Metallic properties of magnesium point contacts

We present an experimental and theoretical study of the conductance and stability of Mg atomic-sized contacts. Using mechanically controllable break junctions (MCBJ), we observed that the room temperature conductance histograms exhibit a series of peaks, which suggests the existence of a shell effect. Its periodicity, however, cannot be simply explained in terms of either an atomic or electronic shell effect. We also found that at room temperature, contacts of the diameter of a single atom are absent. A possible interpretation could be the occurrence of a metal-to-insulator transition as the contact radius is reduced, in analogy with what is known in the context of Mg clusters. However, our first principles calculations show that while an infinite linear chain can be insulating, Mg wires with larger atomic coordinations, as in realistic atomic contacts, are always metallic. Finally, at liquid helium temperature, our measurements show that the conductance histogram is dominated by a pronounced peak at the quantum of conductance. This is in good agreement with our calculations based on a tight-binding model that indicated that the conductance of a Mg one-atom contact is dominated by a single fully open conduction channel.


Introduction
The understanding of conductance at the very small scale of a single atom has advanced greatly over the last decade due to a joined effort from both experiment and theory [1].In this regime the total conductance, G, is described by the Landauer formula: G = G 0 N i=1 τ i , where G 0 = 2e 2 /h is the quantum of conductance and τ i is the value of transmission for the i-th electronic mode (channel).For many metals the conductance of a one-atom contact does not match the value of a single quantum of conductance.Values in excess of 2e 2 /h, found for transition metal contacts, indicate that the conductance has to be determined by multiple channels.A pivotal contribution in the understanding of these single-atom contacts was given by Scheer et al. [2].They showed that the maximum number of modes to be considered was limited by the number of valence orbitals of the central atom [3].
In the case of s 1 metals, such as the alkali and noble metals, the conduction is given by one single electronic mode.Experimentally, the transmission of this "channel" is found to be close to one, resulting in a conductance close to 2e 2 /h or a resistance of 13 kΩ.For sp metals the total conduction is governed by three channels, but this does not lead to a conductance of 3 G 0 , since the electronic states in general have a transmission τ < 1.For the transition metals the number of conduction channels even increases to five, due to the d orbitals.
When we shift our attention to Mg, however, it is not so easy to predict what the conductance through the single atom will be.An isolated atom of this s 2 metal has the s-shell fully filled, but the p-shell still empty.It is therefore not straightforward to predict which electronic states are of importance to the conductance of a single-atom contact.A comparable electronic structure can be found in Zn, where the 4s-shell is completely filled, while the 4p-shell is empty.Häfner et al. [4] have shown that the conductance of a single-atom contact in this case was between 0.8 and 1.0 G 0 , where the transmission was largely given by a single channel.
The fact that bulk Mg is a metal, results from the hybridization of the 3s and the 3p bands that originates from the large coordination number (12 in a hcp structure).However, the situation might be different when reducing this coordination.Indeed, it is well known that Mg clusters exhibits unusual properties (see reference [5] and references therein).In particular, it has been shown that the nonmetallic-to-metallic transition in Mg clusters is non-monotonic and clearly slower than, for instance, in alkali metals [6].This is due to the dramatic change in the electronic structure of Mg with the number of interacting atoms.In the extreme case of a dimer, the bonding is due to weak van der Waals interactions.In the thinning of an atomic contact there is a progressive reduction of the coordination of the atoms and, with it, the hybridization.In this sense, the first question we want to address in this work is whether this reduction is enough to induce a metal-to-insulator transition for small Mg contacts.
Besides the cross-over from insulating to metallic behaviour, cluster experiments also demonstrated a shell structure [6].One speaks of a shell structure if the mass spectrum shows a periodicity as a function of the radius [7].The first observation of a shell structure in magnesium clusters was found to be due to the closing of facets and the crystalline arrangement [8].The hexagonally close-packed lattice leads to the periodic occurrence of highly stable icosahedrons.Later experiments performed with magnesium in supercold helium droplets showed shell structure due to the delocalized electrons that display electronic level bunching [9].
Analogies to cluster shell structures were already found in quantum point contacts of alkali metals [10] and subsequent work showed that these are not limited to this chemical group [11,12].For metallic contacts to demonstrate shell structures the atoms require sufficient thermal energy in order to find the local minima in energy.For sodium the necessary temperature for observing the shell effects was found to be around 80 K [13].Since magnesium has a much higher melting temperature (922 K versus 371 K for Na), one expects the necessary temperature in this case to be above 200 K.The second question we want to address, therefore, is whether magnesium point contacts demonstrate shell structures at room temperature.The fact that small clusters are insulating, while small contacts have their electronic structure influenced by the leads, makes the comparison between the two manifestations of the shell structures all the more interesting.
The rest of the paper is organized as follows.In the next section we describe the experimental technique used in this work to study the mechanical and transport properties of Mg atomic-sized contacts.Then, in section 3 we discuss our observations at room temperature, which suggest the existence of a shell effect.Section 4 is devoted to the theoretical analysis of Mg nanocontacts.In particular, we present first principle calculations of the electronic structure of Mg infinite wires of different thicknesses as well as conductance calculations for Mg one-atom contacts based on a tight-binding model.In section 5 we discuss our experimental results for the conductance of Mg point contacts at liquid helium temperature.Finally, we summarize in section 6 the main conclusions of our work.

