Magnetotransport in atomic-size bismuth contacts

The room-temperature resistance of the pristine Bi samples is in the order of 1 kΩ and may slightly increase or decrease towards low temperature; this variation is typical for Bi nanostructures [7, 17]. The breaking curves, i.e. measurements of the conductance as a function of elongation of the bridge, recorded at 4.2 K, reveal no or only very short plateaus at 1 G₀, but multiple plateaus at fractions of 1 G₀. The exact shape of the individual breaking curves varies from curve to curve as the atomic arrangement changes, but the general shape, e.g. not-very-well-defined plateaus with substructure and curvature, are reproducible [1]. This behaviour is typical for low-temperature measurements in particular, but not exclusively when contacts are formed with the help of lithographic MCBJs [1, 3, 18–20]. Examples of Bi breaking curves are given in figure 2(a), where a pronounced upward slope and curvature of the conductance on the individual plateaus are observed. This behaviour has been reported before for low-temperature measurements on Bi contacts formed in an STM, and explained by a distortion of the lattice and the formation of a metallic-like state with heavy electrons [8]. The transition between the steps is attributed to changes in the atomic configuration that alter the cross-section by one or a few atoms. In contrast to [8], where curved plateaus were found only above G = 0.15 G₀, in our samples they also occur reproducibly at conductance values as low as 0.015 G₀. Apparently, in our thin-film samples a different metallic-like state can be stabilized. We argue that this state arises from hole transport, because according to the x-ray analysis the <111 > direction hosting the hole states may be aligned with the transport direction. Furthermore, the assumption of hole states naturally explains an increase of conductance within a plateau when stretching it further: the size of the hole pockets at the Fermi surface and their overlap increase when the lattice is elongated. Another example of plateaus with increasing conductance is aluminium [35], which also has hole pocket states at the Fermi surface (there located in the 3rd brillouin zone [34]). In Al, at the plateaus maximum, the saturation transmission of one channel is reached, and it has been suggested that a resonant state develops by the formation of a dimer structure [36, 37]. Because of the small Fermi surface of Bi, such a resonance is not expected to develop fully. However, already a slight distortion of the lattice by the stretching process will gradually increase the band overlap, and thereby enhance E_F and the transmission.

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Regardless of the sign of the charge carriers, the observation of these curved plateaus confirms the formation of a metallic-like state, which is formed only in very clean contacts, of a few atoms' size. The metallic character is further confirmed by mostly linear current-voltage characteristics and the observation of quantum-interference effects; see the next section. However, it is not straightforward to determine the conductance of the smallest contact from individual opening traces. We therefore calculate conductance histograms from repeated openings. Figures 2(b) and (c) show conductance histograms calculated from the same 80 opening traces with varying bin size and in different scaling. As a result of the non-suspended sample geometry, smaller bending radii are necessary, and thus the maximum number of opening-breaking repetitions is reduced because often the elastic deformation limit has to be overcome to stretch the wires to rupture. Figure 2(b) shows a linear plot for all data recorded in the range from 0 to 3.5 G₀ with a bin size of 0.05 G₀. No pronounced peaks are observable at 1 G₀ or its multiples. Figure 2(c) shows the same data in a logarithmic conductance scale, revealing broad maxima below 0.1 G₀ and several narrower ones between 0.1 and 1.2 G₀. The broad maxima at very low conductance arise from the curved plateaus described above. Because within a plateau the conductance may change by a factor of 2 or more, no narrow peaks can be expected. From this histogram we conclude that the smallest stable contacts feature conductance values in the order of 0.02 G₀. Figure 2(c) shows several peaks around 0.04 G₀,0.07 G₀ and 0.15 G₀ and its even multiples. The structures at 0.04 G₀ and 0.07 G₀ are absent in histograms of other samples, and are therefore discarded here. We then consider the peak at 0.15 G₀ to be the lowest robust superstructure peak (besides the structures mentioned above around 0.015 G₀). We attribute the 0.15 G₀ peak and its multiples to somewhat larger, symmetric structures that form, e.g., when crystal planes slide along each other as proposed in [8] and [9]. The fact that the odd multiples are suppressed may indicate bilayer sliding, as suggested in [9]. Due to the film growth with columnar structure, the planes might not develop the topologically protected surface state with perfect transmission, but instead contribute with the smaller value 0.15 G₀ to the conductance. Revealing the exact mechanism and origin of these super-structure peaks needs further investigation. We note that the peak at 0.15 G₀ is visible in histograms recorded on several samples, although not in all. An alternative interpretation of the data could thus be that the fundamental peak is at 0.3 G₀ and all multiples would occur. On the basis of our data we cannot distinguish between these two scenarios. Below, in section 4, we will show that the magnetic field and bias dependence of the transport changes characteristically around 0.2 G₀, in agreement with both possibilities.

