Dynamical Coulomb Blockade as a Local Probe for Quantum Transport

Quantum fluctuations are imprinted with valuable information about transport processes. Experimental access to this information is possible, but challenging. We introduce the dynamical Coulomb blockade (DCB) as a local probe for fluctuations in a scanning tunneling microscope (STM) and show that it provides information about the conduction channels. In agreement with theoretical predictions, we find that the DCB disappears in a single-channel junction with increasing transmission following the Fano factor, analogous to what happens with shot noise. Furthermore we demonstrate local differences in the DCB expected from changes in the conduction channel configuration. Our experimental results are complemented by ab initio transport calculations that elucidate the microscopic nature of the conduction channels in our atomic-scale contacts. We conclude that probing the DCB by STM provides a technique complementary to shot noise measurements for locally resolving quantum transport characteristics.

antum uctuations are imprinted with valuable information about transport processes. Experimental access to this information is possible, but challenging. We introduce the dynamical Coulomb blockade (DCB) as a local probe for uctuations in a scanning tunneling microscope (STM) and show that it provides information about the conduction channels. In agreement with theoretical predictions, we nd that the DCB disappears in a single-channel junction with increasing transmission following the Fano factor, analogous to what happens with shot noise. Furthermore we demonstrate local di erences in the DCB expected from changes in the conduction channel con guration. Our experimental results are complemented by ab initio transport calculations that elucidate the microscopic nature of the conduction channels in our atomic-scale contacts. We conclude that probing the DCB by STM provides a technique complementary to shot noise measurements for locally resolving quantum transport characteristics.
An important consequence of the downscaling of electronic circuits towards the atomic limit is the emergence of charge quantization e ects [1][2][3][4][5].
In this Le er, we exploit the DCB in ultra-low temperature scanning tunneling spectroscopy (STS) as a tool to locally identify the quantum transport characteristics of atomicscale junctions all the way from the tunnel to the contact regime. First, we use a junction formed between two single atoms featuring a single dominant transport channel [36]. e DCB is seen at low transmission, but disappears with increasing transmission following the Fano factor for a singlechannel junction [37]. Extending the measurements to a junction between a single atom on one side and two atoms on the other side, we nd a di erent signature in the DCB dip. is indicates a direct in uence of the number of transport channels and their transmission τ i (mesoscopic PIN code or channel con guration) on the DCB. We conclude that DCB measurements in STS below 1 K provide direct access to the mesoscopic PIN code as a technique complementary to shot noise measurements [14,[37][38][39][40].
We rst use the atomic manipulation capabilities of the scanning tunneling microscope (STM) to construct a junction between two single aluminum atoms (see Fig. 1(a)). One atom is placed at the Al tip apex and one on the (100) surface of an Al crystal, as shown in the lower half of Fig. 1(b). By applying a magnetic eld of 20 mT, the superconductivity in Al is quenched and we obtain a normal conducting junction at an experimental temperature T of 15 mK [41]. We can reproducibly and continuously tune the junction conductance up to the quantum of conductance G 0 = 2e 2 /h (with Planck's constant h) by changing the tip-sample distance, as we illustrate in Fig. 1(c).
We start by studying the di erential conductance G(V ) in the tunnel regime at bias voltage V , where the setpoint conductance G N = G 0 i τ i = G 0 τ t and G N G 0 . As we show in Fig. 1(d) for G N = 0.027 G 0 , the conductance exhibits a dip at low bias voltage, which is the typical signature of DCB. To verify this observation we analyze our data using the P(E) theory [26,34,42]. In the P(E) model, the interaction of tunneling charged particles with the environment is taken into account by the environmental impedance Z (ω), as shown schematically in Fig. 1(a). e obtained t is indicated in Fig. 1(d) as an orange line. We nd for the junction capacitance C J = 21.7 fF and for the e ective temperature T e = 84.9 mK. e t con rms that we operate in a lowimpedance regime, where the zero frequency part of the environmental impedance is R env = 377 Ω and much smaller than Approach curve on the Al adatom with an Al tip (both in the normal conducting state) at a bias voltage well above the DCB dip. In (d) the dip in the normal conducting dI /dV curve, prototypical for the DCB, is shown with a P(E) t in the low-conductance limit.
