Journal of Superconductivity and Novel Magnetism

, Volume 26, Issue 1, pp 65–70

Scanning Tunneling Spectroscopic Studies of the Low-Energy Quasiparticle Excitations in Cuprate Superconductors

Authors

    • Department of PhysicsCalifornia Institute of Technology
  • M. L. Teague
    • Department of PhysicsCalifornia Institute of Technology
  • R. T.-P. Wu
    • Department of PhysicsCalifornia Institute of Technology
  • Z. J. Feng
    • Department of PhysicsCalifornia Institute of Technology
  • H. Chu
    • Department of PhysicsCalifornia Institute of Technology
  • A. M. Moehle
    • Department of PhysicsCalifornia Institute of Technology
Original Paper

DOI: 10.1007/s10948-012-1706-y

Cite this article as:
Yeh, N., Teague, M.L., Wu, R.T. et al. J Supercond Nov Magn (2013) 26: 65. doi:10.1007/s10948-012-1706-y
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Abstract

We report scanning tunneling spectroscopic (STS) studies of the low-energy quasiparticle excitations of cuprate superconductors as a function of magnetic field and doping level. Our studies suggest that the origin of the pseudogap (PG) is associated with competing orders (COs), and that the occurrence (absence) of PG above the superconducting (SC) transition Tc is associated with a CO energy ΔCO larger (smaller) than the SC gap ΔSC. Moreover, the spatial homogeneity of ΔSC and ΔCO depends on the type of disorder in different cuprates: For optimally and under-doped YBa2Cu3O7−δ (Y-123), we find that ΔSC<ΔCO and that both ΔSC and ΔCO exhibit long-range spatial homogeneity, in contrast to the highly inhomogeneous STS in Bi2Sr2CaCu2O8+x (Bi-2212). We attribute this contrast to the stoichiometric cations and ordered apical oxygen in Y-123, which differs from the non-stoichiometric Bi-to-Sr ratio in Bi-2212 with disordered Sr and apical oxygen in the SrO planes. For Ca-doped Y-123, the substitution of Y by Ca contributes to excess holes and disorder in the CuO2 planes, giving rise to increasing inhomogeneity, decreasing ΔSC and ΔCO, and a suppressed vortex-solid phase. For electron-type cuprate Sr0.9La0.1CuO2 (La-112), the homogeneous ΔSC and ΔCO distributions may be attributed to stoichiometric cations and the absence of apical oxygen, with ΔCO<ΔSC revealed only inside the vortex cores. Finally, the vortex-core radius (ξhalo) in electron-type cuprates is comparable to the SC coherence length ξSC, whereas ξhalo∼10ξSC in hole-type cuprates, suggesting that ξhalo may be correlated with the CO strength. The vortex-state irreversibility line in the magnetic field versus temperature phase diagram also reveals doping dependence, indicating the relevance of competing orders to vortex pinning.

Keywords

Cuprate superconductivityPseudogapCompeting ordersQuasiparticle excitationsScanning tunneling spectroscopy

1 Introduction

The low-energy quasiparticle excitations of cuprate superconductors exhibit various spectral characteristics that differ from those of simple Bogoliubov quasiparticles for pure superconductors because of the presence of competing orders (COs) in the ground state of under- and optimally doped cuprates [110]. Some of the best known unconventional spectral characteristics include: the presence (absence) of pseudogap and Fermi arc phenomena in hole-type (electron-type) cuprates [412]; dichotomy of the quasiparticle coherence for momentum near the nodal and anti-nodal parts of the Fermi surface [9, 12, 13]; pseudogap (PG)-like spectral features inside vortex cores [47]; and non-universal spectral homogeneity among different types of cuprate [4, 5, 1418].

In this work we investigate the effects of varying doping levels and magnetic fields on the spatially resolved low-energy quasiparticle excitations of hole- and electron-type cuprate superconductors. Our experimental results suggest that the PG phenomena are closely related to COs, and that the correlation of superconductivity (SC) and PG with different types of disorder may account for the varying degrees of spatial homogeneity in the quasiparticle spectra. We also demonstrate the effect of hole doping on the vortex-state irreversibility line, which suggests the relevance of competing orders to vortex pinning.