Because of the small Fermi surface of Bi, moderate magnetic fields are sufficient to arrive at the regime where the Fermi body is filled with a small number of Landau levels only [28]. There is a general agreement that for bulk crystalline samples of Bi, when the field is applied along the trigonal axis, the lowest Landau level is driven out of the Fermi surface around B = (8.85  ±  0.2) T-i.e., at the limit of our magnetic field range [23, 24]. For all other orientations this limit is reached at even lower fields. In polycrystalline film samples as they are investigated here, no dependence on the orientation is expected. Figure 3(a) shows two MC curves recorded at a pristine sample for T = 4.2 K and T = 120 mK. In figures 3(b), at (T = 4.2 K) and 3(c) (T = 120 mK) the background MC has been subtracted and the data is plotted versus 1/B separately for both field directions to reveal the high symmetry of the G(B) curves. At T = 4.2 K a sawtooth-like behaviour is observed for both sweep directions of B, while at T = 120 mK no regular pattern arises, but aperiodic conductance fluctuations (CF) govern the pattern. The origin of these CF will be discussed in the next section. The sawtooth pattern is not equidistant in 1/B, as would be expected for Shubnikov-de Haas (SdH) oscillations for a 3D Fermi surface with a constant cross-section [27–29]. However, when approaching the ultra-quantum regime, deviations from the usual functional behaviour are expected, and were indeed observed for Bi single crystals [23, 24]. Here we observe an abrupt increase in the periodicity around 1/B = 0.27 1/T. Since the periodicity of the SdH oscillation is a measure of the extreme cross-sections of the Fermi surface perpendicular to the applied field, this signals a reorientation or reshaping of the Fermi surface at around 4 T. This transition becomes even more apparent when plotting dG as a function of B; see figure 3(d), where we show two subsequent sweeps recorded at 4.2 K, one for increasing and one for decreasing field. A periodic variation of the conductance as a function of B would indicate a physical phenomenon, e.g. the Aharonov Bohm effect, which is related to real space rather than in k space. Also in this scaling, the peaks are not equidistant and one observes a transition from a longer period at low field (|B| < 3.5 T) via a region were two periods seem to be superimposed (3.5 T < |B| < 5.5 T) to a shorter period (|B| > 5.5 T). Although we cannot definitely reveal the physical origin of the periodicity changes, they suggest that the magnetic field, in combination with the low dimension, initiates a reversible structural change in the constriction that gives rise to significant changes in the charge transport behaviour. This interpretation is supported by the observation of a pronounced change in the CF in a similar field range, as we will show in the next section. Although this is a rather unlikely scenario for conventional metals, the extreme sensitivity of the electronic properties of Bi to minor geometrical changes may make this a possible scenario for Bi [50, 51].

