Kondo Effect in Coupled Quantum Dots
A. M. Chang∗†+ , J. C. Chen+
∗
Department of Physics, Duke University, Durham, NC 27708-0305
†
+
Institute of Physics, Academia Sinica, Taipei, Taiwan
Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan 30043
Abstract
We discuss Kondo systems in coupled-quantum-dots, with emphasis on the
semiconductor quantum dot system. The rich variety of behaviors, such as
distinct quantum phases, non-fermi liquid behavior, and associated quantum
phase transitions and cross-over behaviors are reviewed. Experimental evidence for such novel characteristics is summarized. The observed behaviors
may provide clues as to the relevance of the 2-impurity Kondo (2IK) effect, and
the 2-channel Kondo (2CK) effect to the unusual characteristics in stronglycorrelated systems, such as the heavy fermion system.
1
Contents
I
INTRODUCTION
II
THE KONDO EFFECT
A Single Impurity Kondo Effect . . . . . . . . . . . .
B Connection to other Physical Systems and Models
C Two Impurity and Two-Channel Kondo Effects . .
1 The two-impurity Kondo Effect . . . . . . . . .
2 The two-channel Kondo effect . . . . . . . . . .
III
3
QUANTUM DOT MODEL SYSTEMS
A 1CK Model in a single Quantum Dot . . .
B 2IK Model in Coupled-Quantum-Dots . . .
1 Series-coupled Double-Quantum-Dots .
2 Parallel-coupled Double-Quantum-Dots
C 2CK Model in Coupled-Quantum-Dots . . .
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IV
ELECTROSTATIC MODELS OF SINGLE AND DOUBLE QUANTUM DOT SYSTEMS
23
A Electrostatic Model of a Single Quantum Dot . . . . . . . . . . . . . . . . 25
B Electrostatic Model of a Double-Quantum Dot . . . . . . . . . . . . . . . 26
V
DQD CHARACTERIZATION AND EXPERIMENTAL STABILITY
DIAGRAMS
A Quantum Dot Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Weak-coupling to leads . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Strong-coupling to leads-Kondo limit . . . . . . . . . . . . . . . . . . .
31
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VI
DATA PRESENTATION-SEMICONDUCTOR COUPLED QUANTUM DOTS
36
A Experimental Results-2IKE . . . . . . . . . . . . . . . . . . . . . . . . . . 36
B Experimental Results-2CKE . . . . . . . . . . . . . . . . . . . . . . . . . 46
VII
METALLIC SYSTEMS
48
VIII SUMMARY
50
2
I. INTRODUCTION
The study of quantum phase transition is an active area of research in condensed matter
physics and in quantum field theory. A quantum phase transition (QPT) refers to a transition between quantum ground states of differing characteristics driven by the change in
some parameter(s) in the Hamiltonian of the physical system, rather than by a change in
temperature [1]. In many instances, condensed matter systems are particularly well-suited
to investigating the properties of novel quantum phase transition due to the tunability of
parameters, which determine the Hamiltonian. Thus, it is often possible to experimentally
access distinct phases of differing quantum ground states by tuning various experimental
knobs.
Quantum phase transitions in Kondo systems is one such example, where a rich variety of
distinct phases exhibiting novel properties can arise. In recent years, the study of the Kondo
effect in various forms and shapes, e.g. in semiconductor quantum dots [2–19], metallic quantum dots [20–23], tunnel junctions containing magnetic impurities [24], atomic/molecular
systems [25–33], and various pseudo-spin systems [34,35] has enjoyed a resurgence and has
further stimulated interest in this correlated many-body effect.
In its simplest form, the Kondo effect deals with the screening of an impurity spin by a
sea of conduction electron spins. It is a non-trivial many-body effect, in which the notion of
asymptotic freedom, i.e. bound state at low energies, and free spin at high energies, is known
to play a key role in determining the physical behavior of the system. Among the diverse
physical systems exhibiting the Kondo effect, the semiconductor quantum-dot (QD) system
has produced wide-ranging impact. The reason is two fold. Firstly, from the experimental
side the tunability of semiconductor QD systems via electrostatic gates is essential. This
tunability enables diverse behaviors to be explored in differing regimes. The electrostatic
gating allows control of the coupling to external leads and between coupled-QDs, etc., which
in turn controls the Kondo energy scale, and control of parameters such as the impurity
-impurity interaction. Secondly, theoretically a great deal is known about the properties of
the Kondo system in the context of the quantum dot. This makes possible direct test of key
concepts, as well as opportunities for new discoveries.
Following the original treatment by Jun Kondo using perturbation theory methods [36],
much of the fundamental theoretical predictions on equilibrium properties have come through
the use of sophisticated machinery such as renormalization group methods [37–40], exact solutions from Bethe-Ansatz methods [41–44], and conformal field theory methods [45–48].
These have provided detailed predictions, testable in experiment. Recent advances are extending into the non-equilibrium situations, when a current is being passed through the
systems, specifically in the context of the QDs [10,11,49,50]. Many of these predictions have
either undergone rigorous experimental tests, or are actively being tested.
It is now well established that the problem of a single S=1/2 impurity coupled to a single
electron sea, the so-called 1-channel Kondo problem (1CK) exhibits only one stable phase,
namely a low temperature (low energy) phase, characterized by a ground state configuration
of a fully screened bound-state, where a Kondo cloud of conduction electrons cooperatively
3
screen the impurity moment, forming a many-body singlet state. In the renormalization
group sense, this singlet bound state is associated with a stable low temperature (energy)
fixed point. In contrast, at high energy, the spin is asymptotically free.
The situation can become even more interesting when two different impurity spins are
present in the conduction electron fermi sea, each interacting with the other through an antiferromagnetic exchange, or when two channels of fermi seas are coupled to a single S=1/2
impurity spin, and each attempts to independently screen the impurity moment. In both of
these situations, different phases can arise depending on parameters. In particular, critical
behavior with non-fermi-liquid (nFL) characteristics emerges. The nFL behavior in the two
models have been found to be similar in certain respects [51,52]. The nFL fixed points (in
the renormalization group flow diagram) associated with the critical behavior, are rather
delicate, and are experimentally accessible under stringent conditions, e.g. when certain
symmetries are preserved. In recent years, advances on coupled-QD systems have enabled
the first realizations of these sophisticated Kondo systems. These new experiments are
opening a new direction for gaining an understanding of these systems, and for ascertaining
the relevance of the Kondo type correlations in diverse physical scenarios. Coupling between
Kondo spin impurities, or coupling with more than one channel of conduction electrons,
such as believed to be occurring in heavy fermions systems, represent possible scenarios for
bringing about the observed non-fermi liquid behaviors in such strongly-correlated systems.
Understanding Kondo systems and the associated QPT can have ramification beyond
magnetic systems. Numerous apparently disparate condensed matter systems, and models systems in quantum field theory, are intimately connected to the Kondo problem. For
instance, the physics of 1-dimensional interacting systems, e.g. Hubbard model, Luttinger
liquids, 1-d superconductivity, etc., or 1+1 dimensional quantum field theory systems, such
as the Gross-Neveu fermionic model with quartic interactions, or the related Thirring model,
exhibit a great deal of commonality with the Kondo system. Such commonality can perhaps
be most easily seen in their connection with 1+1 dimensional conformal field theories [45,46].
Moreover, a more thorough understanding of the notion of asymptotic freedom and its ramifications may even contribute to our understanding of the asymptotic freedom properties in
SU(3) gauge theories of the strong interaction, despite fundamental differences in the two
systems.
In this review, our main goal is to provide an up-to-date summary of the recent developments in the experimental investigations of quantum phases in the Kondo systems, realized
in semiconductor coupled QDs. Along the way, we hope to also provide a larger perspective
on the significance of the novel quantum phases, and the transitions (crossovers) between
phases.
This review is structured as follows: Section II contains a qualitative description of the
Kondo systems. Starting from the one-channel Kondo (1CK) system and its relation to
other physical systems, such as systems where the notion of asymptotic freedom is of relevance, this is followed by a description of the two-impurity Kondo (2IK) and two-channel
Kondo (2CK) systems. Section III discusses the implementation of the Kondo systems in
the context of the semiconductor quantum dots. Section IV deals with the electrostatics of
4
coupled-quantum-dot systems, specifically, the double quantum-dot (DQD), and its characterization. Characteristics of devices used in experiment as well as the methodology for such
characterization are provided in Section V. In Section VI, the main experiments on Kondo
effect in semiconductor based coupled-QDs, are reviewed, followed by a brief summary of
related works in metallic systems, in Section VII. Concluding remarks and future prospects
are discussed in the last Section VIII. For an experimental perspective, Sections II, III,
and IV may largely be omitted without loss of continuity.
5
II. THE KONDO EFFECT
A. Single Impurity Kondo Effect
The single impurity Kondo effect is a many-body effect, which exhibits the notion of
asymptotic freedom. When the coupling between a localized impurity spin and the spins of
the sea of conduction electrons is anti-ferromagnetic (AF), the screening of the impurity spin
is controlled by a many-body energy scale, the Kondo energy EK , or equivalently the Kondo
temperature, TK = EK /k, where k is the Boltzmann constant. At temperatures much above
TK , the impurity behaves much like a free spin–an indication of asymptotic freedom at high
energies, while below TK , a bound resonance develops, as the impurity becomes completely
screened by the conduction electrons as the temperature approaches zero.
The Kondo Hamiltonian introduced by Kondo in 1964 is given by [36], the so-called single
channel spin 1/2 (S = 1/2) Kondo (1CK) effect:
H=
X
kσ
ǫk c†kσ ckσ + JS · s(0).
