Kondo Versus Fano in Superconducting Artificial High-Tc Heterostructures

Abstract

Recently, the quest for high-Tc superconductors has evolved from the trial-and-error methodology to the growth of nanostructured artificial high-Tc superlattices (AHTSs) with tailor-made superconducting functional properties by quantum design. Here, we report the growth by molecular beam epitaxy (MBE) of a superlattice of Mott insulator metal interfaces (MIMIs) made of nanoscale superconducting layers of quantum confined-space charge in the Mott insulator La₂CuO₄ (LCO), with thickness L intercalated by normal metal La_1.55Sr_0.45CuO₄ (LSCO) with period d. The critical temperature shows the superconducting dome with Tc as a function of the geometrical parameter L/d showing the maximum at the magic ratio L/d = 2/3 where the Fano–Feshbach resonance enhances the superconducting critical temperature. The normal state transport data of the samples at the top of the superconducting dome exhibit Planckian T-linear resistivity. For L/d > 2/3 and L/d < 2/3, the heterostructures show a resistance following Kondo universal scaling predicted by the numerical renormalization group theory for MIMI nanoscale heterostructures. We show that the Kondo temperature, T_K, and the Kondo scattering amplitude, R_0K, vanish at L/d = 2/3, while T_K and R_0K increase at both sides of the superconducting dome, indicating that the T-linear resistance regime competes with the Kondo proximity effect in the normal phase of MIMIs.

1. Introduction

It is known that high-Tc cuprate superconductors can now be engineered by relying on the quantum design of nanoscale artificial high-Tc superlattices (AHTSs) of quantum wells [1,2,3,4] that capture the key features of cuprates. The growth of these novel nanoscale non-conventional heterostructures has been guided by the quantum material design of an artificially modulation-doped [5] correlated electron gas. This design relies on the predictions of the Bianconi–Perali–Valletta (BPV) theory [6,7,8] describing the amplification of the critical temperature driven by Fano–Feshbach shape resonance [9,10,11,12] in two-gap superconductors. This occurs in the presence of spin–orbit coupling (SOC) at the interfaces of nanostructured materials (NsMs) exhibiting quantum-size effects. Recently, this approach has been applied to the material design of the AHTS samples studied in this work [1,2]. These three-dimensional AHTS heterostructures are formed by superconductor layers (S) of a stoichiometric modulation-doped Mott insulator La₂CuO₄ (LCO) of thickness L, intercalated with potential barriers of a normal metal (N) made of chemically overdoped non-superconducting La_1.55Sr_0.45CuO₄ (LSCO) with ten repeats of period d variable within the 2.97 < d < 5.28 nm nanoscale range. The normal N units play the role of charge reservoirs and transfer the interface space charge into the superconducting Mott insulator units, forming a 3D superlattice of a 2D Mott insulator–metal interface (MIMI) of period d. The internal interface electric field at the SNS junctions induces Rashba SOC in the superconducting interface space charge in the S layers, which is split into two electronic components by quantum-size effects, where the lowest sub-band shows a large cylindrical Fermi surface with high Fermi velocity, while the upper sub-band exhibits a low Fermi velocity and an unconventional extended van Hove singularity generated by the SOC at the interface [1,2,3,4]. It has been established that these artificial samples exhibit the superconducting dome predicted by the BPV theory [1,2]. According to this theory, the critical temperature, T_C, is a function of the geometrical parameter, L/d, with a maximum at the magic ratio, L/d = 2/3. At this ratio, the theory predicts that the Fano–Feshbach resonance enhances the superconducting critical temperature.

