DIRECT OBSERVATION OF BALLISTIC ANDREEV REFLECTION
T. M. Klapwijka'b* S. A. Ryabchunhc
Kavli Institute of NanoScience, Faculty of Applied Sciences, Delft University of Technology 262S, CJ Delft, The Netherlands
bLaboratory for Quantum Limited Devices, Physics Department, Moscow State Pedagogical University
119992, Moscow, Russia
cMoscow Institute of Electronics and Mathematics, National Research University Higher School of Economics
109028, Moscow, Russia
Received June 9, 2014
An overview is presented of experiments on ballistic electrical transport in inhomogeneous superconducting systems which are controlled by the process of Andreev reflection. The initial experiments based on the coexistence of a normal phase and a superconducting phase in the intermediate state led to the concept itself. It was followed by a focus on geometrically inhomogeneous systems like point contacts, which provided a very clear manifestation of the energy and direction dependence of the Andreev reflection process. The point contacts have recently evolved towards the atomic scale owing to the use of mechanical break-junctions, revealing a very detailed dependence of Andreev reflection on the macroscopic phase of the superconducting state. In present-day research, the superconducting inhomogeneity is constructed by clean room technology and combines superconducting materials, for example, with low-dimensional materials and topological insulators. Alternatively, the superconductor is combined with nano-objects, such as graphene, carbon nanotubes, or semiconducting nanowires. Each of these "inhomogeneous systems" provides a very interesting range of properties, all rooted in some manifestation of Andreev reflection.
Cwitribvtiwi for the JETP special issue in honor of A. F. Andreev's 75th birthday
DOI: 10.7868/S0044451014120013
1. INTRODUCTION
The 50-year-old concept of Andreev reflection fl], published in May 1964, arose originally in the context of ballistic transport in inhomogeneous crystalline materials with parts in the superconducting phase intermixed with parts in the normal phase. The difference between the electrical and thermal conductivities, already observed in the early 1950s by Mendelssohn and Olseri [2], and Hulm [3] was not resolved by the 1959 microscopic theory of the thermal conductivity by Bardeen et al. [4]. Subsequent experimental work by Zavaritskii [5] in 1960, and by Strassler and Wyder [6] in 1963 led Andreev to the analysis of electron transport at the interface between the normal and the superconducting phase in the same crystal. He identified the unique process of the conversion of an electron
E-mail: t.m.klapwijk'ffltudelft.nl
into a hole which retraces the path of the incident electron, accompanied by the simultaneous process of a charge of 2e being carried away by the superconducting condensate. This process facilitated charge transport but it did not allow for energy transport and the observed thermal boundary resistance was a natural consequence fl, 7].
An interface between a normal metal and a superconductor is an example of an inhomogeneous superconducting system. Since the early 1950s, the natural framework for dealing with a position-dependent superconducting order parameter was provided by the Ginzburg Landau theory [8]. The original BCS theory [9, 10] assumed a uniform superconducting state. By developing a formulation in 1958 of the microscopic theory [11], which allows for spatial variations, Gorkov [12] showed in 1959 that the Ginzburg Landau theory can be derived from the microscopic theory. The Ginzburg Landau theory is only valid close to the critical temperature Tc, whereas the difference in thermal
and electrical conductivities was primarily manifest at temperatures much lower than Tc. A conceptual framework for inhomogeneous superconductors was needed, which included the spectral properties of the superconducting state, which is available in the original Gorkov theory [11]. The Bogoliubov De Gennes equations, which are now commonly used, are a limit case of these Gorkov equations, suitable for treating ballistic transport .
