^{1,a)}, M. Tortello

^{1}, P. Pecchio

^{1}, V. A. Stepanov

^{2}and R. S. Gonnelli

^{3}

### Abstract

Point-contact Andreev-reflection spectroscopy (PCARS) has demonstrated to be one of the most effective experimental tools for the investigation of fundamental properties of superconductors such as the superconducting gap and the electron–phonon (or, more generally, electron–boson) coupling. By reviewing relevant examples reported in literature and presenting new results, in this paper we show that when the direction of the interface with respect to the crystallographic axes can be controlled (as in single crystals and epitaxial films) PCARS can provide invaluable information about the anisotropy of the pairing wavefunction or—in the case of multiband superconductors—on the number, amplitude and symmetry of the energy gaps. Moreover, the analysis of PCARS results within a suitable 3D generalization of the BTK model allows obtaining qualitative information about the topology of the Fermi surface.

We acknowledge the financial support of the European Community through to the Collaborative EU-Japan Project “IRON SEA” (NMP3-SL-2011-283141), and of the Italian Ministry of Research through the PRIN Project No. 2008XWLWF9-005. Our warmest thanks to all the people who contributed to the research summarized in this paper: G. A. Ummarino, S. Galasso, G. Profeta for help in theoretical analysis and bandstructure calculations; J. Karpinski, N. D. Zhigadlo, Z. Bukowski, J. Jiang, J. S. Kim, K. Iida, and F. Kurth for providing us with crystals and films used for PCARS measurements. R.S.G. acknowledges R. K. Kremer and the Max Planck Institute for Solid State Research in Stuttgart, where the measurements on Ca(Fe,Co)_{2}As_{2} were carried out.

I. Introduction

II. The 3D BTK model

III. Directional PCARS in magnesium diboride

IV. Directional PCARS in Ca-intercalated graphite

V. Directional PCARS in Co-doped Ca122

VI. Directional PCARS in Co-doped Ba122

VII. Conclusions

### Key Topics

- Fermi surface
- 67.0
- Superconductivity models
- 30.0
- Superconductors
- 22.0
- Anisotropy
- 19.0
- Charge injection
- 19.0

## Figures

(a) Fermi surface of MgB_{2}. (b) Model Fermi surface used for the calculation of the theoretical spectra within the two-band 3D BTK model. (c) Theoretical PCARS curves generated at *T* = 4.2 K by using the two-band 3D BTK model with the gaps calculated in Ref. ^{ 7 } , i.e., Δ_{π} = 2.70 meV and Δ_{σ} = 7.09 meV and the model Fermi surface depicted in (b), for *I*||*ab* (bottom) and *I*||*c* (top). The bottom (blue) and the top thin (red) curves were obtained by using the same parameters, i.e., *Z* _{π} = 0.37, *Γ* _{π} = 1.65 meV, *Z* _{σ} = 0.85, *Γ* _{σ} = 2.10 meV, that allow reproducing the *ab*-plane spectrum of panel (d); the top thick (red) curve has parameters optimized to reproduce the shape of the *c* axis spectrum of panel (d), i.e., *Z* _{π} = 0.17, Γ_{π} = 0.5 meV, *Z* _{σ} = 0.85, Γ_{σ} = 0.4 meV. (d) Symbols: two examples of PCARS spectra taken in MgB_{2} single crystals, with current injected along the *ab* plane (bottom) and along the *c* axis (top). The vertical scale is on the left axis; the top curve has been offset by 1 for clarity. Lines: two examples of STS spectra taken in different grains of a polycrystal (from Ref. ^{ 24 } ). The vertical scale is on the right axis and the top curve has been offset for clarity. Vertical lines show the correspondence of the gap features in PCARS and STS spectra along the different directions.

(a) Fermi surface of MgB_{2}. (b) Model Fermi surface used for the calculation of the theoretical spectra within the two-band 3D BTK model. (c) Theoretical PCARS curves generated at *T* = 4.2 K by using the two-band 3D BTK model with the gaps calculated in Ref. ^{ 7 } , i.e., Δ_{π} = 2.70 meV and Δ_{σ} = 7.09 meV and the model Fermi surface depicted in (b), for *I*||*ab* (bottom) and *I*||*c* (top). The bottom (blue) and the top thin (red) curves were obtained by using the same parameters, i.e., *Z* _{π} = 0.37, *Γ* _{π} = 1.65 meV, *Z* _{σ} = 0.85, *Γ* _{σ} = 2.10 meV, that allow reproducing the *ab*-plane spectrum of panel (d); the top thick (red) curve has parameters optimized to reproduce the shape of the *c* axis spectrum of panel (d), i.e., *Z* _{π} = 0.17, Γ_{π} = 0.5 meV, *Z* _{σ} = 0.85, Γ_{σ} = 0.4 meV. (d) Symbols: two examples of PCARS spectra taken in MgB_{2} single crystals, with current injected along the *ab* plane (bottom) and along the *c* axis (top). The vertical scale is on the left axis; the top curve has been offset by 1 for clarity. Lines: two examples of STS spectra taken in different grains of a polycrystal (from Ref. ^{ 24 } ). The vertical scale is on the right axis and the top curve has been offset for clarity. Vertical lines show the correspondence of the gap features in PCARS and STS spectra along the different directions.

