**2. Josephson effect and Josephson junctions**

Modern voltage metrology is based on the Josephson effect which is a macroscopic quantum effect of inherent fundamental precision occurring in a weak link between two superconductors.

The essential properties of superconductive material are that the resistance to a continuous electrical current becomes zero and a magnetic field cannot penetrate inside, up to a critical value, which is characteristic of each material. Superconductivity is therefore described by the conditions electrical resistance *R*=0 and magnetic induction *B* =0. The superconductive, non dissipative state, is characterized by an advantageous energetic condition respect to the normal state, since in some materials below a critical temperature *T*c and a critical applied magnetic field *H*c, all the electrons condensate at the same energy, which is separated by the Fermi level from a gap *Δ*, in the range of about 1 mV.

This energetically favorable superconductive state establishes a long range order extending through the material, where the charge carriers, the Cooper pairs, flow without dissipation. The Cooper pairs are a particle which has twice the electric charge of the electron and a virtual radius corresponding to the coherence length of the superconductor, of the order of tenths of micrometer. They are formed at low temperature by the mutual attraction occurring between two electrons exchanging virtual phonons related with vibrations of the lattice [1].

Fig. 1. Picture of the Cooper pair binding mechanism inside the atomic lattice.

Concerning the Josephson effect, it was firstly theoretically predicted and observed for tunnel junctions with two superconductive electrodes separated by a very thin insulating

The main focus of section 3, Current junction technologies and fabrication issues, is on the different type of Josephson junctions employed so far in circuits for programmable and AC voltage standard, with details on the link between fabrication technology and device

In a last section, Special issues for next generation standard and possible solutions the present and future applications of these standards to measurement are mentioned, with

Modern voltage metrology is based on the Josephson effect which is a macroscopic quantum effect of inherent fundamental precision occurring in a weak link between two

The essential properties of superconductive material are that the resistance to a continuous electrical current becomes zero and a magnetic field cannot penetrate inside, up to a critical value, which is characteristic of each material. Superconductivity is therefore described by the conditions electrical resistance *R*=0 and magnetic induction *B* =0. The superconductive, non dissipative state, is characterized by an advantageous energetic condition respect to the normal state, since in some materials below a critical temperature *T*c and a critical applied magnetic field *H*c, all the electrons condensate at the same energy, which is separated by the

This energetically favorable superconductive state establishes a long range order extending through the material, where the charge carriers, the Cooper pairs, flow without dissipation. The Cooper pairs are a particle which has twice the electric charge of the electron and a virtual radius corresponding to the coherence length of the superconductor, of the order of tenths of micrometer. They are formed at low temperature by the mutual attraction occurring between

two electrons exchanging virtual phonons related with vibrations of the lattice [1].

Fig. 1. Picture of the Cooper pair binding mechanism inside the atomic lattice.

Concerning the Josephson effect, it was firstly theoretically predicted and observed for tunnel junctions with two superconductive electrodes separated by a very thin insulating

main stress on the routes to overcome present limitations to these challenges

**2. Josephson effect and Josephson junctions** 

Fermi level from a gap *Δ*, in the range of about 1 mV.

performances.

superconductors.

barrier (of the order of nm) [2]. In this first model, for a Josephson junction realized as a weak link between two superconductors, a quasiparticle inside the barrier, in order to penetrate in one of the superconductors must get an energy greater than the superconductor energy gap, which is not possible by direct charge transfer.

But another process may occur: in the so called Andreev reflection [3] an electron impinging at the interface of the weak link with the superconductor, is converted into an hole moving in the opposite direction, thus creating a Cooper pair. Similarly, at the other side of the weak link, the reflection process annihilates a Cooper pair from the other superconductor. The result is the tunneling of the Cooper pairs through the barrier of the weak link. Nowadays, it was generalized to a variety of junctions, the most accepted interpretation is based on the Andreev reflection process.

This tunneling is not dissipative up to a critical value of current *Ic*, whose amplitude is determined by the geometry of the junction, the materials and the operating temperature. Below this value the current depends only from the phase difference φ of the two wave functions describing the superconductive electrodes of the junction.

Fig. 2. Josephson junction general scheme. When a voltage drop V appears between the two superconductive electrodes an oscillating signal of frequency f is emitted by the junction.

