Extracting the Transport Channel Transmissions in Scanning Tunneling Microscopy using the Superconducting Excess Current

Transport through quantum coherent conductors, like atomic junctions, is described by the distribution of conduction channels. Information about the number of channels and their transmission can be extracted from various sources, such as multiple Andreev reflections, dynamical Coulomb blockade, or shot noise. We complement this set of methods by introducing the superconducting excess current as a new tool to continuously extract the transport channel transmissions of an atomic scale junction in a scanning tunneling microscope. In conjunction with ab initio simulations, we employ this technique in atomic aluminum junctions to determine the influence of the structure adjacent to the contact atoms on the transport properties.

At the heart of the modern theory of quantum transport lies the concept of transport channels [1,2].Similar to transverse electromagnetic modes in an optical waveguide, charge can be transported between two baths through a set of distinct pathways arising from individual quantum states.ese quantum states are rooted in the electronic structure of a device and can be manipulated by changes in materials or geometry [3].Each transport channel is further characterised by a transmission value, which speci es the probability for an electron that enters the channel from one bath to be transmi ed through it to reach the other bath.
e number of transport channels and their respective transmissions, o en referred to as the mesoscopic PIN code (in analogy to the personal identi cation number), characterises a transport conguration.
In this article, we show that, if at least one of the electrodes involved in transport is superconducting, the excess current can be used to determine the PIN code in a simple and rapid fashion.Using an ultra-low temperature scanning tunnelling microscope (STM), we study transport properties of superconducting tunnel junctions from the deep tunnelling regime to atomic contact and extract the PIN code as a continuous function of the junction conductance.By using ab initio transport calculations, we are able to elucidate the mi-croscopic nature of the conduction channels.
In addition to quasi-particle tunnelling, superconducting contacts support another mode for charge transport through Andreev re ections [28].An electron incident onto the superconductor is re ected as a hole, thereby transferring a charge of 2 into the superconductor and forming a Cooper pair.Higher orders of this process occur in superconductorsuperconductor junctions and are referred to as multiple Andreev re ections (MARs).
Even at bias voltages  outside of the superconducting gap ( 2), the lowest order Andreev re ection continues to contribute to the current leading to a constant o set from the expected single particle current [29].Formally, this excess current is de ned as where  is the superconducting gap parameter, the subscripts S and N refer to the superconducting and normal state, respectively, and  is the elementary charge.A non-linear dependence of the excess current on the channel transmission allows an extraction of the junction PIN code.Assuming  Tip =  Sample = , the excess current across a superconductor-insulator-superconductor junction at zero temperature can be calculated as where   is the transmission of the  th electronic transport channel and ℎ is Planck's constant [3].We assume that the   are independent of each other and of the bias voltage  , a good approximation in the small energy window around zero bias relevant to transport in atomic contacts.As can be seen from Eq. (2), the excess current depends on the   and , but not on the bias voltage  applied to the contact.If only few open transport channels are present, a single data point is thus enough to fully determine the PIN code, greatly facilitating and expediting data acquisition.
All measurements are performed in a custom-built STM placed in a dilution refrigerator and operating at a base temperature of 10 mK [30].
e Al(100) sample was cleaned by bombardment with Ar ions followed by stepwise annealing in UHV from 480 • C to 460 • C to 435 • C. We extracted individual Al atoms from the substrate and placed them on the pristine Al(100) surface to create a simple transport con guration which serves as a model system for our PIN code analysis [31].
We approach the adatom with the STM tip to measure conductance spectra at constant height and  ()-curves.e situation is schematically shown in Fig. 1(a).Conductance spectra acquired at low conductance show the characteristic signature of superconductor-superconductor tunnelling with an energy gap around zero bias, anked by coherence peaks on either side (see Fig. 1(b)).
