FIELD OF THE INVENTION

The present invention relates to the field of deter- mining electric currents, in particular to devices and methods for determining DC electric currents for use as current standards.

BACKGROUND OF THE INVENTION

In International System of Units (SI) the unit of electric current (Ampere) is one the seven basic units. The definition is based on the Ampere's law re ^¬ lating the force F12 between two electric wires 1 and 2, carrying the electric currents II and 12 (Fig. 1) .

Being set at the turn of 19th to 20th centuries, the SI system reflected the state-of-the-art at that mo ^¬ ment. The above definition of Ampere, still formally constituting the SI, is extremely inconvenient from the practical points of view. The accuracy of such a standard of electric current is low and by no means corresponds to the needs of electric metrology in 21st century. Hence, there is a strong need for a current standard of electric current of high accuracy.

SUMMARY OF THE INVENTION

It is a purpose of the present invention to provide novel means for providing a current standard of elec ^¬ tric current with high accuracy.

At present moment, the frequency f of electromagnetic radiation is the electric parameter which can be set with the highest possible accuracy. Hence, a desired device and method - providing the standard of electric current - eventually should provide an output current l _{out /} which is uniquely related to the input frequency f input · However, to our best knowledge at present moment there are no provisions for such a hypothetical de ^¬ vice, which can directly provide at its output the de ^¬ sired l out (f input ) signal of sufficient accuracy. More realistic alternative is a device where several elec ^¬ tric parameters, including the current and frequency, are interlinked. For example, I _out (V _inpu t , f input ) , where Vinput is the additional input parameter. In this case, the definition the desired value of the output current l out _/ can be based on registration of a certain pecu ^¬ liar' point on the I _out (V _inpu t , fi„ _put =const ) dependency.

The present invention is characterized by what is pre ^¬ sented in claims 1 and 10.

A nanostructure for determining a DC electric current I _n for use as a current standard is formed on an insu ^¬ lating substrate and comprising a nanowire, electrodes with impedance higher than the impedance of the nan- owire arranged in a 4-probe measurement configuration and optionally being electrically connected to contact pads, optional electrode for external radiation input, and optional ground plates. The presence of the electrodes and their property (high impedance) is an important part of the inven ^¬ tion. The electrodes are the part of the long circuit: narrow part on chip, wide part on chip and the connecting wires from the whole nanostructure on the chip to the laboratory electronic. They lie on the sub ^¬ strate. The narrowest part of the electrodes (closest to the nanowire and with high impedance) should locat ^¬ ed not further than ~ 100 micrometers from the super ^¬ conducting nanowire.

Preferably, the nanowire is fabricated of a supercon ^¬ ducting material. The cross section of the nanowire is preferably small enough to provide the rate of quantum fluctuations comparable to the energy gap of the superconductor of which the nanowire is fabricated.

In one preferred embodiment, the material of the nan ^¬ owire is a superconductor with material parameters en ^¬ abling the rate of quantum fluctuations comparable to the energy gap of the superconductor within the technologically achievable small cross sections of the nanowire .

In one embodiment, the material of the nanowire is disordered superconductor enabling formation of a one- dimensional chain of Josephson junctions.

Preferably, the electrodes are fabricated of a high- Ohmic material.

In a preferred embodiment, the electrodes comprise on- chip cooling fins.

In one embodiment the electrodes are fabricated of one-dimensional chain of Josephson junctions.

In one embodiment, the electrodes are fabricated of one-dimensional chain of superconducting quantum devices (SQUIDs) .

The method for determining a DC electric current I _n for use as a current standard is based on measuring the current-voltage characteristic of a superconduct ^¬ ing nanostructure as specified above when exposed to an external electromagnetic radiation. Without application of radiation with frequency f, the current voltage characteristic I (V) and its derivative dI/dV(V) are sufficiently smooth. When the radiation with frequency f is applied on these characteristics appear singularities (= "not smooth dependence", or more accurate: where the derivative changes its sign) . These points are indicated in fig. 3 and 4 with arrows corresponding to I _n =(2e)x fxn. The current I _n , where these points are, is the desired current of high accu- racy, to be considered as the output of the electric current standard.