2 Experimental

The primary experimental technique employed in this work is cryogenic scanning tunneling spectroscopy (STS). Details of the experimental setup, surface preparations and methodology of data analysis for the STS studies have been described elsewhere [57, 14, 15]. The hole-type cuprates investigated in this work include optimally and under-doped Y-123 single crystals with SC transition temperatures Tc=93 K, 85 K and 60 K, which correspond to hole-doping levels of p=0.15, 0.13 and 0.09, respectively; and over-doped (Y1−xCax)Ba2Cu3O7−δ epitaxial films grown by pulsed laser deposition with x=0.05,0.10,0.125,0.20 and 0.30, and the corresponding Tc determined from magnetization measurements were 68, 64, 59, 42 and 74 K, respectively. We note that the p value for a given Ca-doping level x depends on the oxygen annealing process [19], and that there is a maximum Tc value for a given x,Tc,max(x), which were empirically determined to be 93.5, 89.0, 82.9 and 82.9 K for x=0,0.10,0.20 and 0.30, respectively [19]. Hence, the p values of Ca-doped Y-123 samples are estimated from Tc,max(x) and the empirical formula [20] Tc(x,p)=Tc,max(x)[1−82.6(p−0.16)2], yielding p=0.216,0.218,0.214,0.238 and 0.19 for x=0.05,0.10,0.125,0.20 and 0.30. The electron-type cuprate studied in this work is an optimally doped infinite-layer system Sr0.9La0.1CuO2 (La-112) with Tc=43 K [7, 15]. All samples had been characterized by x-ray diffraction and magnetization studies to ensure single-phased structures and superconductivity. In addition to microscopic STS studies, the effects of Ca-doping on macroscopic vortex dynamics were investigated by measuring the irreversibility temperature Tirr(H,x,p) from the zero-field-cool (ZFC) and field-cool (FC) magnetization (M) vs. temperature (T) curves [21].

3 Doping-Dependent Quasiparticle Tunneling Spectra and Vortex Dynamics

Spatially resolved tunneling conductance (dI/dV) vs. energy (ω=eV) spectra for the quasiparticle local density of states (LDOS) maps at T=6 K were obtained on aforementioned Y-123, Ca-doped Y-123, and La-112 samples in zero and finite magnetic fields (H). For H=0, the tunneling spectra revealed long-range spatial homogeneity in under- and optimally doped Y-123 and optimally doped La-112 samples [47], which differ from the strong spatially inhomogeneous tunneling spectra observed in Bi-2212 [17, 18]. In contrast, for Ca-doped Y-123, the zero-field LDOS spectra revealed spatial homogeneity only within a limited range (up to ∼102 nm in length); variations in the spectral characteristics appeared over a long range, which may be attributed to disorder in Ca-doping.

