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Single-electron tunnelling effect

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Quantum metrology and fundamental constants: international school at Les Houches Physics Centre, 1st to 12 October 2007.

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Presentation of the study

The objective of this LNE study is to develop a quantum standard of current using devices based on the single-electron tunnelling (SET) effect. Once this current standard has been characterized and the required uncertainty level attained, it will be combined with the electromotive force standard (Josephson effect, JE) and resistance standard (quantum Hall effect, QHE) in the context of the metrological triangle experiment. The goal of this research is to test, with a relative uncertainty level of 1.10-8,the consistency of the constants involved in the three quantum phenomena (SET, JE and QHE), in particular the constants RK and KJ used for conservation of the ohm and the volt.

Metrological triangle

For the last 15 years, the units of electromotive force and resistance of the International System of Units (SI) have been maintained in France's national metrology laboratories by applying standards based on the quantum Hall effect (QHE) and the Josephson effect (JE). The measurement of voltage, UJ, supplied by Josephson junction array, and the measurement of Hall resistance of the ith plateau, RH(i), involve the universal constants KJ and RK respectively. In order to facilitate inter-laboratory comparisons, the comité international des poids et mesures (CIPM) fixed the values of these two constants KJ-90 and RK-90 in 1990. Over the last ten years, however, numerous international comparisons involving these two standards quantum have shown a consistency level of a few parts in 10-9 for the Josephson effect and a few parts in 10-9 for the quantum Hall effect. This is at least two orders of magnitude better than the estimated uncertainties, according to SI units, for the values KJ-90 (4.10-7) and RK-90 (1.10-7). It is clear that this situation cannot continue.
Moreover, for physicists the constants KJ and RK correspond to the following ratios of fundamental constants:

KJ 2e / h (1)      RK h / e² (2)

These two expressions are pleasing because they represent (1) the inverse of the flux quantum and (2) the resistance quantum, and could be used to replace the current SI units by fundamental constants (h, e, α ..). However, they cannot be used in a metrological context without giving an estimation.
LNE therefore decided to complete the metrological triangle experiment (fig. 1), which consists in applying Ohm's law with quantities generated by the quantum Hall effect and the Josephson effect and by a current standard based on the principle of the single-electron tunnelling effect:

UJ = RH(i).N.I (3)

I represents the current at the output of an electron pump (single-electron device), which is a multiple of the charge quantum and may be written as:

I = n2.e.f2 (4)

n2 is an integer and f2 denotes electron pumping frequency. As with RK and KJ, a new estimator Qx may be defined as:

Qx e

In order to compare the voltage at the Hall resistance terminals with UJ, the current intensity I must be amplified (4). For this, we use a cryogenic current comparator (CCC) with a winding ratio N.
The main aim of this experiment is to verify, with an uncertainty of 10-8, the consistency of the constants and their values in the three quantum phenomena.

Metrological triangle


Figure 1 : The metrological triangle with the three material standards linked to the three quantum effects:
Josephson, quantum Hall and single-electron tunnelling.

Note: the image of the electron pump, showing the microlithographed pattern, was obtained with a scanning electron microscope.
The Josephson junction array and Hall bar photographs are overall views of the samples, hence the difference in scale.

Principle of single-electron devices

Figure2 : metal island

Figure 2 : Diagram of a metal island formed
by two tunnel junctions of capacitance C.
Its charge state is controlled by a gate electrode.
This device is called a SET transistor.

LNE is working on developing a current standard with the aim of closing the metrological triangle. Since current is defined as the quantity of charge that flows per unit of time, the simplest idea for realizing a current standard is to create a system capable of controlling the transport of the electrons one by one with a known frequency. This is now possible following the development of single-electron tunnelling (SET) devices such as the SET transistor or the electron pump. These SET devices are based on the Coulomb blockade principle. This phenomenon occurs when a section of conductor called a "metal island" (see figure 2) is electrically isolated from the rest of the circuit by two tunnel junctions of capacitance C.