Preferably, the current-voltage characteristic of the nanostructure is measured with the on-chip electrodes with the impedance higher than the impedance of the nanowire .

The frequency of the external radiation f preferably corresponds to the value of the desired current I _n through the relation I _n =(2e)xfxn, where e is the electron charge and n=l,2,3... is an integer number.

Preferably, the value of the desired current I _n corre ^¬ sponds to the positions of the step-like peculiari- ties on the current-voltage characteristic I (V) or spike-type peculiarities on the derivative of the cur ^¬ rent-voltage characteristic dI/dV(V).

Preferably, the measuring lines leading from the room- temperature electronics to the nanostructure should are filtered against electromagnetic interference.

The body of the nanostructure (=superconducting nan ^¬ owire) is connected to the laboratory equipment with measuring electric lines. Very close to the nanowire are these (i) high-impedance very narrow electrodes, then they become (ii) wider leading to contact pads on the chip. Length of part (i) must be smaller than 100 micrometers, length of part (ii) is not essential (not part of the invention) and is about 2-3 mm. Then from the chip (insulating substrate containing the nanostructure) larger wires of length up to several meters lead to laboratory electronics. The requirement of filtering applies to these long wires which inevitably act as antennas picking-up all undesired elec ^¬ tromagnetic noise. They should contain RLC filters re- ducing the contribution of this electromagnetic noise. The requirement is that the electron temperature of the thinnest part of the nanostructure Tel should be much smaller (at least 10 times) than the critical temperature T _c of the superconductor, constituting the nanowire.

Preferably, the temperature of the nanostructure and the electrodes is smaller than the critical tempera ^¬ ture of the superconducting transition of the material the nanostructure is fabricated of.

Any superconductor is characterized by its critical temperature T _c (synonym to" superconducting transition of the material") . The effect of quantum fluctuations is mostly pronounced in materials with (i) low criti ^¬ cal temperature and (ii) having high resistivity in the normal state (=at temperature above T _c ) . Another (iii) requirement that the nanowire should have small diameter (in our case < 40 nm) . As the minimum diame- ter is limited by capabilities of nanofabrication, conditions (i) and (ii) are preferred.

The present invention follows the path described above. The ^x peculiar' points I _ou t,n on the I _out (Vi _nput , ) correspond to the relation between the current and the frequency through fundamental constant - the charge of electron e: I _out , _n =n* (2e) * finput, where n is just an integer number n=l,2,3... , corresponding to the number of the peculiarity on the I _out (Vinput / ^ in-

_Put =const) dependency. In physics an electric current is defined as the amount of charge Aq transferred during the dime At: I=Aq/At. Hence, if to ^x organize' a periodic transfer of charges with frequency finput _/ one formally may ob ^¬ tain at the output of such a device the desired cur- rent desired I _OUT ( finput) · Several attempts along this road [N. Zimmerman, Phys . Today 63, No. 8, 68 (2010)] indeed demonstrated principal possibility, but the ac ^¬ curacy of such devices is still far from the needs of practical metrology.

In the present invention, a qualitatively different approach is used. Let us consider a metal wire con ^¬ nected to external current source providing DC current I _DC - Then there is the corresponding dependence between the applied DC current and the DC voltage drop V _DC along the wire: V _DC (I _DC ) · In the embodiments of the in ^¬ vention :

(1) The wire is preferably fabricated of super ^¬ conducting material.

(2) The diameter of the wire is preferably small to enable the effect of quantum fluctuations.

(3) The superconducting nanowire is preferably connected to external electronics with high-resistive electrodes located ^λ οη chip' within < 100 ym from the nanowire.