The key result of this work is the compelling evidence that the temperature-dependent sheet resistance of various AHTSs in the normal phase above the top of the superconducting dome, i.e., within the 50 K < T < 270 K range, evolves as the geometric L/d ratio changes. At the top of superconducting dome for the magic ratio L/d = 2/3 for optimum Tc [1,2], the normal phase of the AHTS exhibits T-linear resistivity, similar to what is observed in slightly overdoped cuprate perovskites [13,14]. On either side of the superconducting dome for AHTS samples with L/d < 2/3 and L/d > 2/3, the temperature-dependent normal electrical resistance is predicted and experimentally probed for the Kondo proximity effect in Mott insulator–metal interfaces with the single variable T/T_K, where T_K is the Kondo temperature. While other factors such as weak localization or electron–electron interactions may also increase resistance at low temperatures, the proximity Kondo effect is particularly notable for its universal scaling behavior, as described by the numerical renormalization group (NRG) theory expressed by the universal function. This Kondo effect has been found in heterostructures made of metal–Mott insulator interfaces, where spin–orbit coupling driven by a potential drop at the interface and the coexistence of itinerant and localized electronic components generate a unique proximity Kondo effect. This effect appears in the nanoscale heterostructures studied in this work, which are made of Mott insulator–metal interfaces [15,16,17,18,19]. While the large charge correlation gap due to the local Coulomb repulsion U prohibits the tunneling of electrons from the metal into a Mott insulator, the resonant spin-flip scattering opens a new channel for tunneling. In this manner, the metal ‘eats’ itself layer by layer into the Mott insulator. The peculiar temperature dependence of the proximity Kondo in MIMI resistivity was observed in two-component electronic systems like, e.g., SrTiO₃/LaTiO₃/SrTiO₃ heterostructures [20].

Evidence of the Kondo scattering of itinerant electrons by localized electrons has been reported in systems with logarithmic van Hove singularities [21], kagome local-moment metals [22], twisted bilayer graphene [23], and cuprates [24,25], including Rashba SOC [26]. In our MIMI superlattices, the superconducting transition temperature was amplified by the Fano–Feshbach shape resonance between two superconducting gaps tuned by quantum-size effects in the doped Mott insulator nanoscale layers of thickness L due to resonant scattering between the closed and open scattering channels. The Fano–Feshbach shape resonance in nanoscale heterostructures has been shown to compete [27] and coexist [28,29] with the Kondo effect.

The competition between the proximity Kondo scattering in the normal phase and the Fano–Feshbach shape resonance in MIMI superlattices can be tuned by changing the chemical potential in superlattices with variable L/d. While the electronic structure of the 2D electron gas at cuprate oxide interfaces with the associated interlayer phase separation has attracted high interest [30,31], we focused on quantum-size effects due to the nanoscale confinement of the interphase-correlated electron gas [1]. We first synthesized MIMI superlattices made of quantum wells with a period in the range of 1.98 < d < 5.28 nm, tuned at the resonant geometry where the pure Mott insulators LCO layers of thickness L, free of chemical substitutions, are intercalated with the metallic LSCO layers of thickness W, so that L/d = 2/3, where d = L + W, which form a superlattice of Mott insulator–metal interfaces (MIMIs). These superlattices exhibit T-linear resistivity in the metallic phase from 50 K to 270 K. By changing the conformational geometry parameter, L/d, away from the shape resonance, in the range 0.3 < L/d < 0.9, we observed a Kondo-like temperature dependence on resistivity, with the Kondo temperature, T_K, being minimal at L/d = 2/3, where the superconducting critical temperature is maximal and the amplitude of the Kondo effect in the resistivity, R_0K, is minimal.

2. Results

The nanoscale LSCO/LCO superlattices of alternating overdoped LSCO layers and undoped LCO layers were on a LaSrAlO₄ substrate using molecular beam epitaxy (MBE). These superlattices formed a 2D electron gas (2DEG) at the interface space charge and exhibited 2D high-Tc superconductivity. The overall superlattice structure has a periodicity represented by parameter d, as shown in Figure 1a.

Our experimental approach involves the manipulation of the 2DEG interface space charge layer, which extends approximately 2.6 nm into the LCO layer from the LSCO–LCO interface. This layer experiences quantum confinement between the two LSCO potential barriers, of which the width is denoted as W within the superlattice. As a result, two artificial sub-bands emerge, allowing us to modulate their energy splitting or the transparency of the potential barrier by adjusting the thickness ratio, L/d. Figure 1a depicts a typical LSCO–LCO superlattice with a period of d = 3.96 nm. In this arrangement, the heterostructure consists of four LCO layers (MLs), with thickness L = 2.64 nm, while the LSCO layer has a thickness of two MLs, W = 1.32 nm. The superconducting critical temperature, Tc, as a function of L/d assumes a dome shape where, at L/d = 2/3, the T_C reaches its maximum values [8] (see Figure 1b).