Experimental Technique
In order to investigate Mg contacts experimentally, we have used Mechanically Controllable Break Junctions (MCBJ) at both room and liquid helium temperature.We start with a magnesium wire (purity better than 99.9 %) of 125 µm diameter and about 15 mm length, which we give a small incision in the middle.
For the low temperature measurements, the wire is glued on top of an insulating Kapton layer, that covers a phosphor bronze bendable substrate.The glue (Stycast Epoxy 2850FT with curing agent 24LV) is positioned on either side and as close to the incision as possible.The bendable substrate is clamped in a three-point bending configuration inside a vacuum pot which is brought to a pressure < 10 −5 mbar.The pumping is done by an oil-free diaphragm pump in combination with a turbo molecular pump to reduce possible contamination, especially by hydrogen.This is important considering the strong chemical affinity of magnesium to hydrogen [14].After pumping, the vacuum pot is submerged in liquid helium, cooling the sample down to a temperature close to 4.2 K.An important benefit of this technique is that it maintains the sample in a cryogenic vacuum and minimizes the possibility of contamination.Using a mechanical axis for the coarse movement, the wire is broken at the incision.In this way the two electrodes are formed by freshly exposed surfaces, which is critical for the study of reactive materials such as magnesium.A stacked piezo element is used for the fine adjustments of the contact diameter during the experiment.By relaxing the bending of the substrate the two clean surfaces can be brought into contact again, adjusting the size of the contact to the level of single atoms.
The set-up we use to study magnesium at room temperature is comparable to the one used at low temperatures [15].Instead of one substrate, we now use two, fixed in line with a small gap as separation.In this way, both the substrates can act as separate electrodes, avoiding the need for insulation of the wire.The magnesium wire is stretched across the gap at the top and fixed by small clamps on either side.One thus avoids the use of glue, which can cause problems in the combination with ultrahigh vacuum (UHV).The piezo is positioned below the gap and presses the two substrates upward and outward, effectively breaking the wire.The ratio between the upward movement of the piezo and the outward stretching of the sample wire (commonly called the reduction ratio) is comparable to the low temperature configuration, resulting in similar stability.The sample holder with the wire is mounted on a stainless steel insert, which is inserted from the top into the UHV chamber.The chamber is pumped down to 10 −7 mbar with a turbo pump, after which it is baked at 450 K.In combination with an ion pump the pressure can be brought down to 1 × 10 −10 mbar.During the measurement, however, the turbo pump is shut down to minimize the mechanical vibrations coupling to the sample.The base pressure during the experiments is 4 × 10 −10 mbar.
The level of contamination in both these setups is too low to have an influence on the results.This can be concluded from the data being reproducible during a period of several days.Since the sample configuration is comparable in both setups their difference is limited to the temperature.The surface diffusion of atoms at low temperatures is frozen out, forcing the contact to the nearest low energy state in configuration space.At room temperature, however, thermal diffusion of the atoms allows the contact to probe a wider range of configurations.Subtle energy differences such as the one caused by the shell effect are then probed more effectively.

