Andreev–Saint-James reflections: A probe of cuprate superconductors

Electrical transport through a normal-metal/superconductor contact at biases smaller than the energy gap can occur via the reflection of an electron as a hole of opposite wave vector. The same mechanism of electron-hole reflection gives rise to low-energy states at the surface of unconventional superconductors having nodes in their order parameter. The occurrence of electron-hole reflections at normal-metal/superconductor interfaces was predicted independently by Saint-James and de Gennes and by Andreev, and their spectroscopic features were discussed in detail by Saint-James in the early sixties. They are generally called Andreev reflections but, in view of the literature, will here be referred to as Andreev–Saint-James (ASJ) reflections. This review presents a historical review of ASJ reflections and spectroscopy in conventional superconductors, and reviews their application to the high-$T_c$ cuprates. The occurrence of ASJ reflections in all studied cuprates is well documented for a broad range of doping levels, implying that there is no large asymmetry between electrons and holes near the Fermi level in the superconducting state. In the underdoped regime, where the pseudogap phenomenon has been observed by other methods such as nuclear magnetic resonance (NMR), angular-resolved photoemission spectroscopy (ARPES), and Giaever tunneling, gap values obtained from ASJ spectroscopy are smaller than pseudogap values, indicating a lack of coherence in the pseudogap energy range. Low-energy surface bound states have been observed in all studied hole-doped cuprates, in agreement with a dominant $d$-wave symmetry order parameter. Results are mixed for electron-doped cuprates. In overdoped $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }(\delta <0.08)$ and ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$, ASJ spectroscopy is consistent with the presence of an additional imaginary component of the order parameter. Results of ASJ spectroscopy under applied magnetic fields are also reviewed. A short section at the end is devoted to some recent results on spin effects.

Figure 1

Density of states for a normal-metal/superconductor $(\mathrm{N}∕\mathrm{S})$ contact for different values of the normal layer thickness $a$. The thickness is given in terms of a normalized parameter $(2a∕\pi \xi )$. 105.

Figure 2

(Color in online edition) Schematic representation of an Andreev–Saint-James (ASJ) cycle for a $d$-wave superconductor coated with a normal-metal layer, the interface being oriented perpendicular to a nodal direction.

Figure 3

(Color in online edition) Schematic representation of a Sharvin point contact having a size $s$ much smaller than the coherence length $\xi$ and the mean free path $l$. At biases of the order of the gap, the current density at the contact is of the order of the depairing value, but at distances of the order of $(\xi ,l)$, it is reduced much below that value. The condition $s⪡(\xi ,l)$ avoids heating effects and quenching of superconductivity in the vicinity of the contact.

Figure 4

Buildup of the superfluid density $\mathrm{S}$ and decrease of the quasiparticle density $QP$ in the superconductor side of an $\mathrm{N}∕\mathrm{S}$ contact. ASJ reflections build up progressively over the distance $\xi$.

Figure 5

Conductance characteristics of $\mathrm{N}∕\mathrm{S}$ contacts for various values of the barrier parameter $Z$. 21.

Figure 6

Conductance characteristics of $\mathrm{Au}∕\mathrm{YBCO}$ point contacts for the antinodal orientation. The increase in the conductance at low bias is about 50%. The edge is at about 20 mV. Note the absence of a conductance peak at the gap edge. Data from three different locations on the same sample. 62.

Figure 7

Calculated conductance characteristics for a $d$-wave order parameter in an antinodal orientation, at different values of the barrier parameter $Z$ (0; 0.2; 0.3; 0.5; 0.7; 1.0). Note for $Z>0.5$ the clear V shape at low bias and the sharp decrease in the conductance at the gap edge.

Figure 8

(Color in online edition) Conductance characteristics of $\mathrm{Au}∕\mathrm{YBCO}$ film contacts fitted to a $d$-wave order parameter for different values of the barrier parameter $Z$ (from left to right: 0.68; 0.49; 0.34). The fit is for an antinodal direction. Note the pronounced V shape at low bias for the highest-$Z$ contact. 71.

Figure 9

Measured conductance characteristic of a $\mathrm{Au}∕{\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$ in-plane contact to a bulk-oriented sample. Note the low-bias V shape and the sharp drop at about 20 mV. The fit is for an antinodal direction and $Z=0.8$. 92.

Figure 10

Calculated conductance characteristics for a contact to a $d$-wave superconductor in a nodal direction, for different values of the barrier parameter $Z$ (0; 0.5; 1.0; 2.0; 3.0). Contrary to the case of antinodal contacts, there is no sharp structure at the gap bias, only a smooth return to the normal-state conductance.