e values of  for tip and sample, required for a quantitative analysis based on Eq. ( 2) or numerical simulations, can be extracted from a t (see Supplemental Material for details).For the present case, we nd Junctions such as the one described here have been reported to exhibit exceptional stability [32].Fig. 1(c) shows the height-dependent normal-state conductance  N of a typical junction calculated from an  ()-measurement at 1.5 mV, outside the gap, in units of the quantum of conductance,  0 = 2 2 /ℎ.We de ne the point  = 0 to be at the maximum of conductance.At rst, the conductance increases exponentially at a rate of roughly one order of magnitude per 100 pm, as expected from theory, until reaching a maximum value near  0 .If the tip is approached further, the conductance decreases again at a rate which depends on the microtip.In some cases, a reduction to about 0.2  0 has been observed.Remarkably, no jump to contact occurs [33] and there is no hysteresis in the current when retracting the tip [31].We conclude that the junction remains unchanged a er the approach-retract cycle.
While the general shape of the height-dependent conductance is consistent across measurements with di erent microtips, the value of the maximum conductance  max N attained and the magnitude of the drop past  max N varies significantly [31].
ese di erences must be rooted in the details of quantum transport between tip and sample, as characterized by the mesoscopic PIN code.Monitoring the PIN code continuously as a function of  promises deeper insights into the quantum properties of the junctions.
As can be seen from Eq. ( 2), the complete PIN code is contained in the excess current.e  ( )-characteristic of a superconducting tunnel junction evolves to rst order in  for  > 2, but it does not pass through the origin.It instead intersects the current axis at a constant value, referred to as the excess current (see Fig. 1(d)).It arises from the lowest- order Andreev re ection, which contributes to the total current even outside the gap.Indeed, an established method of extracting the PIN code in superconducting junctions is to analyse the subharmonic gap structure due to MARs in a junction at high conductance.MARs lead to a series of features at integer fractions of 2 which characterize the channel con guration.e excess current, having the same physical origin, contains identical information.
To show that an analysis of the excess current is a pertinent way of determining the junction PIN code, we supplement our continuous  ()-measurements of the excess current with full MAR spectra at selected points in the  ()-curve.Representative data sets from two distinct junctions are shown in Fig. 2(a) and (b).e superconducting gap is visible as a step in the  ( )-curve at 2 = 360 μeV, while the MARs are manifested as shoulders at  = 2/2, 2/3, ....In addition to the MARs, the Josephson e ect, the coherent tunnelling of Cooper pairs, is visible as a sharp rise in current close to zero bias.Strong -dependent variations of the sub-gap structure are clearly apparent in both data sets, pointing towards changes in the channel con guration.
We use a well established model based on the relation between MARs and PIN code [34,35] to analyze the channel con guration of the di erent junctions and their transmission dependence.Owing to its partially lled -shell, we assume that an atomic contact of Al may sustain up to three distinct transport channels.Due to this construction, we nd the two transport channels having -symmetry to be degenerate.We t the MAR data using values for  and the Dynes broadening parameter [36], which we both extract from a quasiparticle t such as the one shown in Fig. 1(b) and capture the environmental broadening by a convolution with a Gaussian [32].e voltage range between ±50μV, dominated by Josephson transport, is excluded from the analysis.e results of this channel analysis are presented in Fig. 2(c) and (d).
e non-linear dependence of the di erent orders of Andreev re ections on the channel transmission allows us to extract the junction PIN code.Comparing this with the total conductance, which is well reproduced by our calculations, we con rm the predominant single-channel nature of the contacts.
We now turn toward the task of extracting the excess current from the  ()-traces.As can be seen from Fig. 1(d), the excess current may be regarded as an integration constant when calculating the current from conductance.We de ne the normal state current at bias voltage  as where  N is the di erential conductance which is measured directly by means of a lock-in ampli er (see Supplemental Material for details).e excess current is then the di erence between the experimentally detected current and the normal state current, see Eq. ( 1).
Representative results from the excess current determination are shown in Fig. 3 for the same two junctions which have been discussed in the context of Fig. 2. Figure 3 e blue and orange markers show the same quantities as derived from the spectra shown in Fig. 2