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We now turn to the description of the magneto-conductance (MC) measurements recorded for selected contacts. As an example, figure 4(a) shows the differential conductance dI/dV versus B of a contact with G(B = 0 T) = 1.38 G₀ recorded at T = 150 mK. It shows a bell shape with a plateau region between −1.5 T and 1.5. T and a saturation above |B| = 7 T at G ~ 0.76 G₀. The red curve is a moving average calculated over a field range of ΔB = 1 T, which is used for subtraction of the long-range field dependence for analysis of the CF. Figure 4(b) shows the CF signal dG after subtraction; figure 4(c) summarizes the symmetry analysis: for negative field direction we plot the anti-symmetrized part (dG(B) − dG(−B))/2, and for positive field direction, the symmetric part (dG(B) + dG(−B))/2. In a two-point geometry, and if the transport is time-reversal symmetric, the CFs are expected to be symmetric with respect to field inversion [25, 26, 38]. The analysis reveals that this is fulfilled for Bi atomic-size contacts. The remaining anti-symmetric signal shows the characteristics of white noise and reflects the high signal-to-noise ratio of the measurement with a voltage noise that amounts to ~ 1 nV. Figure 4(d) shows the anti-autocorrelation function

$$\begin{eqnarray}&&{\rm{AACF}}(\Delta B) = \lim\limits_{B \to \infty } \frac{1}{{2B}}\int_{ - B}^B {{\rm{d}}G({B^\prime } + \Delta B){\rm{d}}G( - {B^\prime }){\rm{d}}{B^\prime }} ,\end{eqnarray}$$

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from which we determine the characteristic properties of the CF-i.e. the fluctuation amplitude G_rms is the square-root of the main maximum ${{G}_{\text{rms}}}=\sqrt{\text{AACF}\left(0\right)}$, and the correlation field B_c is the half-width at half maximum $\text{AACF}\left({{B}_{\text{c}}}\right)=\frac{1}{2}{{G}_{\text{rms}}}^{2}$ [25, 26, 38]. In the fully phase-coherent diffusive regime for metals with strong spin-orbit scattering, a universal amplitude of G_ucf = 0.25 G₀ is expected [25, 26, 38].

Figure 5 summarizes typical MC curves recorded when stretching a sample starting at roughly 5 G₀ down to 0.014 G₀. The bell-shaped background MC is subtracted and the traces are offset for clarity; the zero-bias conductance (ZBC) values for each trace are indicated in the legend. The most prominent observations are as follows: for high conductance there is a pronounced peak around zero-field that diminishes with decreasing conductance and is indistinguishable from the CFs below ZBC of 0.2 G₀. We attribute this peak to weak antilocalization, as is expected for elements with high atomic numbers and therefore high spin–orbit coupling [39–41]. The analysis of the fitting to the 1D theory [40, 41] yields a rough estimate for the phase coherence length l_ϕ = 150 nm for contacts above 1 G₀. This value is somewhat smaller than the findings by Rudolph and Heremans [42] on Bi wires patterned from thin films, who observed a width-dependence of l_ϕ. We note that our data could not be described with the 2D formula (as was used in [42]), which indicates that in our experiment l_ϕ was limited by the sample geometry rather than by intrinsic length scales. Also, the disappearance of the antilocalization peak for the contacts with very small conductance hints towards geometry limited transport phenomena. In this sense the obtained value for l_ϕ represents a lower limit. This analysis also indicates a lower diffusion coefficient in our columnar films than the smooth films investigated in [42].