(1)
where ǫk is the energies of the k state in the conduction band, c†kσ is the creation operator
of the k-state of spin σ, J > 0 the anti-ferromagnetic (AF) coupling between the impurity
spin S and the conduction electron spin s(0) at the location of the impurity at r = 0. Kondo
sought to explain the anomalous behavior in the resistivity observed in metal systems doped
with a dilute amount of magnetic impurities. To third order in J, Kondo showed that the
temperature dependent resistivity behaves as follows:
ρ(T ) = ρo + aT 2 + bT 5 + c{J 2 + J 3 D(EF )ln(EBW /T )},
(2)
where ρo is the T = 0 limiting resistivity caused by non-magnetic static disorder, aT 2 the
usual fermi liquid contribution, bT 5 the phonon contribution, D(EF ) the conduction electron
density of states at the fermi energy EF , and EBW the conduction electron bandwidth.
The logarithmic divergence in the third order term J 3 ln(EBW /T ) at low temperatures is
indicative of the type of infrared divergence well known in this type of problem, and signals
the breakdown of perturbation theory below a Kondo energy scale
EK = kTK = EBW exp[−1/JD(EF )].
(3)
This logarithmic term is responsible for the up-turn of the resistivity below the Kondo scale.
To properly account of the logarithm divergence at low temperatures, it is necessary to use
renormalization group techniques [37–40] to approach and reach the strong coupling fixed
point at zero energy (or temperature). These methods showed that in the case of isotropic
coupling as in the Hamiltonian of Eq. 1, the effective coupling strength scales (inverse)
logarithmically with energy (at high energies):
J(E) = EBW
1
,
[1 − JD(EF )ln(E/EBW )]
6
(4)
and diverges at EK = kTK , signaling the formation of the bound Kondo spin- singlet state as
the impurity spin becomes screened. This scaling is thus only correct at energies exceeding
the Kondo scale. At low energies, numerical renormalization techniques developed by Wilson
are needed to produce the correct behavior toward the low energy fixed point. The main
results of the renormalization group analysis were confirmed by exact Bethe-Ansatz solutions
[43] and boundary conformal field theory methods [46]. Interestingly, the above scaling form
is identical to the scaling of g 2 in the 3+1 dimensional Yang-Mills guage theory in the high
energy (small distance) limit, where g is the coupling constant for the Yang-Mills non-Abelian
guage fields.
The formation of the Kondo singlet causes the remaining electrons in the fermi sea to
experience a scattering phase shift of π/2. In the original case of a magnetic impurity, e.g.
Mn, Fe, Co, etc., in a noble metal, this resonance gives rise to a an increase in the resistivity
as the temperature T is decreased toward TK as indicated in Eq. 2 in this regime where
perturbation theory is valid, attaining a saturated value as T approaches zero below TK . In
the complementary case of a Kondo state in a quantum dot, on the other hand, this Kondo
resonance, which occurs at the fermi level of the lead, gives rise to enhanced conduction in a
Coulomb blockade valley at zero source-drain bias [53]. At low temperatures, the electrical
transport coefficients deviate from their respective T = 0 value in a quadratic fashion in T ,
as expected for a fermi-liquid. A universal scaling curve is available for finite voltage bias,
between the source and drain in the QD case, as will be discussed in Section III C.
Aside from the contribution to the transport characteristics, the Kondo scattering also
modify the thermodynamic quantities, such as the specific heat, cv , and the magnetic susceptibility, χ. In bulk metals doped with magnetic atoms, the impurity contributions are
again fermi-liquid-like, with cv ∝ T , and χ ∝ constant. In addition, the Wilson ratio,
RW = (δχimp /χ)/(δcv,imp /cv ), measuring the relative change contributed by the impurity in
χ to total χ divided by the relative change in heat capacity to its total, is a universal number
equaling 2 [39,46].
B. Connection to other Physical Systems and Models
The single impurity Kondo effect is often described as the first many body system to
exhibit the notion of asymptotic freedom. In (3+1) dimensions, asymptotic freedom is an
important property of the Yang-Mills, non-abelian gauge theories, which currently form
the basis for the understanding of the strong interaction, embodied in the SU(3) theory of
quantum chromodynamics. Although the form of the Hamiltonian for the Kondo system
is fundamentally different from the Yang-Mills, non-abelian gauge theory, studies of the
Kondo system may potentially help elucidate the behaviors of the general class of theories
exhibiting asymptotic freedom. For example, the 1+1 dimensional fermionic model with
quartic interaction, namely the Gross-Neveu or the related SU(2) Thirring [54] models,
is closely related to the Kondo problem. Its ultraviolet behavior and asymptotically free
properties have been analyzed in great detail following the seminal work of Gross and Neveu.
By now, it has been rigorously established that the Kondo problem can be mapped to the 1+1
7
dimensional Gross-Neveu model [55,56] with 4-fermion (quartic) interaction in the fermion
field.
Furthermore, boundary critical Conformal Field Theories (CFTs) has now been shown to
be extremely useful in analyzing the Kondo systems, including the single and multi-channel
Kondo effect, as well as the 2IK system [46,48]. This different perspective, enabled Affleck
and Ludwig to analyze the Kondo problem in terms of boundary critical phenomenon in 1+1
dimensional conformal field theories (CFT). The spatial dimension 1 arises because of the
delta-function approximation for the scattering potential, which leads to an s-wave channels
as the only channel with a Kondo coupling. This delta-function approximation is reasonable
given that the relevant Kondo length scale is
lK ∼ vF
h̄
≫ (ao , λF ),
kTK
(5)
where ao is the lattice size and λF the fermi wavelength of the conduction electrons. The
mapping into 1+1 dimension (1 spatial, 1 temporal) enables an immediate connection with
other 1d systems, such as Luttinger liquids, and 1-d superconductors. It is thus possible that
investigations of such correlated model systems can shed light to diverse condensed matter
systems, and even to quantum field theories of relevance of high energy physics, such as
string physics.
On a more speculative direction, even in SU(3) gauge theories of quarks, which necessarily are of fundamentally different character since quarks are never free, one might attempt to
draw parallels. The formation of the spin-singlet at low energies might inspire a phenomenologically similar situation in SU(3) of an ultra-heavy quark and anti-quark pair in a light
quark sea, with tunable heavy-quark/anti-heavy-quark interaction, where the tuning can
either be artificial or can be accomplished by varying their separation [57]. Here, the linear
attractive potential (linear in distance) responsible for confinement plays a similar role as
the J in the Kondo problem. When the heavy-quark/anti-heavy-quark interaction is weak,
each is screened by the light quark sea, forming separate bound states. As the interaction
increases, the heavy-quark and anti-heavy-quark form a bound state and drop out of the
continuum of light quark sea, much like the two-impurity Kondo situation.
8
C. Two Impurity and Two-Channel Kondo Effects
The Kondo effect, as it is, is already an extremely interesting many body phenomenon,
as evidenced by the large body of experimental and theoretical work devoted to this topic
in the past 40 years. Since the 1980’s, more exotic Kondo systems have become the focus
of investigation, spurned on by theoretical analyses in relation to possible explanations of
unusual nFL behaviors in strongly correlated, heavy-fermion systems. This interest came
in part as a result of the theoretical discovery that even in the simplest extensions of the
original Kondo model of a single impurity coupled to one fermi sea (1CK), new behaviors
can emerge. These extensions include models where: (i) 2-impurities are coupled to the
fermi sea while at the same time coupled to each other anti-ferromagnetically [58–60], and
(ii) a coupling of the impurity moment to more than one channel of fermi sea [40,44,46].
In the first case, there is a competition between ground states where in one state each spin
moment is separately screened by the fermi sea, and in the other, the two impurity moments
form a singlet between them, and the impurity moment disappears. In the second case, the
independent channels each attempts to screen the impurity moment leading to over-screening
of the impurity moment. The new behaviors are a consequence of the emergence of a new
fixed point, which has a character distinctqfrom the fermi liquid. These nFL fixed points
yield a spectral density, which behaves as |ω| at low energies h̄ω measured relative to the
fermi surface, as opposed to the ω 2 correction of a fermi liquid. This modification leads to
low temperature dependences of various physical quantities, e.g. the resistivity (in a metallic
system), or conductivity in QD systems, and the impurity contribution to the cv and χ, which
differ from a fermi liquid.
√ In particular,2 the low temperature correction to the transport
coefficients behaves as T rather than T , while the specific heat and susceptibility for the
2CK system are modified from the fermi-liquid T dependence by a multiplicative factor of
ln(T /EBW ). In the 2IK system, interestingly, there is no modification in the functional forms
of these thermodynamic quantities, but the Wilson ratio is no longer universal in this case.
The emergence of the semiconductor QD system for studying the Kondo effect commenced in the mid 1990’s, following the experimental realization of a Kondo system in a
semiconductor quantum dot [2–9]. It has since become technically feasible to utilize the
large degree of tunability and control in semiconductor quantum dots to realize such more
complex Kondo systems. In this respect, the semiconductor quantum dot system offers a
unique and hereto-fore unavailable system for studying such exotic new physical behavior.
The tunability of the coupled-QD system affords many different regimes to be accessed, enabling the investigation of a rich varieties of physical behaviors in the Kondo effect. This
opens a door for investigating the quantum phase transitions in such systems due to the
tunability of parameters. Different coupled-Kondo systems have recently been realized. Of
particular interest are models of the 2IK and 2CK effects. The underlying physical behaviors,
accessible to varying degrees in the different cases, are generally not accessible to the heavy
fermion systems themselves, since metallic systems are in most instances difficult to tune,
or at least have a more restricted range of tunability. In other respects though, they can
be more versatile. For example, both ferromagnetic (FM) and AF coupling can be realized
9
with greater ease.
In semiconductor coupled-QDs, it is thus possible to: (a) tune into a Kondo valley by
filling with an odd number of electrons, (b) optimize the coupling of each QD to the fermi
sea in the leads to enhance the Kondo temperature scale, and (c) tune the coupling between
the quantum dots. Such tunability, in conjunction with other knobs available in a transport
measurement, such as non-equilibrium measurement at finite source-drain voltage bias, and
the application of a magnetic field in different orientations, has enabled meaningful attempts
to realize and examine the 2IK and 2CK systems. These pioneering experimental studies
have succeeded in yielding a substantial amount of information about these rather complex
systems and the transition between different quantum states as the relevant parameters are
tuned.
Below, we discuss the 2IK and 2CK models, followed by the implementation of such
models in the context of coupled-quantum-dots.
