o 1
I
I-
-I-
-~ \..
.,.-~ ~
-4 3 2
o 1
-1
0.5
-0.3 -0.2 -0.1 0.0 0.1 V (mV)
~ 0.5
_ _ u - _..._.L...._L-...J_---L_...L_-L._-'-_..._.L...._L----LJ
Fig. 1.4. Top panel: Conductance of tunnel junctions F1 , F2 and F3 , placed respectively 100, 200 and 800 nm away from the NS interface, normalized by the junction conductance at voltageV = 0.3 mY. The conductance is proportionnal to the DOS in the copper wire in good contact with the aluminum wire. Inset:
normalized differential conductance of a tunnel junction between a normal probe and a superconducting aluminum wire. All measurements were performed at T =30 mK. Bottom panel: predicted DOS using the theory of the proximity effect, calculated with a spin-flip scattering time ofTsf =65 ps.
1.4 Coherent transport at an NS boundary: the NS-QUID
1.4 Coherent transport at an NS boundary: the NS-QUID
The subgap (or Andreev) current through a normal metal/superconductor tunnel junction
IS another indicator of the pair correlations in the normal metal. Indeed, this current is exclusively due to pairs of normal electrons tunneling into the superconductor. Since this tunneling of a pair is a second order process in barrier transmission, the current across opaque barriers should be negligible. However, tunneling attempts by pairs of electrons in time-reversed states add up coherently, in contrast with the incoherent tunneling attempts of a single electron (see Fig. 1.5). Therefore the Andreev current should be enhanced in a metal where impurities or boundaries confine the electronic trajectories near the NS interface, and all the more so as the coherence time in the normal metal is long.
N
2e
s
Fig. 1.5. Semiclassical representation of the mechanism responsible for constructive interference in the tunneling of pairs of normal electrons. Two weakly localized electrons in the normal electrode with nearly time-reversed wave functions tunnel through the barrier at different points with the same total phase. Ifthe order parameter of the superconductor is uniform, the tunnel amplitudes at these different points contribute constructively to the total current.
Inorder to probe the quantum coherence of electrons in the normal metal, we have devised an interference experiment with two superconducting/normal tunnel junctions in parallel. The relative phase of the two superconducting electrodes is controlled by applying a magnetic field perpendicular to the plane of the loop they form (see Fig. 1.6 for the electron micrograph of three such NS-QUIDs, which differ only by the length of normal wire separating the two tunnel
19
-Chapter 1 Introduction
junctions). The interference pattern is the conductance of the structure, which is sinusoidally modulated by the field. Figure 1.7 shows theIVcharacteristics of three NS-QUIDs, measured
Fig. 1.6. Scanning electron micrograph of a sample containing three NSQUIDs: each device is made of an open superconducting aluminum loop (upper shadow of the loop), oxidized to form two tunnel junctions with the normal copper wire (lower grey shadow of horizontal wire). The three devices differ only by the distance between the two tunnel junctions.
respectively with no magnetic flux (maximal subgap current), and one flux quantum (minimum subgap current) in the loop.
The modulation of the current by the magnetic field, measured at one point of the IVcurve of one NS-QUID, is shown in the panel below. The modulation is perfectly sinusoidal. In addition, in all three NS-QUIDs, the magnitude of the modulated current (difference between the current with a superconducting phase difference of 0 and 1f) is of the order of the total current through the structure. The intensity of the modulated current as a function of voltage is shown in the right panel of Fig. 1.7. The maximal current at low voltages, and the decrease in current modulation at high voltage illustrate the loss of coherence between electron pairs with non negligible energy difference. The difference in modulation intensity between the three NS-QUIDs at low energy demonstrates the existence of inelastic processes, such as scattering
1.4 Coherent transport at an NS boundary: the NS-QUID
Fig. 1.7. Top panel: IV curves of three interferometers with tunnel junctions separated by 410, 620 and 785 nm respectively; with zero magnetic field (maximal subgap current) and one half flux quantum (minimal current) in the loop. The three sets of curves have been offset vertically for clarity. Lower panel: modulation of the current through an interferometer, for a given voltage, by a magnetic field H applied perpendicularly to the loop of surface A (symbols). The continuous curve is a cosine fit to the data. Right pannel: measured modulated current (symbols) compared to the current computed from the semi-classical probability to diffuse from one junction to the other (continuous lines). All measured curves were taken at T=30 mK.
21
-Chapter 1 Introduction
by magnetic impurities, which limit the coherence of electron pairs in the normal metal. From the measured curves, a coherence time of about 100 ps, corresponding to a coherence length of about 1 f.Lm, is inferred. The specific shape of the modulated current of all three devices can be deduced from an Andreev rate given by the Fermi golden rule. This current can also be calculated with the theory of the proximity effect. In that framework, the current is due to the existence of pair correlations induced in the normal metal by the presence of the superconductor. This experiment illustrates how Andreev reflection and the proximity effect are two aspects of the same phenomenon.
REFERENCES
[1] D. Pines, P. Nozieres, The theory of quantum liquids (W.A. Benjamin, New York, 1966).
[2] M. Tinkham, Introduction to Superconductivity (Me Graw Hill, New York, 1985), chapter 3.
[3] P. G. de Gennes, Superconductivity of metals and alloys (W. A. Benjamin, New York, 1966).
[4] V. T. Petrashov, V. N. Antonov, P. Delsing, and T. Claeson, Phys. Rev. Lett. 70, 347 (1993); See also Procedings of the NATO Advanced Research Workshop on Mesoscopic Superconductivity, F. W. J. Hekking, G. Schon, and D. V. Averin, Editors (Elsevier, Amsterdam, 1994).