Room temperature results
The evolution of conductance when breaking the wire at room temperature has a characteristic that can be seen in the example of figure 1(a).We see a step-like lowering of the conductance towards zero with stretching of the wire.Relatively stable intervals or plateaus occur when the mechanical tension on the contact gradually builds up.When this tension is released by a mechanical reconfiguration of the contact, its diameter and conductance are reduced stepwise [16].The plateaus in conductance appear flat and the jumps during the mechanical reconfigurations are large.This is an indication that some values of conductance are stabilized when they are related to structures with a local minimum in the energy.
In order to verify that these plateaus occur at reproducible values we use a statistical description by means of a conductance histogram [17].This histogram is constructed by dividing the conductance interval of interest in equal sub-intervals or bins.For each measured data point we determine the corresponding bin, resulting in a probability distribution for the conductance during breaking.An example of such a distribution or histogram for Mg is given in figure 1(b).We varied the speed of the electrode separation between 1.0 and 5.0 × 10 1 nm/s, but this had no significant effect on the conductance histograms.
The result in figure 1 indeed shows a series of peaks at higher conductance values.The positions of these peaks were determined automatically by a numerical procedure registering changes in the sign of its numerical derivative.The obtained values are indicated by arrows in the same figure.These could indicate the presence of a shell structure, but in order to investigate whether the peaks in figure 1 are indeed periodic in the radius we need to plot the histogram as a function of radius, R. Since the range of conductances in this graph corresponds to the ballistic conductance regime we can obtain the R from G via the corrected Sharvin formula where k F represents the Fermi wave vector [18,19].
For the specific case of figure 1 the calculated radius for the peaks is plotted in figure 2. The obtained data points are clearly described by two linear fits, indicating a crossover between two shell effects with different periods [10].For this specific histogram the period changes from 1.6 (k F R) −1 for small contact diameters to 3.2 (k F R) −1 for larger contact diameters.
For comparing these values to the periodicity expected for shell structure due to atomic packing, one starts from a simple packing model for a close-packed hexagonal lattice and increases the contact diameter facet by facet.One then obtains a periodicity of 2.3 (k F R) −1 , which is only slightly influenced when considering other crystal structures such as face (body) centred cubic.In figure 2 this period is represented by the dotted line, clearly deviating from the obtained data.
Possible electronic shell structures have been studied for metallic point contacts as well.Both, calculations using jellium models [20] and nanoscale free-electron models [21] applied to aluminium point contacts [12] give frequencies up to 1 (k F R) −1 .As the frequencies for Mg are suggested to be the same as for Al [21], we represent this period by the dashed line in figure 2. Since none of the lines provides a good description of the data, the shell effects seen in the experiment are still without satisfactory explanation.A remaining possibility is that the contact is governed by competing effects of the atomic packing and the electronic free energy simultaneously.Similar mixed structures were also found for Mg clusters [22].
In the total of our experiments, more than 90% of the measured histograms for all six samples at room temperature show a periodicity in k F R, but the observed frequencies in the total conductance interval vary in a range of 1 to 3.5 (k F R) −1 .In 50% of the histograms we obtained a value around 1.7 (k F R) −1 for lower conductance values.At higher conductance values there is almost always a crossover to higher frequencies.The frequency at this second conductance interval varies strongly and often there are multiple frequencies superimposed.This indeed makes it likely that the shell effect is the result of multiple properties of the metal.
Another feature of the histogram in figure 1, that is even more remarkable, is the absence of plateaus below 5 G 0 .This minimal value is much higher than for all other metals studies with this technique [11,12,15].Below this threshold value we only found a smooth exponential decrease in conductance, typical for tunneling behaviour.In principle, this suggests that smaller contacts are not stable.However, from the analysis of the data, we cannot exclude that a metal-to-insulator transition is taking place.As we explained in the introduction, as the contact radius is reduced, the overlap between the s and p states of the atoms in the constriction decreases.Such a decrease could potentially lead to the opening of a gap in the Mg density of states.In the next section we present a theoretical analysis to elucidate this issue.