The first Josephson equation, called DC Josephson effect then states:

$$I = I\_c \sin \eta \tag{1}$$

When the biasing current I is > Ic , a voltage V appears across the junction and this voltage generates on its own an oscillating current, whose frequency f is related to the voltage from the second Josephson equation.

$$V = (\text{ly} \, 2e) \times f \tag{2}$$

The last relation is at the basis of the modern Voltage metrology. The quantity *h/2e* is called *Φ0*, and represents the unity flux quanta.

As stated, equation 2) is independent from any device characteristic or any operating condition, and does not have any drift in time.

In the practical experiment, the Josephson frequency, which is in the THz range, is modulated by a microwave signal, producing voltage steps at fixed, constant interval *n (h/2e) × f*, with *n* integer 1, 2,.. These steps, called Shapiro steps, are used as reference voltage in the measurements, and their amplitude in current depends on the power of the microwave signal coupled to the junction, according with good approximation to the relatioship: (3) These steps, which are what is used as reference voltage in the measurement, are called Shapiro steps, and their amplitude depends on the power of the microwave signal coupled to the junction.

$$I\_s = I\_c \sum\_{n=0}^{\alpha} (-1)^n J\_n \left( \begin{matrix} V\_1 \\ \end{matrix} \Big/ \begin{matrix} V\_1 \\ \Phi\_0 f\_1 \\ \end{matrix} \right) \sin \left[ \begin{matrix} \varphi\_0 + \frac{2\pi}{\Phi\_0} V\_0 t - 2\pi n f\_1 t \\ \end{matrix} \right] \tag{3}$$

where: *Is* step current width, *Ic* critical current*, Is* step current width, *V0* (*V1*) dc (ac) voltage bias, *f1* ac bias frequency, and J*n* Bessel function of order *n*.

Experiments where the voltage provided by two different Josephson devices radiated by different microwave signals was detected through a Superconductive Quantum Interference Device, SQUID, the present most sensitive flux and current detector (resolution of the order of. φ0 = 2.068 × 10-15 Wb), showed no difference also in measurement repeated along the time, at the level of 10-16.

The device realizing the relations 1) and 2), the Josephson junction, can be obtained in different ways, according to what mentioned above. In particular, for the application to Voltage standard, after first point contact junctions, which were not reproducible, and generated voltages around 1 mV, thin film junctions have been used, exploiting the development of microelectronics processes.

The motion equation of the Josephson junction, describing the current flow is the electrical equivalent of the mechanical pendulum, and gives the total current in the junction as sum of three terms

$$I = I\_c \sin q\mathbf{p} + \mathbf{V}\mathbf{R} + \mathbf{C} \, dV / dt,\tag{4}$$

where *R* and *C* are equivalent resistance and capacitance of the junction

Defining the parameter *βc = (2πIcR2C) / φ<sup>0</sup>* as the damping factor of the junction in analogy with the pendulum, the current voltage, I V characteristic of a Josephson junction can be single valued for *βc <* 1, and the junction is called non-hysteretic or overdamped, or multiple valued, for *βc* > 1, and the junction is called hysteretic or underdamped.

Correspondingly, the response to a microwave signal produce two type of staircase, as will be reported in the next section.

#### **3. Voltage standards: From DC to AC**

A DC voltage *V* applied to a Josephson junction equals exactly the rate of single flux quanta *fΦ*0, transferred along the junction: *V* = *Φ*0*f*. If the flux transfer is phase-locked to an external oscillator with the highly stabilised frequency *f*, the transfer rate is kept constant over a certain range of current through the junction: this leads to constant voltage steps in the DC characteristic at *V*n = n*Φ*0*f.* Due to the nonlinearity of the DC charcteristic, this occurs not only at the fundamental frequency f but also at the higher harmonics (n = 2, 3, 4,….). For a

In the practical experiment, the Josephson frequency, which is in the THz range, is modulated by a microwave signal, producing voltage steps at fixed, constant interval *n (h/2e) × f*, with *n* integer 1, 2,.. These steps, called Shapiro steps, are used as reference voltage in the measurements, and their amplitude in current depends on the power of the microwave signal coupled to the junction, according with good approximation to the relatioship: (3) These steps, which are what is used as reference voltage in the measurement, are called Shapiro steps, and their amplitude depends on the power of the microwave signal coupled to the junction.