is establishes the DCB in the tunneling regime at low conductances. However, as we approach the tip to the adatom on the sample, the conductance increases, and we observe a clear reduction in the DCB. e experimental data is shown in Fig. 2(a) for di erent conductance values ranging from 0.03 G 0 close to 1 G 0 . e spectra have been normalized to the setpoint conductance G N in the voltage range outside of the DCB dip. e reduction in conductance at zero bias voltage δG(0) gradually decreases until it disappears at the highest conductance. is suppression of the DCB as the channel transmission approaches the ballistic limit of perfect transmission (τ 1 → 1) has been observed in other types of quantum point contacts [30][31][32]44]. It can be understood by considering the suppression of uctuations in the number of transmi ed electrons through the junction with increasing transmission towards the ballistic limit, which is captured in the Fano factor F = i τ i (1 − τ i )/ i τ i . e relative change in conductance δG(V )/G N for weak coupling to the environment Z (ω) and at zero temperature was derived for a singlechannel system in Ref. [37] and for multiple channels in Ref. [45]: e integral in Eq. (1) shows that for a generally small environmental impedance ReZ (ω) R Q , as realized in the  Ref. [37]. Parameters were determined by the P(E) t in Fig. 1(d), the color code corresponds to (a). (c) e dI /dV reduction at zero bias δG(0)/G N dependent on junction conductance, plo ed as blue circles for the single atom and as a yellow diamond for the dimer. We added a linear t to the data assuming a single-channel junction τ t = τ 1 , where the dip reduces with (1−τ 1 ) from its value in the tunneling limit. For comparison, a dashed line (1−τ t /2) is shown, representing the behavior of a corresponding junction with two equal channels STM, the change in conductance will be comparatively small. In Fig. 2(b) we model the transmission-dependent DCB dip based on the theory in Ref. [37] for one transmission channel τ 1 (see also SM [43]). We use the same parameters for the environmental interaction as before in the P(E) t depicted in Fig. 1(d) and nd good agreement with the data. e decrease of the experimental DCB dip with increasing conductance G N is shown in Fig. 2(c) as blue circles. It follows a (1 − τ 1 ) dependence as veri ed through the linear t. is nding of pronounced single-channel characteristics in a junction between two Al atoms is consistent with previous experimental results obtained using the subgap structure of the current in the superconducting state [36].
In order to understand the observation of a single channel and to eludicate its origin, we have performed quantum transport calculations within the Landauer-Bü iker approach for coherent transport using a method that combines density functional theory (DFT) with nonequilibrium Green's function (NEGF) techniques. In particular, this approach makes it possible to optimize the junction geometries, to compute their electronic structure and transport characteristics, including the transmission eigenchannels [46]. As in the experiment the Al sample is modeled as a (100) surface with an additional Al adatom. e structure of the tip oriented along a (100) direction, and the sample are displayed in Fig. 3(a). e channel transmissions τ i were extracted as a function of tip-sample distance, as is visible in Fig. 3(b). We can clearly see that the calculations reproduce the singlechannel nature of the atomic Al contact. e transmissions of the second and third channel τ 2 and τ 3 are about two orders of magnitude smaller than those of the dominant channel τ 1 over the full range of z-values considered, in contrast to the situation in break junction experiments [47][48][49]. Since higher order channels contribute even less, we focus on τ 1 τ 2 , τ 3 [50] in the following, corresponding to the valence states of Al [49,51]. Further insight can be obtained by calculating the complex-valued sca ering-state wave functions of the transmission channels, as shown in Fig. 3(c). In the plots colors encode the phase, while absolute values are visualized through the isosurface [52,53]. For an electron wave impinging on the contact from the substrate, we observe that the dominant rst transport channel is of σ symmetry. In comparison, the second and third channels have a π shape when viewed along the transport direction. us, the theoretically calculated PIN code is (0.575, 0.003, 0.001), which implies that the rst channel provides 99.3% of the total transmission. Similar theoretical results were obtained for a junction geometry with an atomically sharp tip oriented along the (111) direction (see SM [43]). From the experimental data at higher transmission, we estimate that channels beyond the rst contribute no more than 3% to the total transmission at 0.99 G 0 , which agrees nicely with the theoretical results.
Exploiting the local atomic resolution and manipulation capabilities of the STM, we can build more complicated atomic structures on the surface such as a dimer of Al atoms. is is visualized in Fig. 4(a), where the dimer is marked in purple. We placed two Al atoms on face-centered cubic lattice sites parallel to the atomic rows, separated by one site. Approaching the tip over the bridge position of the dimer, we anticipate more than one signi cant transport channel in the junction. e DCB spectrum for the dimer is shown in Fig. 4(b) as a blue line together with a measurement on a monomer. Both of them are taken at a total conductance of 0.58 G 0 . e characteristic dip at zero bias voltage is clearly visible. Comparing the dI /dV curve on the dimer with the one on the monomer, we nd that the DCB dip for the dimer is much more pronounced. From the experimental data on the dimer we extract a conductance reduction at zero voltage of δG(0)/G N = −5.4 %, whereas the reduction on the monomer at the same G N value is δG(0)/G N = −3.7 %. Considering the identical total conductance, this is only possible if the number of transmissive channels has changed, such that the rst channel has a lower transmission, which leads to a more pronounced DCB dip. Analyzing the dimer DCB dip, we consider two contributing channels and experimentally nd a PIN code of (0.46, 0.12), with an estimated uncertainty of ±0.05 for each channel.