1. The two-impurity Kondo Effect
The 2IK problem was first introduced by Jones et al., in the mid-1980 [58–60]. The
model was investigated as a first attempt to understand the classic strongly-correlated heavy
fermion system. Such a system is often characterized as a Kondo lattice system, with roughly
an impurity moment in each unit cell on the site of ions contain f-orbital. Classic examples
include intermetallic compounds containing cerium (Ce), uranium (U), ytterbium (Yb), or
neptunium (Np), and possibly in transition metals ”heavy-fermion” systems contain transition metals with d-orbital electrons [61]. In the classic case of f-electrons, Kondo physics
is believed to be relevant, as the Fermi sea of itinerant electrons attempt to screen the moments, forming a Kondo lattice system [62–64]. The physics associated with the Kondo
lattice is often-invoked to help explain the exotic behaviors observed in such heavy fermion
systems [65–67]. Surprisingly, the simple model of two coupled Kondo impurities yielded
fascinating non-fermi-liquid characteristics, and generated a great deal of interest as a result
of its potential relevance to the strongly correlated Kondo systems.
The 2IK Hamiltonian first studied by Jones et al. is given by:
H = Ho + Himp ,
(6)
with
Ho =
X
ǫk c†kσ ckσ ,
(7)
kσ
and
Himp = Jo [s(r1 ) · S1 + s(r2 ) · S2 ] + J1 S1 · S2 .
(8)
Here Ho refers to the Hamiltonian of the conduction electrons, with energy dispersion ǫk ∝ k
(after linearization about the fermi point), and electron creation operators, c†kσ , with σ
10
labeling the spin. The interaction term Himp involving the impurity spin moments Si , i =
{1, 2}, at locations ri , respectively, contains an exchange coupling to the conduction electron
spin at their respectively positions, with coupling constant Jo , as well as an impurity-impurity
exchange with coupling J1 .
The Hamiltonian can be simplified by projecting into the s-wave channel [46], and the
remaining channels become decoupled from the spin interaction. For the case of rotationally invariant electron dispersion, averaging over angles produces an effective 1-dimensional
model, and the interaction term becomes [58,59,48]:
Himp = J+ (S1 + S2 ) ·
+ J− (S1 + S2 ) ·
+ Jm (S1 − S2 ) ·
X
(c†k′ e σcke + c†k′ o σcko )
(9)
k′ k
X
k′ k
(c†k′ e σcke − c†k′ o σcko )
X
k′k
(ic†k′ e σcko + h.c.) + J1 S1 · S2 .
The creation operators c†ke,o denote operators of even and odd parity, respectively, constructed
from the c†kσ operators, based on the symmetry about the mid-plane of the two impurity
moments [58]. The exchange couplings J+,− = (Je ± Jo )/2, and Jm = (Je Jo )1/2 , with the
even and odd couplings given by Je,o = (Jo /2)(1 ± sinkF R/kF R), R being the separation
between the moments. To arrive at these values, the k-dependent couplings are replaced by
their values at the fermi wave-number, kF , as it is believed that the k-dependence does not
change the qualitative behavior of the system.
The first three terms involving exchange between the localized moments and the conduction electrons generates an effective exchange between the localized moment in second
order perturbation. In combination with the explicit exchange coupling J1 , a total effective impurity-impurity coupling J is present. By taking J+ , J− , Jm , and J1 as independent
parameters in Eq. [9], it is thus possible to tune the impurity-impurity coupling from ferromagnetic (FM) to anti-ferromagnetic (AF), spanning the entire range from J → −∞ (FM),
to J → +∞ (AF).
The main new features for this model results from an anti-ferromagnetic coupling J
between the two local moments. The inclusion of this impurity-impurity coupling leads to
a competition between two possible quantum ground state configurations: When the AF
coupling J is weak compared to the Kondo energy scale of each spin moment arising from
their respective coupling to the conduction electron fermi sea, the system behaves as two
”separate” Kondo systems, one in even channel, the other in the odd channel, where the
even and odd channels refer to the even and odd linear combinations of the conduction
electron sea in the right and left leads. On the other end, when the coupling J is strong, the
two impurity spins lock into a spin singlet, and thus the system no longer has a moment,
and Kondo screening therefore no longer takes place. In the absence of additional on-site
potential scattering at the impurity sites, which breaks electron-hole (e-h) symmetry of the
Hamiltonian interaction, this competition gives rise to an unstable fixed point at a critical
value of the parameter jc = J/kTK ∼ 2.2. The staggered spin susceptibility and the heat
11
capacity, cv , were found to diverge at the fixed point as 1/(j − jc )2 . In terms of the low
temperature dependence, however, there is no significant deviation from the fermi-liquid
functional dependence. However, the Wilson ratio, RW , measuring the fractional change
(due to the impurities over the overall metallic values) of cv to that of the susceptibility
(not staggered) is not universal, as opposed to the case of 1CK or 2CK models for a single
impurity [60,48].
Due to the fact that on both sides of the quantum phase transition, the symmetry of
the ground state is the same, i.e. a singlet, the fixed point must be protected by electronhole symmetry [58–60,48]. This fixed point is rather delicate, and it is not entirely clear
whether it could be accessed in real systems. In the absence of e-h symmetry, the situation
changes into a cross-over behavior, which is nonetheless still of significant interest. Due to
the intractability of the 2IK model via analytic methods, much of the theoretical analyses
were carried out either by numerical renormalization group techniques, or mean-field theory,
and thus were subjected to some degree of controversy. Affleck et al. have argued that such
a fixed point must exist, as the phase shift changes from (π/2, π/2) in the two independent
channel limit, to (0, π) when the impurity-impurity singlet is formed. Furthermore, an ansatz
within the framework of the boundary CFT analysis have confirmed much of the results of
the numerical calculations. More recently, a more realistic model, which includes on-site
potential scattering, and thus breaking e-h symmetry, has been analyzed [68,48]. It was
found that the presence of the J− term in Eq. [9] leads to a rounding of the sharp quantum
phase transition associated with the novel nFL fixed point.
2. The two-channel Kondo effect
The 2CK model represents another direct extension of the original 1CK model, yielding
non-trivial nFL behavior. The 2CK model is a special case of the multi-channel model, first
discussed by Nozieres and Blandin in a pioneering work, where the situation was analyzed in
the limit of a number of channels [40]. The 1CK for a single impurity involves an AF coupling
between the impurity spin and the spins of the fermion sea, which provides the screening.
In general in such AF Kondo systems, the conduction electrons attempt to screen the local
moment. The resultant behavior of the system depends on the number of components
associated with the spin degree of freedom of the impurity moment, e.g. 2 for S = 1/2 in
the familiar case, and also on the number of independent channels (or flavors) C of electron
seas. The different scenarios can then be classified by the degree to which screening takes
place. There are three possibilities, under-screening, 2S > C, exact screening, 2S = C, and
over-screening 2S < C. The first case yields an under-screened Kondo behavior as a net
moment of magnitude (2S − C) remains after the screening takes place. In the second case,
the usual Kondo scenario arises, and at low temperatures the impurity moment becomes
completely screened. In the third case, an over-screened multi-channel Kondo effect takes
place with nFL behavior. Here, the different channels of fermi seas independently attempt
to screen the impurity, resulting in an over-screened situation.
12
The simplest multi-channel model is that of an S = 1/2 impurity coupled to two independent channels of conduction electron fermi seas. This is the well known S = 1/2 2CK
problem. Here a new non-fermi-liquid fixed point controls the behavior of systems. This
fixed point, however, is unstable to an asymmetry of the coupling of the impurity to the two
different channels,
and thus is also delicate. Similar to the 2IK case, the spectral density
q
behaves as |ω| at low energies. The low temperature χ and cv are modified from the fermi
liquid behavior by an addition logarithmic T factor. Here, the Wilson
√ ration RW is universal
and equals 8/3 [69,46]. The correction to the resistivity behaves as T at low temperatures,
rather than as T 2 for a fermi liquid. In this respect, the correction is identical in form as the
2IK effect (2IKE).
Without reproducing the complex computations to deduce the green functions and spectral density, one may gain a plausible explanation of the cause of the nFL behavior in this
system, by appealing to the boundary conformal field theory type analysis [45,46]. Affleck
and Ludwig demonstrated that the multi-channel Kondo Hamiltonian can be cast in terms
of current densities for the charge, SU(2) spin, and SU(k) flavor degrees of freedom. These
currents obey non-abelian Kac-Moody algebra. The presence of the impurity at the boundary at x = 0 imposes conditions at the boundary in the (t, x) plane, which leads to anomalous
boundary dimensions that determine the power law exponent in the spectral density. For the
familiar ordinary Luttinger liquid, an abelian Kac-Moody algebra yields a power-law spectral density, with an exponent determined by the conductance parameter contained in the
Kac-Moody commutation relation. Here, to determine the self-energy in the two-point green
function, it is necessary to rely on fusion rules for the various boson fields corresponding to
the currents. In the bulk, far from the boundary, the fusion rules give rise to a composite,
which ensures that the fermi liquid behavior is recovered. However, near the boundary, the
fusion rules are relaxed, and as a result, and it is perhaps not surprising that non-fermi liquid
anomalous dimensions can arise.
13
III. QUANTUM DOT MODEL SYSTEMS
A. 1CK Model in a single Quantum Dot
Schrieffer and Wolff showed in the 1960’s [70] that the Kondo model (Eq. 1) can be
obtained from the Anderson model of an impurity embedded in a host metal, in the appropriate region of parameter space after elimination of terms linear in the impurity-conduction
electron hoping. The Anderson impurity model is given by [71]:
H = Ho + Himp + HI ,
(10)
where
Ho =
X
ǫk c†kσ ckσ ,
(11)
kσ
is the Hamiltonian of the conduction electrons, with dispersion ǫk , and electron creation
operator c†kσ .
