SCIENTISTS AND technologists depend on being able to measure accurately. Now, more than ever, advanced technology demands finer and more consistent measurements. The accuracy depends on a pyramid of instrument calibrations and standards, each storey of the pyramid representing a more accurate measurement. At the apex are the calibrations made at a national measurement laboratory. In Britain, this is the National Physical Laboratory (NPL) at Teddington near London.
Each national laboratory must ensure that its standards are consistent with those of other national laboratories. Mechanical measurements are not a problem. Each national laboratory bases its standards on three fundamental units – the metre, the kilogram and the second. These are defined in the International System of Units, or SI units. The metre is derived from the defined speed of light, the prototype kilogram is kept at the International Bureau of Weights and Measures at Sevres in France and the second is derived from the frequency of radiation emitted by the caesium ‘atomic clock’. We can copy these standards very accurately – more accurately than engineers and physicists need.
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Defining accurate electrical standards, however, is a problem. The basis of all electrical measurements is the SI definition of the ampere. This unit is defined as the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in a vacuum, would produce between these conductors a force equal to 2 X 10-7 newtons per metre of length. Until recently, we could transform this definition into a usable electrical standard only with the aid of a current balance. In this instrument, the impossible infinite straight conductors are, in practice, ordinary copper wire wound into two coils of carefully measured dimensions, one of which hangs from the end of a balance arm inside the other. We can then deduce the value of the current in the coils directly from the definition of an ampere by weighing the magnetic force between them.
Current balances are accurate to about six parts in a million, but this is no longer good enough because we would now like to make some scientific, and even commercial, electrical measurements, with an accuracy of better than one part in a million. Moreover, as a result of the limited accuracy of current balances, some countries adopted slightly different values for their electrical units which in the past has led to much needless confusion.
Fortunately, metrologists have now developed some new standards that are much more reliable than those obtained from the current balance. They depend on two remarkable discoveries in physics made during the past 30 years. These are phenomena that show up at the quantum level at extremely low temperatures – the Josephson effect and quantum Hall effect (see Boxes 1 and 2). Brian Josephson won the Nobel Prize for Physics in 1973 for the first discovery and Klaus von Klitzing was awarded his Nobel prize more recently, in 1985, for the second effect.
Both effects depend on the subtle behaviour of the fundamental unit of electricity, the electron. According to quantum mechanics, the electron can adopt only certain velocities or energies which depend on two fundamental physical constants, the elementary electrical charge e and the Planck constant h. This means that physical phenomena, such as the Josephson and quantum Hall effects, which are directly related to the energy of the electron in electric and magnetic fields, also lead to discontinuous values of voltage and resistance based on h and e. The Josephson and quantum Hall effects, therefore, provide a way of measuring electrical properties in terms of these two constants.
Calibrating electrical standards in this way means that not only can national laboratories make electrical measurements a 100 times more accurately than before but also they can at last agree on universal electrical standards.
Exploiting the Josephson effect involves relating the frequency of microwave radiation (10 to 100 gigahertz) that impinges on a narrow gap between two superconductors, called a Josephson junction, to increments in the voltage developed across the gap when a superconducting current ‘leaks’ through it. The voltage increments are equal to multiples of the quantity h/2e times the frequency of the radiation absorbed by the device. So the accuracy of measuring the voltage depends on how accurately you can measure this frequency. Fortunately, the accuracy of frequency measurements, which depends on comparing them with a particular frequency in the spectrum of caesium (the caesium atomic clock), is better than one part in a million million. But because the value of h/2e is only about 0.000002 volts per gigahertz, it is quite difficult to measure the resulting voltage of a few millivolts precisely enough to use it for calibrating more practical electrical standards.
Within the past few years, this voltage measurement has become much easier to make, mainly because it is now possible to mass-produce Josephson junctions, each only a few tens of micrometres across, as thin films. This means that many thousands of such junctions can be connected in a long line. If all the junctions are sufficiently alike, then they will all absorb equal amounts of energy from microwaves travelling down the line. The total voltage produced depends on the number of junctions included in the line. This approach allows us to obtain measurable voltages of between 1 and 10 volts.
We can then compare these voltages with those of the usual laboratory standards – Weston cells, Zener diodes, and so on – with a resolution of better than one part in a 100 million. This precision is equivalent to comparing the diameter of a human hair with the cruising height of a modern jet. There are very few laboratory voltage standards that are stable enough to need calibration to this accuracy, however.