Like for the monomer, we simulated the junction with the dimer to gain further insight into the microscopic origin of the transport channel con guration. e wave functions for the channels 1, 2 and 3 are displayed in Fig. 4(c) for G N = 0.58 G 0 . e simulations yield a PIN code of (0.543, 0.029, 0.003) for a (100)-oriented tip and (0.540, 0.034, 0.004) for a (111)-oriented tip, in acceptable agreement with the experimental ndings (details see below). For these con gura-tions, 93.6% and 93.1% of the total transmission is carried by the rst channel, respectively. is is in contrast to the simulations of the monomer at the same conductance [(100)-tip orientation: (0.575, 0.003, 0.001); (111)-tip orientation: (0.576, 0.002, 0.002)], where both con gurations contribute more than 99% to the total transmission (see SM [43]). Hence, the transport channel con guration has clearly changed between the monomer and the dimer. Even if our calculations predict that the transport between the dimer and tip is dominated by the rst channel, the transmission of the second channel is enhanced by one order of magnitude with respect to the monomer. For this reason we regard the dimer-tip system as a two-channel junction. e experimentally observed more pronounced DCB dip on the dimer than on the monomer is in agreement with these predictions.
antitative di erences between theory and experiment may arise from a tip con guration that deviates from a perfect single-atom apex. Such deviations are visible as a small distortion of the dimer in Fig. 4(a). Considering that a change of the tip orientation in the calculations, which is hardly expected to be visible in the topography, already yields a 14% change of τ 2 demonstrates the sensitivity of our method.
To test the range of applicability of this technique, we measured the DCB also in the high-temperature limit. is data was taken on the crystal surface at 1.32 K and 0.13 G 0 , see Fig. 5(a). We model it with the same values of the parameters describing the electromagnetic environment in the P(E) t of the DCB in Fig. 1(d), only changing the temperature. While we nd overall consistency between low-and hightemperature data and modeling, the dip at high temperature only reduces the conductance by about 1 %, making it more challenging to detect changes. To reduce the error bar on these measurements, the strength of the DCB needs to be signi cantly increased. is can be achieved by changing the junction capacitance, since a smaller C J yields a more pronounced dip. To illustrate the e ect, we model the DCB within an experimentally relevant range of C J between 1 and 60 fF and temperatures between 10 mK and 1.5 K based on the P(E) model [33]. All other parameters are kept at the values used above. e obtained dependence is representative for the tunneling regime (τ t 1) and is plo ed in Fig. 5(b). Our calculation shows that even in the high-temperature limit, small-capacitance junctions should yield a reasonable δG(0)/G N . e junction capacitance can be changed by adjusting the macroscopic tip geometry [33]. erefore, we surmise that a number of experiments would pro t by probing local PIN code variations using the DCB. e trade-o in energy resolution due to the reduced capacitance is likely not an issue at higher temperatures (of around 1 K) due to dominating thermal broadening [33]. In this sense using the DCB to extract the transport characteristics becomes a viable, complementary alternative to shot-noise measurements.
In summary, we have shown an alternative path to access transport properties on the atomic scale based on the DCB, applicable with standard measurement electronics. Apart

SAMPLE PREPARATION
e experiment was conducted in a scanning tunneling microscope (STM) at 15 mK [1]. e tunnel junction consists of an atomically sharp polycrystalline Al tip and a single Al atom placed on the (100) surface of an Al crystal, resulting in a single dominant channel [2]. In order to work in the normal-conducting state of Al, the superconductivity in tip and sample was quenched by a magnetic eld of 20 mT. e surface of the Al crystal was prepared by several cycles of Ar ion spu ering and annealing. e aluminum tip was cut from a wire (1 mm diameter), spu ered with Ar ions, treated by eld emission and then dipped into the sample surface until it yielded an atomically sharp topography. is tip was then used to extract single Al atoms from the crystal and place them on the surface.

