis well satisfied here (∆ (cid:39) G ( ω )= L k ω ,where L k is the kinetic inductance of the superconductordue to the inertia of Cooper pairs [5]. In our experiment, L k corresponds to a sheet inductance (i.e for a squaresample). Below T c the total inductance is given by L t ( T ) = L L k ( T ) L + L k ( T ) corresponding to the kinetic term L k in parallel with the constant SMD inductance L . Noticethat in our circuit, the geometric inductance of thesample is negligible compared to the kinetic one. As for T < T c , L k decreases when lowering the temperature,the superconducting transition observed in dc resistancefor positive gate voltages, coincides with a continuousshift of ω towards high-frequency (Fig. 3). Microwave reflection coefficient In a microwave circuit, the reflection coefficient at a dis-continuity of a transmission line is defined as the ratioof the complex amplitude of the reflected wave A out ( ω )to that of the incident wave A in . When the transmissionline is terminated by a load of impedance Z L ( ω ), it isgiven by [6]Γ( ω ) = A out ( ω ) A in ( ω ) = Z L ( ω ) − Z Z L ( ω ) + Z (2) where Z = 50Ω is the characteristic impedance of astandard microwave line. The measurement of Γ( ω )allows therefore to access directly to the load impedance Z L ( ω ) or equivalently its admittance G L ( ω ) = 1 /Z L ( ω ),commonly called complex conductance. In this work,a LaAlO /SrTiO heterostructure is inserted betweenthe central strip of a coplanar waveguide guide (CPW)transmission line and its ground, and is electricallyconnected through negligible contacts impedance. Thehigh dielectric constant of the SrTiO substrate atlow temperature (i. e. (cid:15) (cid:39) C STO in parallel with the 2-DEG whichhas to be correctly subtracted to extract the dynamictransport properties of the 2-DEG. This problem canbe overcome by embedding the LaAlO /SrTiO het-erostructure in a RLC resonating circuit whose inductorL =10nH and resistor R =70 Ω are Surface Mountedmicrowave Devices (SMD), and whose capacitor C STO is the SrTiO substrate in parallel with the 2-DEG(Figure 1a and 1c.). A directional coupler allows to sendthe microwave signal from port 1 to the sample througha bias-tee, and to separate the reflected signal whichis amplified by a low-noise cryogenic HEMT amplifierbefore reaching port 2 (Fig. 1b). Such type of microwavereflection set-up has been widely used in the quantumcircuit community.After cooling the sample to 3K, the back-gate voltage isfirst swept to its maximum value +50V while keepingthe 2-DEG at the electrical ground, to insure that nohysteresis will take place upon further gating [7]. Thetransmission coefficient S ( ω ) between the two ports ismeasured with a vector network analyzer. The reflectioncoefficient Γ( ω ) taken at the discontinuity betweenthe CPW line and the circuit formed by the sampleand SMD components (see Fig 1) is given by Eq. (2)where Z L = 1 /G L and G L is obtained by summing upall the admittances in parallel in the RLC circuit ofFig 1c. Loses of SrTiO substrate are not included inthis model as they only renormalize the amplitude ofthe absorption deep without modifying the resonancefrequency (or equivalently G ( ω )). Standard microwavenetwork analysis relates Γ( ω ) to the measured S ( ω ),through complex error coefficients representing thereflection tracking, the source match and the directiv-ity coefficient of the set-up [6]. A precise calibrationprocedure requiring three reference impedances, usuallyan open, a short and a match standard, allows acomplete determination of these error coefficients. Inthis experiment, the microwave set-up was calibrated byusing as references, the impedances of the sample circuitin the normal state of the 2-DEG for different gate values. References for Methods section [1] Ganguly, R., Chaudhuri, D., Raychaudhuri, P. andBenfatto, L. Slowing down of vortex motion at theBerezinskii-Kosterlitz-Thouless transition in ultra-thin NbN films. Phys. Rev B , 054514 (2015).[2] Kitano, H., Ohashi, T. and Maeda, A. Broadbandmethod for precise microwave spectroscopy of super-conducting thin films near the critical temperature. Rev. Sci. Instrum. , 074701 (2008).[3] Scheffler, M.and Dressel, M. Broadband microwavespectroscopy in Corbino geometry for temperaturesdown to 1.7 K. Rev. Sci. Instrum. , 074702 (2005).[4] Mattis, C. and Bardeen, J. Theory of the Anoma-lous Skin Effect in Normal and SuperconductingMetals. Phys. Rev. , 412 (1958).[5] Tinkham, M. Introduction to SuperconductivitySecond Edition, Dover Publications, Inc., Mineola,New York (2004). [6] Pozar, D. M. Microwave engineering 4th edition,John Wiley & Sons (2012).[7] Biscaras, J., Hurand, S., Feuillet-Palma, C., Ras-togi, A., Budhani, R. C., Reyren, N., Lesne, E.,Lesueur, J. and Bergeal, N.