The measurements made using the quantum Hall effect are similar. In this case, we pass a current through a semiconducting, rather than a superconducting, device, which is also cooled to a very low temperature. A voltage generated across the device assumes a constant ratio to the current for certain ranges of the applied magnetic field. According to Ohm’s law, the voltage divided by the current has the dimensions of resistance. You can, therefore, compare it with the ratio of voltage to current produced by a standard resistor. In fact, the values of resistance generated by the quantum Hall effect are whole number fractions of about 25812.8 ohms, which present theories calculate to be exactly equal to h/e2.
Metrologists carry out the actual comparison by adjusting the ratios of the currents passing through the resistor and quantum Hall effect device until the voltages across them are the same. This is done by arranging for the two currents to flow in opposite directions through two coils wound together. Each coil will then produce a magnetic field. Ideally if the currents are identical, the fields should cancel each other. But, in practice, the fields depend on the exact position and shape of the coils. One way of getting around the problem is to put the whole apparatus in a superconducting tube. A superconducting material does not allow a magnetic field to penetrate it (the Meissner effect). Instead the tube ‘guides’ the magnetic fields so that they cancel on the outside of the tube. Any tiny differences in magnetic flux created by current imbalance can be detected by another device that also works on quantum principles at low temperatures, the SQUID, or superconducting quantum interference device. This device is extremely sensitive to minute magnetic fields.
By using measuring equipment of this kind, which is totally dependent on the effects of superconductivity observed at liquid helium temperatures, we can make a resistance equal to the quantum Hall voltage-to-current ratio with an accuracy of a few parts in a thousand million.
Now, how can we measure the values for the volt and the ohm obtained from the Josephson and quantum Hall effects in terms of the basic SI units of mass, length and time? We have been able to establish the ohm quite accurately for the past 20 years by first calculating the electrical capacitance of an arrangement of conducting surfaces from their dimensions measured in metres. An extremely elegant theorem in electrostatics discovered by Australian physicists Mel Thompson and Douglas Lampard in 1957 allows us to calculate a capacitance from just one measurement of length. Using a laser, we can measure this with optical interference techniques to a part in a thousand million(See Figure 3).FIG-mg17154403.GIF
We do lose a little accuracy in the electrical measurements needed to scale up the extremely small calculated capacitance to a more measurable value. We lose a little more accuracy when relating the impedance of this capacitance, at a known frequency of the alternating current, to that of a resistance. Nevertheless, we can use this resistance to calibrate the quantum Hall effect with an accuracy of a few parts in a hundred million. We can then measure any other resistance in true SI ohms to the same accuracy, in terms of the calibrated quantum Hall effect.
The second breakthrough came when researchers at the National Physical Laboratory developed a method of deriving the watt from mass, length and time. The method uses what has come to be known as a moving-coil apparatus. It is a considerable improvement on the current balance, whose accuracy is limited by the difficulty of measuring the dimensions of the coils and of weighing the rather small magnetic force between them.
In a moving-coil apparatus, a strong magnet replaces the fixed coil of a current balance. When a current flows through a coil hanging from a balance arm so that the coil is in the field of the magnet, it creates a much larger force of about a kilogram. Instead of measuring the dimensions of the coil as is necessary when using a current balance, the coil is moved at a velocity (known from the rate of passage of optical fringes in an interferometer) through the position it occupied for weighing. We can then measure the voltage induced in the suspended coil by this movement. The current times this induced voltage then equals the weighed force, which is being caused by the current, times the velocity, the first product being electrical power and the second, being mechanical power. The current, voltage, velocity and force can all be measured to a few parts in a 100 million, which is at least 100 times better than the accuracy achievable with a conventional current balance.
So now we can measure the product of a voltage and a current in terms of the kilogram, metre and second by using a moving-coil apparatus and we can also measure a resistance, which is a voltage divided by a current, in terms of a length and a frequency by using a calculable capacitor. By combining these two results we obtain SI voltage and resistance units – the volt and the ohm – separately in terms of the metre, kilogram and second with an accuracy of better than a part in 10 million. From the volt and the ohm all other electrical SI units can then be derived.
One outcome of all this is that because we have measured h/2e in SI units via the Josephson effect, and h/e2 similarly via the quantum Hall effect, we can also calculate the individual values of the important fundamental constants e and h with unsurpassed accuracy. The values obtained in this way at the NPL are: e = 1.60217635 coulombs and h = 6.62606821 joule-seconds. These measurements are accurate to about 8 and 14 parts in 100 million respectively.
