^{1,a)}and Elina Tuuli

^{2}

### Abstract

We have investigated Andreev reflection at interfaces between superconducting indium (*T _{c} * = 3.4 K) and several normal conducting nonmagnetic metals (palladium, platinum, and silver) down to

*T*= 0.1 K as well as zinc (

*T*= 0.87 K) in its normal state at

_{c}*T*= 2.5 K. We analyzed the point-contact spectra with the modified one-dimensional BTK theory valid for ballistic transport. It includes Dynes' quasiparticle lifetime as fitting parameter Γ in addition to superconducting energy gap 2Δ and strength

*Z*of the interface barrier. For contact areas from less than 1 nm

^{2}to 10 000 nm

^{2}the BTK

*Z*-parameter was close to 0.5, corresponding to transmission coefficients of about 80%, independent of the normal metal. The very small variation of

*Z*indicates that the interfaces have a negligible dielectric tunneling barrier. Also Fermi surface mismatch does not account for the observed

*Z*. The extracted value

*Z*≈ 0.5 can be explained by assuming that practically all of our point contacts are in the diffusive regime.

Elina Tuuli acknowledges a 2-yr grant from the Graduate School of Materials Research (GSMR), 20014 Turku, Finland. We thank Yu. G. Naidyuk for discussions and the Jenny and Antti Wihuri Foundation for financial support.

I. Introduction

II. Experiments and results

III. Discussion

IV. Conclusion

### Key Topics

- Superconductivity
- 22.0
- Superconducting metals
- 20.0
- Interface structure
- 16.0
- Contact resistance
- 15.0
- Dielectrics
- 13.0

## Figures

Schematics of a point contact between two different metals in momentum space with *k _{F} *

_{1}>

*k*

_{F}_{2}, see Refs.

^{ 31 and 32 }. The vertical line symbolizes the interface. At low temperatures and no applied bias voltage only electrons near the Fermi surfaces, indicated by the two circles, take part in transport processes. Flow from left to right requires

*k*> 0 for electrons of the left-hand sphere. When the size of the Fermi surfaces differs like in the figure, only electrons in the highlighted region can travel through the contact and find states in the highlighted region of the smaller right-hand sphere. The others are normal reflected. In opposite direction, normal reflection does not occur since all electrons from states of the left-hand sphere with negative

_{z}*k*find states on the left-hand side within an angle Θ

_{z}_{ c }around the negative

*k*axis. The critical angle is Θ

_{z}_{ c }= arcsin(

*k*

_{F}_{2}/

*k*

_{F}_{1}) to satisfy conservation of parallel momentum

*k*.

_{p}Schematics of a point contact between two different metals in momentum space with *k _{F} *

_{1}>

*k*

_{F}_{2}, see Refs.

^{ 31 and 32 }. The vertical line symbolizes the interface. At low temperatures and no applied bias voltage only electrons near the Fermi surfaces, indicated by the two circles, take part in transport processes. Flow from left to right requires

*k*> 0 for electrons of the left-hand sphere. When the size of the Fermi surfaces differs like in the figure, only electrons in the highlighted region can travel through the contact and find states in the highlighted region of the smaller right-hand sphere. The others are normal reflected. In opposite direction, normal reflection does not occur since all electrons from states of the left-hand sphere with negative

_{z}*k*find states on the left-hand side within an angle Θ

_{z}_{ c }around the negative

*k*axis. The critical angle is Θ

_{z}_{ c }= arcsin(

*k*

_{F}_{2}/

*k*

_{F}_{1}) to satisfy conservation of parallel momentum

*k*.

_{p}Differential resistance spectra *dV*/*dI* versus bias voltage *V* of contacts with In. Normal metal counter electrode and temperature are indicated: an Andreev-reflection double-minimum structure. The two pairs of sharp side peaks make the analysis difficult (a); one pair of side peaks and slightly structured around zero bias (b); no side peaks but a minimum at zero bias, which appears to consist of two separate minima (c). The zero-bias minima in (b) and (c) could result from the proximity effect, while the side peaks in (a) and (b) stem from the self-magnetic field exceeding a critical value in the contact region.