(4) Then due to this intrinsic process of quan ^¬ tum fluctuations, there is the internal process of pe ^¬ riodic charge transfer resulting in periodic component of current I _A c = locsin ) , where the character- istic frequency of this internal process (2e) . (5) If to irradiate such a system with external electromagnetic radiation with frequency f inp _{ut /} there should be the resonance effect at matching points fi _n - _put =n* f _B i _oc h _/ n=l,2,3,... leading to peculiarities on the V _DC (I _DC ) dependency at particular points I _DC (n) =n* (2e) f inp _ut · These peculiar points can serve as the standards of electric current.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 illustrates the basic definition and determi ^¬ nation of electric current formally constituting the ampere definition in SI system of units.

Figure 2 shows a scanning electron microscope image of a typical sample with high-Ohmic contacts enabling four-probe transport measurements and introduction of RF radiation.

Figure 3 shows results of an all-titanium sample with the nanowire with length L=20 ym, effective diameter σ ^{1 2} =40 ± 2 nm and the probe resistance ~ 15 kOhm. First derivative dV/dl (I) in presence of external RF radiation with frequency f _RF —1 GHz measured at tempera ^¬ ture T=70 mK. Arrows indicate the positions of the ex ^¬ pected current singularities corresponding to the re ^¬ lation I _n =2e* f _RF *n, where n=l, 2, 3... Inset: posi ^¬ tions of the first current singularity II (left axis, circles) and the corresponding voltage VI (right axis, stars) as function of the RF frequency f _RF . Note the acceptable linear fit (solid line) for current singu ^¬ larities, while absence of any rationality for the VI (f _RF ) dependency, which one might expect in case of a conventional Shapiro effect observed in Josephson j unctions . Figure 4 shows results of titanium nanowires with length L=20 ym, the effective diameter below 20 nm and high- Ohmic bismuth contacts, (a) Sample#3 with effec ^¬ tive diameter σ ^{1 2} =15 ±3 nm and resistance 1.5 MOhms . Zoom of the I (V) characteristic at the transition point from insulating to current-carrying state in presence of external radiation with frequency f _RF =50 MHz at two levels of the RF signal. Arrows indicate the positions of the expected current singularities corresponding to the relation I _n =2e* f _RF *n, where n=l, 2, 3... Note the characteristic back-bending shape of the I (V) dependence: ^x Bloch nose'. Left inset: whole scale I (V) characteristics of the three adjacent sec ^¬ tions of the same nanostructure . Arrows indicate the directions of the I-V dependence recording. Right in ^¬ set: first derivative dV/dl (I) in presence of exter ^¬ nal RF radiation with frequency f _RF =350 MHz. (b) Sam- ple#l with effective diameter σ ^{1 2} =12 ±4 nm and resistance 2.5 MOhms. Magnified view of the current sin- gularities at RF frequency f _RF =3.8 GHz. Arrows indicate the positions of the expected current steps. One can distinguish both the Cooper pair (2/1 and 4/1) and single electron (1/1 and 3/1) singularities, as well as the first single electron sub-harmonic (1/2) . The inset shows the dependence of the singularity width DV on the RF magnitude for the first Cooper pair singu ^¬ larity (2/1) and the Coulomb gap (0/1) . Note that for the Coulomb gap the scale is reduced by a factor of 10. (c) Positions of the current singularities I _n as function of the RF frequency f _RF .

Figure 5 illustrates schematically a nanostructure ac ^¬ cording to one embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

The principles behind and embodiments of the invention are described in the following in more detail. Small current-biased tunnel junction should exhibit coherent oscillations qualitatively described by simi ^¬ lar expressions as motion of Bloch electrons in peri- odic potential of a crystal lattice. Weak singulari ^¬ ties on I-V dependencies of RF irradiated Josephson junctions (JJs) have been attributed to the effect - Bloch oscillations. Here we report the experimental evidence of the phenomenon in ultra-narrow supercon- ducting titanium nanostructures . Small cross sections of the nanowires suggest that the quantum phase slip ^¬ page might be responsible for the observation of the pronounced Bloch singularities at quantized values of electric current. The phenomenon is dual to the well- known Shapiro effect, routinely observed in Josephson systems, and originates from the equivalence of the quantum dynamics of a JJ and a superconducting nan- owire governed by quantum fluctuations. In the present invention, the effect is utilized to implement an im- portant metrological application the quantum standard of electric current.