From the experimental point of view, another very important step was taken almost simultaneously in 19C5 by Sharvin [13] by the invention of mechanically constructed metallic point contacts. This allowed the study of electrical transport between two dissimilar materials, with electrical transport governed by classically ballistic electrons. The application of this concept of ballistic transport to normal-metal superconductor contacts provided the framework, introduced by Blonder et al. [14], to measure the energy dependence of the Andreev scattering process very directly. The Sharvin point contacts also stimulated a new approach to the description of electrical transport on the nanoscale level by using the scattering matrix approach, introduced already in 1957 by Landauer [15] and generalized and applied to phase-coherent normal transport in nanoscale objects by Biittikcr in 1985 [16]. Rather than relying on a general theory for inhomogeneous systems, it focuses on simplified experimental systems in which the phase-coherent transport problem can be split into three pieces. It selects the class of problems in which two equilibrium reservoirs can be defined, usually at a different chemical potentials or temperatures, which serve as emitters or absorbers of quantum particles and a scattering region in which the interesting physical processes occur and which can be characterized by a scattering matrix with certain symmetry properties.
The experimental progress in constructing nano-ob.jects with the clean room technology, now universally available, has led to many experiments based on nano-objects connected to superconducting rather than normal-metal reservoirs. This leads to a large variety of objects and observations in which the challenge is to discover new phenomena and at the same time establish through transport experiments what has actually been made in the clean room. In some cases, the general theory of inhoniogcncous iionequilibrium superconductivity is used to interpret these specific cases. At the same time, the perceived unique nature of these nano-objects has led to an application of the scattering-matrix approach, in which the superconducting contacts serve as equilibrium reservoirs that communicate with the scattering region through the Andreev reflection pro-
cess. An experimental challenge is to determine which framework is appropriate for the actual nano-objects emerging from the clean room and where theoretical innovation is needed.
In what follows, we attempt to summarize the developments in the subject over the past 50 years. The focus is on experimental observations, which provide a direct demonstration related to ballistic Andreev reflection. The main attention is paid to the demonstration of the reversal of direction, as well as of the charge, and the spectroscopically important dependence on energy. Furthermore, a third important aspect is the dependence on the macroscopic quantum phase, which manifests itself when more than one superconductor is used. It leads to the concept of Andreev bound states, which carry the Joscphson current. Since the field has become large, a further selection was applied by focusing on experiments that are sufficiently well-defined, such that a quantitative description turns out to be possible. Needless to say, many experiments are not included, in particular those in which diffusive scattering is the dominant ingredient. The section headings give an indication of the subject. They are supplemented with the dates in which, in our view, the most significant developments for this subject took place.
2. INHOMOGENEOUS SUPERCONDUCTIVITY CLOSE TO Tc: 1950-1957
After the discovery of superconductivity by observing zero resistance by Kanicrlingh Onncs in 1911, it took until 1933 for a second fundamental property to be identified by Mcissner and Ochscnfcld, and called perfect dianiagnctisni. An early explanation was provided by Fritz and Heinz London in 1935 by a modification of the Maxwell equations inside a superconducting material. It was known that these properties were very nicely observed in pure crystals of tin, aluminium, and mercury. However, it was also known that many superconducting alloys did not obey these basic relations. In particular, perfect dianiagnctisni was not observed although the material provided zero resistance. Apparently, magnetic flux was not completely excluded and the magnetization curve was not reversible but showed clearly hysteretic effects. The first theory capable of handling inhoniogcncous systems was the Ginzburg Landau theory, introduced in 1950. It was used by Abrikosov in 1957 to analyze what would happen with a superconductor if the magnetic penetration depth Al known from the London theory exceeded another cliar-
actoristic length now called the Ginzburg Landau coherence length.
By niininiizing the expression for the free energy in a volume in which the order parameter can vary with position, we find the two celebrated Ginzburg Landau (GL) expressions
1
2m*
-ifiV ■
e*A c
4> + ai> + ß\4fi> = 0 (1)
and
2im*
IDC
-n>A. (2)
These two equations allow calculating the order parameter as a function of position in the presence of a magnetic field, including the distribution of the current. The magnetic field H is the locally present field strength. And, of course, it is assumed that the order parameter ip is complex with a phase <f>, which can also be position dependent.
The most ideal inhomogeneous system is one in which we have a clearly defined boundary between a piece of atomic matter in the superconducting state and a piece of the same atomic matter in the normal state. In such a system, no barrier would be encountered for normal electronic transport, because the material is uniform in its atomic arrangement. Obviously, this is not the case in the many na
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