Theory
The numerous existent theoretical results on Mg clusters (see reference [22] and references therein) cannot be used to resolve the problem of whether or not a Mg point contact can become insulating.A metal-to-insulator transition is only welldefined for truly infinite systems.Thus, in order to shed light on this problem, we have studied the stability and electronic structure of a series of Mg infinite wires with small coordination numbers.To be precise, we have carried out density functional theory (DFT) calculations of the structural and electronic properties of Mg wires ranging from the smallest possible coordination, two in the case of a chain, up to 6.In the insets of figure 3 we show two examples of the configurations of the studied wires.In the upper panel one can see a linear chain, while the lower one shows an infinite wire with a cross section of 3 atoms (coordination number equals 6) grown along the c-axis of a hcp bulk structure and keeping the bulk relative positions.
For these ab initio calculations we have used a standard implementation of DFT [24,25] with a plane waves basis set and ultrasoft pseudopotentials [26].The Perdew-Burke-Ernzerhof approach (PBE) [27] has been chosen for the exchangecorrelation contribution.The calculations were performed with the CASTEP v4.2 code [23].The plane wave cutoff used (375 eV) in all our calculations assures wellconverged structural and electronic properties.Convergence criteria for the atomic relaxations involved in the different calculations are: 0.01 eV/ Å for the mean value of the forces, 0.001 Å for the atomic positions and 10 −6 eV for the total energy.We have used a cubic supercell of 4 atoms for the linear chain and 2 atomic layers for the wire adding a vacuum of 10 Å in the directions perpendicular to the wires.We have optimized, minimizing the system energy, the interatomic distances of the wires keeping the original symmetry with a Monkhorst-Pack (MP) k-sampling mesh of 1 × 1 × 64 [28].
The DOS has been calculated with a mesh of 1 × 1 × 1024.Turning to the results, in the two systems shown in figure 3, we have found interatomic distances (3.1 Å) slightly smaller than the bulk value (3.2 Å) in accordance with calculations of Mg clusters [22].In figure 3(a) we show the DOS of the linear chain for the optimized interatomic distance §.A small gap of 0.3 eV is observed; consequently the linear chain does not show metallic behaviour.However, even for this wire with the lowest coordination number, the broadening due to the 3s-3p hybridization has nearly closed the gap.Indeed, we have found that all the wires with larger coordination than the linear chain are metallic.The DOS of a wire with a cross section of 3 atoms is shown in figure 3(b) and does not exhibit any gap.In fact, up to 5 different bands cross the Fermi level.We have investigated whether the application of some additional stress could modify these results, but we have found that even increasing the interatomic distances by a 10%, the wires remain metallic.So, in short, from this analysis we do not expect the formation of insulating Mg wires in the last stages of the breaking of Mg contacts.
The results above show that low-coordinated Mg structures are in principle stable, which suggests that the formation of few-atom contacts might be possible, at least at low temperatures.Moreover, since these structures are metallic, the following natural question arises: What is the expectation for the linear conductance of the smallest Mg contacts?In order to answer this question we have computed the conductance of Mg oneatom contacts within the Landauer formalism.For this purpose, we have combined the tight-binding parameterization of reference [29] with non-equilibrium Green's functions techniques.This approach has been very successful describing the transport properties of a great variety of metallic atomic-sized contacts [30,31,32,33].We proceed to briefly explain it.In this approach the electronic structure of the atomic contacts is described § We have studied the stability of the linear chain allowing the 4 atoms of the supercell to relax starting form a zig-zag configuration.For cell lengths larger than 10.5 Å the final configuration is a lineal chain slightly distorted, with a energy minimum corresponding to an interatomic distance of 3.125 Å and a zig-zag angle of 177 • .The stable solution for smaller supercell lengths is a compressed zig-zag chain where the atoms have 4 nearest-neighbors with an interatomic distance of 3.1 Å.
in terms of the following Hamiltonian written in a nonorthogonal local basis where i and j run over the atomic sites, α and β denote different atomic orbitals, σ is the spin and H iα,jβ are the on-site energies (i = j) or hopping elements (i = j).Additionally, we need the overlap integrals S iα,jβ of orbitals at different atomic positions.We take these matrix elements from the tight-binding parameterization of reference [29], which is designed to accurately reproduce the band structure of bulk materials.The atomic basis is formed by 9 valence orbitals, namely the s, p, and d orbitals which give rise to the main bands around the Fermi energy.In this parameterization both the hoppings and the overlaps are functions of the atomic distances, which have a cutoff radius that encloses up to 13 nearest-neighbour shells.Finally, in order to take into account the low coordination in the smallest atomic contacts, we impose local charge neutrality through the self-consistent variation of the on-site energies of the atoms in the constrictions.With the help of Green's functions techniques, one can translate the information on the electronic structure contained in the Hamiltonian of equation ( 2) into the conductance of these atomic junctions (see references [31,33] for details).As explained in the introduction, this low-temperature conductance adopts the form of the Landauer formula where G 0 = 2e 2 /h denotes the quantum of conductance, E F the Fermi energy, and τ n the transmission of the n-th transmission eigenchannel at E F .We now turn to the analysis of the results for the linear conductance of some ideal, and yet plausible, one-atom geometries, which have been chosen to simulate what may happen at the last conductance plateau before the rupture of the nanowires.In the inset of figure 4(a) we show an example of an one-atom contact grown along the [0001] direction (c-axis) which contains a dimer in its central part.Different molecular dynamics simulations of atomic contacts of various metals have suggested that this type of geometry is realized very frequently at the last plateau [31,33,34].This particular geometry is constructed starting with the dimer and choosing the nearest-neighbours in the next layers.Finally, the leads are modelled as infinite surfaces grown along the same direction (yellow atoms in the inset of figure 4(a)).In this ideal case, all interatomic distances are fixed to their bulk values, which is justified by the ab initio calculation of Mg clusters [22] and our results for infinite wires presented above.The total transmission and the individual transmission coefficients, τ i , for this contact are shown in figure 4(a) as a function of energy.The first thing to notice is the fact that the system is metallic, in accordance with our expectations based on the DFT results described above.Furthermore, as one can see, the conductance, which is determined by the transmission at E F , is very close to 1 G 0 and it is completely dominated by a single fully open channel.The second and third channels, which are degenerate due to the symmetry of the contact, have transmissions below 10 −3 .One can get a deeper insight into these results and, in particular, into the nature of the conduction channels by analyzing the local density of states projected onto the different orbitals of the two central atoms (not shown here).Such analysis indicates that the dominant channel is formed by a symmetric (bonding) combination of the s and p z orbitals of the central atoms (z is the transport direction), while the p x and p y orbitals are responsible for the second and third channels.A fourth channel, which in this case has a transmission below 10 −5 , is formed by the antisymmetric combination of the s and p z orbitals.Such anti-bonding combination is basically orthogonal to incoming states (from the leads) and therefore does not contribute significantly to the transport.So in short, these results resemble very much what happens in the case of Zn one-atom contacts [4], and also in the case of the final stages of the last plateau of Al contacts [35,36].This is, after all, quite reasonable since in all these cases the electronic structure at the Fermi energy is governed by the s and p orbitals.
Since we do not exactly know the growth direction of the Mg atomic contacts, we have studied the conductance of geometries with different crystallographic orientations.In figure 4(c) we show another example of a dimer contact, but this time grown along the [11 20] direction (a-axis).Notice that the total conductance and the transmission coefficients are similar to those of the [0001] case, the main difference being the larger values for the second and third channels and the lifting of their degeneracy.While the lack of degeneracy reflects the lower symmetry of the contacts in [11 20] direction, the larger transmission values can be attributed to the larger apex angle of those contact geometries and in consequence a stronger coupling of the dimer atoms and next layers.
In principle, the contact geometries should be determined from molecular dynamic simulations, but this is computationally very demanding.Instead, and in order to test the robustness of our results, we have studied the role of disorder in the atomic positions.For this purpose, starting from the ideal geometries of figure 4(a) and (c), we have changed randomly the positions of the atoms in the constriction region (those highlighted in brown in figure 4(b) and (d)) with a maximum amplitude of ±5% of the nearest-neighbour distance.Then, we have computed the total transmission and the transmission coefficients of the disorder geometries and the results are shown in the form of histograms in figure 4(b) and (d).As one can see, in both cases the conductance is still dominated by a single channel that is almost fully open.Therefore, these results confirm our basic conclusion, namely the fact that a Mg one-atom contact is expected to have a conductance close to 1 G 0 dominated by a single channel.

