Standards Based on Fundamental Quantum Effects

1. Standards Based on Fundamental Quantum Effects

2. Realization versus Representation Two distinct classes of dissemination of the units: Realization of a unit is a physical experiment or artifact based on well-established principles (which may be impossible to make!) Representation of a unit is an experiment of artifact which produces the quantity which can be routinely compared to other standards. Examples: Realization of the volt flows through the equivalence of electrical and mechanical units. One realization is from force balance which uses the voltage across the capacitor to balance the gravitational force: a complicated and slow experiment Representation of the 1.0183 Volt based on Weston cell

3. SI definition of Volt The volt is defined as the potential difference across a conductor when a current of one ampere dissipates one watt of power. Hence, it is the base SI representation: 1 V= 1m2 � 1kg �1 s-3 �1 A-1, which can be equally represented as one joule of energy per coulomb of charge, J/C This is not something one can easily compare the measured voltage with This is the reason to use standard voltage cells, however imperfect they are Is there a way to implement a voltage standard based upon fundamental quantum phenomena? Since frequency can be so accurately measured, would not it be nice if we can generate voltage based on some frequency?

4. Josephson Effects The Josephson effect is a quantum-mechanical phenomenon occurring when two superconductors are weakly joined to form a Josephson tunnel junction. If two superconductors are separated by a thin insulating barrier a few nm thick), it is possible for electron pairs to pass through the barrier without resistance (and dissipation!) Current in the superconductor is represented by a global wavefunction The expression for the supercurrent through the junction (phase difference of the supercurrent wave function between two superconductors: it is changing from 0 to 90 degrees, then voltage starts to appear )��� The Josephson effect in particular results from two superconductors acting to preserve their long-range order across an insulating barrier. With a thin enough barrier, the phase of the electron wavefunction in one superconductor maintains a fixed relationship with the phase of the wavefunction in another superconductor. This linking up of phase is called phase coherence. It occurs throughout a single superconductor, and it occurs between the superconductors in a Josephson junction. Phase coherence - or long-range order - is the essence of the Josephson effect. " The Josephson effect in particular results from two superconductors acting to preserve their long-range order across an insulating barrier. With a thin enough barrier, the phase of the electron wavefunction in one superconductor maintains a fixed relationship with the phase of the wavefunction in another superconductor. This linking up of phase is called phase coherence. It occurs throughout a single superconductor, and it occurs between the superconductors in a Josephson junction. Phase coherence - or long-range order - is the essence of the Josephson effect. "

5. AC Josephson effect If I > IC, then a voltage V appears across the junction, and the suppercurrent oscillates with a frequency derived from equation df/dt=2(e/h) �V Phase changes linearly in time: f =[2(e/h) �V]�t +const So that suppercurrent I = IC�sin f (t) oscillates with a frequency wJ= 2eV/h Thus the junction irradiates electromagnetic waves of that frequency. This is the AC Josephson effect. when current through the junction exceeds certain limit (critical current) then a for a current to flow a quantum of energy has to be irradiated from the junction. The Josephson effects = macroscopic quantum effects (currents are defined by the phase of the wave function). The Josephson effects = macroscopic quantum effects (currents are defined by the phase of the wave function).

6. Inverse AC Josephson effect when an RF electromagnetic field of frequency f �is applied across a junction formed by two superconductors, the IV characteristics of this junction shows voltage steps with zero slope (Shapiro steps) The quantized voltages are given by the equation V(n) = n f /(2e/h) [Volts] where f is the applied frequency in GHz range (10 to 100 GHz); n is an integer and KJ=(2e/h)= 483 597.9 GHz/V. where KJ is Josephson constant.

7. More about inverse Josephson effect If the phase takes the form f(t) = f0 + n?t + asin(?t), the voltage and current will be The DC components will then be

8. Voltage standard based on Josephson effect It has been demonstrated that KJ is the same in every junctions between superconductors in ample ranges of the operating parameters (materials, temperature, fields, etc.) to the level of 3 parts in 1019. Since this relationship of voltage to frequency involves only fundamental constants and since frequency can be measured with extreme accuracy, the Josephson effect is a natural phenomenon suitable for establishing reproducible and invariant voltage standard. The Josephson junction has become the standard voltage measurement in 1990 (KJ-90). For a typical microwave frequency of 94 GHz derived from Gunn oscillator, the step voltage is seen to be 194.4 mV. Exactly one volt can be defined as the EMF of a Josephson junction when it is irradiated with electromagnetic radiation of 483 597.9 GHz. But! Josephson junction will not work if it is irradiated with 620 nm light ( f= 483 597.9 GHz). Why? ~1THz is the superconducting gap characteristic frequency Stability of frequency is crucial, so typically Gunn oscillator is phase-locked to an ultra-stable frequency source (e.g., cesium beam clock).