LOW-CONDUCTANCE DYNAMICAL COULOMB
BLOCKADE MEASUREMENTS e dynamical Coulomb blockade (DCB) e ect in the tunneling limit, where τ i 1 for all i, is not signi cantly inuenced by the number of transmissive transport channels. In Fig. S1 a DCB measurement on an adatom (dark blue) at low conductance is compared with several measurements on various surface positions (light blue, green, yellow), using microscopically di erent tips. It is clearly visible that in this transmission regime the DCB has the same e ect on the conductance around zero bias. Furthermore, it is apparent that the dI /dV signal next to the dip is not entirely at and varies to some extent, indicating a slight modulation in the densities of states of microscopically di erent tips and at di erent sample positions. ese modulations lead to some uncertainty in the determination of the depth of the DCB dip. We approximate this uncertainty with 1 % of the setpoint conductance G N .

DETAILS ON THE DFT-NEGF APPROACH
We analyze the elastic transmission of our Al atomic contacts by means of microscopic theoretical calculations. e Al sample was modelled such that it features a (100) surface. For the monomer-tip system an additional Al atom placed on top. To reproduce the experimental situation as closely as possible, we assume two di erent orientations for the Al tip, namely (100), as displayed in the main text, and also (111) for comparison. e calculations involve a combination of density functional theory (DFT) and nonequilibrium Green's function (NEGF) techniques. ey are used to obtain optimized junction geometries, their electronic structure and transport properties in the phasecoherent regime, including information on transmission eigenchannels [3][4][5]. Fig. S2(a) shows the computed τ i (∆z) curves of the monomer-tip junction with a (100)-oriented tip in a larger range than in the manuscript, where ∆z is the change of distance between tip and sample. Additionally, the transmission of channels 2 and 3 in relation to the total transmission τ t is plo ed in Fig. S2(b) to highlight their small contribution. In addition, we studied a (111)-orientation of the tip, see Fig.  S2(c). We nd overall the same behavior of a strongly dominating rst channel, which is highlighted again in Fig. S2(d).
In comparison to break junction experiments channels two and three are signi cantly reduced [6][7][8].
DCB IN THE TUNNELING LIMIT -P(E) MODEL e interaction of quantum uctuations of the phase ϕ and its conjugate variable charge Q with the electromagnetic environment Z (ω) can be described by the probability of energy exchange in terms of the P(E) model [9][10][11][12][13]. In addition to describing the DCB e ect in the tunneling limit, the P(E) model also describes the Josephson e ect in the DCB regime and the broadening of spectral features at temperatures below 1 K [14][15][16]. e P(E) model is based on the phase correlation function with the resistance quantum R Q = 1/G 0 and the total junction impedance given by the junction capacitance C J and the transmission line impedance Z (ω): Here, ω 0 describes the principal mode of the STM tip, which behaves as a λ/4-monopole antenna and α is an e ective damping parameter [17]. R env is the e ective d.c. vacuum impedance R env = 376.73 Ω. e interaction with the electromagnetic environment during the tunneling process of a charged particle yields a probability distribution for its nal energy given by with the reduced Planck constant . Additionally, the e ect of the temperature-dependent capacitive noise on the junction needs to be considered. We do this by means of a Gaussian P(E) function [11] with the charging energy e convolution of both P N (E) and P 0 (E) functions captures the in uence of the electromagnetic environment on a measurement For details on the computation see Ref. [16].
e tunneling rate in one direction between tip and sample is then given by extending the standard tunneling rate [18,19]. Assuming a at density of states in tip and sample, at the tunneling conductance G N the tunneling rate in one direction is where f is the Fermi function. e tunneling rate in the opposite direction ← − Γ (V ) is obtained by exchanging electrons and holes in Eq. (9). Consequently the tunneling current is For the t of the DCB in the main text an e ective temperature of T e = 84.9 mK and a junction capacitance of C J = 21.7 fF were used. e e ective temperature takes into account residual noise broadening that is not explicitly included in the model and, therefore, is higher than the measured temperature. e damping factor is α = 0.4 and the resonance energy is ω 0 = 70 µeV.

TRANSMISSION-DEPENDENT DCB MODEL
e transmission dependence of the DCB and its similarity to the behavior of shot noise was derived in Ref. [20].
ere, a theory was developed describing the in uence of a macroscopic impedance Z (ω) on the transport through a single-channel quantum point contact with changing transmission τ 1 . e link between current uctuations, shot noise and the DCB in the low impedance limit Z (ω) 1/G 0 was demonstrated [20,21]. Our results are obtained by a numerical evaluation of Eq. (7) in Ref. [20] δI (V ) =