A conventional Josephson effect is observed in systems with high coupling energy Ej= (R _Q /R _N ) (Δ/2)>> E _c and high conductance G≡l /R _N >>1 /R _Q , where the quantum resistance R _Q = h/ (2e) ² = 6.47 kOhm with e being the elementary charge. In this limit the superconducting phase φ be ^¬ haves as a nearly classical variable. Application of external RF radiation with frequency f _RF leads to the well-known Shapiro effect - formation of quantized voltage steps on the I-V characteristics: V _n =h (f _RF /2e) *n, where n=l, 2, 3.... In the opposite limit, Ej<<E _c and R _N >>R _Q , the quasicharge q rather than φ is the nearly classical quantity, and the corre- sponding Coulomb effects should take over the Joseph- son coupling. The experimental condition for observation of the charge phenomena in Josephson systems requires ful ^¬ fillment of two conditions. First, the capacitance C of the JJ should be very small providing the high charging energy E _c =(2e)2/2C>> Ej. Second, to enable the quasicharge q be a well-defined quantity, the system should be current biased through the on-chip high- Ohmic probes with resistance R _P >> R _N >> R _Q . The peri ^¬ odic charging/discharging of the junction leads to Bloch-type oscillations resulting in a peculiar back- bending shape of the I-V characteristic - the so called Bloch nose. Applied external RF radiation can be synchronized with the internal charge oscillations manifesting as singularities of the I-V characteristic at quantized values of the current I _n =(2e)* f _RF *n ³ . The effect can be formally used as a quantum standard of electric current. However, only rather broadened n=lcurrent singularities (1st Bloch step) were ob ^¬ served so far in JJ having little practical interest for metrological applications.

During the last decade it has been a remarkable pro ^¬ gress in understanding the physics of quasi-one- dimensional supercondcutors . In particular, it has been shown that in ultra-narrow superconducting channels the resistance remains finite even at tempera ^¬ tures well below the critical T<<T _C due to the essen ^¬ tially nanoscale phenomenon - the quantum phase slips (QPS) . It has been theoretically demonstrated that a JJ is dual to a superconducting wire governed by QPS10. If the parameter S _QPS =A (R _Q /R _N ) (L/ξ) »1 , where R _N is the normal state resistance of the wire with length L and superconducting coherence length ξ, and A is the numerical parameter of the order of unit, then the system should exhibit superconducting properties. In the opposite limit S _QPS <1 the quantum fluctuations de- stroy the superconducting order and the system is dual to a JJ demonstrating Coulomb effects.

To test the hypothesis we fabricated titanium nanostructures embedded in high-Ohmic environment Fig. 2. This particular material has been selected as it has been already demonstrated that in titanium nan- owires (with low-Ohmic probes) at diameters σ ^{1 2} ≤ ~ 40 nm the QPS dramatically broaden the R(T) dependencies. The relatively thick samples with σ ^{1 2} ~ 40 nm in not- too-high-Ohmic environment -15 kOhm indeed demonstrate the weak Coulomb blockade with the zero-bias conduc ^¬ tivity lower than at a finite bias at a given tempera ^¬ ture T<T _C , but larger than in the complete normal state at T>T _C (Fig. 3) . The observation indicates that in these samples with parameter 1< S _QPS <10 the residu ^¬ al superconductivity still ^x wins' over the Coulomb ef ^¬ fects. Application of external RF radiation stimulates nonlinearities , clearly resolved only at dV/dl depend- encies, with positions correlating with the expecta ^¬ tion I _n =(2e)* fRF*n with the quantum number as high as n=4 (Fig.3) . Note that the corresponding positions in voltage scale V _n (f _RF ) do not form any rational pattern (Fig. 4, inset) indicating the absence of the Shapiro effect to be observed in a conventional Josephson sys ^¬ tem.