9. How to make a 1V standard The Josephson junction arrays (JJA) consist of connected in series several thousand Josephson junctions. They are fabricated to form a microwave transmission line. Which quantum step step, n, the JJs are at? To solve this problem: Note that step is the same, because all JJs are connected in series obtain a stable array output voltage for known frequency f1 vary the dc bias current until the array jumps to the next step level adjust the microwave frequency down to f2 until the same voltage is obtained Then� nf1 = (n+1) f2, and therefore � n = INT( f2/(f1 - f2)) For JJA with N� JJs in series� exposed to the same microwave frequency and operating at step n �

10. Tips on Josephson Voltage Standards The DC current should be chosen carefully to exclude jumps between steps. This is the primary source of uncertainty of JJA voltage standard. JJAs are utilized in voltage standards and are used now to maintain voltage standards both in national and industrial laboratories. JJAs can generate potentials up to 10 volts and are accurate to parts in 109� (few ppb) or better. For example: f=71 GHz, V(1 junction) = 146.8 mV V(6930 junctions) = 1.018 V � remember the Weston cell voltage? NIST has produced a chip with 19000 in-series junctions to measure voltages on the order of 10 volts with the accuracy of 0.1 ppb. (Calculate the frequency)

11. Example of Experimental Realization of Josephson Voltage Standard The microwave frequency is ~ 16 GHz Voltage per junction is V = f / KJ-90 = 33.08 mV. The smallest cell in the device consists of 128 junctions, which makes the least significant-bit (LSB) value to be about 4 mV. Better output resolution is obtained by adjusting the frequency f to cover the voltage range between LSB's. This allows the JVS to provide gap-free coverage of the voltage range from 50 mV to 1.1 V (and from -50 mV to -1.1 V), where the voltage resolution is limited only by the resolution of the frequency source.

12. Experimental details of 1V JVS The precision output voltage from the Josephson array must be brought from liquid helium temperature to room temperature with a thermal offset voltage less than a few parts in 108, and a drift rate in that offset of a few parts in 109 per hour or less.

13. Developments in Voltage standards 1V JVS consisting out of 3020 junctions Operates at

14. Resistance, Ohm An ohm is a resistance that produces a potential difference of one volt when a current of one ampere is flowing through it. 1 O = 1 V/A = 1 m��kg�s�3�A�2 "Legal Ohm " (1884) Column of 1 cm2 ultra-pure mercury 106 cm in length, measured at 0oC. Poor. High dR/dT, affected by impurities, terminals, non-uniformity in the diameter of the glass tubing. Adjusted in 1893 to become 106.3 cm of Hg, and Hg must weight 14551g. "Absolute Ohm" (1920) : a derived quantity based on the fundamental units of length, mass, and time. Was very hard to determine, sample from sample variation was large.

15. Resistance Standard Based on Quantum Hall Effect What is Hall effect? What is quantum Hall effect? Why it makes an excellent resistance standard?

16. Hall Effect In 1880 E. H. Hall discovered that when a magnetic field H is applied perpendicular to the direction of a current I flowing through a semiconductor or metal, a voltage VH is developed in the direction perpendicular to the current flow. This happens due to the deflection of the charge carriers towards the edge of the sample by the Lorentz force in H field. The magnitude of this discovery is even more impressive considering how little was known about electricity in Hall's time. The electron, for instance, was not identified until more than 10 years later

17. Elementary theory of Hall effect Equilibrium is achieved when the Lorenz force is balanced by the electrostatic force from the build up of charge across the sample: eEy = evxBz. where e is electron charge, vx is the carrier velocity in X direction. The Hall coefficient is defined as RH = Ey / (Bzjx ). where current density jx is �jx = vxNe , then for a single type charge carriers RH =1/Ne. �RH can thus be used to find N - the carrier concentration in the material Also Hall effect can be used for magnetic field measurements: � UH=RHBzIx/t, where t is the thickness, and Ix is current. Hall resistance is simply the product RH�B

18. Quantization of Hall Resistance In a two-dimensional high mobility semiconductor the Hall effect is also observed, but at low temperatures a series of steps appear in the Hall resistance as a function of magnetic field instead of the linear monotonic increase. Moreover, these steps occur at incredibly precise values of resistance which are the same no matter what sample is investigated. The resistance is quantized in units of h/(ie2). The value of Quantum Hall resistance only depends on the fundamental constants of physics: e, the electric charge and h, Plank's constant. It is accurate to 1 part in 100,000,000 (10 ppb). The QHE can be used as primary a resistance standard, 1 klitzing = 25,813 ohm.

19. Nobel prize 1985 It was not expected, however, that the quantization rule would apply with a high accuracy. It therefore came as a great surprise when in the spring of 1980 von Klitzing showed experimentally that the Hall conductivity exhibits step-like plateaux which follow this rule with exceptionally high accuracy, deviating from an integral number by less than 0.000 000 1.

20. Explanation of Quantum Hall effect In the absence of magnetic field the density of states in 2D is constant as a function of energy, but in field the available states shrink into Landau levels separated by the cyclotron energy, with regions of energy between the LLs where there are no allowed states. As the magnetic field is swept the LLs move relative to the Fermi level.