3.1 Doping-Dependent Zero-Field LDOS of Hole-Type Cuprates

A representative zero-field LDOS of the optimally doped Y-123 in the top panel of Fig. 1a shows a set of coherent peaks at ωΔSC and shoulder-like features at ±Δeff. Both features exhibit long-range spatial homogeneity, as manifested by the histogram in the bottom panel of Fig. 1a. We attribute the two features to the consequence of coexisting SC and CO in the ground state of the under- and optimally doped hole-type cuprates [410]. Briefly, the LDOS \(\mathcal{N}(\omega)\) is associated with the spectral density function A(k,ω) and the Green function G(k,ω) by the relation \(\mathcal{N}(\omega) = \sum_{\mathbf{k}} A(\mathbf{k},\omega) = -\sum_{\mathbf{k}}\) Im[G(k,ω)]/π, and G(k,ω) may be obtained from diagonalizing the mean-field Hamiltonian \(\mathcal{H}_{\mathrm{MF}} = \mathcal{H}_{\mathrm{SC}} + \mathcal{H}_{\mathrm{CO}}\), which consists of coexisting SC and a CO, where \(\mathcal{H}_{\mathrm{SC}}\) is given by [410]
https://static-content.springer.com/image/art%3A10.1007%2Fs10948-012-1706-y/MediaObjects/10948_2012_1706_Equa_HTML.gif
Here the SC pairing potential is given by ΔSC(k)=Δd(coskx−cosky)/2 for pure \(d_{{x}^{2}-{y}^{2}}\)-wave pairing and ΔSC(k)=Δd(coskx−cosky)/2+Δs for (\(d_{{x}^{2}-{y}^{2}}+s\))-wave pairing [14], k denotes the quasiparticle momentum, ξk is the normal-state eigen-energy relative to the Fermi energy, c and c are the creation and annihilation operators, and α=↑,↓ refer to the spin states. For \(\mathcal{H}_{\mathrm{CO}}\), there is a CO energy ΔCO and a density wave vector associated with a given CO [410]. In the case of charge density waves (CDW) being the relevant CO, we have a Q1 parallel to the CuO2 bonding direction (π,0)/(0,π) [410]. The LDOS thus obtained for the optimally doped Y-123 is shown by the solid line in Fig. 1a, where Δeff≡[(Δd)2+(ΔCO)2]1/2. We further note the occasional occurrence of a zero-bias conductance peak (ZBCP) for tunneling along the {100} direction, as exemplified in the main panel of Fig. 1a, which is the result of the atomically rugged {100} surface so that Andreev bound states near {110} can contribute to the tunneling spectra, as detailed in Ref. [14].
https://static-content.springer.com/image/art%3A10.1007%2Fs10948-012-1706-y/MediaObjects/10948_2012_1706_Fig1_HTML.gif
Fig. 1

Doping-dependent zero-field STS in Y-123 taken at T=6 K: (aUpper panel: Normalized LDOS of optimally doped Y-123 (Tc=93 K) with k∥{100} and {001} (inset). The solid lines represent theoretical fittings [410] by assuming coexisting SC and CDW with parameters ΔSC=29 (21) meV for k∥{100}({001}), ΔCDW=32 meV and QCDW=(0.25π±0.05π,0)/(0,0.25π±0.05π), and the zero-bias conductance peak in the main panel is due to atomically rugged {100} surface so that some of the Andreev bound states near {110} are detected and fit with the BTK theory [14]. Lower panel: Histograms for ΔSC and ΔCDW. (bUpper panel: Normalized LDOS of under-doped Y-123 (Tc=60 K) with k∥{100} and {001} (inset, Tc=85 K). Lower panel: Histograms for ΔSC and ΔCDW. (c) Normalized LDOS of Ca-doped Y-123 with k∥{001}, Tc=74 K and p=0.19. The parameters are (Δd, Δs, ΔCDW) = (16, 3, 27) meV. (d) Zero-field ΔSC and ΔCDW of Y-123 vs.p (Color figure online)

For under-doped Y-123, the zero-field LDOS also reveals similar spectral features (Fig. 1b, upper panel), except that Δd is reduced and Δeff evolves from shoulder-like features to peak-like features separated from the SC coherence peaks. Both Δd and Δeff remain spatially homogeneous (Fig. 1b, lower panel).

In the case of Ca-doped Y-123, the pairing symmetry evolves from pure \(d_{{x}^{2}-{y}^{2}}\) to (\(d_{{x}^{2}-{y}^{2}}+s\))-wave with ΔSC(k)=Δd(coskx−cosky)/2+Δs [14, 22, 23], and the spectral characteristics are homogeneous only over smaller areas ∼(102×102) nm2 probably due to disordered Ca-doping. As exemplified in Fig. 1c for a Ca-doped Y-123 with x=0.3 and p∼0.19, two sets of coherent peaks appear at ω=±(Δd+Δs) and ±(ΔdΔs) [14], and the shoulder-like features correspond to Δeff=[(ΔSC)2+(ΔCO)2]1/2, where ΔSC≡max{ΔSC(k)}. The doping-dependent ΔSC and ΔCO for the Y-123 system is shown in Fig. 1d, showing a dome-like ΔSC(p) similar to that of Tc(p) and a decreasing ΔCO(p) similar to the PG temperature T(p).