Significantly thinner nanowires with the effective di ^¬ ameter σ ^{1 2} <20 nm and the corresponding parameter S _QPS ~1 demonstrate very pronounced back-bending of the I-V characteristic (Fig. 4a) with the RF induced singular ^¬ ities traceable up to n=8 (Figs 4a and 4b) . However, the large value of the Coulomb gap (Fig. 3a, left in ^¬ set) , which do not disappear above the T _c of supercon- ducting titanium, leads to a conclusion that, though unintentionally, some weak links were formed. It should be emphasized that all RF-induced singularities are observed only at temperatures below T*~170 mK. Note that the critical temperature of titanium is size dependent and for the thinnest nanowires the T _c is ex ^¬ pected to be below 200 mK. At higher frequencies of external RF radiation quite rich structure develops at the I-V dependencies (Fig. 4b) . The positions of all these singularities form the regular pattern (Fig. 4c): I (n, m) =e (n/m) * f _RF , where principal steps with m=lcan be associated with single electron transport n=l,2,3,4, while the even steps (n=2, 4, 6) might also originate from Cooper pairs. First single electron sub-harmonic n=l and m=2 (step 1/2) can be also re ^¬ solved. Co-existence of superconducting and single electron Bloch steps has been earlier reported in JJ, though only for quantum number n=l . Remarkably the dependence of the step width δν on the amplitude of the RF signal V _RF is essentially non-monotonous (Fig. 4b, inset) following the theoretical prediction δν _η ~ (- 1 ) ⁿ J _n (V _RF ) sin (q) , where J _n is Bessel function.

The whole set of our data indicates that the observed singularities on the V-I dependencies indeed can be associated with Bloch oscillations. From the theory point of view the quantum dynamics of a JJ (or a chain of JJs) and a nanowire governed by QPS is essentially identical and qualitatively indistinguishable. Never ^¬ theless one may ask a practical question: what system out of these two our titanium nanowires represent? Mi ^¬ croscopic analysis of all studied samples (SEM and AFM) could reveal no imperfections clearly associated with pronounced constrictions or/and breaks: all sam ^¬ ples have very smooth surface with roughness ± 2 nm. For the ^x thick' samples in relatively low-Ohmic envi ^¬ ronment (Fig. 3) both the normal state sheet re- sistance RQ~10 Ohm, and the very shape of the I-V characteristic (rather weak Coulomb blockade) might indicate the absence of weak links and the nanowire can be considered as a homogeneous metallic structure. Hence in these structures the existence of tunnel junction (s) is highly improbable and the observation of the RF-induced singularities might indeed originate from the QPSs providing some kind of ynamically driven' equivalent of a JJ.

Interpretation of data for the high-Ohmic samples (Fig. 4) is less straightforward. From one side, the normal state sheet resistance ¾ of the thinnest nan- owires, measured well above T _c , ranges from 400 Ohm to 1.9 kOhm being on the metallic side of dirty titanium. Note that the Coulomb effects have been observed in deliberately anodically oxidised titanium nanowires with noticeably higher resistivity. From the other side, the large value of the Coulomb gap surviving above the T _c of titanium (Fig. 4a, left inset) re ^¬ quires the existence of weak link(s) - for example, tunnel barrier (s). Presumably, those tunnel barriers were unintentionally formed at the overlapping regions between the titanium nanowire and bismuth contacts. Making a rough estimation that the tunnel junction is a plane capacitor with the area σ~15 nm ^χ 15 nm sepa ^¬ rated by a barrier with thickness d=l nm and the die- lectric constant ε ranging from 1 (vacuum) to 80 (crystalline T1O ₂ ) one comes to a conclusion that a chain of -10 of such junctions might simulate the Cou ^¬ lomb blockade in high-Ohmic samples. Note that for ^¬ mation of parallel tunnel junctions (not a ID chain, but rather a 2D array) does not alter the value of the Coulomb gap, and, hence, can be disregarded in estima ^¬ tions. Given the rather small number of serially con ^¬ nected tunnel junctions, the major part of the 20 ym long samples (Fig. 2) can be considered as a homogene- ous metallic nanowire, where the QPS effect should take place. From this point of view particularly interesting is the data presented in the inset in Fig. 4b. If it would be the same number of junctions re ^¬ sponsible for the Coulomb blockade and for the Bloch oscillations, then the values δν for the step 0/1 and 2/1 should be comparable, which is not the case. One may conclude that several serially connected junctions are responsible for the (large) Coulomb gap, while much smaller number of ^x active' elements for the Bloch steps. If the case, then those serially connected junctions act as an additional high- Ohmic environment favoring observation of the Bloch oscillations. Given the equivalence of the quantum dynamics of a JJ and a QPS wire, our data cannot distinguish whether that ^x active' element is a static JJ, or a driven by quan ^¬ tum fluctuations ^Λ dynamic' QPS junction. However, whatever is the case, our experiments is a clear evi ^¬ dence of Bloch oscillations. The effect is expected to lead to an important metrological application - the quantum standard of electric current. Methods