Himp =
X
ǫd d†σ dσ + Un↑ n↓ ,
(12)
σ
is the localized impurity Hamiltonian with energy level ǫd measured from the chemical potential (more precisely the electro-chemical potential) of the conduction electron fermi sea,
impurity electron creation operator d†σ , and on-site repulsion U for double occupancy by an
up- and a down-spin electron, and
HI =
X
Vk [c†kσ dσ + h.c.]
(13)
k
arising from coupling between the conduction electrons and the local impurity site. When
P
U, the on-site repulsion, is large, i.e. U ≫ (ǫd , Γ), with Γ = k |Vk |2 δ(ǫk − EF ), being the
impurity level broadening, a Schrieffer-Wolff transformation eliminates terms linear in Vk .
In the limit where a local moment exists on the impurity site, given by the condition [72]:
−ǫ∗d = −[ǫd + Γ/πln(EB W/Γ)] ≫ Γ,
(14)
the resultant Hamiltonian has the Kondo form, with an exchange coupling between the
impurity spin and the conduction electron spin at the impurity site given by:
J ≈ 2|Vk |2
U
.
|ǫd |(ǫd + U)
(15)
At low temperatures, below the Kondo scale:
TK =
√
UΓexp[−
π|ǫd |(ǫd + U)
],
2UΓ
14
(16)
the AF coupling leads to a Kondo resonance associated with the screening of the impurity
spin (S = 1/2) by the conduction electrons.
The situation in a QD is essentially the same as that of the Anderson impurity [10,11].
To enable electrical transport measurements to be performed, the QD is typically connected
to two (or more) leads via tunnel coupling. In the two-lead case, With a coupling VU (VL )
to the upper (lower) lead, respectively, we have:
H = HU + HL + HQD + HI ,
(17)
where
X
H(U,L) =
ǫ(U,L)k c†(U,L)kσ c(U,L)kσ ,
(18)
kσ
are the Hamiltonians of the conduction electrons in the upper and lower leads, with electron
creation operators c†(U,L)kσ , and identical dispersion ǫk .
HQD =
X
ǫd d†σ dσ + Un↑ n↓ ,
(19)
σ
is the QD Hamiltonian with energy level ǫd measured from the equilibrium chemical potential
of the conduction electron fermi sea, impurity electron creation operator d†σ , and on-site
repulsion U for double occupancy of an up and a down spin electron, and
HI =
X
VU kσ [c†U kσ dσ + h.c.] +
k
X
VLkσ [c†Lkσ dσ + h.c.]
(20)
k
from the tunnel coupling between the upper and lower leads and the QD. Although two leads
are coupled, it is possible to make the transformation:
c†+kσ = cosθc†U kσ + sinθc†Lkσ
c†−kσ = −sinθc†U kσ + cosθc†Lkσ
(21)
where tanθ = VU /VL , to decouple the c†−kσ channel from any tunnel coupling to the QD.
This yields an interaction:
HI =
Xq
2
VU2kσ + VLkσ
[c†+kσ dσ + h.c.].
(22)
k
Thus, we retain the behavior of the Anderson impurity model under equilibrium conditions,
with the same chemical potential in both the upper and lower leads.
B. 2IK Model in Coupled-Quantum-Dots
The 2IK model in the coupled-QD context can be realized in several related configurations, differentiated by the geometry in which the coupling to the electrical leads is
implemented, as well as the different methods to couple the two QDs to each other. The
configurations that have been investigated include: (i) series-coupled quantum dots with
a direct tunnel coupling between QDs [12], (ii) parallel-coupled DQDs with direct tunnel
coupling [13], (iii) parallel-coupled DQDs with RKKY coupling mediated through a third
large QD [14], and (iv) parallel-coupled DQDs with capacitive coupling [19].
15
FIGURES
U
(a)
ΓU
QD1
QD1
t
QD2
QD2
L
ΓL
L
U
(c)
U
(b)
(d)
QD1
QD2
L
U
QD1
QD2
L
FIG. 1. Various configurations of coupled double-quantum-dots with tunnel coupling (V(U,L) ) to the
upper (U) and Lower (L) leads: (a) series-coupled with the inter-dot tunnel coupling (t) in line with the
conduction path through the DQD, (b) parallel-coupled with direct tunnel coupling between dots, (c) parallel-coupled with RKKY coupling via a third large dot, and (d) parallel-coupled with capacitive coupling
between dots. Note that the level broadenings are related to the tunnel coupling via Γ = |Vk |2 D(EF ).
These configurations are illustrated in Fig. 1. Although all of these have been experimentally realized, the last configuration involving inter-dot capacitive coupling has yet to
yield novel behaviors in the Kondo regime. This to a large part is due to the inability to
tune the coupling. This configuration will not be discussed further from the experimental
perspective, although from a theoretical perspective, it is an important system in which the
quantum critical point is expected to be reachable, as opposed to a crossover behavior. In
what follows, we discuss the first three configurations.
1. Series-coupled Double-Quantum-Dots
To date non-equilibrium transport studies of the first three configurations for coupled
double-quantum-dot (DQD) have yielded useful insight into their Kondo behavior relevant
to the 2IK model. We begin with the series-coupled configuration with an upper QD and
lower QD, each QD is coupled separately to its own lead, respectively. A tunnel coupling
connects the two QDs. The Hamiltonian for this system is given by:
H = HU + HL + HU QD + HLQD + HU I + HLI + HDQD ,
(23)
where as in the single QD case,
H(U,L) =
X
ǫ(U,L)k c†(U,L)kσ c(U,L)kσ ,
kσ
16
(24)
are the Hamiltonians of the conduction electrons in the upper and lower leads, with electron
creation operators c†(U,L)kσ , and identical dispersion ǫk .
H(U,L)QD =
X
σ
ǫ(U,L)d d†(U,L)σ d(U,L)σ + Un(U,L)↑ n(U,L)↓ ,
(25)
are the upper (U) (lower (L)) QD Hamiltonians, respectively, with energy levels ǫU d (ǫLd )
measured from the equilibrium chemical potential of the conduction electron fermi sea in
each lead, impurity electron creation operator d†U σ (d†Lσ ), and on-site repulsion UU (UL ) for
double occupancy of an up and a down spin electron, and
H(U,L)I =
X
V(U,L)k [c†(U,L)kσ d(U,L)σ + h.c.]
(26)
k
from the tunnel coupling between the upper lead and upper QD (lower lead and lower QD),
respectively. In addition to these terms, which give rise to the Kondo coupling of each QD
to its respective leads, a coupling between the upper QD and lower QD is also present:
HDQD = t
X
[d†U σ dLσ + h.c.)].
(27)
σ
Defining the level broadening for each dot due to coupling to the respective leads, ΓU and
ΓL , as:
Γ(U,L) =
X
k
|V(U,L)k |2 δ(ǫ(U,L)k − EF ),
(28)
under the conditions that each dot is tuned so that a moment is localized on each dot and
that the inter-dot tunneling is minimal (t ∼ 0), each dot has its own Kondo temperature
scale given by Eq. 16:
TK =
q
U(U,L) Γ(U,L) exp[−
π|ǫ(U,L)d |(ǫ(U,L)d + U(U,L) )
],
2U(U,L) Γ(U,L)
(29)
If the coupled DQD is further tuned to a fully symmetric situation, with all parameters
for the two QDs as well as the couplings to the respective leads being identical, i.e. with
UU = UL = U, ΓU = ΓL = Γ, theory indicates that the parameter t/Γ delineates two separate
regimes as the inter-dot tunneling, t, is tuned from zero. The theoretical calculations based
on approximate methods such as numerical renormalization group, and slave-boson mean
field theory (appropriate as U → ∞, produced the following interesting new behaviors in
different regimes [73,50,74–79]. For t/Γ < 1, the effective Anti-ferromagnetic coupling arising
from the tunneling, J = 4t2 /U plays a role analogous to the AF coupling between impurity
moments in the 2IK model in a metal. Thus for J/TK ≤ 2.5, the system behaves much like
a system of individual impurities each screened by its respective fermi seas, i.e. two separate
Kondo impurities. For J/TK > 2.5 exceeding the critical value, on the other hand, the two
impurity moments lock into a singlet, and thus the net moment disappears altogether, and
Kondo physics no longer takes place.
17
In the opposite limit of t/Γ > 1, J no longer plays a role as the large tunnel rate between
QDs give rise to a coherent superposition of the many-body Kondo state of each dot. Here,
the bonding state, corresponding to a singlet configuration, represents the ground state. This
singlet state is an entangled state, and is believed to be related to the spin singlet formed
under the condition of t/Γ < 1 and J/TK < 2.5. The different phases in the parameter space
is summarized in the phase diagram in Fig. 2.
U3
Coherent Bonding
U2
t/Γ
Kondo Singlet
1.0
U1 < U2 < U3
Stot = 0
Kondo
No Local
Regime
Moment
Atomic-like
Kondo
j(t) for U1
J=4t 2/U
2.5
j = J/TK
FIG. 2. Phase diagram for the 2IK system in a series-coupled DQD with a direct tunnel-coupling, t,
plotted in the t/Γ − j plane. In a DQD, t and j = J/TK are not independent, with J = 4t2 /U , being the
effective anti-ferromagnetic coupling between the local spin moments of the two dots. Here, U is the charging
energy on each dot site, TK the Kondo temperature (of each dot), and Γ the level-broadening of each dot
due to its coupling to the respective lead. TK and Γ are fixed in the diagram. The dashed line at t/Γ = 1
delineates a separation between regimes with different quantum ground states of atomic and molecular Kondo
correlations (Atomic-like Kondo and Coherent Bonding Kondo Singlet, respectively), while for t/Γ < 1 the
vertical line at j = 2.5 denotes the boundary between the Kondo regime and a local moment Stot = 0 singlet
regime discussed by Jones et al. [58]. This local moment singlet forms as the anti-ferromagnetic coupling
J between two QD spin moments becomes large, causing the total impurity moment to disappear from the
fermi sea. The curves with different values of U indicate the system trajectory as t is varied. In the case of
the parallel-coupled DQD, the situation is similar to the above when the source drain bias is small.