0 1 0 0

*<sup>V</sup> II J V t nf t <sup>f</sup>*

where: *Is* step current width, *Ic* critical current*, Is* step current width, *V0* (*V1*) dc (ac) voltage

Experiments where the voltage provided by two different Josephson devices radiated by different microwave signals was detected through a Superconductive Quantum Interference Device, SQUID, the present most sensitive flux and current detector (resolution of the order of. φ0 = 2.068 × 10-15 Wb), showed no difference also in measurement repeated along the

The device realizing the relations 1) and 2), the Josephson junction, can be obtained in different ways, according to what mentioned above. In particular, for the application to Voltage standard, after first point contact junctions, which were not reproducible, and generated voltages around 1 mV, thin film junctions have been used, exploiting the

The motion equation of the Josephson junction, describing the current flow is the electrical equivalent of the mechanical pendulum, and gives the total current in the junction as sum of

Defining the parameter *βc = (2πIcR2C) / φ<sup>0</sup>* as the damping factor of the junction in analogy with the pendulum, the current voltage, I V characteristic of a Josephson junction can be single valued for *βc <* 1, and the junction is called non-hysteretic or overdamped, or multiple

Correspondingly, the response to a microwave signal produce two type of staircase, as will

A DC voltage *V* applied to a Josephson junction equals exactly the rate of single flux quanta *fΦ*0, transferred along the junction: *V* = *Φ*0*f*. If the flux transfer is phase-locked to an external oscillator with the highly stabilised frequency *f*, the transfer rate is kept constant over a certain range of current through the junction: this leads to constant voltage steps in the DC characteristic at *V*n = n*Φ*0*f.* Due to the nonlinearity of the DC charcteristic, this occurs not only at the fundamental frequency f but also at the higher harmonics (n = 2, 3, 4,….). For a

<sup>2</sup> <sup>1</sup> sin <sup>2</sup> *<sup>n</sup>*

00 1

*I = Ic sinφ + VR+ C dV/dt,* (4)

 

(3)

<sup>1</sup>

where *R* and *C* are equivalent resistance and capacitance of the junction

valued, for *βc* > 1, and the junction is called hysteretic or underdamped.

*sc n n*

bias, *f1* ac bias frequency, and J*n* Bessel function of order *n*.

time, at the level of 10-16.

be reported in the next section.

**3. Voltage standards: From DC to AC** 

three terms

development of microelectronics processes.

drive frequency of 70 GHz, the fundamental voltage step amounts to about 135 µV. As the frequency can be stabilised to a very high degree, such a voltage step is a perfect voltage reference with a reproducibility of up to a few parts in 1010.

Josephson junctions used nowadays in DC voltage standard applications are based on hysteretic SIS junctions with zero crossing steps i.e. voltage steps whose current range spans positive and negative values, including the condition of zero DC bias. The choice of this technique dates back to the first attempts in series-connecting the thousands of junctions needed to reach output levels of 1 V and above, as required in metrological applications. The voltage steps are generated in the sub-gap part of the DC characteristi and all voltage steps between the maximum output voltages, e.g. -10 V and +10 V are biased by the same current. Exploitation of the zero crossing steps, fist suggested by Levinsen [4] eventually allowed to overcome many technological difficulties and made it possible to realize arrays with reproducible overlapping steps.

Fig. 3. Operating principle of programmable Josephson arrays (right) and IV characteristics (left) of the INRiM 10 V standard observed at the oscilloscope.

Fig. 4. Operating principle of programmable Josephson arrays. By controlling the bias currents I1 -I4 it is possible to change the output voltage Vout.

To increase the output reference voltage to practical values, large series arrays of strongly under-damped Josephson tunnel junctions of the SIS-type (Superconductor-Insulator-Superconductor) – in this case Nb-Al2O3-Nb – with hysteretic DC characteristic were developed. On the one hand, this allows to make use of higher harmonics steps up to n = 6 on a average per junction and generate 10 V output voltages with the relatively low number of 12000 to 14000 junctions. The availability of 10 V standards with quantum accuracy has led to dramatic improvements in DC voltage metrology, and it is now possible in primary DC voltage calibrations at 10 V to attain relative uncertainties as low as 10-11.

More recently, the interest in voltage standard research has moved to the investigation of techniques for extending the application of Josephson arrays to AC quantum standards and to standards for arbitrary time-varying signals. To this aim, junctions with non hysteretic behavior were suggested to allow changing the output voltage through control of the bias current. The substantial difference, from the application viewpoint, in using non hysteretic junctions, is that their IV curve (voltage *vs.* current relationship) under irradiation is a one to one staircase, thus the output voltage is univocally defined by the current feed through the bias circuit. This is not the case for hysteretic junctions used in DC standards, were steps are overlapping and all share approximately the same interval of currents.