(a) Fermi surface of CaC_{6}. The almost spherical surface originates from the nearly-free electron band (interlayer band) while the warped cylinders are mainly arising from the carbon π bands. As shown in Ref. 30, this distinction is however not strict. (b) Model Fermi surface used for the calculations within the 3D BTK model. For simplicity, we assumed two isotropic gaps of different amplitudes (indicated by different colors) to reside on the spherical surface and on the warped cylinders (here represented by a hyperboloid of revolution). Note that in the cell of panel (a) there are 7 cylinders for each spherical surface. (c) Theoretical spectra at *T* = 400 mK calculated by using the 3D BTK model and the schematic Fermi surface of panel (b). Apart from the gap values Δ_{1} = 1.7 meV and Δ_{2} = 1.3 meV that reside on the two portions of the FS, the parameters *Z* _{1} = 0.60 and *Z* _{2} = 0.95 were used. Note that these values are the same for both *I*||*c* and *I*||*ab*. Finally, the broadening parameters were set to zero. (d) Symbols: experimental PCARS spectra measured at 400 mK with current injected along the *c* axis (top) and along the *ab* plane (bottom). Thin lines are the theoretical spectra calculated as in panel (c) but using the broadening parameters Γ_{1} = 0.8 meV and Γ_{2} = 0.5 meV.

(a) Fermi surface of CaC_{6}. The almost spherical surface originates from the nearly-free electron band (interlayer band) while the warped cylinders are mainly arising from the carbon π bands. As shown in Ref. 30, this distinction is however not strict. (b) Model Fermi surface used for the calculations within the 3D BTK model. For simplicity, we assumed two isotropic gaps of different amplitudes (indicated by different colors) to reside on the spherical surface and on the warped cylinders (here represented by a hyperboloid of revolution). Note that in the cell of panel (a) there are 7 cylinders for each spherical surface. (c) Theoretical spectra at *T* = 400 mK calculated by using the 3D BTK model and the schematic Fermi surface of panel (b). Apart from the gap values Δ_{1} = 1.7 meV and Δ_{2} = 1.3 meV that reside on the two portions of the FS, the parameters *Z* _{1} = 0.60 and *Z* _{2} = 0.95 were used. Note that these values are the same for both *I*||*c* and *I*||*ab*. Finally, the broadening parameters were set to zero. (d) Symbols: experimental PCARS spectra measured at 400 mK with current injected along the *c* axis (top) and along the *ab* plane (bottom). Thin lines are the theoretical spectra calculated as in panel (c) but using the broadening parameters Γ_{1} = 0.8 meV and Γ_{2} = 0.5 meV.

(a) Fermi surface of Ca(Fe_{1.94}Co_{0.06})_{2}As_{2}. The hole-like FS sheets in the center of the Brillouin zone is on the verge of a topological transition and is splitting into two closed pockets centered at the *Z* points. (b) The model Fermi surface used in the 3D BTK model, shown here only in the upper half of the Brillouin zone. Matt surfaces are the Fermi surface sheets, while the gridded surfaces indicate the amplitude of the relevant gap, which is isotropic on the electron-like FS but anisotropic on the hole-like one. (c) Theoretical curves calculated for *I*||*ab* (top) and *I*||*c* (bottom) using the Fermi surface of panel (b) and the gaps indicated in the labels. Here “1” and “2” refer to the holelike (electron-like) FS, respectively. The other parameters were: for curve *1*, Γ_{1} = 1.25 meV, *Z* _{1} = 0.05, Γ_{2} = 6.15 meV, *Z* _{2} = 0.145; for curve *2*, Γ_{1} = 1.7 meV, *Z* _{1} = 0.05, Γ_{2} = 6.5 meV, *Z* _{2} = 0.23. In both cases the angle between the normal to the interface and the *a* axis was taken to be α = π/8. The latter set of parameters was also used to calculate the *c*-axis spectrum (curve *3*). (d) Two examples of experimental curves measured in single crystals of 6% Co-doped CaFe_{2}As_{2}, with the current injected along the *ab* planes.