Low temperature results
In order to test this latter conclusion of our theoretical analysis, we repeated the experiments at low temperatures.Here we broke the contact by ramping the piezo element continuously with a speed of 1.8 × 10 3 V/s (corresponding to 1.0 × 10 2 nm/s electrode separation).This resulted in traces as the one shown in figure 5(a).The plateaus in conductance appear more structured and the jumps during the mechanical reconfigurations are smaller than those measured at room temperature.In the conductance histogram, presented in figure 5(b), this results in a flat distribution at higher conductance values.The shell structures found at room temperature are thus absent at low temperatures, as we expected.
When focusing on the trace at low conductance values, the staircase of plateaus indeed continues down to values below 5 G 0 .The histogram, the shape of which was reproduced over a set of ten different samples, shows its lowest conductance peak close to 1 G 0 .For a minority of the histograms a shoulder down to values as low as 0.8 G 0 is seen.In rare cases this shoulder even grows to form the primary peak.Although the conductance does not reveal information on the individual values of transmission, τ i , of the electron states, the strong asymmetry and the closeness of the peak to 1 G 0 do suggest that its conductance is given by only one, almost fully opened, channel.For multiple channels the value of 1 G 0 would not form a fundamental limit.This result agrees nicely with our calculations for the conductance of Mg singleatom contacts.The properties of Mg at low temperatures are therefore very close to the results obtained previously for Zn [4].The most noticeable differences are the absence of data points between 0.5 G 0 and tunneling and the higher relative intensity of the peak close to 1 G 0 in the case of Mg.The presence of this peak indicates that the instability of small point contacts at room temperature is not related to a metal-toinsulator transition.