In the coupled DQD context, the inter-dot tunneling matrix (t) is a relevant perturbation
for the critical behavior of the 2IK model [80,81,73]. Thus the quantum phase transition
discussed in the 2IK model by Jones et al. cannot be reached, and there is instead a
cross over behavior between the singlet, coherent-bonding Kondo state and the state with
separate, atomic-like Kondo states one on each dot. The cross-over behavior was investigated
by numerical methods [80,81]. Using numerical RG calculations Sakai and Izumida found
that for instance, in the cross-over regime, the susceptibility no longer diverges as j ≡ J/TK
approaches the critical value of jc ∼ 2.5. Instead, the divergence is cutoff as:
χo
.
(30)
χ” ∼
(j − jc )2 + (1.5t/23TK )2
18
In terms of transport, calculations based on the slave-boson mean-field-theory method
predict a splitting of the Kondo peak in the differential conductance dI/dV as a function
of source-drain bias, when the coherent bonding singlet Kondo state is formed at large t, or
when J dominates over TK when t/Γ < 1. At small t, when the atomic-like individual Kondo
states are present, a single peaked resonance center at zero bias is expected. Thus, by tuning
t, a cross-over between double- and single-peaked behavior should take place. Under ideal
conditions rarely accessible in experiment, theory also predicts a dramatic negative differential conductance behavior at large source-drain bias, occurring when |eVbias | is increased
beyond the DQD Kondo temperature scale. This reduction of conductance with an increase
of the bias results from decoherence effects.
Other outstanding predictions are differences in the low temperature correction to the
conductance near the critical point, indicating nFL type behavior. The extent to which this
type of behavior is observable depends on how closely the critical point can be approached.
The novel, nFL universal scaling behavior as the temperature and source-drain bias are
varied will be presented in Section III C below, in conjunction with the 2CK effect.
2. Parallel-coupled Double-Quantum-Dots
In the case of parallel-coupled DQDs, each QD is coupled to the upper and lower leads at
the same time. The inter-dot coupling has been implemented in two ways, either by direct
tunneling [13]–as in the series-coupled configuration discussed above [12], or via an RKKY
coupling through a third large dot. While the direct tunneling leads to an anti-ferromagnetic
exchange between the spin moments, in the case of RKKY coupling, the exchange could in
principle be either ferromagnetic or anti-ferromagnetic. The ferromagnetic case leads to an
S = 1 impurity at low temperatures, and there will be an under-screening of the impurity
at low temperature. This behavior is well known and will not be discussed further.
Focusing on the AF case, this system behaves in a manner similar to the series-coupled
case, particularly when the parameters for the two dots are fully symmetric. Under equilibrium conditions in the absence of source-drain bias, this equivalence between the series- and
parallel-coupled cases is evident: In the parallel-coupled geometry implemented in experiment, the coupling of the DQD to leads takes place at the two spatially separated sites. Thus
this situation is similar to the metallic case considered by Jones et al. [58–60] where the two
impurities reside at spatially separated sites. The difference compared to the series-coupled
case is in the effective coupling to leads, leading to a level broadening: Γ = ΓU + ΓL . This
Γ enters in the expression for the Kondo temperature. This effective coupling comes about
because of the transformation given in Eq. 21, which enables the two leads (U and L) to be
transformed into one effective channel containing Kondo coupling to a QD, while a second
channel becomes decoupled. With small asymmetry in the dot parameters, however, the
transform is approximate.
There are notable differences in the transport behavior for the series and parallel cases.
For instance, under idealized condition of zero temperature, the series case has a conductance
approaching the unitarity value of 2e2 /h near the critical point where jc ∼ 2.5. On the other
19
hand, if a singlet is formed for j = J/TK > 2.5 when t/Γ < 1, or if the coherent bonding
Kondo singlet forms when t/Γ > 1, the conductance at zero bias rapidly reaches zero.
Moreover, even when t/Γ < 1, in the limiting case of t → 0, for which J → 0, at finite
T the conductance clearly must approach zero as the series path is cutoff. In contrast, in
the parallel-coupled case, this cutoff does not occur, due to the fact that the conduction
path does not pass through the pincher, which controls the inter-dot coupling t. Even in
the limit of t → 0, there is still a measurable conductance due to the presence of the Kondo
resonance in each QD running in parallel. In fact, the idealized conductance should reach
4e2 /h. When a singlet forms, however, the Kondo resonance at zero bias is quenched. This
leads to a reduction in the conductance. From a practical point of view, the parallel geometry
is more suitable for studying the behavior of a transition in the quantum ground state. In
real semiconductor QDs, the unitarity limit is rarely attainable, and the conductance is
highly sensitive to the value of t in the series geometry. In fact, it was found that in the
series-coupled geometry, a slight reduction in t already pushed the differential conductance
down to the noise limit [12,13].
U
ΓU
SD
LD
ΓLD
ΓL
L
FIG. 3. Schematic diagram of the coupled small and large dots proposed by Oreg and Goldhaber-Gordon
[82], for implementing the 2CK system. Here, the level broadenings of the small dot (SD) Γ(U,L) , arise from
coupling to the (upper, lower) leads, and ΓLD , from coupling to the large dot (LD), respectively, with
Γ(U,L) = |V(U,L) |2 D(EF ).
In Section III B 1 it was pointed out that in the presence of a finite t it is not feasible
to fully access the quantum critical point of the 2IK model in this coupled- DQD setting.
To reduce the rounding effect of the cross-over and approach the critical point more closely,
several variations of the geometry have been proposed. The key is to suppress the tunneling
coupling, which leads to charge exchange pass through the interconnection between the
DQD from one lead to another. One way is to utilize capacitive coupled between the DQD,
rather than tunnel coupling [50]. Another is to insert additional QDs between the two local
moments [51,52]. However, these proposals are technically even more challenging than the
experiments described in this review, and have yet to be fully implemented.
20
C. 2CK Model in Coupled-Quantum-Dots
In order for a 2CK system to be realized, it is necessary to have two independent channels
of fermi seas to be coupled to the impurity moment. The transformation in Eq. 21 shows
that having 2 different leads coupled to a QD in a semi-conducting device is not sufficient. It
is always possible to decouple one effective channel, leaving only one channel with AF spin
coupling to the moment. Adding more leads does not change this situation. In a seminal
paper, Oreg and Goldhaber-Gordon proposed a clever scheme to circumvent this difficulty
[82,83].
They added a ”third lead” fermi sea (Fig. 3). However, this third lead is in reality a
large quantum dot, with a small charging energy uLD and level spacing ∆LD . The charging
energy is small due to the large size and associated large capacitance of the dot to its
surroundings. The first two leads (U, L) are used for a transport measurement. For these, the
usual transformation (Eq. 21) yields one effect channel (fermi sea). The second independent
channel comes from the large dot fermi sea. At temperatures below the charging energy
kT ≪ uLD , but still sufficient large to far exceed the level spacing of the large dot, uLD ≫
kT ≫ ∆LD , the near continuum of state in the large dot act as a fermi sea for screening,
but the charging energy uLD prevents charge fluctuations, provided this large dot is tuned
to favor an integral number of charges on it. This suppression of charge fluctuation on the
large dot prevent spin-flip processes involving the transfer of charge between it and one of
the open leads from taking place. Consequently, the transformation of Eq. 21 no longer is
able to decouple the channel, leading to two independent channels of fermi sea for screening
the impurity moment. This charging energy is contained in the total Coulomb energy of
the system Hamiltonian, E(NSD , NLD ), with NSD electrons on the small dot (SD), and NLD
electrons on the large dot (LD):
1
C g Vg
1
C g Vg
E(NSD , NLD ) = ESD (NSD − SD SD )2 + ELD (NLD − LD LD )2
2
|e|
2
|e|
C g Vg
C g Vg
+EDQD (NSD − SD SD )(NLD − LD LD ),
|e|
|e|
where E(SD,LD) =
e2
C(SD,LD)
1
2
1−CDQD
/CSD CLD
, EDQD =
e2
CDQD
1
2
CSD CLD /CDQD
−1
(31)
, and
C(SD,LD) is the total capacitance of the small dot (SD) (large dot (LD)) to its surroundings, Cg(SD,LD) the capacitance of the respective dot to its control gate, Vg(SD,LD) the applied
gate voltage, and CDQD the capacitance between the SD and LD. From this expression, one
obtains:
1
uLD = EDQD
2
(32)
This coupled-QD system with a small and a large dot is versatile in that it is possible
to obtain manifestation of both the 1CK effect (1CKE) and the 2CK effect (2CKE). At
intermediate temperatures, where kT > uLD , the 1CKE is recovered, as charge fluctuations
21
on the large dot becomes no longer suppressed. Moreover, the 2CK fixed point requires
that the coupling to the impurity spin to be symmetric between the two channels. Any
asymmetry will also drive the system to the fermi liquid fixed point. Thus the fixed point is
delicate. Nevertheless, in principle it can be reached in an actual realization in coupled-QD
system [82,83,16]. In terms of the coupled-QD shown in the diagram of Fig. 3, this means
that the diagonal exchange JSD and JLD , given by [70]:
and
JSD = ΓSD [E(2, 0) − E(0, 1)]−1 + [E(0, 0) − E(0, 1)]−1 ,
(33)
(34)
JLD = ΓLD [E(1, 0) − E(0, 1)]−1 + [E(2, −1) − E(0, 1)]−1 .
must be equal to each other.
To experimentally establish nFL behavior in a transport measurement, it is therefore
necessary to tune to the symmetry point where JSD = JLD , and examine the low temperature
correction to the small QD conductance in the regime of relevance. In the QD context, the
Kondo resonance at the fermi energy increases the density of states for tunneling. This in
turn increases the conductance.
√
Theory predicts a T correction at zero source-drain bias. This stands in contrast to
the T 2 dependence of a fermi liquid, arising from second order processes. This
q correction
is related to the spectral density near the fermi energy EF , which behaves as |ω| and ω 2 ,
respectively for the two cases. This dependence also governs the quasi-particle decay rate for
the single-particle states near EF . The enhance decay for the 2CK case as ω → 0 indicates
that quasi-particles are no longer well-defined objects, and the fermi-liquid description is
breaking down.