(a) and (b).
e excess current itself is plo ed in Fig. 3(c) and (d).e behaviour of  Exc is starkly di erent for both junctions.While  Exc decreases sharply past the point of maximum conductance in junction A, it is nearly symmetric around the point of maximum conductance in juntion B. We shall now show that the source of these di erences lies in the height-dependent evolution of the PIN codes of both junctions.
For speed and convenience, we build a look-up table based on Eq. ( 2) containing the excess current at a large number of transmission values and nd the best point-by-point match to the  ()-curve using the same constrains as in the MAR analysis above.
Fig. 4 shows the resulting transmissions   for the dominant (orange) and degenerate (blue) channels for junctions A and B. As a control, the results of the full MAR analysis are superimposed as open circles.e analysis of the excess current nds virtually the same channel con guration as the established MAR analysis at those points where data from both methods is available.is agreement is expected since the physical processes at the heart of both methods are the same.e measurement can, hence, be simpli ed from a full  ( )-characteristic to a single data point with no apparent loss of accuracy.
Owing to the ease and speed of the measurement, the excess current provides a very re ned picture of the channel evolution as a function of .In general, the dominant channel increases as the tip is approached until reaching nearly unity transmission, at which point  1 drops sharply.
e transparency of the secondary channels, when present, increases rather monotonously for smaller .A similar drop of the total transmission has been observed in Al junctions before and was a ributed to varying channel transmissions under elastic deformation of the contact [37].A comparatively detailed decomposition into individual transport channels as we report here has not been achieved until now.
To understand the observed PIN code variations between di erent junctions and tie them to microscopic origins, we performed ab initio simulations using density functional theory (DFT) for two microtips terminated by a (100) and (111) facet, facing a Al(100) surface with a tip atom on top.
e junction geometries are optimized at each tip height, and their transport characteristics, including the transmission channels, are computed from the electronic structure through non-equilibrium Green's function (NEGF) techniques [31,38].
e PIN codes obtained from the simulations of the two di erent tips are shown in Fig. 5(a) and  (b).
e simulations qualitatively reproduce the experimental observations, with the total conductance reaching a maximum value and subsequently decreasing upon closer approach.
e magnitude of the drop-o is dependent on the crystal facet exposed by the tip.As it is unlikely that the tip apex in the experiment is perfectly crystalline, an exact reproduction of the experimental data is not expected.It is clear, though, that changes in PIN code evolution can be traced to structural properties of the microtip.
e DFT simulations also allow us to extract the coupling strength between the tip and sample electrodes, which rises continuously as a function of .
e rise does not necessarily translate into a higher transmission, though.is behaviour can be qualitatively understood in a one-dimensional tight-binding model.We thus study two semi-in nite atomic chains with uniform nearest-neighbour hopping amplitude  0 and coupled to each other by the hopping .In the limit   0 , itinerant charges in the chain are likely to be backsca ered from the point of contact.In the other limiting case,   0 , electrons are likely to pass back and forth between the two sides of the junction, thereby blocking it for transport.is kind of system has been described theoretically in Ref. [3]. e transmission between the two chains is where  is the coupling between the two leads and  = 1/(  (  )) is an energy scale related to the density of states at the Fermi level. is model allows us to understand the essential physics of single atom tunnel junctions in a minimal se ing.In the weak coupling limit, the small inter-chain hopping acts as a potential barrier that limits the charge transfer between the two leads.However, when the inter-chain coupling becomes larger than the intra-chain hopping, a bound state forms between the leads, which also inhibits charge transfer.ese results are in qualitative agreement with our experimental results, see Fig. 5(c).e PIN code is the central quantity for understanding quantum transport in mesoscopic contacts.It is a general property of a transport con guration and does not change between the superconducting and normal states.When at least one of the electrodes participating in transport is superconducting, the PIN code can be derived from Andreev processes.e excess current has its origin in MARs, which continue to contribute to the total current even as  > 2.It, therefore, contains the same physical information as the sub-gap structure, while being both easier and faster to measure.
In summary, we use the excess current to measure the continuous evolution of the PIN code in tunnel junctions.Control experiments using a full MAR analysis con rm the accuracy of our measurements.e ability to extract complex quantum properties such as the PIN code from such a simple measurement hold the promise of understanding transport as a function of an external control parameter, or to relate atomic structure to transport in much greater detail than hitherto possible.

SAMPLE PREPARATION
e Al(100) sample was prepared by cycles of Ar-ion sputtering and annealing.
e tip was cut from 1 mm diameter aluminum wire (purity of 99.9999 %) and spu ered in vacuum to remove the surface oxide.
e tip was further prepared by eld emission in proximity of the Al(100) surface and repeated indentations into the substrate until achieving an atomically sharp topography.All measurements were performed at the base temperature of the STM system of 15 mK.
We developed an experimental procedure, allowing us to reproducibly extract single Al atoms from the surface.e tip is rst positioned at the height corresponding to 10 mV bias and 100 pA tunnelling current.A er turning o the feedback loop, we approach the tip by −400 pm towards the pristine surface and retract it back to the reference height.Following this approach-retract procedure, we observe a single vacancy defect at the point of approach and an individual Al adatom nearby.e adatom can then be moved to a desired position on the surface by atom manipulation.Fig. S1 shows the two Al adatoms of junctions A and B from the main text, which we produced by this technique.