The samples were fabricated using electron beam li ^¬ thography followed by the multi-angle ultrahigh vacuum deposition of the metals and the lift-off technique. The ^x body' of the sample - titanium nanowire - was al ^¬ ways deposited at zero angle and relatively high speed to enable good quality of the film. Two types of con ^¬ tact probes were used. First ones were all-titanium utilizing evaporation of the (titanium) probes at a high angle and slow speed. The obtained electrodes were relatively low-Ohmic with the probe resistance RP below 20 kOhm. In the second type of structures (Fig. 2) slowly deposited at a poor vacuum bismuth nano- probes were used. The overall resistance of the up to 100 ym long bismuth probes could be made as high R _P ~50 MOhm enabling reliable current biasing of the sample. Low energetic Ar+ ion beam milling was used to gradu ^¬ ally and non-invasively reduce the nanowire cross sec ^¬ tion σ. Utilization of Ar+ ions at acceleration voltages ≤~ 1 keV can be considered as virtually introduc- ing no defects as the ion penetration depth ~ 2 nm is comparable to the thickness of the naturally grown ox ^¬ ide. Additionally, the sputtering provides polishing effect eroding small scale imperfections of the lift ^¬ off fabricated nanostructures .

Scanning electron (SEM) and atomic force (AFM) analysis revealed the expected smooth surface with rough ^¬ ness of about ±2 nm and absence of any pronounced de ^¬ fects, which could be associated with constrictions and/or breaks. Slightly higher error up to ±4 nm, specifying the effective diameter of the ion milled samples, originates from the uncertainty in determina ^¬ tion of the position of the interface between the met ^¬ al and the (also ion milled) substrate. High resolu- tion transmission electron microscopy (TEM) analysis of co-fabricated films revealed the expected non- single-crystalline structure of the samples. The grains were compactly packed with inevitable disloca ^¬ tions and associated stacking faults. However, no sig- nature of any sort of inclusions (e.g. foreign materi ^¬ al clusters) or distortion due to ion implantation in the ion milled samples was found. Elemental analysis of the samples, fabricated using the same vacuum cham ^¬ ber as in present work, revealed the highest concen- tration of foreign elements inside the metal matrix being associated with 0.3 at. % of oxygen. The high- resolution TEM images of the nanowires and wide 2D films, fabricated under the same conditions, were in ^¬ distinguishable between themselves, and what is even more important - indistinguishable before and after the ion milling. The observation supports the state ^¬ ment that the utilized reduction of the cross section of the nanowires by low energy ion milling does not introduce any structural defects.

After each step of reducing the cross section, the samples were cooled down to temperatures well below the critical temperature of titanium using He3/He4 di ^¬ lution refrigerator. The measurements were performed inside electromagnetically shielded room using battery powered front-end amplifiers and carefully filtered input/output lines. Four-stage RF filtering enabled us to keep the increase of the electron temperature δΤ _θ ≤20 mK above the base temperature of the refrigerator

Tbath .