Exotic Kondo effects

Contact Andrew Keller (ajkeller@) for more information.
Most conducting materials can be understood in terms of effectively non-interacting electrons. However, inducing strong interactions between electrons and their environment can give rise to novel ground states. Our approach to understand this behavior is to use semiconductor nanostructures to control and manipulate systems composed of a few electrons. In particular, we induce complex interactions between manufactured 'islands' of electrons (quantum dots) and their electrical leads (Fermi reservoirs), which we characterize by measuring transport. We primarily study interactions that may be described as variants or analogues of the Kondo effect: a many-body interaction wherein a localized degeneracy is screened by itinerant electrons. We are presently studying a parallel double quantum dot system where both spin and orbital degrees of freedom may be important, and uniquely we can achieve pseudospin resolution of a Kondo effect at an orbital degeneracy [1].

Kondo Effect

The Kondo effect is a paradigm in correlated electron physics. Normal metals such as copper conduct better at lower temperatures, since atoms in their lattice vibrate less, and are less likely to deflect electrons. Remarkably, tiny concentrations of magnetic impurities cause a reversal of this trend: at very low temperatures, resistance rises with further cooling. In 1964, decades after this resistance increase first puzzled researchers, Kondo posited that mobile electrons in the metal like to align their spins opposite to that of a nearby magnetic impurity [2]. At low temperatures, an electron moving past an impurity would then tend to flip its spin and be simultaneously deflected from its path, qualitatively explaining the observed resistance rise. But why should a conduction electron even care about the spin of a magnetic impurity atom?

Several years earlier, Anderson had proposed modeling a magnetic impurity as a single quantum state for an electron bound to a local site [3]. According to the Pauli exclusion principle, the site can be occupied by up to two electrons (with opposite spins) but repulsion could make it energetically favorable for only a single electron to reside there. Then that electron's spin would be free to point in any direction. In 1966, Kondo's phenomenological model was shown to emerge from Anderson's more microscopic one: tunneling of electrons on and off the local site makes delocalized electrons want to make their spin opposite that of the local electron. At low temperature, a single delocalized electron pairs with the local electron to form a state of total spin zero: a singlet. The basic physics behind this is the same that drives the two electrons in a H2 molecule to take on opposite spins.

More generally, Kondo effect need not involve the screening of a localized spin, but rather the screening of any localized degeneracy. For instance, the Kondo effect has also been observed when an even number of electrons occupy a quantum dot, at a singlet-triplet degeneracy [4]. In carbon nanotubes, the orbital and spin degrees of freedom are screened as a unified "hyperspin," giving rise to a highly symmetric SU(4) Kondo effect [5, 6]. It is then perhaps not surprising that the Kondo effect should occur at an orbital degeneracy of a double quantum dot, as is the case in the system we are currently studying.

Kondo Effect and Quantum Dots

With nanotechnology, experimental physicists can design and fabricate single artificial magnetic impurities, such as semiconductor quantum dots, molecules, and nanotubes. In each of these systems, conducting leads attached to the artificial impurity play the same role as the host metal of a traditional magnetic impurity. Electron flow from one lead to the other through the magnetic impurity serves as a powerful probe of the local electronic states. ?Our approach is to use an AlGaAs/GaAs heterostructure that contains a high mobility two-dimensional electron gas (2DEG) about 100nm below the surface to form a semiconductor quantum dot.

Quantum dots are electrons confined in a small potential well, such that they are described by a discrete level spectrum and a charging energy. The discrete levels arise from the quantum mechanics of zero-dimensional confinement, and hence the quantum dot may be thought of as an artificial atom. The charging energy is a purely classical phenomenon that arises because of the tiny capacitance of the quantum dot to its environment. It represents the energy one must pay to add another electron to the quantum dot.

To fabricate a quantum dot, we first anneal ohmic contacts into a 2DEG so that we can measure transport in the buried 2D sheet. We then use both electron beam and optical lithography to pattern metal gates on the insulating surface of the heterostructure. When negative voltages are applied to the gates, the electrons in the 2DEG below are repelled, creating a shadow of the gates in the conducting 2DEG. In this way we can form narrow constrictions (quantum point contacts) or separate islands of electrons (quantum dots), such as in the figure below.

We typically place a quantum dot in close proximity to a reservoir of mobile conduction electrons-metallic leads-so that electrons can tunnel back and forth between the quantum dot and the metal. Since electrons may be added controllably to the system one at a time, the impurity can be converted from even occupancy (paired electrons and thus non-magnetic) to odd occupancy (unpaired electron and thus magnetic). This system [7, 8, 9] has been used to study the Kondo effect in an idealized single, highly tunable "impurity", testing theoretical work on both artificial atoms and bulk Kondo systems, as shown below.

Capacitively-coupled Double Quantum Dots

When two quantum dots are placed in close proximity, their coupling can result in a rich array of physical behaviors. If the barrier between the dots is such that electrons may tunnel between the two, then the dots are said to be tunnel coupled. Most work with double quantum dots has been done in this regime, where current is measured from one metallic lead, through both quantum dots in series, and out to another metallic lead. In the strong tunnel coupling regime, the quantum dots can hybridize, resulting in molecular orbital-like behavior. Strongly interacting dots may also be used to explore the physics of interacting quantum impurities.