Differential resistance spectra *dV*/*dI* versus bias voltage *V* of contacts with In. Normal metal counter electrode and temperature are indicated: an Andreev-reflection double-minimum structure. The two pairs of sharp side peaks make the analysis difficult (a); one pair of side peaks and slightly structured around zero bias (b); no side peaks but a minimum at zero bias, which appears to consist of two separate minima (c). The zero-bias minima in (b) and (c) could result from the proximity effect, while the side peaks in (a) and (b) stem from the self-magnetic field exceeding a critical value in the contact region.

Differential resistance spectra *dV*/*dI* versus bias voltage *V* of an In–Zn junction at the indicated temperatures. At *T* = 0.1 K the Josephson-type anomaly at zero bias as well as multiple Andreev reflection within the superconducting gap are clearly visible. At *T* = 1.5 K the little dip at zero bias could indicate proximity-induced superconductivity in Zn. It is completely suppressed at *T* = 2.5 K. The spectra of this contact show a side peak (arrow), which moves slightly with temperature and is possibly caused by the self-magnetic field. The inset displays the details of the differential resistance around zero bias with emphasis on the spectra at 1.5 and 2.5 K, respectively.

Differential resistance spectra *dV*/*dI* versus bias voltage *V* of an In–Zn junction at the indicated temperatures. At *T* = 0.1 K the Josephson-type anomaly at zero bias as well as multiple Andreev reflection within the superconducting gap are clearly visible. At *T* = 1.5 K the little dip at zero bias could indicate proximity-induced superconductivity in Zn. It is completely suppressed at *T* = 2.5 K. The spectra of this contact show a side peak (arrow), which moves slightly with temperature and is possibly caused by the self-magnetic field. The inset displays the details of the differential resistance around zero bias with emphasis on the spectra at 1.5 and 2.5 K, respectively.

Selected differential resistance spectra *dV*/*dI* versus bias voltage *V* of In–Ag contacts at *T* = 0.7 K (thin lines). The underlying grey curves are fits with the modified BTK model. ^{ 37 } Fit parameters are shown in Table II .

Selected differential resistance spectra *dV*/*dI* versus bias voltage *V* of In–Ag contacts at *T* = 0.7 K (thin lines). The underlying grey curves are fits with the modified BTK model. ^{ 37 } Fit parameters are shown in Table II .

Selected differential resistance spectra *dV*/*dI* versus bias voltage *V* of In–Pd contacts at low temperatures (thin lines). The underlying grey curves are fits with the modified BTK model. ^{ 37 } Temperatures and fit parameters are shown in Table II .

Selected differential resistance spectra *dV*/*dI* versus bias voltage *V* of In–Pd contacts at low temperatures (thin lines). The underlying grey curves are fits with the modified BTK model. ^{ 37 } Temperatures and fit parameters are shown in Table II .

Selected differential resistance spectra *dV*/*dI* versus bias voltage *V* of In–Pt contacts at low temperatures (thin lines). The underlying grey curves are fits with the modified BTK model. ^{ 37 } Temperatures and fit parameters are shown in Table II .

Selected differential resistance spectra *dV*/*dI* versus bias voltage *V* of In–Pt contacts at low temperatures (thin lines). The underlying grey curves are fits with the modified BTK model. ^{ 37 } Temperatures and fit parameters are shown in Table II .

Selected differential resistance spectra *dV*/*dI* versus bias voltage *V* of In–Zn contacts at *T* = 2.5 K (thin lines). The underlying grey curves are fits with the modified BTK model. ^{ 37 } Fit parameters are shown in Table II .

Selected differential resistance spectra *dV*/*dI* versus bias voltage *V* of In–Zn contacts at *T* = 2.5 K (thin lines). The underlying grey curves are fits with the modified BTK model. ^{ 37 } Fit parameters are shown in Table II .