3.2 Vortex-State LDOS of Hole-Type Cuprates

In the vortex state of conventional type-II superconductors, SC inside the vortex core is suppressed by the supercurrents surrounding each vortex, giving rise to enhanced local density of states (LDOS) peaking at ω=0 near the center of each vortex [24]. In contrast, the vortex-state LDOS of Y-123 exhibits several important differences. First, despite spatially homogeneous zero-field LDOS, field-induced vortices are relatively disordered and the radius of the vortex “halo” (ξhalo∼10 nm) appears much larger than the SC coherence length ξSC∼1.2 nm, (Fig. 2a). Further, the vortex-state LDOS remains suppressed inside the vortex core (Figs. 2c–2d), with PG-like features appearing at the same energy ΔCO as that derived from theoretical analysis of the zero-field LDOS. Moreover, density-wave like constant-bias conductance modulations are apparent, as exemplified in Fig. 2b. The histogram of the spectral evolution from ΔSC to ΔCO and another sub-gap feature at Δ′ with increasing H is shown in Fig. 2e, which is in stark contrast to the vortex-state spectral evolution of conventional type-II superconductors [4].
https://static-content.springer.com/image/art%3A10.1007%2Fs10948-012-1706-y/MediaObjects/10948_2012_1706_Fig2_HTML.gif
Fig. 2

Spatially resolved STS studies of the vortex state of Y-123 at T=6 K [5, 6]: (a) Tunneling conductance power ratio rG map over a (75×40) nm2 area for H=4.5 T, showing aB=(23.5±8.0) nm. Here rG at each pixel is defined by the ratio of (dI/dV)2 at V=(ΔSC/e) to that at V=0. (b) The LDOS modulations of Y-123 at H=5 T over a (22×29) nm2 area, showing patterns associated with density-wave modulations and vortices (circled objects) for ω=−9 meV∼−Δ′ and ω=−23 meV∼−ΔSC. (c) Conductance spectra along the white line in (a), showing SC peaks at ωΔSC outside vortices and PG features at ωΔCO inside vortices. (d) Spatially averaged intra- and intervortex spectra for H=2.0 T, 4.5 T and 6 T from left to right. (e) Energy histograms for the field-dependent spectral weight derived from the STS data for H=0,2,4.5, and 6 T, showing a spectral shift from ΔSC to ΔCO and Δ′ with increasing H (Color figure online)

To obtain further insights, we perform Fourier transformation (FT) of the LDOS at constant energies (ω). As shown in Fig. 3a for the FT-LDOS in the reciprocal space for spectra integrated from −1 to −30 meV, various spectral peaks are apparent, which may be divided into two distinct types: One is associated with the ω-independent wave vectors that may be attributed to COs of charge-, pair- and spin-density waves (CDW, PDW and SDW) and the (π,π) magnetic resonance, as shown in Figs. 3b, 3d and summarized in Fig. 3g [46]. The other type consists of ω-dependent quasiparticle interference (QPI) wave vectors [5, 6, 17], as exemplified in Fig. 3e and summarized in Fig. 3f. The spectral intensity of these ω-independent wave vectors exhibits interesting evolution with H that further corroborates the existence of COs, as exemplified in Fig. 3c [46].
https://static-content.springer.com/image/art%3A10.1007%2Fs10948-012-1706-y/MediaObjects/10948_2012_1706_Fig3_HTML.gif
Fig. 3