We measured the temperature dependence of the resistance in eighteen LSCO–LCO superlattices with varying values of d, L, and W. The changes in resistivity with temperature, as shown in Figure 1c, display curves that both approach and deviate from a linear trend (thick black line). The resistances that most closely follow a linear pattern are reported in Figure 1d, where we present the resistance of two significant samples. These samples have the same L/d parameter (L/d = 2/3) but exhibit different d-periodicities. At this L/d value, where the superlattices approach the maximum T_C, in accordance with [8], the Fano–Fashback shape resonance drives up the critical temperature; we clearly observe the compelling experimental evidence that normal resistivity in the normal phase shows the T-linear behavior in our artificial MIMI superlattices in this figure.

To investigate the interplay of Fano resonance and Kondo scattering, away from the top of the superconducting dome, we considered the sheet resistance as a function of temperature and L/d in the measured eighteen LCO–LSCO superlattices listed in Figure 2. These samples were classified by different d-values, corresponding to different colors and different L/d values. The sheet resistance of all eighteen superlattices, as a function of temperature, is shown in Figure 2a, while Figure 2c depicts the normalized sheet resistance in units of R(T = 150 K) on a semilogarithmic scale.

The R(T) curves of all samples were fitted by a coexisting contribution of the Planckian T-linear resistance and the universal resistance function for the Kondo proximity effect at Mott insulator–metal interfaces [16,17,19,20]:

$${\displaystyle \frac{R\left(T\right)}{R\left(T=150\mathrm{K}\right)}}=r_0+{\displaystyle \frac{T}{150}}+A{T}^2+B{T}^5+{\displaystyle \frac{R_{0K}}{{\left\{1+\left(2^{\frac{1}{s}}-1\right){\left(\frac{T}{T_K}\right)}^2\right\}}^s}}$$

(1)

for experimental R(T) values measured at T > 50 K. The Kondo temperature, T_K, represents a characteristic temperature below which the coupling between the impurity and conduction electrons leads to the screening of the impurity spin. R_0K is the amplitude of the Kondo-like contribution, while r₀ is the residual contribution at T = 0 that represents the baseline resistivity of the material in the absence of Kondo physics, including contributions such as impurities and lattice defects. The term AT² accounts for electron–electron scattering, while BT⁵ represents phonon scattering. Both AT² and BT⁵ terms are non-Kondo contributions. The best-fitted curves are represented by continuous lines. The three different representations (Figure 2, panels a, and b) of the sheet resistance well visualize the fact that resistivity decreases as the temperature is lowered, reaching a minimum. At even lower temperatures, the electrical resistivity of the system increases (logarithmically, in the original perturbative Kondo treatment, or as a power law, beyond the perturbation theory). We observed how the behavior of the resistance deviates from linearity as the L/d value moves away from 2/3, where Fano resonance prevails. A phase diagram of normalized sheet resistance as a function of both temperature and the geometrical parameter L/d is given by the 3D color map in Figure 2c. In Figure 2d, we visualize a projection of this phase diagram on a T vs. L/d plane, highlighting the correspondence between L/d and the charge <δ> = 0.45 × (1 − L/d).

We used a least-squares fitting algorithm, extracting parameters A, B, R_0K, and T_K. The evolution of these parameters as a function of L/d is shown in Figure 3. The exponent, s, is a material-dependent parameter depending on the metal–Mott insulator interfaces, and, here, was found to be in the [0.10–0.18] range. In the sheet resistance measured at 150 K in all 18 measured samples with different L/d values, we observed an exponential increase for L/d > 2/3, indicating a metal insulator transition triggered by the geometrical conformational parameter. The evolutions of the fitting parameters A, B, r₀ + R_0K, and T_K in Equation (1) are shown in Figure 3b–e. All parameters show a minimum at L/d = 2/3. In the scatter plot of T_K vs. R_0K, the positive correlation between these two parameters is made evident in Figure 3f.

Finally, in Figure 4, we show the anticorrelation between T_C and T_K. We report the theoretical calculations of superconducting critical temperature as a function of the geometry conformational parameter L/d, using the BPV theory in the case of superlattices with period 3 and 4 nm. This clearly shows the agreement between the theory and experiments, where the maximum T_C can be controlled by the FFSR amplification [8].