Superconducting energy gap 2Δ_{0} = 2Δ (T→0) of In extracted from the point-contact spectra using the modified BTK theory ^{ 37 } versus normal state resistance *R.* Different symbols mark separate measurement series. For In–Ag contacts we have two measurement series at *T* = 0.7 K (open symbols) and one at *T* = 0.1 K. All In–Zn contacts were measured at *T* = 2.5 K. The In–Pd and the In–Pt contacts were measured at low temperature down to *T* = 0.1 K. The solid lines at 2Δ_{0} = 1.20 meV serve as a guide to the eye.

Superconducting energy gap 2Δ_{0} = 2Δ (T→0) of In extracted from the point-contact spectra using the modified BTK theory ^{ 37 } versus normal state resistance *R.* Different symbols mark separate measurement series. For In–Ag contacts we have two measurement series at *T* = 0.7 K (open symbols) and one at *T* = 0.1 K. All In–Zn contacts were measured at *T* = 2.5 K. The In–Pd and the In–Pt contacts were measured at low temperature down to *T* = 0.1 K. The solid lines at 2Δ_{0} = 1.20 meV serve as a guide to the eye.

Dynes lifetime parameter Γ of In in contact with the indicated normal metals extracted from the point-contact spectra using the modified BTK theory ^{ 37 } versus normal state resistance *R*. Different symbols mark separate measurement series. For In–Ag contacts we have two measurement series at *T* = 0.1 K (open symbols) and one at *T* = 0.1 K. All In–Zn contacts were measured at *T* = 2.5 K. The In–Pd and the In–Pt contacts were measured at low temperature down to *T* = 0.1 K.

Dynes lifetime parameter Γ of In in contact with the indicated normal metals extracted from the point-contact spectra using the modified BTK theory ^{ 37 } versus normal state resistance *R*. Different symbols mark separate measurement series. For In–Ag contacts we have two measurement series at *T* = 0.1 K (open symbols) and one at *T* = 0.1 K. All In–Zn contacts were measured at *T* = 2.5 K. The In–Pd and the In–Pt contacts were measured at low temperature down to *T* = 0.1 K.

*Z*-parameter of normal reflection of contacts between In and the indicated normal conductors extracted from the point-contact spectra using the modified BTK theory ^{ 37 } versus normal state resistance *R*. Different symbols mark separate measurement series. For In–Ag contacts we have two measurement series at *T* = 0.7 K (open symbols) and one at *T* = 0.1 K. All In–Zn contacts were measured at *T* = 2.5 K. The In–Pd and the In–Pt contacts were measured at low temperature down to *T* = 0.1 K. The solid lines at *Z* = 0.5 serve as a guide to the eye.

*Z*-parameter of normal reflection of contacts between In and the indicated normal conductors extracted from the point-contact spectra using the modified BTK theory ^{ 37 } versus normal state resistance *R*. Different symbols mark separate measurement series. For In–Ag contacts we have two measurement series at *T* = 0.7 K (open symbols) and one at *T* = 0.1 K. All In–Zn contacts were measured at *T* = 2.5 K. The In–Pd and the In–Pt contacts were measured at low temperature down to *T* = 0.1 K. The solid lines at *Z* = 0.5 serve as a guide to the eye.

Schematics of a diffusive point contact between two different metals. The contact (hatched area) has a spatial extension L ≫ *l* _{0}. Its elastic electron mean free path *l* _{0} is not necessarily the same as in the bulk electrodes.

Schematics of a diffusive point contact between two different metals. The contact (hatched area) has a spatial extension L ≫ *l* _{0}. Its elastic electron mean free path *l* _{0} is not necessarily the same as in the bulk electrodes.

## Tables

Distribution of contact type: “Andreev” denotes contacts that could be analyzed, “proximity-like” contacts look like those where superconductivity has been induced in the normal metal, and “undefined” are all others which cannot be clearly identified.

Distribution of contact type: “Andreev” denotes contacts that could be analyzed, “proximity-like” contacts look like those where superconductivity has been induced in the normal metal, and “undefined” are all others which cannot be clearly identified.

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