Studies of the vortex-state FT-LDOS of Y-123 [46]: (a) FT-LDOS at H=5 T obtained by integrating |F(k,ω)| from ω=−1 to −30 meV. The ω-independent spots are circled for clarity, which include the reciprocal lattice constants, the (π,π) resonance, QPDW and QCDW along the (π,0)/(0,π) directions, and QSDW along (π,π). (b) The ω-dependence of |F(k,ω)| at H=5 T is plotted in the ω-vs.-k plot against k∥(π,0), showing ω-independent modes (bright vertical lines) at QPDW and QCDW. (c) |F(q,ω)| for q=QPDW (red) and QCDW (green) are shown as a function of ω for H=0 (solid lines) and H=5 T (dashed lines). (d) |F(k,ω)| for different energies are plotted against k∥(π,0), showing peaks at ω-independent QPDW, QCDW and the reciprocal lattice constants at (2π/a1) along (π,0). (e) |F(k,ω)| for different energies are plotted against k∥(π,π), showing peaks at ω-independent QSDW along (π,π). Additionally, dispersive wave vectors due to QPI are found, as exemplified by the dispersive QPI momentum q7 specified in (f). (f) The QPI momentum (|qi|) vs. ω dispersion relations derived from FT-LDOS [46]. Lower panel: Illustration of the qi associated with QPI between pairs of points on equal energy contours with maximum joint density of states. (gLeft panel: Illustration of the wave vectors associated with SDW and CDW. Right panel: ω-independent |QPDW|, |QCDW| and |QSDW| [46] (Color figure online)

3.3 LDOS of Electron-Type Cuprates

The zero-field LDOS of electron-type cuprate La-112 exhibited a single set of spectral peaks at ωΔeff (Fig. 4a), and the LDOS revealed long-range spatial homogeneity [7]. Theoretical fitting to the LDOS and the k-dependent spectral density from angle-resolved photoemission spectroscopy (ARPES) yields a \(d_{{x}^{2}-{y}^{2}}\)-wave SC gap with Δd∼12 meV and a SDW with a wave vector of (π,π) and ΔSDW∼8 meV [4, 5, 7]. The vortex-state LDOS of La-112 revealed a vortex-core radius comparable to ξSC∼4.9 nm. The LDOS remained suppressed inside the vortex core, with PG-like features appearing at ΔCO<ΔSC, as shown in Fig. 4c. The fact that ΔCO<ΔSC is consistent with the absence of zero-field PG above Tc in La-112. The histogram of the spectral evolution with H is illustrated in Fig. 4d, which differs from those of Y-123 (Fig. 2e) and conventional type-II superconductors.
https://static-content.springer.com/image/art%3A10.1007%2Fs10948-012-1706-y/MediaObjects/10948_2012_1706_Fig4_HTML.gif
Fig. 4

STS studies of La-112: (a) Normalized tunneling spectra taken at T=6 K (black) and 49 K (red) for H=0. The solid lines represent fittings to the T=6 and 49 K spectra by assuming coexisting SC and SDW, with fitting parameters Δd=12 meV, ΔSDW=8 meV, and QSDW=(π,π) [4, 5, 7]. (b) A spatial map of the rG ratio over a (64×64) nm2 area for H(∥c)=1 T and T=6 K, showing vortices separated by an average vortex lattice constant aB=52 nm, comparable to the theoretical value of 49 nm. The average radius of vortices is ξhalo=(4.7±0.7) nm, comparable to ξSC=4.9 nm. (c) Spatial evolution of (dI/dV) along the black dashed line cutting through two vortices in (b) for H=1 T, showing an intra-vortex PG smaller than the intervortex SC gap. (d) Energy histograms of La-112 (left), which differ from those of conventional type-II SC (right) (Color figure online)