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For the analysis of the CF the zero-field peak is cutout of the calculation of the AACF because the integration range is normalized accordingly to obtain the correct CF amplitude. The amplitude of the CF diminishes with decreasing G, so the lowest two traces in figure 5 have been multiplied by 10 for better visibility. We also note that for contacts above 1 G_0, the fluctuation amplitude is considerably reduced for magnetic fields above |B| > 3 T. This indicates a change in the trajectories that contribute to the interference pattern. In combination with the results from the SdH oscillations described above, we suggest a change of the transport mechanism above this field range. For the further analysis we calculated the anti-autocorrelation functions of all CF traces and determined the rms amplitude G_rms and the correlation field B_c from its main maximum, as described above [25, 26]. Figure 6 shows the results of 35 contacts investigated in the same sample, at similar temperature T < 150 mK. The amplitude G_rms increases steadily from very low values G_rms << 0.001 G₀ at low conductance and levels off to roughly G_rms ~ 0.025 G₀ at around 3 G₀, with rather large standard deviation. Also, this saturation value is one order of magnitude smaller than expected for universal conductance fluctuations in the diffusive regime [25, 41]. A reduction of the CF was reported for ballistic metal contacts [43], and is explained by a reduced probability of the charge carriers being scattered back through the constriction and contributing to the interference pattern G_rms/G_UCF ~ l/a. For Bi, with its very long λ_F and its medium elastic mean free path l, this ratio is very small, and an even smaller amplitude G_rms could be expected here. The same reduced backscattering probability may explain the reduced antilocalization peak for the smallest contacts. Similarly, it has been shown for atomic contacts of noble metals that the CF amplitude as a function of bias is connected with the interference of electronic trajectories being backscattered from the atomic contact and those being backscattered from the reservoir [44]. This means the CFs of atomic-size contacts are sensitive to the local structure at the contact region itself. When the transmission probability of the contact is small, the fluctuation amplitude is strongly reduced. A quantitative comparison between the model of [44] and our experiment cannot be made because of the complex Fermi surface of Bi.

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The correlation field B_c shows no discernible trend. Within the numerical uncertainty range it is constant <B_c > = 90 mT ± 10 mT. B_c describes the typical averaged 'period' of the aperiodic fluctuations. It is a measure for the length of the electronic paths that contribute to the fluctuation pattern. When the sample is a 2D plane, B_c is translated into an effective path length by l_2D = (h/eB_c)^1/2. For wire-shaped samples with width w smaller than the effective length, it has to be calculated according to: l_1D = h/eB_cw [41]. While in ballistic metal contacts, the phase area contributing to the CF is given by the temperature-independent elastic mean free path l [43]; in the diffusive limit it is given by the minimum of either the thermal length l_T = (hD/k_BT)^1/2 or the phase coherence length l_ϕ. [25, 26, 41]. However, this has only been worked out quantitatively for systems in which λ_F is much shorter than l and which, in turn, is shorter than l_ϕ.. Since for our samples the 2D analysis yields l_2D = 220 nm > w = 120 nm, the 2D approximation is not valid and we have to apply the 1D formula, from which we obtain l_1D ~ 480 nm. This value is higher than the lower estimate for l_ϕ that we obtained from the weak localization analysis above, but larger than the crystal grain structure and the film thickness in the contact area that limit the elastic mean free path. The physical meaning of this length can be revealed by temperature-dependent measurements. Up to our highest measurement temperature of 1 K, the fluctuation amplitude, as well as B_c, showed only a very weak variation (not shown). We therefore assume that temperature averaging is not the limiting quantity in our experiment. We conclude that l_1D has to be identified with l_ϕ, which determines the dominating interference paths of the CFs as for diffusive wires. However, because the theory of CFs has not been developed for the unusual combination of length scales and because of the limited temperature range of our cryostat, we cannot certainly conclude on the physical meaning of the effective length.

To get further inside the transport mechanism, we performed bias- and magnetic field-dependent measurements of the differential conductance. Figure 7 shows typical dI/dV(B,V) maps of eight contacts ranging from dI/dV(0,0) = 0.013 G₀ to 3.022 G₀; selected traces from the two lowermost panels are shown in figure 8. Pronounced variations of G are observed as function of the bias and as function of the magnetic field. They are reproducible when repeating the bias or field sweep, respectively. The CFs are not symmetric with respect to bias inversion in agreement with theory and earlier observations [25, 38, 41, 40]. In addition, we observe the excitation of temporal fluctuations at higher bias levels that give rise to non-reproducible traces and showup in figure 8 as an increased noise level. To distinguish these three effects, we denote the fluctuations as function of bias as CF_V, the fluctuations as function of magnetic field as CF_B, and the temporal fluctuations as CF_t.

