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
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:
H1CK = Js0·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
H2CK= 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 . 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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