3. Discussion

Many authors have observed that cuprate superconductors exhibiting high-Tc superconductivity display unconventional transport properties in the normal phase above the critical temperature. These properties are associated with quantum tunneling characterized by T-linear resistivity, the Planckian limit of the scattering rate assigned to quantum criticality, strong electronic correlation, and the coexistence of localized and itinerant states [32,33,34,35,36,37,38,39,40,41,42,43] in a homogeneous strange metal.

In this work, we provided further experimental evidence that artificial high Tc superlattices, made of modulation-doped quantum wells formed by the superconducting (S) layers of a chemically stoichiometric Mott insulator, La2CuO4, of thickness within the range of 2–3 nanometers intercalated by a normal metal (M), with the role of charge reservoir and a potential barrier of 500 meV [1], encompass the key feature of natural chemical-doped La₂-xSrxCuO₄. We confirm that AHTSs exhibit first (i) the superconducting dome of the critical temperature versus modulation doping, and second (ii), the T-linear Planckian sheet resistance around optimum modulation doping reported in reference [1]. The present results support the physical paradigm describing unconventionally high Tc cuprate perovskites characterized by intrinsic functional-arrested chemical nanoscale-phase separation observed by local and fast experimental probes [44,45,46,47,48,49,50], called the “superstripes” [46] or “swiss cheese” [48] scenario near the BEC–BCS crossover.

4. Materials and Methods

The MBE synthesis of artificial high-Tc superlattices: AHTS based on normal metal LSCO alternating with superconducting space-charge layers in LCO thin layers were synthesized via an ozone-assisted MBE method (DCA Instruments Oy, Turku, Finland) on LaSrAlO₄ (001) substrates (compressive strain for La₂CuO₄ on LaSrAlO₄ is +1.4%). The superlattice growth was controlled by the in situ reflection high-energy electron diffraction (RHEED). This method is characterized by the sequence of deposition of single atomic layers and minimal kinetic energy of impinging atoms (about 0.1 eV). The substrate temperature, Ts, according to a radiation pyrometer reading, was 650 °C, and the chamber pressure p ≈ 1.5 × 10⁻⁵ Torr of mixed ozone, and atomic and molecular oxygen. At the end of the procedure, the samples were cooled down to Ts.

Resistance measurements: The temperature dependence of resistance was determined in a four-point van der Pauw configuration with alternative DC currents ± 10 μA in a temperature range from room temperature to 4.2 K (liquid helium). The temperature-dependent resistance was measured by using a motorized custom-made dipstick in a transport helium dewar with temperature rate < 0.1 K/s.

5. Conclusions

The key remarkable result, shedding further light on the mechanism of high-Tc superconductivity, is that in the T-linear regime of sheet resistance in the normal phase of the AHTS made of MIMI at the magic ratio L/d = 2/3 and proximity Kondo scattering at a low temperature, vanishes, together with electron–electron Fermi liquid scattering. The Fano–Feshbach resonance between the two superconducting gaps, determined by quantum-size effects of nanoscale units, were optimized by quantum design at the geometrical conformational parameter of the superlattice, L/d = 2/3, at the top of the superconducting dome. Here, the Kondo temperature, T_K, reached a minimum of the order of 4 Kelvin and the Kondo amplitude R_0K vanished, as shown in Figure 3. On both sides of the superconducting dome where the average doping was in the L/d < 2/3 or L/d > 2/3 regime, the Kondo temperature, T_K, increased and became higher than the superconducting critical temperature. These results are supported by the clear experimental anticorrelation between T_K and R_K. Finally, it is remarkable that where the resonant quantum tunnelling is tuned at the Fano–Feshbach for optimum resonant superconductivity [1,2], i.e., the residual T = 0 K resistance, the amplitude of the electron–electron Fermi scattering (T² term) and the electron–phonon (T⁵ term) vanished in the normal phase.

The observed competition between the Fano–Feshbach resonance in multi-gap AHTS superconductors [6,7,8,51] can be expected since the Kondo effect [52] is based on the Anderson impurity model [53] and the atomic Fano resonance [54]. In fact, the Anderson model used in Kondo physics is similar to the Fano–Feshbach resonance near the quasi-flat band, although their band-mixing terms are different (i.e., one-particle and two-particle scatterings) [55,56,57,58,59].