However, the dots need not be tunnel coupled in order to interact. Two quantum dots that have only an electrostatic (capacitive) coupling will also exhibit interesting physics. We are currently studying a double dot system where the tunnel coupling may be tuned negligibly small. At orbital degeneracies where the energy to add an electron to either dot is the same, we observe dramatically enhanced conduction through each dot. Such an effect was first observed by Hübel, et al. in 2008 [10]. We have recently verified that a zero-bias anomaly is observed at the orbital degeneracy, a hallmark of the Kondo effect. In addition, because there are spatially separated source and drain leads for each dot, current through each dot may be measured independently. This is important because the two dots constitute a pseudospin system, and pseudospin resolution is achieved by measuring through either dot independently [1]. To put this in context, one example of why spin resolution is important in studying the Kondo effect is because it allows one to address how the Kondo resonance splits for small magnetic fields [11]. In our system, pseudomagnetic fields may be easily introduced as an orbital detuning of the energy levels of the two dots.

However, we have neglected real spin in the discussion above. In the absence of a Zeeman field to explicitly break spin degeneracy, how do the spin and orbital degrees of freedom each contribute to the Kondo effect we observe? We are currently addressing this question.

Two-Channel Kondo Effect

The two-channel Kondo effect (2CK) occurs when two independent reservoirs of electrons (or channels) attempt to screen a two-fold degenerate system, such as a spin-1/2 impurity. The situation is described by the Hamiltonian H2CK= J1 s1.S + J2 s2.S. Since the reservoirs do not communicate, each attempts to screen the local spin, resulting in overscreening. Unlike single-channel Kondo (1CK), this system exhibits fascinating low-energy non-Fermi-liquid behavior [12, 13]. The two-channel Kondo ground state has residual entropy even at zero temperature. A stable fixed point for the ground state of 2CK effect is formed when the two independent channels (or reservoirs) are equally coupled to the magnetic impurity, i.e. J1 = J2 [14]. However, any channel anisotropy that occurs will force the system away from the non-Fermi liquid 2CK fixed point and towards the 1CK fixed point.

The possibility of 2CK has been invoked to explain phenomena like zero-bias conductance anomalies in metal constrictions [15, 16] and anomalous specific heat of heavy fermion metals [12, 13, 17, 18, 19, 20], though no conclusive observations were made. In recent years, 2CK has attracted renewed interest because of proposals (including our own) to implement it with coupled quantum dots [21, 22]. We realized our proposal in 2007, as evidenced by measuring the expected universal scaling law [23].

Our experimental system contains two non-interacting leads (blue) and a large dot (red) attached to a single-level small dot. The spin of the (singly-occupied) small dot provides the local degeneracy needed for 2CK. When the dot is occupied by a single electron, it can flip its spin by virtually hopping the electron onto either large dot (red) or the leads (blue), and then returning an electron with opposite spin to the small dot. The large dot (red) and the leads (blue) serve as the two distinct screening channels required to produce the 2CK effect. Stabilization of the 2CK fixed point requires fine control of the electrochemical potential in each droplet, which can be achieved by adjusting voltages on nearby gate electrodes. Creating and studying a single 2CK system will not only benefit solid state physics by providing an intimate connection between experiment and many-body non-Fermi liquid theory, but may also allow the 2CK ground state to be applied in explaining a variety of other, more complex metallic systems.

In the future, we hope to probe the 2CK state further using complementary methods to access transmission phase information. We have recently fabricated new devices where the previous 2CK geometry is incorporated into an Aharonov-Bohm interferometer. An early version of the design is depicted below, where the color scheme (artist's rendition) is the same as before.

Fabricated by Andrew Keller in the Submicron Center at the Weizmann Institute of Science

[1] S. Amasha, et al., Phys. Rev. Lett. 110, 046604 (2013).
[2] J. Kondo, Prog. Theor. Phys. 32 1, 37-49 (1964).
[3] P. W. Anderson, Phys. Rev. 124 1, 41-53 (1961).
[4] S. Sasaki, et al., Nature 405, 764-7 (2000).
[5] P. Jarillo-Herrero, et al., Nature 434, 484-8 (2005).
[6] A. Makarovski, J. Liu, G. Finkelstein, Physical Review Letters 99, 066801 (2007).
[7] Goldhaber-Gordon, D., et al., Nature 391, 156 (1998).
[8] Cronenwett, S.M., et al., Science 281, 540 (1998).
[9] Goldhaber-Gordon, D., et al., Phys. Rev. Lett. 81, 5225 (1998).
[10] A. Hübel, et al., Phys. Rev. Lett. 101, 186804 (2008).
[11] T. A. Costi, Phys. Rev. Lett. 85 7, 1504-7 (2000).
[12] Affleck I. and Ludwig A.W.W., Phys. Rev. B 48 7297 (1993).
[13] Cox D.L. and Zawadowski, A., Advances in Physics 47 599 (1998).