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Superconducting point contacts have been used for measuring magnetic polarizations, identifying magnetic impurities, electronic structures, and even the vibrational modes of small molecules. Due to intrinsically small energy scale in the subgap structures of the supercurrent determined by the size of the superconducting energy gap, superconductors provide ultrahigh sensitivities for high resolution spectroscopies. The so-called Andreev reflection process between normal metal and superconductor carries complex and rich information which can be utilized as powerful sensor when fully exploited. In this review, we would discuss recent experimental and theoretical developments in the supercurrent transport through superconducting point contacts and their relevance to sensing applications, and we would highlight their current issues and potentials. A true utilization of the method based on Andreev reflection analysis opens up possibilities for a new class of ultrasensitive sensors.

Since its first discovery over a hundred years ago [

Over the past decade there are mainly two very significant landmarks in the applications of PCAR spectroscopies. The first one is the measurement of magnetic polarization [

In order to have a meaningful physical understanding of the PCAR physics, we shall also present a detailed discussions of the theoretical aspects in both semiclassical and quantum pictures. The theoretical discussions in this review shall be divided into two parts. The first part is the summary of the semiclassical treatment based on the famous Blonder–Tinkham–Klapwijk (BTK) theory [

The technique of PCAR spectroscopy has been used for measuring the polarization of ferromagnetic materials [_{c}

The PCAR method is based on the fact that the current through the PC differs when the tip is superconducting compared to when it is in normal state. The PCAR method is based on the behaviour of the conductance at very low bias where the current is most dependent on the polarization _{NS}/_{NN} = 2. This ratio is called the normalized conductance. When the normal metal is a ferromagnet with perfect polarization, _{NS}/_{NN} = 0. A simple linear interpolation between these two extremes gives, _{NS}/_{NN} = 2(1 −

Mazin _{2} [_{2} system reveals only 0.02 difference in the polarization measurements, which is about the accuracy of the PCAR method [

Hundreds of related works on PCAR magnetic measurements appeared following these main experimental and theoretical achievements ever since. For instance, Pérez-Willard _{c}^{2}^{2}) of the contacts and concluded that smaller contact diameters are necessary to achieve truly ballistic transport, and to obtain a reliable PCAR measurements contact sizes around 10 nm or smaller are generally preferable. PCAR can also be used to measure spin diffusion lengths. For example Geresdi

The widespread use of the BTK theory extension for PCAR spectroscopy has been questioned by Xia

The second important landmark in the applications of the PCAR method is to determine the individual transmission coefficients of an atomic point contact (APC) [_{n}_{0}_{n}_{0} is the quantum conductance given by _{0} = 2^{2}/

Since the transmission coefficient of each channels can take value between zero and unity, the conductance of a single channel is mostly less than _{0} despite the fact that statistically the conductance of an APC tends to be quantized. The quantitative information on individual conductance channels has been inaccessible through normal conductance measurements, but for superconducting systems this can be extracted due to the sensitivity of the so-called sub-gap structures (SGS) of the superconductor at low bias to small changes of each conductance channels. The SGS originates from multiple Andreev reflection (MAR) [_{n}

PCAR spectroscopy has also been used to detect and identify magnetic impurities on superconducting surfaces. Yazdani _{c}

Ji

Excitations of vibrational modes by traversing electrons have been observed in metallic electrodes attached to nanostructures and molecules such as carbon nanotubes [^{th}_{n}_{n}_{n}

The measurements were performed at various temperatures from well below T_{c}_{c}

At the heart of the supercurrent transport mechanism is the so-called Andreev reflection (AR) process which can take place when a superconductor is in contact with a normal metal [

To illustrate the MAR process, we can use the following arguments: initially an electron from the interface between N and S on the left is accelerated by the external field toward the right, but unable to enter due to the energy gap. This would result in a reflection of a hole moving back to the left. The charge of 2e (one from the electron, the other from the hole moving in opposite direction) increase the supercurrent. The process is repeated until the particle gains sufficient energy to overcome the gap. Octavio

Now we shall summarize the derivations of the phenomenological treatments for transport through a normal-superconducting (NS) interface of the famous BTK theory [

The Bogoliubov de Gennes equation [_{F}_{c}_{F}_{F}_{N}_{F}_{S}

The original BTK theory solves the scattering conditions to obtain reflection and transmission probabilities at the interface between normal metal and superconductor using the simplest possible assumptions. First, BTK theory assumes equal Fermi energy between normal metal and superconductor. Second, the superconducting gap Δ(

The wave-numbers _{N}_{S}_{F}

This allows for the solutions of the coefficients and therefore the probabilities, ^{2}, ^{2},

After we know the probabilities

The integration is mainly over a small energy region around the Fermi level since the term [_{F}

The conductance defined as

In order to extend the BTK theory to measure the spin polarizations of ferromagnets, Mazin

Complete tabulations of

In this section we shall summarize a model based on quantum Hamiltonian theory, whose origin can be traced back from the early work by Bardeen who proposed a microscopic Hamiltonian approach for tunneling junction problems [

In quantum Hamiltonian theory, a system with two metallic leads can be represented by two independent Hamiltonians, _{L}_{R}_{T}_{C}_{C}

The leads are governed by the mean field BCS theory [_{iσ}_{α}_{α}^{(†)} annihilate (create) particle on their respective leads, while operators ^{(†)} do the same for the quantum dot. The time dependent phase is the consequence of the AC Josephson effects in finite bias, and it is incorporated into the tunneling terms following a gauge transformation suggested by Rogovin