3.4 Doping-Dependent Vortex Dynamics

In addition to the LDOS, we investigate how vortex dynamics may evolve with different doping. By applying H parallel to the CuO2 planes of Y-123 and Ca-doped Y-123 and determining the irreversibility temperatures Tirr(H,p,x) from the ZFC and FC M-vs.-T curves [21], we find that the normalized irreversibility line that separates the vortex solid from the vortex-liquid initially decreases with increasing Ca-doping x for nearly constant p (Fig. 5), suggesting suppressed SC coherence due to Ca-induced disorder [23]. As the hole doping further increases, the trend eventually reverses (Fig. 5), probably due to vanishing COs and therefore enhanced SC stiffness and reduced vortex-state fluctuations [21].
https://static-content.springer.com/image/art%3A10.1007%2Fs10948-012-1706-y/MediaObjects/10948_2012_1706_Fig5_HTML.gif
Fig. 5

Vortex irreversibility lines vs. (T/Tc) for Hab in Y-123 systems, where the irreversibility field Hirr(T,x,p) for each sample is normalized to its respective theoretical upper critical field [Hc2(0)]ab by the relation [Hc2(0)]ab=Φ0/[2πξab(0)ξc(0)]. Using the mean-field gap relation 2Δd=4.3kBTc for d-wave pairing and the BCS relation Δd=hvF/[2πξab(0)], where h is the Planck constant and vF is the Fermi velocity (∼105 m/s in the cuprates), we obtain [Hc2(0)]ab∼(0.18 Tesla/K2)×(Tc)2 for Tc measured in K by using the empirical anisotropy ratio γ=ξab(0)/ξc(0)∼7 [21]. For x<0.2, the vortex-solid phase below the normalized irreversibility line \(H_{\mathrm{irr}}(T,x,p)/(0.18T_{c}^{2})\) is suppressed with increasing Ca-doping if p is kept nearly constant. However, for sufficiently large hole-doping levels (such as for p>0.23), the trend is reversed due to vanishing CO and increasing SC stiffness

4 Discussion

In addition to the doping dependence of ΔSC and Δeff, it is interesting to address the issue of spatial homogeneity of LDOS in different cuprates. Comparing our empirical findings with the highly inhomogeneous quasiparticle spectra in Bi-2212 [17, 18], it appears that the spatial homogeneity in the LDOS depends on the type of disorder: For optimally and under-doped Y-123, the long-range spatial homogeneity in both ΔSC and Δeff may be attributed to the stoichiometric cations and ordered apical oxygen. In contrast, for Ca-doped Y-123, the substitution of Y by Ca contributes to excess holes as well as disorder in the CuO2 planes, giving rise to spatial variations in ΔSC and Δeff. In the case of under and optimally doped Bi-2212, the non-stoichiometric Bi-to-Sr ratio results in disordered Sr and apical oxygen in the SrO layer [25], leading to highly disordered PG features. Finally, for optimally doped electron-type cuprate La-112, the homogeneous LDOS may be attributed to stoichiometric cations and the absence of apical oxygen.

5 Conclusion

Scanning tunneling spectroscopic studies of various cuprate superconductors as a function of doping level, doping type and magnetic field (H) reveal that their low-energy excitations consist of not only the Bogoliubov quasiparticles but also bosonic excitations associated with COs, leading to a PG above Tc for H=0 and inside the vortex core for TTc if ΔCO>ΔSC, as in the under- and optimally doped Y-123. In contrast, for ΔCO<ΔSC as in the electron-type cuprate La-112 and strongly over-doped Y-123, PG is absence above Tc for H=0 and is only revealed inside the vortex core for TTc. The vortex-core size ξhalo for cuprates with ΔCO>ΔSC is much larger than ξSC, whereas ξhalo for cuprates with ΔCO<ΔSC is comparable to ξSC, suggesting that the vortex-core states are sensitive to the CO strength. The microscopic doping-dependent spectral characteristics are found to be relevant to the macroscopic vortex dynamics, as manifested by the doping-dependent vortex irreversibility lines of Y-123.

Acknowledgements

We acknowledge funding provided by NSF Grant #DMR0907251, by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation, and by the Kavli Nanoscience Institute with support of the Kavli Foundation. ZJF acknowledges the support from China Scholarship Council during his visit to Caltech.

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