Exotic Kondo effects: Two channel Kondo

Contact Ileana Rau (igrau@) or Mike Grobis (mkgrobis@) for more information.

Advances in semiconductor nanostructures enable control and manipulation of systems composed of a few electrons. In particular, we study complex interactions between manufactured 'islands' of electrons (quantum dots) and their electrical leads (Fermi reservoirs) by measuring transport. We are currently pursuing experiments to realize a two channel Kondo effect, which may allow studies into the center of modern solid state theory such as quantum phase transitions, highly correlated electron physics, and quantum criticality.

Kondo Effect

The explanation of the Kondo effect is a paradigm of modern progress in solid state 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 rise first puzzled researchers, Kondo posited that mobile electrons in the metal like to align their spins opposite to that of a nearby magnetic impurity. 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. 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.

Kondo Effect and Single Electron Transitors

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.

We choose to use a AlGaAs/GaAs heterostructure which contains a high mobility two dimensional electron gas (2DEG) about 100nm below the surface to form the artificial magnetic impurity. Using both electron beam and optical lithography, metal gates are patterned on the insulating surface of the heterostructure (annealed ohmic contacts to the 2DEG are fabricated in an earlier lithography step). 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.

In particular, we place an artificial atom near a reservoir of mobile conduction electrons---an artificial metal---so that electrons can tunnel back and forth between the "impurity atom" and the "metal". Since electrons could be added to the system one at a time, the impurity could be converted from even occupancy (paired electrons and thus non-magnetic) to odd occupancy (unpaired electron and thus magnetic). This system [1, 2, 3] 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.

Motivation for Two-Channel Kondo Effect

Two-channel Kondo physics has also been used to explain specific heat anomalies in certain heavy fermion materials [4, 5, 6]: the unusual two-channel Kondo ground state has residual entropy even at zero temperature. Recently the two-channel Kondo system has attracted new interest because of proposals (including our own) to implement it in a modified single electron transistor [7, 8]. In the next section, we describe our experimental effort to create a single two-channel Kondo system, whose study may both advance many-body theory and help explain behavior of other candidate realizations of two-channel Kondo.

Semiconductor nanostructures should provide a clean experimental system for exploring the properties of the two-channel Kondo (2CK) Hamiltonian, whose unusual non-Fermi liquid fixed point has attracted substantial theoretical interest. While there have been no conclusive experimental observations of 2CK, it has been used to explain phenomena ranging from zero-bias conductance anomalies in metal constrictions [9, 10] to anomalous specific heat of heavy fermion metals [11,12, 13].

We use a modified single-electron transistor with two spatially-separated sets of confined electrons to study 2CK behavior at experimentally-accessible temperatures. 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. We have derived the conditions for obtaining this type of 2CK behavior and its experimentally-observable consequences. 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.

Two-channel Kondo Effect

The original Hamilitonian used by Kondo to describe the single-channel Kondo effect involved antiferromagnetic coupling between conduction electron spins so and a local spin S:

H_1CK = J_s0·S,

where J, the coupling constant, must be positive (antiferromagnetic) for Kondo screening to occur. In 2CK, a twofold degenerate system such as a local spin is antiferromagnetically coupled to not one but two independent electron reservoirs. The situation is described by the Hamiltonian

H_2CK= J₁ s₁·S + J₂ s₂·S.

Since the reservoirs do not communicate, each attempts to screen the local spin, resulting in overall overscreening. Unlike single-channel Kondo (1CK), this system exhibits fascinating low-energy non-Fermi-liquid behavior [11,12].

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]. Any channel anisotropy that occurs will force the system away from the non-Fermi liquid 2CK fixed point and towards the 1CK fixed point. We have designed a system to have precise control of the parameters determining the coupling constants J₁ and J₂, while maintaining the independence of the two channels.

Experimental Progress Towards Two-Channel Kondo

In order to observe this effect, we have modified a dilution refrigerator to achieve extremely low electron temperatures (<15mK as determined by Coulomb blockade peak width), and have constructed a low noise electrical measurement setup.

As shown in figure below, the 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 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.

Using this system, we may have already observed the quantum phase transition associated with asymmetric coupling to the two reservoirs described above. We are now working on a new set of measurements to further qualify this assertion, along with studying the non-Fermi liquid behavior when the two reservoirs are equally coupled to the magnetic impurity.

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