In these conditions, (, total island capacitance) is the electrostatic energy needed for one electron of the "source" electrode to tunnel across the metal island and reach the "drain" electrode. If this energy barrier is appreciably higher than the thermal fluctuation energy KBT, the island remains sensitive to the addition of just one extra electron to the millions it already contains, and precisely one extra electron can be added in certain polarization conditions of the SET device. If there is not enough electron energy, the transfer will be blocked (blockade state, see fig. 3). This phenomenon is called the Coulomb blockade. The charging energy of the island may be modified, however, by means of a third "gate" electrode coupled to the island through a capacitor, and the effect of the energy barrier can be cancelled ("open" state, figure 3).

Diagram of energy band before and after modifying the gate voltage of a SET transistor

Figure 3 : Diagram of energy band before and after modifying the gate voltage of a SET transistor
(blockade state / open state).

Electron pump

Figure 4: Scanning electron microscope image of
R-type three-junction pump built by PTB.
The tunnel junction capacitance values are in the range
of 100 atto-farad (10-18 F), for 0,01µm² surface areas
made by e-beam nanolithography techniques.

Diagram of pump and control circuit

Diagram of pump and control circuit

The electron pump is a Coulomb blockade device designed to control current intensity electron by electron. LNE uses R-type three-junction pumps designed by Physikalisch-Technische Bundesanstalt (PTB, Germany) in which the three junctions connected in series form two metal islands linked capacitively to two gate electrodes (figure 4). By adding a harmonic signal to the DC values of the gate voltages (, , with close to 90°), and adjusting the gate voltages and , the system can be regulated so that it relaxes to the lowest possible energy states by inducing the transfer of a single charge carrier through the tunnel junctions. The pump is thus able to generate a current whose intensity is proportional to the frequency f and the unit of charge e : I = e.f.
A quantum phenomenon called the co-tunnelling effect can produce uncontrolled coherent transfers of electrons across one or more junctions, reducing the accuracy of the pumping current. In order to minimize this effect, resistors are placed at each end of the SET device. This is called an R- type pump.

In pumping mode, when the polarization voltage varies at the pump terminals we can observe current steps. The presence of these plateaux is a result of current quantization. For metrology purposes the steps must be as "flat" as possible.
We can now show usable steps up to 150 MHz (I ˜ 24 pA).

          


Left: current steps measured at a temperature of 30 mK at different pumping frequencies. Current intensity is quantized (I = e.f) on a current plateau between 200 and 400 µV. The plateau shrinks as the frequency applied increases. The deviation from quantization on some steps is due to offsets that we cannot as yet eliminate.
Right: close-up of a pump measurement at 10 MHz.

Allan deviation on electron pumping measurements at frequency f = 50 MHz


Figure 7: Measurement lasted over 36 hours. The linear evolution of the Allan deviation over time is characteristic of white noise indicating non-correlation between measurement samples.

Mean value: 8.21 pA

White noise level: 63 fA/Hz1/2

Experimental standard deviation of the mean (36.75 h) : 1.32x10-4pA (relative uncertainty: 1.59x10-5)

Allan deviation (9 h): 3.13x10-4pA (relative uncertainty: 3.81x10-5)

Statistical analysis based on the Allan deviation [1 - 2] is used to estimate the A type uncertainty on current measurement.

Producing samples

Samples are produced by nanolithography followed by evaporation at three angles (one for the Cr line and two for Al, with an intermediate oxidation stage) on a GaAs substratum. The results presented here were obtained with devices supplied by PTB (Germany). Three years ago LNE set up a partnership with CNRS/LPN (the National Scientific Research Centre's photonics and nanostructures laboratory) to produce single-electron devices. The photograph below is a scanning electron microscope image of an LPN/LNE R-type three-junction pump.