Another interesting possibility is that if we could improve the accuracy of the moving-coil apparatus by, say 10 times, we could use it to redefine mass in terms of fundamental constants. That is, by redefining the SI unit of resistance in terms of the quantum Hall effect and the SI unit of voltage in terms of the Josephson effect, we could derive a more accurate unit of mass. A long-standing goal of metrologists is to define all units in terms of fundamental physical phenomena and the kilogram is the only SI unit left that depends on an actual artifact – the prototype kilogram at Sevres.
A more concrete result of all this activity is that the national measurement laboratories have all agreed on the same conventional values for the conversion constants with which they obtain resistance values from the quantum Hall effect and voltage values from the Josephson effect. All electrical calibrations will henceforth be on the same basis. These agreed conventional values took effect from 1 January 1990, when the voltage ascribed to every laboratory standard in Britain was overnight decreased by about eight parts in a million, and the resistance value of every laboratory resistance standard was decreased by about one and a half parts in a million.
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1: The Josephson effect – a strange oscillator
IN THE EVERYDAY world of large objects and ordinary temperatures, quantum mechanical effects are blurred out into familiar ‘classical’ physics, where physical laws seem to conform to common sense. But small (atom-sized or less) objects, or those at low temperatures where gases become liquids obey the unfamiliar laws of quantum physics.
In superconducting materials, moving pairs of electrons constitute a current that does not meet any resistance.
If we combine two superconductors into a junction so that they are separated by a tiny insulating gap a few atoms across, and then apply a voltage, we find that something curious happens. The superconducting pairs can leak, or tunnel, across the insulator. This is allowed by quantum mechanics which describes the position of small objects such as electrons in terms of ‘probability’ waves. This means that electrons have a small but finite probability of being where you would not normally expect them to be. In this case, on the other side of an electrical insulator. The current passing through the junction, however, is not smooth as you might expect but alternates at a microwave frequency, f. This is called the AC Josephson effect and is related to the voltage, V, by the formula: V = (h/2e)×f. The combination, 2e/h, of the fundamental physical constants h, Planck’s constant and e, the electronic charge, is called the Josephson constant.
Conversely, if we then apply an external electromagnetic field that also alternates at a microwave frequency to the junction, the voltage across it increases in discrete steps as the current through it is increased continuously (see Figure 1). These voltage increments are incredibly precise, and are again related to the applied frequency by the above formula. By comparing the voltages given by two junctions connected back to back, we can show that the effect is accurate to at least 1 part in 1016.FIG-mg17154401.GIF
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2: The quantum Hall effect – a peculiar resistor
METALLIC conductors at room temperature have electrical resistance and obey Ohm’s law – the current is proportional to the applied voltage and inversely proportional to the resistance. A strong magnetic field applied in a direction perpendicular to the direction of the current produces a small voltage across the conductor at right angles to both the current and field. This transverse, or Hall voltage, is proportional to the current multiplied by the strength of the magnetic field. The Hall effect is caused by the sideways ‘drift’ imposed on the motion of the electrons by their passage through the magnetic field.
If, however, we constrain the electrons to move in a two-dimensional flatland by confining them in a conductor that is only a few nanometres thick and in which the temperature is very low, the Hall effect acquires some peculiar characteristics caused by the laws of quantum mechanics which now govern the motion of the electrons.
In fact, certain types of semiconducting devices can confine electrons in two dimensions. This is done by arranging layers of various semiconductors in such a way that external electric fields confine the electrons in a potential valley. For certain values of a strong magnetic field applied perpendicularly to the layer, electrons flow without any loss of energy along the valley in the layer in the direction of an applied voltage. This means that the voltage drop in this direction is small and tends to zero as the temperature is lowered towards absolute zero. At the same time, a transverse voltage appears as in the ordinary Hall effect, but now there are exceedingly precise constant ratios of transverse voltage Vt to the current I for some ranges of magnetic field. These ratios are: Vt/i=(h/e2)/i, where i is a whole number (see Figure 2). Because Vt/I has the dimensions of resistance, the values of (h/e)/i are in ohms.FIG-mg17154402.GIF
You can measure Vt/I very accurately, which means that you can use the quantum Hall effect to calibrate a conventional resistance standard. The results are highly reproducible.
Bryan Kibble and Tony Hartland are physicists in the division of electrical science of the National Physical Laboratory at Teddington near London.
Further reading ‘Quantum standards for electrical units’, A. Hartland, Contemporary Physics, vol 29, 1988, p 477.