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Condensed-matter physics

Quantum dots and the Kondo effect

Nature volume 526, pages 203204 (08 October 2015) | Download Citation

Nanotechnology studies explore the extreme properties of strongly interacting electronic systems through conductance measurements, and probe quantum phase transitions close to absolute zero temperature. See Letters p.233 & p.237

Magnetic impurities, in the form of atoms that have spin angular momentum (spin) of ½, can drastically modify the electrical resistivity of metals. In the presence of such atoms, the resistivity drops to a minimum at a certain low temperature a few kelvin above absolute zero, but then increases as the temperature falls further1,2. This behaviour at low temperatures is called the Kondo effect, and it has been modelled in nanosystems known as quantum dots (QDs) that have been engineered to behave as spin-½ artificial atoms3.

To model the Kondo effect, a QD is connected by means of quantum tunnelling to two electron reservoirs through electrodes that form electron-transport channels — a set-up known as the one-channel Kondo model. Writing on pages 233 and 237 of this issue, respectively, Iftikhar et al.4 and Keller et al.5 report on experiments with QDs based on gallium–aluminium–arsenic interfaces, in which they implement an improved set-up called the two-channel Kondo model6. Keller et al. go on to test fundamental properties of quantum phase transitions in their systems near absolute zero temperature7.

In metals, the mechanism responsible for the Kondo effect at low temperatures is the coupling between a magnetic impurity and the spins of conduction electrons. At temperatures lower than the Kondo temperature TK, the electrons interact strongly with the impurity (a process known as screening), forming a collection of heavy quasiparticles1,2 called a Fermi liquid. In the one-channel QD set-up, electrons in the reservoirs enter a state of superposition that forms a single electron channel, which strongly couples with the QD's spin near the Fermi level (the energy of the highest filled electronic level of a system at absolute zero temperature). The conductance through the channel reaches a maximum value below TK (which is much lower than 1 kelvin in QDs3), in contrast to the increased resistivity in metals with magnetic impurities.

The two-channel Kondo model involves two sources of electrons that form two separate electron channels which compete to screen the impurity, thereby producing non-trivial low-temperature effects predicted by Kondo theory8. Iftikhar and colleagues observe the two-channel effect using a micrometre-scale QD (Fig. 1a) that comprises an 'island' of several billion electrons4. On this scale, the QD behaves like a metal, but its charge is quantized; this means that, to add one electron to the QD, a finite amount of charging energy is required. The QD's ½ spin results from a resonance between two macroscopic charge states that have the same energy9. When electrons enter or exit the island, the QD's spin changes direction.

Figure 1: Strongly interacting electronic systems.
Figure 1

a, Iftikhar et al.4 model the one- and two-channel Kondo effect using a device consisting of a micrometre-scale semiconductor known as a quantum dot, or island, that has a pseudo-spin angular momentum ½ and is connected to two electron reservoirs through nanometre-scale constrictions (orange) called quantum point contacts (QPCs). The authors place the device perpendicular to a strong magnetic field and measure how the conductance at each QPC changes at extremely low temperatures as single electrons enter or exit the quantum dot (red arrows). b, Keller and colleagues' device5, which probes the difference between the one- and two-channel effects, consists of a nanometre-scale spin-½ quantum dot (red) that is coupled (red arrows) to a metallic electron island, forming one electron-transport channel, and to source and drain electrodes, which together form a separate channel. The authors measure the conductance between the electrodes and investigate transitions between different quantum phases of matter at temperatures close to absolute zero. (b Adapted from ref. 5.)

The authors connect the QD to external electron reservoirs through two macroscopic electrodes and apply a strong magnetic field (3.9 tesla). The field polarizes the spins of individual electrons, which restricts the number of quantum electron channels to two. The authors demonstrate the two-channel effect through two quantum point contacts (QPCs), which are nanoconstrictions separating the QD from the electron reservoirs.

Iftikhar et al. obtain a first indication that two-channel Kondo physics is at work in their system from an analysis of conductance measurements, which yield TK as a function of the transmission probability of a single electron through the QPCs. The increase of the conductance below TK that they observe is in agreement with Kondo theory10. At each of the two QPCs, the conductance shows a maximum value of e2/h, where e is the electron charge and h the Planck constant. Taking both QPCs into account, the conductance through the system becomes e2/(2h), by analogy to two resistors connected in series10.

The authors then observe how the conductance changes with temperature, and expose the fragility of the two-channel effect: they find that this effect occurs only for a narrow range of conductances and temperatures; outside that range, it transitions to the one-channel effect, in agreement with theory10,11. The channel that couples most strongly to the QD screens the QD at the low-temperature limit (14 millikelvin). An open question is why the conductance of the most strongly coupled QPC slightly exceeds e2/h — the quantum upper limit.

Keller and colleagues explore the difference at the quantum level in the conductance between the one- and two-channel effects using highly controllable nanotechnology5 and theoretical and numerical arguments. The authors fabricate a quantum device12 consisting of a nanometre-scale spin-½ QD that has an odd number of electrons3 and two electron channels: a pair of source and drain electrodes together form one 'delocalized' channel, and a metallic electron island acts as a separate channel (Fig. 1b). In contrast to Iftikhar and colleagues' system, the number of electrons in the metallic island cannot be modified in this experiment.

The microscopic origins of transitions between different quantum phases of matter at zero temperature are not always understood and are often debated. According to the theory of the two-channel Kondo effect8, the temperature (or conductance) domain close to the quantum phase transition is well understood and can be studied not only by tuning the strength of the coupling of each (macroscopic) quantum channel with the spin-½ impurity, but also by adjusting external parameters such as the temperature, the magnetic field or the bias and gate voltages of nanosystems. This leads to a complex behaviour in the electron-transport properties across the quantum phase transition. When the two channels couple equally to the spin-½ QD, the electron transport for temperatures below TK (corresponding to thermal energies less than about 50 microelectronvolts) cannot be interpreted by the standard quasiparticle picture, leading to the emergence of non-Fermi-liquid behaviour.

In their system, Keller et al. observe that, when the spins of the two electron channels couple to the nano-QD in such a way that the energy exchange between the channels and the QD (which acts as the magnetic impurity) is different in the two cases, the more strongly coupled channel pairs with the QD. The conductance measurements then show that the system undergoes a quantum phase transition, reverting to a Fermi liquid of quasiparticles (whose wavefunction acquires a 90° phase shift). At a temperature above absolute zero, the smooth transition between the non-Fermi-liquid and Fermi-liquid regimes involves a distinct energy scale (T*), and the authors' conductance measurements are in agreement with precise theoretical predictions of that energy scale13,14. Keller et al. report a quadratic dependence of T* on the gate voltage, and confirm the exact theoretical description of the transition between the strongly correlated non-Fermi-liquid and Fermi-liquid states.

Keller and co-workers' device essentially provides a sophisticated 'nanoscope' with which the authors explore domains close to quantum phase transitions in an extremely narrow physical-parameter space. Iftikhar and colleagues' impressive experiment demonstrates the two-channel Kondo effect on the basis of the quantum charge states of a microscopic QD in an electric circuit. Follow-up research could involve more electron channels by increasing the numbers of QPCs in an effort to investigate other similar systems that are best described in terms of strongly interacting entities (strongly correlated physics).

Notes

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  1. Karyn Le Hur is at the Centre for Theoretical Physics (CPHT) at the Ecole Polytechnique, and at CNRS, Université Paris-Saclay, 91128 Palaiseau, France.

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