Conclusions
In conclusion, magnesium contacts at room temperature demonstrate shell effects at multiple frequencies.The most frequent and intense frequency of 1.7 (k F R) −1 did not match with either the expected frequency for atomic packing or the previously calculated frequencies for electronic shell structures.This value therefore remains without satisfactory explanation.A possibility is that both effects play an important role at the same time.
The histograms at low temperatures exhibit a first peak close to 1 G 0 .The strongly asymmetric shape of the peak, with a small weight above 1 G 0 suggests this conductance is dominated by a single channel.Our calculations confirm this and identify this channel to be a symmetric combination of the s and p z orbital of the central Mg atom.This behaviour is similar to the results obtained for Zn [4], although the tendency for the channel to be fully open is stronger.
At room temperature contacts of the diameter of a single atom are absent.The instability of these smaller contacts is not caused by a metal-to-insulator transition at lower coordination.From the appearance of the peak at 1 G 0 at low temperatures, we can conclude that the metal-insulator transition is absent even in the smallest of contacts.Future experiments at intermediate temperatures may prove valuable in understanding these different behaviours.Our first principle calculations suggest that an infinite linear chain of Mg could be an insulator, but they also show that as soon as the coordination is larger than 2, as in any realistic contact at low temperature, the system is metallic.

Figure 1 .
Figure 1.(a) The conductance evolution during the stretching of a magnesium contact at room temperature in UHV.The conductance was measured at a bias voltage of 20 mV.(b) A conductance histogram resulting from 1.2 × 10 4 of these traces.The resolution of this histogram was set to 10 bins per G 0 .The conductance peaks, obtained by an automatic procedure, are indicated by arrows at the top of the panel.

Figure 2 .
Figure2.Graph showing the value for k F R obtained for the conductance peaks in figure1.The data points are successfully described by two linear fits, indicating a periodicity of 1.6 (k F R) −1 at smaller contact diameters and 3.2 (k F R) −1 at bigger contact diameters.Two additional lines indicate the expected periodicity for atomic (dotted) and electronic (dashed) shell effect as described in the text.

Figure 3 .
Figure3.Density of states (DOS) of Mg wires calculated with CASTEP[23] for: (a) a linear chain with an optimal interatomic distance of 3.125 Å; and (b) an infinite wire with a 3 atoms section.The wire is built up along the c-axis keeping the bulk symmetry as shown in the inset.The interatomic distance that minimizes the total energy is 3.1 Å.

Figure 4 .
Figure 4. (a) Total transmission and transmission coefficients as a function of energy for the contact shown in the inset.This geometry is grown along the [0001] direction (c axis), it contains a dimer in its central part and the interatomic distances are fixed to the bulk values.The yellow atoms correspond to atoms in the semi-infinite surfaces that are used to model the leads.The transmission coefficients at the Fermi energy (set to zero and indicated with a dotted vertical line) are τ 1 = 0.993, τ 2 = τ 3 = 0.76×10 −3 and the total conductance is ∼ 0.995 G 0 .(b) Histograms of the transmission coefficients for 2000 disorder realizations (see text) of the one-atom contact of the inset of panel (a).The brown atoms are those which have been randomly displaced.The vertical dotted lines indicate the values of the transmission coefficients for the ideal geometry.The transmission values of the second and third channels remain below 2 × 10 −3 and are not shown.(c) The same as in panel (a), but for the contact grown along the [11 20] direction (a-axis) shown in the inset.The total conductance is ∼ 1.01 G 0 with τ 1 = 0.9640, τ 2 = 0.024, τ 3 = 0.021 and τ 4 = 0.3 × 10 −3 .(d) The same as in panel (b) for the contact in the inset of panel (c).

Figure 5 .
Figure 5. Figure comparable to figure 1, but for a measurement performed at liquid helium temperature.Panel (a) shows the conductance evolution measured at a bias voltage of 100 mV.Panel (b) shows a conductance histogram resulting from 3 × 10 3 of these traces.The resolution of this histogram was set to 115 bins per G 0 .