Figure 11

Calculated conductance characteristics for a contact to a $d$-wave superconductor in a nodal direction with $Z=5$, at different openings of the tunneling cone. Return to the normal-state conductance always occurs at about the gap value. The curves have been calculated using the weight function $\exp [-{(\theta ∕{\theta }_M)}^2]$, with ${\theta }_M$ values 90°, 57°, 33°, 23°, and 18°.

Figure 12

Measured conductance characteristic of a $\mathrm{Au}∕{\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ single-crystal underdoped contact. The data were fitted for a nodal direction, giving $Z=0.3$, and a gap of 5 meV. 33.

Figure 13

Measured conductance characteristic of a $\mathrm{Au}∕{\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ single-crystal contact near optimum doping. Note the small split of the conductance peak at small bias. The line is an attempt to fit the data with an $s$-wave gap. Arrows indicate high-bias structures possibly related to phonons. 1.

Figure 14

$I\left(V\right)$ and conductance characteristics of an in-plane contact to a ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$ single crystal. Note the return to the normal-state conductance at about 10 meV. 110.

Figure 15

Measured scanning tunneling microscope (STM) conductances at various positions on an YBCO grain having a (001)-oriented upper surface. The shape of the characteristics goes from V shape on top of the grain (b, c), to U shape near a (100) face (d), to inverted V near a (110) face (e, f). 106.

Figure 16

Tunneling characteristics of an $\mathrm{In}∕\mathrm{YBCO}$ junction on a (110)-oriented film at increasing magnetic fields of up to 6 T. Inset: measurements in decreasing fields. 31.

Figure 17

Tunneling characteristics of an $\mathrm{In}∕\mathrm{YBCO}$ junction in increasing fields. The YBCO film has the (110) orientation, and the field is oriented parallel to the surface of the film and parallel to the $c$ axis (itself in-plane oriented). Note (inset) that the splitting does not take place when the field is applied perpendicular to the $c$ axis. 13.

Figure 18

Field dependence of the zero-bias conductance peak for in-plane junctions to YBCO films. The fit for the $\mathrm{YBCO}∕\mathrm{Cu}$ contact is from the theory of 30. 30.

Figure 19

Scanning tunneling microscope characteristics measured on an underdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$ single crystal, at temperatures ranging from 4.2 K up to room temperature. A conductance dip persists up to the highest temperature. Note, however, the change in the dip amplitude when going from 123 to 151 K. 103.

Figure 20

Comparison between an ASJ (inside graph) and a Giaever characteristic measured on similarly underdoped YBCO samples. Note the difference in energy scales: about 15 meV for the ASJ reflection edge, and 50 meV for the Giaever gap. 38.

Figure 21

Behavior of ASJ and Giaever (or ARPES) energy scales as a function of doping in different cuprates. The ASJ scale $\left({\Delta }_c\right)$ follows the same behavior as does $T_c$ while the Giaever scale $\left({\Delta }_p\right)$ keeps rising as the doping is reduced. 38.

Figure 22

(Color in online edition) Split of the zero-bias conductance peak in a number of $\mathrm{In}∕\mathrm{YBCO}$ junctions on (110)-oriented films of different thickness, measured in decreasing fields. The split follows a square-root law, with a coefficient of the order of $1\mathrm{mV}∕{\mathrm{T}}^{1∕2}$. 14.

Figure 23

Dependence of the spontaneous zero-bias conductance-peak split in $\mathrm{In}∕\mathrm{YBCO}$ junctions on (110)-oriented films, as a function of a parameter proportional to the doping level. Overdoping was achieved by increasing the oxygen content. A spontaneous split is only found in overdoped samples. The inverse of the initial slope of the zero-bias conductance-peak field splitting, or susceptibility, is also shown. The susceptibility diverges near optimum doping. The behavior of the spontaneous splitting and of the susceptibility is indicative of the presence of a quantum critical point near optimum doping, beyond which the order parameter develops a small imaginary component. 31.

Figure 24

(Color in online edition) Variation with the barrier parameter $Z$ of the $d$-wave component and of the $is$-wave component at $\mathrm{Au}∕\mathrm{YBCO}$ contacts fitted to the (100) orientation. The $is$ component becomes of the order of the $d$-wave one for high-transparency barriers, suggesting that it is due to a proximity effect. 71.