Figure 8: LPN/LNE R-type three-junction pump

Figure 8: LPN/LNE R-type three-junction pump

Measuring instruments

For junction capacitances approaching 100 aF, the total energy in play is in the order of Ec/KB≈3K. In other words, the Coulomb blockade phenomenon disappears above 3K. The SET device and the measurement circuit wiring are therefore fixed to a dilution refrigerator (see figure 6), making it possible to work at 50 mK. Current is amplified by means of a cryogenic current comparator (CCC). For electron pump measurement, a CCC is used with a 10,000 gain on the current and a measured noise of 4fA/(Hz) at around 1 Hz. For 5 fA

          


Figure 6
Left: specific experimental set-up for measuring single-electron devices at LNE
Right: diagram of wiring and current amplification system (CCC) mounted on a dilution unit.

References

  • [1] D.W. Allan, IEE T. Instrum. Meas., vol. 36, no.2, p.646, 1987
  • [2] T.J. Witt, IEE T. Instrum. Meas., vol. 50, no.2, p.445, 2001

Contacts

Laurent Devoille
Tel : (33) 1 30 69 21 55

Nicolas Feltin
Tel : (33) 1 30 69 21 60

Publications

  • F. Piquemal, G. Genevès, "Argument for a direct realization of the quantum metrological triangle", Metrologia, 37, 207-211, 2000
  • F. Piquemal, "L'effet Hall Quantique en métrologie", Bulletin du BNM, 116, 1999
  • J.P. Lo-Hive, S. Djordjevic, P. Cancela, F. Piquemal, R. Behr, C. Burroughs, H. Seppä, "Characterization of binary Josephson series arrays of different types at BNM-LNE and comparisons with conventional SIS arrays", IEEE T.I.M., Special issue CPEM 2002, Vol. 52, pp. 516-520, April 2003
  • Y. de Wilde, F. Gay, F. Piquemal, G. Genevès, "Measurements of single electron transistor devices combined with a CCC: progress report" IEEE T.I.M. special issue, Vol. 50, no. 2, pp.231-234, 2001
  • F. Gay, Y. De Wilde, F. Piquemal, G. Genevès, "The quantum metrological triangle experiment at BNM-LCIE: a progress report", document for meeting of Comité Consultatif d'Electricité et de Magnétisme (CCEM), September 2000
  • F. Gay, "Un comparateur cryogénique de courants pour la réalisation d'un étalon quantique basé sur l'effet tunnel mono-électronique", doctoral thesis, CNAM, Paris, December 2000
  • F. Gay, F. Piquemal, G. Genevès, "Ultra low noise amplifier based on a cryogenic current comparator", Rev. Sci. Instrum., Vol. 71, no.12, pp. 4592-4595, December 2000
  • Y. de Wilde, F. Gay, F. Piquemal, G. Genevès, "Measurements of single electron transistor devices combined with a CCC: progress report", IEEE T.I.M., Special issue CPEM 2000, vol. 50, pp. 231-234, April 2001
  • F. Piquemal, A. Bounouh, L. Devoille, N. Feltin, O. Thevenot, G. Trapon, "Fundamental electrical standards and the quantum metrological triangle", Compte-Rendus de l'Académie des Sciences Physique 5, pp. 857-879, 2004
  • N. Feltin, L. Devoille, B. Steck, F. Piquemal, C. Ulysse, Y. Jin, "Un nouvel outil pour la métrologie électrique : le dispositif à un électron", Revue française de métrologie, vol. 2005-2.
    Download (french - pdf - 4,1 Mo)

Projects

  • Design of single-electron devices in normal or super state, in collaboration with the photonics and nanostructures laboratory (LPN) of the National Scientific Research Centre (CNRS) and the "Quantronics" research group at the Atomic Energy Commission (CEA)
  • Trimet metrological triangle project: determination of fundamental constants, financed by the National Research Agency ANR (Dec 2005 to Dec 2008), coordinated by Laurent Devoille
  • Participation in the Poesi integrated silicon electron pump project, financed by the National Research Agency ANR (Dec 2006 to Dec 2009), coordinated by X. Jehl of the Atomic Energy Commission (CEA) Grenoble (Statistical Physics, Magnetism and Superconductivity Section, Department of Fundamental Research into Condensed Matter DRFMC/SPSMS, in collaboration with the Atomic Energy Commission's Information Technology Electronics Laboratory (CEA-Leti).