ABSENCE OF HYSTERESIS
e single atom junctions we investigated are exceedingly stable and remain unchanged during the approach-retract experiments.
ere is no jump to contact in the -range explored in our experiment, up to ca. −100 pm past the point of maximum conductance.We observe no hysteresis between the approach and retraction of the tip.An example of a typical approach-retract curve is shown in Fig. S2. e tip was placed at the reference position of 10 pA current at 15 mV bias.A er disengaging the feedback loop, the tip is pushed -350 pm toward the surface and retracted to the starting position.ere is no jump to contact and no hysteresis between the approach and retract paths.

ADDITIONAL DATA
We applied the excess current method developed in this manuscript to further single atom contacts.e results from three additional junctions, which are not shown in the main text, are displayed in Fig. S3.All three contacts show a qualitatively similar behaviour to junctions A and B of the main text.e conductance is dominated by a single transport channel.e transmission through the dominant channel peaks at the maximum total transmission and decreased therea er.
e secondary transport channels are negligible up until the maximum conductance is reached and only gain importance upon further approach to the surface.

DFT+NEGF RESULTS
We simulated single atom junctions akin to those in the experiment using a combination of density functional theory (DFT) and non-equilibrium Green's function (NEGF) techniques.
ese calculations yield the optimized geometries and the electronic structure of the junctions as well as forces [1,2].We model the experimental situation by an Al(100) slab with an Al adatom and tip structures with (100) and (111) orientation.
ese tip geometries are the closest feasible assumptions for the real, likely amorphous, structure of the tips.e electronic channels in the contact can be represented

FIG. 1 .
FIG. 1.(a) Sketch of the single-atom junction studied in the experiment.e tip height  is adjusted in the course of the experiment.(b) asiparticle spectrum at low setpoint conductance (blue) with corresponding t.(c) Conductance curve  N () recorded above an Al adatom, showing a clear initial exponential increase of the conductance (setpoint 1.5 mV at 1 nA).Inset: Topographic image of an Al adatom on the Al(100) surface.(d)  ( )-curve of the superconducting sample at  N = 0.69  0 .e excess current is the -axis intercept of the curve at  > 2.e dashed vertical line indicates 2 = 360 μeV.

FIG. 2 .
FIG. 2. (a), (b)  ( )-curves at various conductances for two typical Al adatoms.MARs appear as steps in the current at fractions of 2.Fits to the data are superimposed in black do ed lines.(c), (d) PIN code analysis from the MAR model for the junctions in panels (a) and (b), respectively, as circles.epurple curve shows the total transmission, i.e. the conductance  N () in units of  0 .

FIG. 3 .
FIG. 3. (a), (b) Measured current (yellow) and reconstructed  N () (purple) signals for junctions A and B, respectively.e di erence between the two curves results from the excess current.e blue and orange crosses show the same quantities from the MAR point spectra in Fig. 2. (c), (d) Excess current as a function of total transmission  t for junctions A and B, respectively.Note that the curves show the approach curve of the tip, i.e. the arrows indicate the direction of decreasing .
(a) and (b) show the experimentally measured current in yellow and the calculated normal state current according to Eq. (3) in purple.

FIG. 4 .
FIG. 4. (a), (b) Mesoscopic PIN code analysis using the excess current for junctions A and B, respectively, with the dominant channel in orange and the degenerate channels shown in blue.e circles show the result of the full MAR analysis from Fig. 2.
FIG. S1.Topographies of the Al adatoms of junctions A (a) and B (b) in the main text.
FIG.S2.Typical approach-retract curve above an Al adatom.e tip was placed at the reference position of 10 pA current at 15 mV bias.A er disengaging the feedback loop, the tip is pushed -350 pm toward the surface and retracted to the starting position.ere is no jump to contact and no hysteresis between the approach and retract paths.