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The amplitude of the CF_V is reduced when applying higher bias or higher magnetic fields. The field scales at which the reduction becomes apparent depends on the contact size and the applied field. The reduction with magnetic field is in agreement with the observations described above (figure 5). It indicates a loss of phase coherence or reduction of interference path with increasing field, while the reduction of the CF_V amplitude with increasing bias indicates loss of phase coherence due to smearing of the Fermi edge [40]. Alternative explanations, such as inelastic excitations of phonons [52], can be excluded because these would give rise to symmetric features with respect to bias reversal.

The CF_t signal sets in at higher bias level when the CF_V signal is already reduced. We attribute these fluctuations to atomic rearrangements caused by joule heating in the constriction area, although the currentdensity is still relatively low compared to typical currentdensities known to exert atomic fluctuations [45–47]. This interpretation is supported by the occasional occurrence of conductance jumps (see the four topmost curves in figure 8(b)), as has been found for single-atom junctions [37]. In most cases, however, the CF_t amplitudes are small and occur around a constant mean conductance value. This means that the atomic motion seems to be limited to small variations around their stable positions. Exceptions are given by the curves recorded at B = 1 T and B = 3 T in figure 8(b), where small, irreversible changes are observed. Due to the particular sensitivity of the electronic properties of Bi to small changes of the crystal structure, spatially small variations might also give rise to pronounced changes of the conductance [48–51]. Both the CF_V and the CF_t amplitudes are reduced above around |B| = 4 T, indicating that the origin of the CF_t is also field dependent. In other contacts (not shown here), the CF_t amplitude may increase with increasing magnetic field. Such interplay between bias, magnetic field and two-level fluctuations has been observed in mesoscopic Bi samples [49, 51] and was interpreted as originating from field-dependent local double-well potentials created by structural defects. In the samples investigated here, the CF_t do not show a clear two-level structure, presumably because several fluctuators are active. Summarizing this part, we argue that the temporal fluctuations are caused by changes of the electronic interference paths because of dissipation-induced small-scale motion of atoms. They are modulated by the magnetic field because the field affects the phase of the electronic waves. We now discuss the dependence of the CFs on the zero-field conductance. Although for small conductance G < 0.2 G₀ the magnetic field dependence CF_B is much smaller than the bias dependence CF_V, it is opposite for larger conductance. An intermediate regime where field- and bias dependence are of comparable order of magnitude is found in a range from 0.15 to 0.8 G₀. The suppression of the magnetic field dependence CF_B for small contacts is in agreement with a transition from diffusive to ballistic transport: The origin of the pronounced negative MC (i.e. positive magnetoresistance) is the orbital magnetoresistance-i.e. the effective reduction of the mean free path, when the paths of the charge carriers are deflected by the Lorentz force [53]. When the lateral size of a wire becomes comparable to or smaller than the elastic mean free path, this effect is suppressed [54].

We thank C Debuschewitz, C Schirm, V Kunej and C Bacca for valuable and fruitful discussions about this work and for their contributions in the early state of the experiments. We are grateful to A Liebig for performing the x-ray diffractometry. We acknowledge experimental and technical assistance of A Fischer. We gratefully acknowledge financial support from the Krupp foundation and from the Baden-Württemberg-Stiftung in the framework of the Research Network Functional Nanostructures.

HFP fabricated the samples constructed the experimental set-up and performed the measurements. ES initiated the project, and ES and TP wrote the manuscript. All authors analyzed the data and discussed the results. All authors have given approval to the final version of the manuscript.

The project was financially supported by the Alfried-Krupp von Bohlen und Halbach foundation and by the Baden-Württemberg foundation.