For example we can calculate the (retarded) free propagator g^{r}

Evaluations of this term gives [^{N}

Another useful free propagator is the lesser propagator given by,

Time-dependent supercurrent across the junction can be derived from the expectation value of the time derivative of the number operator in any one leads, say the left one for convenience,

The term
_{Li}

The term _{z}

The next step is to express the current in terms of the free propagator of the leads and Green's function of the quantum dot. This can be done through NEGF procedure where the corresponding time-ordered Green's function for

We can then substitute these into ^{<}_{L}_{/}_{R}

The term Γ_{L}^{N}

The Josephson current through SNS QPC oscillates at very high frequency, typically in the terahertz range, which makes the time resolved quantities not so easily compared with the experiments. A more convenient way would be to work with the time averaged quantities derived from the Fourier transformation of the correct intrinsic frequencies of the systems. All dynamic quantities can be expanded as harmonics of the fundamental frequency

The time average current is derived simply from the zeroth order term _{0}. Due to the two-time correlations in the Green's function, we require a transformation that can account them in a consistent manner, and this is done through a so-called double Fourier transform of the Green's functions,

The retarded Green's function is calculated with the Dyson equation in Fourier transformed form, hence the matrices here are in Fourier space and Nambu space, and for the case of multilevel system it would be the tensor product of all,

The advanced function is obtained from the retarded function by ^{a}^{r}^{†}, and the time-average current can then be expressed as the zeroth order component of the Fourier transform,

The sample plot for the time averaged current and differential conductance (_{C}_{C}

The tunneling Hamiltonian directly couples left and right leads. For a single eigenchannel system the hopping term _{0} = _{L}_{R}_{L}_{R}

Another interesting application of the quantum Hamiltonian theory is for studying the interactions with some external electromagnetic radiations. The frequency range of interests in this case would be in the microwave regions, due to the intrinsic energy scale of typical superconducting energy gaps. The interplay between the AC Josephson effect in superconducting junctions under finite bias with the external radiations exhibit the phenomenon known as the _{r}_{ac}_{r}t_{ac}_{r}t

For a superconducting QPC system with a featureless barrier, _{r}

For superconducting QPC with a quantum dot at the centre, the localized energy levels at the quantum dot exhibit another intriguing physics upon exposure to external radiations, at least in two ways. First, in semiclassical limit the external field would oscillate the entire set of localized energy levels in unison. Second, absorptions and emissions of the photons would also stimulate interlevel transitions as the electrons tunnel through the quantum dot, and both would affect MAR process inside the quantum dot and hence the supercurrent behaviours. However, in order to do time averaged analysis, one needs to perform multi-frequency Fourier transformation on the dynamical quantities because of the two frequencies dependence of the phase factor. This is non-trivial particularly when the frequencies are non-commensurate,

External radiations can be modeled semiclassically adopting typical dipole approximations [

Intrinsically small energy gap in superconducting PCAR spectroscopy provides a promising candidate for ultrasensitive sensors, making use of the AR process which carries rich physics at the contacts. AR process in NS systems can be used to probe spin polarizations of ferromagnetic materials with convenience and high precision compared to the conventional methods. Theoretical developments in this area are mainly based on the BTK theory, which had begun earlier and has become a relatively mature theory to be used in spin polarization measurements. However, some problems still remain that relate to various delicate details of the surface properties at the contacts which have been treated phenomenologically.

Atomic contacts such as STM tips and MCBJ have discrete eigenchannels and the quantum Hamiltonian theory combined with NEGF enables rigorous descriptions of the complex transport properties of MAR. The method also has promising potentials to be extended for a fully first principle method if we combine the existing first principle superconductivity theory [

_{x}

_{2}by point contact Andreev reflection

_{c}_{c}

Measured I-V curves for two different Al atomic point contacts having different sets of {_{n}_{n}

Detecting a single atom magnetic impurities of Mn and Cr on Pb surface with a Nb STM tip. (

The schematic view of the atomic configurations for measuring vibrational modes of an Nb dimer fabricated with MCBJ technique [

Multiple Andreev reflection (MAR) process in a symmetric superconductor-normal-superconductor (SNS) system with the normal region sufficiently thin to provide ballistic trajectories. The dark particles (electrons) are the antiparticle of the white particles (holes), and the reflection process repeats until they attain sufficient energy to overcome the superconducting gap Δ. The horizontal axes on the superconductor sides represent density of states.

Band diagram for N (left) S (right) interface for the BTK model. The superconducting energy gap in reality is much smaller to Fermi energy (Δ ≪ _{F}

A resonant tunneling system which consists of two superconducting leads and a quantum dot. The system is represented by three subsystem Hamiltonians, _{L}_{T}_{R}

Plot of the time averaged I-V and _{d} =_{L}_{R}_{b}T

Effects of single mode external radiations on SNN transport in weak coupling limit. (_{1} − _{2}). The main DC resonance at 4Δ splits into two, and the separation between the split is equal to 2

Table for coefficients A (Andreev reflection) and B (normal reflection).

^{2}/[^{2} + (Δ^{2} − ^{2})(1 + 2^{2})^{2}] |
^{2} − 1)/[^{2})]^{2} |

^{2}(1 + ^{2})/[^{2})]^{2}] |