In so-called programmable standards, the junctions bias currents are used to activate/deactivate array sections. Such arrays are typically subdivided in sub-circuits with series connected junctions generating voltages following a power of two sequence. Combining the sections it is then possible to source binary programmed voltages in a way that is very similar to the technique used in electronic digital to analog converters [5]. In order to replace best AC standards, the uttermost accuracy has to be reached and many efforts have been devoted in realizing arrays with performances suited to the tight requirements set by modern primary metrology. Many approaches to junction fabrication have been developed, and several different technologies have proven successful in providing voltages up to 10 V, with good metrological properties. Programmable Josephson arrays are so far the most successful attempt to extend metrological applications of Josephson standards beyond dc. Programmable arrays operating at 1 V have been effectively used for several applications: as traveling standards for international comparisons [6], for generating precisely varying voltages in a watt balance [7], as quantum impedance and power standards [8]. Moreover, only programmable standards can presently provide output voltages up and above 1 V, and even exceeding 10 V [9].

Nonetheless, this technique suffer from some limitations, the most severe is due to the time for step switching, where junctions are not operating in a quantized state. During these transients, the array voltage is not precisely known and the uncertainty of the generated signal increases with the fraction of period spent in the transients. Since the minimum transient time is constrained by technological limitations, programmable arrays can fulfill primary metrology uncertainties only for signals with frequencies up to few hundreds Hz [10].

To overcome limitations of programmable standards, arrays operating with a pulsed, square wave, rf signal have been developed. Using short pulses instead of a sinusoidal rf signal

To increase the output reference voltage to practical values, large series arrays of strongly under-damped Josephson tunnel junctions of the SIS-type (Superconductor-Insulator-Superconductor) – in this case Nb-Al2O3-Nb – with hysteretic DC characteristic were developed. On the one hand, this allows to make use of higher harmonics steps up to n = 6 on a average per junction and generate 10 V output voltages with the relatively low number of 12000 to 14000 junctions. The availability of 10 V standards with quantum accuracy has led to dramatic improvements in DC voltage metrology, and it is now possible in primary DC voltage calibrations at 10 V to attain relative uncertainties as low

More recently, the interest in voltage standard research has moved to the investigation of techniques for extending the application of Josephson arrays to AC quantum standards and to standards for arbitrary time-varying signals. To this aim, junctions with non hysteretic behavior were suggested to allow changing the output voltage through control of the bias current. The substantial difference, from the application viewpoint, in using non hysteretic junctions, is that their IV curve (voltage *vs.* current relationship) under irradiation is a one to one staircase, thus the output voltage is univocally defined by the current feed through the bias circuit. This is not the case for hysteretic junctions used in DC standards, were steps are overlapping and all share approximately the same interval

In so-called programmable standards, the junctions bias currents are used to activate/deactivate array sections. Such arrays are typically subdivided in sub-circuits with series connected junctions generating voltages following a power of two sequence. Combining the sections it is then possible to source binary programmed voltages in a way that is very similar to the technique used in electronic digital to analog converters [5]. In order to replace best AC standards, the uttermost accuracy has to be reached and many efforts have been devoted in realizing arrays with performances suited to the tight requirements set by modern primary metrology. Many approaches to junction fabrication have been developed, and several different technologies have proven successful in providing voltages up to 10 V, with good metrological properties. Programmable Josephson arrays are so far the most successful attempt to extend metrological applications of Josephson standards beyond dc. Programmable arrays operating at 1 V have been effectively used for several applications: as traveling standards for international comparisons [6], for generating precisely varying voltages in a watt balance [7], as quantum impedance and power standards [8]. Moreover, only programmable standards can presently

Nonetheless, this technique suffer from some limitations, the most severe is due to the time for step switching, where junctions are not operating in a quantized state. During these transients, the array voltage is not precisely known and the uncertainty of the generated signal increases with the fraction of period spent in the transients. Since the minimum transient time is constrained by technological limitations, programmable arrays can fulfill primary metrology uncertainties only for signals with frequencies up to few hundreds Hz

To overcome limitations of programmable standards, arrays operating with a pulsed, square wave, rf signal have been developed. Using short pulses instead of a sinusoidal rf signal

provide output voltages up and above 1 V, and even exceeding 10 V [9].

as 10-11.

of currents.