In addition to the prediction of the non-fermi liquid correction, theory is able to predict a
universal scaling form for non-equilibrium transport with finite bias. Zarand et al. [51] have
argued that this scaling form should be identical for the 2IK and 2CK problems, behaving
as [47,83,84]:
( √
3
G2CK (0, T ) − G2CK (Vsd , T )
x − 1 for x ≫ 1
√
= π
(35)
2
0.0758x
for x ≪ 1
T
where x = eVkTsd (note Vsd = Vbias . In the low bias region, x ≪ 1, the scaling may be rewritten
in the form:
G2CK (0, T ) − G2CK (Vsd , T ) = f2CK x2 ,
√
with f2CK (T ) = 0.0758 T for an individual Kondo impurity.
The above can be contrasted with the Fermi-liquid scaling for the 1CK problem:
γ
G1CK (0, T ) − G1CK (Vsd , T )
= 2 x2 ,
2
T
TK
with γ a model dependent numerical factor of order unity.
22
(36)
(37)
IV. ELECTROSTATIC MODELS OF SINGLE AND DOUBLE QUANTUM DOT
SYSTEMS
In the standard, simplified model for the double-quantum-dot system, the Coulomb interaction between electrons within each individual dot is accounted for by a Hartree, charging
energy, with a neglect of exchange and correlation contributions. The filling behavior on
each dot proceeds in an alternating even odd fashion. When the total electron number is
even, all electrons are paired into singlets in occupied orbital levels. In the odd case, an
excess electron occupies the next available (lowest available energy) orbital. Thus, one ends
up with a net spin 1/2 local moment. This impurity moment, when coupled to the fermi sea
of the leads, produces a Kondo resonant state at low temperatures.
In order to create the Kondo system, it is necessary to properly tune and characterize the
filling behavior of the coupled-QD system, to enable the formation of S = 1/2 moments and
produce sizable Kondo and inter-dot couplings. In principle, one can start with an empty
dot, and add one electron to each dot. Technically, however, this turns out to be difficult due
to the spatial proximity of gate electrodes. The non-negligible mutual capacitance between
the metallic electrostatic gates and the two electron puddles contained in the QDs as well
greatly constrains the useful region in the tunable parameter space. The gates are needed
to form the dots, and to control the coupling between the dot and leads and between the
coupled dots. To date, all experiments on Kondo studies in coupled semiconductor QD
systems have been performed in QDs with ∼ 30 electrons (but odd), or larger [12–14,16].
To perform Kondo studies, it is essential to be able to identify odd valleys in the Coulomb
blockade spectrum of the coupled QD system, in the region of sizable coupling to the leads,
where Kondo coupling is strong. This is a tedious and time consuming task, as the contrast
between peak and valley is greatly reduced when the coupling to leads is larger, as opposed
to the situation of DQDs weakly coupled to the leads.
To understand how this identification process is implemented in experiment, we start
with a brief discussion of the QD stability diagram in terms of the electro-static energy of
the system [5,85]. This will be followed by the experimental characterization of the stability
diagram, and identification of the Kondo valleys for each QD in the coupled-DQD system.
23
(a)
Cg Vg =
n|e| (n+1)|e|
(c)
U
(n+1/2)|e|
E
R U , CU
Ec
Q,V
Vg
Cg
N-1
N
(d)
RL , CL
N
N+1 N+2
N+1
L
E
R
C
Ec
Cg Vg
(b)
n-1
U
(e)
n
N
n+1
n+2
|e|
N+1
CU
E
Q,V
Vg
Ec
Cg
∆
CL
n-1
L
n
n+1
Cg Vg
n+2
|e|
FIG. 4. (a) Model of a quantum dot with upper (U) and Lower (L) tunnel junctions to the leads. The
leads are used as source (S) and drain (D) contacts in an electrical measurement. The charge on the dot can
be tuned via the gate, Vg . Inset shows the equivalent circuit for the tunneling junction between the QD and a
lead. (b) Capacitive model of the single quantum dot. (c) Energy parabolas of the single electron transistor
quantum dot versus the number of charge on the dot, in the absence of the contribution from quantum levels,
at fixed gate voltages corresponding to Cg Vg /|e| = n, (n + 1/2) (dashed cureve), and (n + 1), respectively,
where n is an integer. A Coulomb blockade conductance peak takes place when the energy for the Q/|e| = N
and (N + 1) configurations become degenerate (N integer), as is the case for Cg Vg /|e| = (n + 1/2). (d) An
alternative representation of the energy parabolas plotted versus Vg , at fixed charge number, Q/|e| = N
and (N + 1), respectively. At a gate voltage (arrow) where the two parabolas cross, conduction takes place
giving rise to the Coulomb blockade peak. (e) Same as (d) but with the inclusion of quantum level spacing,
∆. Note that the gate voltage position of the energy degeneracy point is shifted for the (N-1) to N case.
24
A. Electrostatic Model of a Single Quantum Dot
We start with the single QD, which is nearly isolated from the upper and lower leads,
i.e. RU , RL → ∞. (See Fig. 4). For this nearly-isolated single dot case. Accounting for
the Coulomb repulsion by the electrostatic charging energy alone, produces the so-called
“constant interaction” model [86,87].
The electrostatic energy, E, of the small electron puddle on the QD is given by:
E=
Q2
Cg
−
QVg ,
2C
C
(38)
where C = CU +CL +Cg is the total capacitance of the dot to the environment, for a resistorcapacitor model of a single quantum dot depicted in Fig. 4(a). Note that to ensure charge is
quantized on the dot, it is necessary for the tunneling resistances, RU and RL to be large. In
the limit of very large tunneling resistances, the model reduces to the situation depicted in
Fig. 4(b). Coulomb blockade arises when the chemical potential of the dot is tuned via Vg
to favor an integral number (n, n+1, etc.) of electrons, i.e. favoring Cg Vg /|e| = n, n + 1, ...,
as depicted in Fig. 4(c) (solid curves). Since charge is quantized in units of e, in this case
the energy can be rewritten in the form:
E=
Cg2 2
(−N|e| + Cg Vg )2
−
V ,
2C
2C g
(39)
where we have Q = −N|e|, i.e. charge is quantized in units of −|e|, yielding a Coulomb
e2
for the addition or removal of an electron. Here, the second
charging energy of 21 EC = 2C
term quadratic in Vg and independent of Q is irrelevant and suppressed in the figures, since
what is important is a comparison of the system energy at different values of the charge, Q,
in integral multiples of −|e|. At temperatures below this charging energy scale, kT < Ec /2,
electron transport through the dot is suppressed. To facilitate charge transport, it is thus
necessary to tune the chemical potential towards a regime where a 1/2 integer number of
electron is favored:
1
Cg Vg = (n + )|e|.
2
(40)
Here the N and (N+1) electron states are degenerate and charge can flow freely through the
dot without the cost of a charging energy [86,88,89] as depicted in Fig. 4(c) by the dashed
curve. Alternatively, we may plot the energy as a function of Vg at fixed number of electrons
as shown in Fig. 4(d). At a Vg value where the parabolas for N electrons and N + 1 electrons
cross, Coulomb blockade is lifted and transport proceeds freely through the dot.
Inclusion of the effect of 0-dimensional quantum confinement introduces an additional
quantum energy level structure on top of the charging energy, EC , as depicted in Fig. 4(e) by
the presence of the quantum level spacing, ∆. In the absence of spin degree of freedom, or
for phenomena in which spin does not play a role aside from doubling the density of states,
and when higher-order virtual tunneling processes are neglected the current (I) through the
25
quantum dot in the limit of coherent tunneling is given by a convolution of the difference
of the Fermi function at the chemical potentials in the two leads and the Breit-Wigner
resonance formula: [90]
I(kT, eVbias ) =
e2
h
Z
dE[f (E + eVbias ) − f (E)]
Γ2
ΓL ΓR
.
+ (E − Eo )2
(41)
where f (x) is the Fermi-Dirac distribution function, ΓL,R denote the level broadening due
to coupling to the left and right leads, respectively, and Γ = 21 (ΓL + ΓR ), and Eo the energy
of the resonant level which may be tuned by the plunger gate, Vg .
(a)
U
(b)
U
CU= C l
R U, C U
C g1 = C g
C g1
Rt , Ct
Ct
Q2,V2
Vg2
Q2,V2
Vg1
Q1,V1
Vg1
Q1,V1
C g2
Vg2
C g2 = C g
CL = C l
RL , CL
L
L
FIG. 5. (a) Model of a series-coupled double quantum dot system with upper (U) and lower (L) tunnel
junctions to the leads used as source (S) and drain (D). A dot to dot tunnel junction (t) is also present. The
charge on each dot can be separately tuned via the gates Vg1 and Vg2 , respectively. (b) Capactive model of
the fully symmetric, series-coupled double-quantum-dot.
B. Electrostatic Model of a Double-Quantum Dot
When two quantum-dots are coupled together, the electrostatics become more complicated as a result of the mutual capacitance between each pair of conductors, including the
quantum dot metallic puddles and the various pincher and plunger gates used for the formation and control of the dots. Here we summarize the simplest scenario based on even-odd
26
filling of each of the two dots. To clearly illustrate the role of the electrostatics, we initially
consider a situation of weak coupling to the leads (Γ ≪ ∆).
This energy diagram can readily be calculated by generalizing the capacitive-resistive
model of a single dot (Fig. 4) to the double dot situation as is depicted in Fig. 5 [85,91–93].
It is most convenient to characterize the system using the capacitance matrix. Starting from
the basic charge-voltage difference relationship on the i-th conductor, the total charge Qi , is
given by:
Qi =
X
qij =
j
X
j
cij (Vi − Vj ).
(42)
Casting this into vector form with Q = (Q1 , ..., Qi , ...QN )T , V = (V1 , ..., Vi , ...VN )T , and
P
defining the capacitance matrix C , where Cij = ( k cjk )δij − cij , yields:
Q = C V,
(43)
and an electrostatic energy for the system, E, of:
1
1
E = VT C V = QT C −1Q.