21. Typical Quantum Hall effect Data Why is it a plateau (not a linear increase rxy~B) in Hall resistance?

22. Explanation of Quantum Hall effect Real Landau levels are NOT d functions, they are smeared out (very important!) In a gap between LLs electrons can not move to other energy states. Thus the transport is dissipationless and the resistance (and conductance too! ) falls to zero. The classical Hall resistance� Rxy = BZRH = BZ/Ne. The number of current carrying states (per unit area) in each LL is Ni = eBZ/h, so when there are i LLs at energies below the Fermi energy completely filled with ieBZ/h electrons, the Hall resistance is Rxy = BZ/( e x (ieBZ/h)) = h/ie2. At integer filling factor this is exactly the same number as in the classical case, but it is not just a point on the curve, but a wide plateau. The difference between HE and QHE is that the Hall resistance can not change from the quantized value when the Fermi energy is in a gap, between Landau levels, resulting in a plateau. Only when the Fermi energy is within the Landau level, can the Hall voltage change and a finite value of longitudinal resistance appear.

23. An example of the setup for QH measurements:comparison with a standard resistance By international agreement, the resistance of the first plateau is taken to be equivalent to 25812.807 W. Note that for i=1this is the same number as required for observation of single electron transport (localization criterion)

24. Metrology Triangle Experiment

25. G. Geneves, F. Piquemal, "Vers une loi d'ohm quantique: le trianglem�trologique," Congr�s International de Metrologie-N�mes,pp. 352-357, Oct. 1995. Left: single electron pump generates I = fe � 1.6 pA. The current is multiplied by a factor of 1000 using precise slave current source (based on CCC) and fed into a quantum-Hall resistance standard. The resulting voltage V = 103 hf/e � 40 mV. Right: 10 MHz reference is multiplied up to 20 GHz so that it drives a single junction of SNS programmable voltage standard.

26. Metrological Triangle Implementation Comparison of the quantized acoustoelectric current with the quantum Hall resistance and Josephson voltage Left part generates current IQ=ef1 Right part : NIQ (N is windings ratio) flows through Hall resistor RH=RK/i (i=1,2,..) Hall voltage VH= RK(IQN)/i = RK(ef1N)/i Cryogenic Current Comparator (CCC) is used to compare VH and Josephson voltage by n junctions: VJ=nf2� (h/2e) RK is measured as RK=(h/e2)�(f2/f1) �(n/2iN) Should enable to resolve the metrological triangle with an accuracy of 1 part in 106

27. Results of current measurements

28. Cryogenic Current Comparator

29. Capacitance Standard based on SET pump SET devices can provide a practical fundamental standard for capacitance Using 1 pA from SET pump in 1 s one can charge 1 pF capacitor to 1 V (which can be measured with metrological accuracy) (Mark Keller, NIST, 1999). C=Q/DV =Ne/DV Typical junction capacitance of SET device is Cj ~ 1e-16 F, so EC ~ k x 10K. Therefore if operated at 50 mK, EC>>kT

30. Single-electron capacitance standard measurement setup

31. How it works?

32. Calibrating External capacitor Configuration B�� (N1 open, N2 closed) is used to� compare C with another capacitor at room temperature using conventional AC bridge: C/Cref=V2/V1. Recent improvements : C = 10 pF, and comparison with Lampard capacitor can be implemented

33. Goals of SI SI has two independent goals: International agreement on a system of units for physical measurements; formation of a coherent system of units. Coherent means that any unit should be related to any other unit by only a multiplicative and divisive combination, with a numerical prefactor of unity. 1 V=1 kg m2/(A s2) SI units versus practical units. A SI unit is a unit which maintains coherency. A practical unit is one that can be maintained by a convenient experiment that provides a useful standard for everyday use.

34. New Fundamental System? The seven base units of the SI are: Meter Kilogram Second Ampere Kelvin Mole Candela All other units are derived which means they can be expressed in terms of seven base units For electrical units only Meter, Kilogram, Second, and Ampere are important. Meter and Second are nowadays defined as what is called "atomic", "fundamental", "quantum" or "natural". The hope is that fundamental units should be the same all the time and in all places. Such fundamental units are clearly preferable to artifacts which may not stay constant. � � Example � the kilogram � not a fundamental unit. Since realization of a ampere requires the kilogram, meter and second � thus realizations of electrical units are based on macroscopic experiments, while highly reproducible JV and QHR experiments are available. The most commonly used practical units are the volt and the ohm, since it is much easier to store and compare voltages and resistances then currents. Example � the kilogram � not a fundamental unit. Since realization of a ampere requires the kilogram, meter and second � thus realizations of electrical units are based on macroscopic experiments, while highly reproducible JV and QHR experiments are available. The most commonly used practical units are the volt and the ohm, since it is much easier to store and compare voltages and resistances then currents.