(a) Fermi surface of Ca(Fe_{1.94}Co_{0.06})_{2}As_{2}. The hole-like FS sheets in the center of the Brillouin zone is on the verge of a topological transition and is splitting into two closed pockets centered at the *Z* points. (b) The model Fermi surface used in the 3D BTK model, shown here only in the upper half of the Brillouin zone. Matt surfaces are the Fermi surface sheets, while the gridded surfaces indicate the amplitude of the relevant gap, which is isotropic on the electron-like FS but anisotropic on the hole-like one. (c) Theoretical curves calculated for *I*||*ab* (top) and *I*||*c* (bottom) using the Fermi surface of panel (b) and the gaps indicated in the labels. Here “1” and “2” refer to the holelike (electron-like) FS, respectively. The other parameters were: for curve *1*, Γ_{1} = 1.25 meV, *Z* _{1} = 0.05, Γ_{2} = 6.15 meV, *Z* _{2} = 0.145; for curve *2*, Γ_{1} = 1.7 meV, *Z* _{1} = 0.05, Γ_{2} = 6.5 meV, *Z* _{2} = 0.23. In both cases the angle between the normal to the interface and the *a* axis was taken to be α = π/8. The latter set of parameters was also used to calculate the *c*-axis spectrum (curve *3*). (d) Two examples of experimental curves measured in single crystals of 6% Co-doped CaFe_{2}As_{2}, with the current injected along the *ab* planes.

(a) Fermi surface of Ba(Fe_{1.92}Co_{0.08})_{2}As_{2}, with the strongly warped hole-like FS sheets around Γ and the electron-like FS sheets at the corners of the Brillouin zone. (b) The model Fermi surface used in the 3D BTK model. Matt surfaces are the Fermi surface sheets, while the gridded surfaces indicate the amplitude of the relevant gap. The drawing refers to the case of isotropic gaps on both the bands. (c) Theoretical curves calculated for *I*||*ab* (*1* and *2*) and *I*||*c* (*3*and *4*) and using the Fermi surface of panel (b). The gap amplitudes indicated in the labels were chosen in order to fit the position of the gap features in the experimental curves of panel (d). The other fitting parameters are the following: for curve *1*, Γ_{1} = 1.85 meV, *Z* _{1} = 0.03, Γ_{2} = 3.6 meV, *Z* _{2} = 0.31; for curve *2*, Γ_{1} = 1.75 meV, *Z* _{1} = 0.08, Γ_{2} = 3.0 meV, *Z* _{2} = 0.245; for curve *3*, Γ_{1} = 2.8 meV, *Z* _{1} = 0.05, Γ_{2} = 1.3 meV, *Z* _{2} = 0; for curve *4*, Γ_{1} = 3.85 meV, *Z* _{1} = 0.1, Γ_{2} = 1.4 meV, *Z* _{2} = 0. (d) Some examples of the experimental curves measured in 8% Co-doped BaFe_{2}As_{2} with *I*||*ab* (top) and *I*||*c* (bottom). The lowest-lying curve was measured in epitaxial films ^{ 52 } with *x* = 0.08 and *T _{c} * = (23.8±0.7) K, the others in single crystals.

(a) Fermi surface of Ba(Fe_{1.92}Co_{0.08})_{2}As_{2}, with the strongly warped hole-like FS sheets around Γ and the electron-like FS sheets at the corners of the Brillouin zone. (b) The model Fermi surface used in the 3D BTK model. Matt surfaces are the Fermi surface sheets, while the gridded surfaces indicate the amplitude of the relevant gap. The drawing refers to the case of isotropic gaps on both the bands. (c) Theoretical curves calculated for *I*||*ab* (*1* and *2*) and *I*||*c* (*3*and *4*) and using the Fermi surface of panel (b). The gap amplitudes indicated in the labels were chosen in order to fit the position of the gap features in the experimental curves of panel (d). The other fitting parameters are the following: for curve *1*, Γ_{1} = 1.85 meV, *Z* _{1} = 0.03, Γ_{2} = 3.6 meV, *Z* _{2} = 0.31; for curve *2*, Γ_{1} = 1.75 meV, *Z* _{1} = 0.08, Γ_{2} = 3.0 meV, *Z* _{2} = 0.245; for curve *3*, Γ_{1} = 2.8 meV, *Z* _{1} = 0.05, Γ_{2} = 1.3 meV, *Z* _{2} = 0; for curve *4*, Γ_{1} = 3.85 meV, *Z* _{1} = 0.1, Γ_{2} = 1.4 meV, *Z* _{2} = 0. (d) Some examples of the experimental curves measured in 8% Co-doped BaFe_{2}As_{2} with *I*||*ab* (top) and *I*||*c* (bottom). The lowest-lying curve was measured in epitaxial films ^{ 52 } with *x* = 0.08 and *T _{c} * = (23.8±0.7) K, the others in single crystals.

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