2
2
(44)
Inclusion of the charge, Qv , and voltage, Vv , of voltage sources, the voltage, Vc , on the
conductors may conveniently be related to its charge, Qc , and the voltage settings on the
sources, and the capacitance submatrices between the conductors, Ccc , and between the
conductors and the voltage sources, Ccv [91–93,85]:
Vc = Ccc −1 (Qc − Ccv Vv ).
(45)
For the series-coupled double quantum dot depicted in Fig. 5(b) –electrostatics for the
parallel-coupled case is similar, such consideration leads to an energy at zero bias of:
1
1
E(N1 , N2 ) = EC1 N12 + EC2 N22 + ECt N1 N2
2
2
1
− [Cg1 Vg1 (N1 EC1 + N2 ECt ) + Cg2 Vg2 (N2 EC2 + N1 ECt )]
|e|
1 2 2
1 1 2 2
Vg1 EC1 + Cg2
Vg2 EC2 + Cg1 Vg1 Cg2 Vg2 ECt ]
+ 2 [ Cg1
e 2
2
1
Cg1 Vg1 2 1
Cg2 Vg2 2
= EC1 (N1 −
) + EC2 (N2 −
)
2
|e|
2
|e|
Cg1 Vg1
Cg2 Vg2
+ECt (N1 −
)(N2 −
),
|e|
|e|
2
2
(46)
1
, and Ci ≡ CU,L + Cgi + Ct is the total
where ECi = Ce i 1−C 21/C1 C2 , ECt = Ce t C1 C2 /C
2
t
t −1
capacitance of the i-th (U or L) dot to its surroundings. Substantial intuition may be
gained by examining the experimentally relevant and simple case of fully identical dots, each
27
with the same capacitive coupling to its respective plunger gate, i.e. CU = CL = Cl and
Cg1 = Cg2 = Cg . Eq. [46] then reduces to the form:
(b)
C t =0, E ct =0
(1,0)
(0,0) (0,1)
C t >0, E ct >0
Vg2
Vg2
(a)
(c)
(1,0)
(0,1) (1,1)
Vg1
Vg1
splitting
for (ii)
E
(d)
(i) (ii)
C t >> C l +C g
E ct ~ Ec
(iii)
Vg2
Cg Vg /|e|
Vg1
FIG. 6. (a) The energy diagram for a fully symmetric series-coupled, double quantum dot, for different
values of the inter-dot capacitance, Ct . The energy at fixed occupancy, (N1 , N2 ), is plotted versus the gate
voltage, Vg = Vg1 = Vg2 . (i) For Ct = 0, ECt = 0, and the two dots are isolated (short dashed curve). (ii) For
Ct > 0 but small compared to (Cl + Cg ), ECt < EC and the single degeneracy point is now split as indicated
by the two short arrows (medium dashed curve). This gives rise to a splitting of the Coulomb peak when
contrasted with the isolated case of Ct = 0. (iii) When Ct dominates, ECt ≈ Ec , and the two dots behave
as a single large dot with a doubling in the frequency of occurrence for the Coulomb peak as a function of
Vg (long dashed curve). (b) Charge stability, honeycomb diagrams in the Vg1 versus Vg2 plane, for the three
cases depicted in (a), where (b) Ct = 0, (c) Ct > 0 but small, and (d) Ct ≫ (Cl + Cg ), respectively. The
situations depicted in (a) correspond to a cut along the diagonal in these three diagrams, respectively.
1
Cg1 Vg1 2 1
Cg2 Vg2 2
E(N1 , N2 ) = EC (N1 −
) + EC (N2 −
)
2
|e|
2
|e|
Cg1 Vg1
Cg2 Vg2
+ECt (N1 −
)(N2 −
),
|e|
|e|
(47)
with C = C1 = C2 and EC1 = EC2 = EC . In the limit of zero inter-dot-coupling, for which
Ct = 0 and ECt = 0, the system behaves as two isolated but identical dots, with energy:
E(N1 , N2 ) =
EC
Cg1 Vg1 2 EC
Cg2 Vg2 2
(N1 −
) +
(N2 −
),
2
|e|
2
|e|
(48)
where EC = e2 /Co and Co = Cl + Cg . In the opposite limit of large inter-dot coupling and
dominant Ct ≫ (Cl + Cg ), so that C ≈ Ct , we have ECt ≈ EC = e2 /2Co and:
28
E(N1 , N2 ) =
EC
[−(N1 + N2 )|e| + Cg (Vg1 + Vg2 )]2 ,
2
(49)
and the system behaves as a single large dot, albeit with a charging energy, EC , which is one
half the isolated case.
If we were to tie Vg1 and Vg2 together so that Vg1 = Vg2 = Vg , and plot the electrostatic
energy at fixed N1 , N2 :
E(N1 , N2 ) =
EC
C g Vg 2
C g Vg 2
C g Vg
C g Vg
[(−N1 +
) + (−N2 +
) ] + ECt (−N1 +
)(−N2 +
),
2
|e|
|e|
|e|
|e|
(50)
versus Vg , for the three cases of: (a) Ct = 0 with ECt = 0, (b) Ct < (Cl + Cg ) with
ECt > 0, and (c) Ct ≫ (Cl + Cg ) with ECt ≈ EC , we see behaviors at the charge degeneracy
point(s) corresponding to two identical isolated dots, the development of a splitting due to
the finite ECt , and one large dot, as shown in Figs. 6(a)(i), (ii), and (iii), respectively, with a
concomitant doubling of the period of the Coulomb blockade conductance peaks versus gate
voltage compared to the isolated-dots case (Ct = 0), when Vg1 and Vg2 are tied together.
Here we have plotted the energy for fixed electron numbers (N1 , N2 ), which denote the excess
occupancy above up-down spin paired occupied quantum levels, with Ni =0,1 indicating the
excess occupancy on dot i, in Fig. 6(a) the parabolas depict the energy curve for the (0,0)
empty state, the (1,1) singly-occupied state on each dot, and the (0,1) and (1,0) states
where one electron occupies one of the two dots. When the inter-dot coupling is turned off,
Ct → 0, the (0,1) and (1,0) parabolas are degenerate so that the condition for which Coulomb
blockade is lifted occurs at one point in the diagram. This is the case of two identical but
isolated dots for which in each dot the Coulomb blockade is removed when it is tuned to favor
a half-integer number of electrons. Note that here the parabola maybe be shifted in energy
by a single particle quantum level spacing, ∆. However, such a shift does not qualitatively
change the picture aside from shifting the gate voltage position where the blockade is lifted.
When coupling is gradually introduced, the now non-zero inter-dot coupling , Ct , a lowering
of the electrostatic energy of the (0,1) and (1,0) states. The interception of the lower curve
with the (0,0) and (1,1) parabolas at two distinct points signals a splitting of the quantum
dot conductance peak into two.
In the more general situation where Vg1 6= Vg2 , it is useful to examine the stability
diagram in the Vg1 versus Vg2 plane. Such a stability diagram can be obtained by first
defining a chemical potential for each dot, µi :
µi ≡ E(N1 , N2 ) − E(N1 − δi,1 , N2 − δi,2 )
Cg Vgi
Cg Vgj
= EC [(Ni −
) − 1] + ECt (Nj −
),
|e|
|e|
(51)
and the addition energy for adding an electron on either dot, Eadd :
Eadd ≡ µi (N1 + δi,1 , N2 + δi,2 ) − µi (N1 , N2 ) = EC ,
29
(52)
where i={1,2} and i 6= j. With the definition of µ = 0 for each lead at zero bias, stability
for occupation (N1 , N2 ) is given by the requirement of µ1 < 0 and µ2 < 0. Such stability
diagrams in this idealized situation are shown in Fig. 6(b)-(d) for different values of Ct and
corresponding ECt . The three scenarios correspond to the situations depicted in the previous
Fig. 6(a)(i)-(iii). The presence of the six-sided polygon for general values of ECt has given
rise to the nomenclature “honeycomb” diagram. Inclusion of the quantum energy level, nonideality in real devices such as residual gate-voltage dependence of EC and ECt , and residual
mutual capacitance between gates lead to distortions of the honeycombs such as variations
in the area of honeycombs and a change in the slope of the domain boundaries. Again, in
the limit of very strong inter-dot coupling, the double-quantum-dot behaves as a single large
dot in accordance with expectation.
In an experiment the stable configuration (N1 , N2 ) is controlled by Vgi . A plot in the
(Vg1 ,Vg2 ) plane can be obtained and represents an extremely useful way to characterize the
DQD system. The above-mentioned limits of zero inter-dot coupling and strong coupling
are shown in Fig. 6(b) and (d), respectively. In available experimental situations, the capacitances are not easily tuned. In addition, ECt is typically small, often smaller than the
quantum dot level spacing, ∆. Instead, a splitting arises from a tunneling-coupling t between
dots. Here, within a quantum-mechanical picture the non-zero inter-dot tunneling splits the
energy of the symmetric and anti-symmetric orbital levels associated with the (0,1) and (1,0)
states. The splitting is proportional to t when the corresponding unperturbed energy levels
for these states are tuned to degeneracy. Such coherent coupling is essential in the use of
quantum dots as qubits for quantum computation. Such an energy splitting due to this coherence have been demonstrated in transport as well as microwave absorption experiments
[94,95]. Again this scenario takes place when coupling to the leads is negligible and the
properties are single-particle in nature.
As in the case of the individual dot, inclusion of quantum level spacing simply shifts the
parabolas by the respective level spacings, ∆. Going beyond single particle properties to
access regimes exhibiting properties of many-body spin correlation may be accomplished by
introducing coupling to the conduction electrons in the leads by increasing Γ to ∼ ∆.
30
V. DQD CHARACTERIZATION AND EXPERIMENTAL STABILITY
DIAGRAMS
A. Quantum Dot Parameters
(a)
(b)
V5
V1
V3
V2
V4
V1
V3
V5
1m m
FIG. 7. Electron micrographs of the series-coupled (a) and parallel-coupled (b) double-quantum-dots
used to study the 2IK model in the works of Jeong et al. [12], and Chen et al. [13], respectively.