[10].

makes it possible to effectively modulate the signal period while keeping junctions phase locked over a wide range of frequencies [11],[12],[13]. Fundamental accuracy follows from the control of the flux quanta transferred through the junctions by the pulsed signal. The output voltage is then exactly calculable in terms of fundamental constants if the number of the flux quanta per unit time, i.e. the pulse repetition rate is known [14]. Pulsed standards allow to synthesize arbitrary waveforms with quantum accuracy based on the sigma-delta technique for digital to analog conversion developed for semiconductor electronics and very high spectral purity [15]. Both operation and fabrication of pulsed standards set very challenging problems. Due to the complexity of pulse waveform, the apparatuses for generation of precisely frequency controlled pulses are sophisticated and expensive, and the design of rf transmission lines is extremely difficult because of the harmonic richness of the signal. In addition, it is extremely difficult to generate a bipolar output with a frequency modulated Josephson array, and very complex AC biasing techniques, involving a sine wave and a pulsed signal, both synchronized, must be used for real AC operation [16]. Generation of the pulse train for proper junction operation requires top-end instrumentation and unavoidably limits the signal fundamental frequency to values much lower than those obtainable with continuous wave sources. Power distribution to array junctions is also of concern, since the usual microwave techniques developed for nearly monochromatic signals are not directly applicable to broadband pulses. Adoption of lumped circuit methodologies seems at present the most viable solution, though highly demanding on the fabrication side. In order to have negligible effects on circuit behavior, the propagation time of rf signal along the array must be smaller than the signal period, i.e. the array dimensions must be smaller than the signal wavelength λ. To guarantee reliable operation, λ/8 is typically considered the maximum value acceptable for array dimensions. Despite all these difficulties, arrays with as many as 10000 junctions have been successfully fabricated, and synthesized voltage signals up to 275 mV rms have been generated with quantum accuracy and extreme spectral purity [17].


Table 1. Comparison of different type of voltage standard.

Fig. 5. Simplified schematic of a Digital to Analog converter realized with RSFQ technology (from [20]).

Fig. 5. Simplified schematic of a Digital to Analog converter realized with RSFQ technology

(from [20]).

A new technique has been recently proposed to overcome some of the difficulties encountered with programmable and pulsed standards. The method, named Pulse Power Modulation (PPM), is based on a controlled activation/deactivation of the driving rf signal, to simultaneously set every junction in the array into either one of two states: zero voltage in absence of rf signal and a quantum defined step voltage upon rf application. Waveform synthesis can then be realized by Pulse Width Modulation of the array voltage [18]. The requirements in junction technology for PPM are different form those set by programmable and pulsed standards, since for proper operation an IV curve where the interval of currents of the relevant steps partially overlaps the critical current is needed. Such a behavior can be obtained if a precise control over junction hysteresis is feasible in the fabrication stage, to provide an intermediate degree of hysteresis, between those of the fully hysteretic SIS for DC and the non hysteretic SNS.

Although limited in space, this overview wouldn't be complete without mentioning RSFQ as an alternative technique to synthesize arbitrary and AC signals with quantum voltage accuracy. RSFQ has been, and still remains, a very active field of research for quantum digital electronics applications [19], yet developments for voltage standards have always been left out of the mainstream of research interests in Metrology. One of the main reasons for that is most likely to be found in the completely different approach, know-how and experiences involved by RSFQ, with respect to the common background of voltage metrologist. In RSFQ standards, accurate voltage signals are generated by controlling flux quantized by a Josephson junction. Flux quanta generation is "triggered" by a pulse sequence, thus can be precisely timed, in analogy with the "phase lock" process exploited in array standards [20]. RSFQ approach is potentially advantageous in that the complex and expensive microwave apparatus needed for ordinary standards is avoided, the drive signal being generated by the superconductive circuit itself. The only requirement for RSFQ circuits is an external accurate frequency reference, typically operating in the MHz range. The simplification in instrumentation must be traded off with a much higher complexity of superconductive circuit, namely an increased number of junctions, additional elements like inductors, and fabrication of junctions with different parameters in the same device. The basic element in RSFQ is the Josephson transmission line, a string of Josephson junctions connected by inductors where the pulse can propagate, like in transmission lines, and even amplified [21]. A basic Digital to Analog converter suitable for voltage standard will include at least some voltage multiplier stages, to increase the output signal to practical values. Coupling between Josephson transmission lines and voltage multipliers is obtained by capacitive coupling [22], but magnetic coupling through transformer-like circuits has proven to be more effective [23]. Nowadays DAC for Metrology are fairly more complex devices, with many digital blocks performing various specialized functions, and correspondingly high power consumption. Successful operation, of a 10-bit RSFQ DAC capable of generating up to 20 mV has recently been reported [24].