The devices used in our studies of the tunnel-coupled DQDs, for both the seriesand parallel-coupled varieties [12,13], were fabricated by electron beam lithography on
a GaAs/Alx Ga1−x As heterostructure crystal (see Fig. 7). The crystal contains a twodimensional electron gas 80nm below the surface, with electron density and mobility of
n = 3.8 × 1011 cm−2 and µ = 9 × 105 cm2 /V s, respectively (see Fig. 7) The lithographic size
of each dot is 170nm × 200nm. If we focus on the parallel-coupled DQD in Fig. 7 (b), the
eight separate metallic gates were configured and operated with five independently tunable
gate voltages. The dark regions surrounding (and underneath) gates 5 represent 120nm thick
over-exposed PMMA, which serve as spacer layers to decrease the local capacitance thus preventing depletion, enabling each electrical lead to simultaneously connect to both dots. The
experiment was carried out at a lattice-temperature of 30mK and ∼ 45mK electron temperature. Standard, separate characterization of each dot in the closed dot regime yielded a
charging energy U of 2.517meV (2.95meV) for the left (right) dot, with corresponding dotenvironment capacitance CΣ ≈ 63.6af (54.3af ), and level spacing ∆ ≈ 219µeV (308µeV ).
Modeling the dot as a metal disk embedded in a dielectric resulted in a disk of radius
re ≈ 70nm(60nm) with 58 (43) electrons.
To characterize the DQD devices and demonstrate their full tunability, the Coulomb
blockade (CB) charging diagram of the conductance versus plunger gates V2 and V4 was
mapped out for different values of inter-dot tunneling, t, controlled by the central pincher
gate V 5, where an increase of the gate voltage increases t [13].
31
1. Weak-coupling to leads
V2(mV)
1
-710
-765
-740
-795
-670
2
-640
3
-860
-720 4
-830
2
G( e /h )
0.008
-765
-750
-795
-785
0
-740
-755
-710
V4(mV)
FIG. 8. Color scale plot of the logarithm of double-dot conductance as a function of plunger gate
voltages V2 and V4, which control the number of electrons on the left and right dot, respectively, in the
parallel-coupled DQD of Fig. 7(b), for increasing values of the inter-dot coupling, t, from panels 1 to 4.
The lead-dot coupling is weak, and correspondingly, the level broadening, Γ, is small compared to the level
spacing, ∆.
Panels (1) - (4) in Fig. 8 show the configuration at small tunneling coupling to the leads
(Γ ≪ ∆) for the parallel-coupled DQD. At weak coupling (t small), the electrons separately
tunnel through the nearly independent dots yielding grid like rectangular domains in Fig.
8(1). With increasing t charge quantization in each dots is gradually lost as the domain
vertices separate and the rectangles deform into rounded hexagon. At large t the two dots
merge into one and the domain boundaries become straight lines (Fig. 8(4)). It is thus
possible to observe the full range of behaviors described in the previous section (Fig. 6), as
the inter-dot coupling, t, is increased. Similar stability diagrams can be obtained for the
series-coupled case, except for the small inter-dot coupling limit, t ≈ 0, since the current
through the series-coupled DQD approaches zero in this instance.
2. Strong-coupling to leads-Kondo limit
The type of stability diagrams shown in Fig. 8 is not adequate for the purpose of studying
the Kondo effect, since it is not possible to distinguish an odd electron numbered from an
even numbered valley. Typically, a quantum dot of lithographic size ∼ 200nm in diameter
32
will contain ∼ 30 − 40 electrons. Ideally, one could imagine adding to an empty dot one
electron at a time, to achieve an odd total. Unfortunately, it is technically unfeasible due
to the complex mutual capacitances in the multi-gate device. For instance, changing the
electron number in the left dot via gate V 2 in Fig. 7(b) changes the coupling (and hence Γ)
to the leads at the same time. One often ends up with a quantum dot configuration, in which
the tunnel coupling becomes extremely sensitive to slight voltage changes, particularly when
the number of electrons is reduced, to the extent of unworkability.
-705
1
V2(mV)
4
3
-725
6
2
x2
5
-745
-650
-630
-610
V4(mV)
FIG. 9. Device characterization in the Kondo regime for the parallel-coupled DQD: Charging diagram
depicted in a grayscale plot of the conductance crest as a function of gate voltages V2 and V4, for a sizable
inter-dot coupling, t < Γ. The differential conductance (dI/dV ) traces in different valleys and the spin
configuration of the highest lying occupied electronic levels on each dot are superimposed. Note that a
single upward-pointing arrow denotes only an unpaired electron, and is not intended to represent the actual
direction of spin alignment [13].
Instead, to identify an odd valley, one relies on the direct observation of the Kondo
resonance. To rule out rare occurrences of a S = 1 Kondo resonance in an even valley, which
occurs when an unusually large exchange-energy is present, it is necessary to tune through the
neighboring valleys of +/- 1 electrons. This process, though conceptually straightforward,
is in fact tedious due to the capacitances, and can be further complicated by temporal drift
caused by nearby charge traps with long relaxation times.
The procedure is then to tune each QD separately to an odd valley, identified by the
presence of the S = 1/2 Kondo resonance, observable in the dI/dV curve versus sourcedrain bias Vbias . This is followed by combining the two QDs. The gate voltage settings
will need to be carefully adjusted to compensate for the shifts brought about by the mutual
capacitances. In the parallel-coupled DQD, further complication can arise from the need
to have a sizable inter-dot coupling, t. This condition is necessary in order to study the
quantum transition (or cross-over) behavior, and has the undesirable effect of reducing the
conductance contrast between the valley and peak regimes.
33
(a)
(b)
FIG. 10. Device characterization in the Kondo regime for the series-coupled DQD: (a) spin configuration, and (b) corresponding charging diagram depicted in a color plot of the conductance crest as a function
of gate voltages V2 and V4 (see Fig. 7), for an intermediate inter-dot coupling, t < Γ [12].
Procedurally, to obtain the desired large t, the center pincher gate V5 is set so that the
zigzag pattern in the charging diagram is barely visible (Fig. 9), ensuring that the Kondo
valleys can be located. To form the Kondo states in both dots, pincher gates V1 and V3
were tuned to give a sizable Γ ≈ ∆ to ensure strong Kondo correlation (Eq. 16), without
introducing complications arising from mixed-valence (Eq. 14) when Γ becomes too large.
In Fig. 9, we also indicate the Kondo resonance in the dI/dV versus Vbias for each dot for the
parallel-coupled case during the separate tuning process. Fig. 10 depicts a similar stability
34
diagram for the series-coupled configuration.
For the parallel case, an estimate for t can be obtained from the charging diagram, which
indicates a configuration close to the limit of a merged single large dot, so that the level
broadening π|t|2 /∆ should be comparable to the level spacing ∆, yielding t ≈ 150µeV , while
Γ is deduced to be ≈ 150µeV from the half-width of the CB peaks. The Kondo temperature
is roughly 500 mK.
(a)
(b)
0.1
dI/dV(e2/h)
0.12
0.20
0.10
0.16
0.08
0.12
0.0
-0.1
0.06
-0.12
(c)
0
0.12
V SD (mV)
0
V SD (mV)
0.1 0.08
-0.12
0
0.12
V SD (mV)
FIG. 11. The differential conductance, dI/dV , versus Vsd , showing the behavior of the Kondo resonance.
The dI/dV exhibits a clear double-peaked structure about zero bias voltage for large values of inter-dot coupling, t: (a) For the parallel-coupled DQD, with t set by different values of V 5 in the vicinity of V 5 ≈ −0.6V .
From top to bottom: V 5 = −0.5995V + ∆V , ∆V = 3, 2.5, 2, 1.7, 1.5, 1.3, 0.8, 0.3, 0mV , first cool-down, offset
by 0.005e2/h for each trace. Note t decreases from top to bottom. (b) For the series-coupled DQD at a fixed
value of t or V 5. (c) In contrast, for small value of t, a single-peaked behavior is observed in the parallel-DQD.
From top to bottom: V 5 = −0.6155V + ∆V , ∆V = 0, −0.5, −1, −1.5, −2, −2.5, −3, −3.5, −4.0, −4.5mV , second cool-down [13].
35
VI. DATA PRESENTATION-SEMICONDUCTOR COUPLED QUANTUM DOTS
In this section, we present the main experimental results on the Kondo effect in semiconductor coupled-QDs. We will present evidence for the existence of rich and complex
behaviors, such as a transition between different quantum ground states, and nFL behavior,
and compare these findings to theoretical predicts. Since much of the theoretical analysis have relied on approximate methods, such as slave-boson mean-field theory, numerical
renormalization group calculations, and ansatz about the fixed point within a conformal
field theory, etc., rigorous experiment tests of the predictions are essential. The coupled-QD
system will be shown to provide a versatile testing ground for such ideas and predictions,
and for exploring novel behaviors.
A. Experimental Results-2IKE
The 2IK model is one of the simplest nontrivial models, which contains a coupling between
Kondo spins, that was found to produce novel behavior beyond the familiar ferm-liquid
paradigm. As was discussed in Section III B, in this system a delicate phase transition
involving a nFL fixed point emerges. In the context of the semiconductor coupled-QD system,
the phase diagram was summarized in Fig. 2 for the case where coupling takes place via direct
tunneling between the QDs. Because the tunneling matrix t is a relevant perturbation in
the renormalization group sense, the transition is not sharp, and a crossover occurs between
different quantum ground states. Here, we focus on establishing the different ground states
as t is tuned. This will be performed by a careful examination of non-equilibrium transport
data, in particular, the evolution of the differential conductance dI/dV as the source-drain
bias is varied. In addition, data on the magnetic field and temperature dependences will be
presented to fully characterize the different regimes.
Several key observations comprise the main experimental findings on the 2IK system in
coupled-QDs. These include:
1) a double-peaked feature in the dI/dV versus Vbias curve at sufficiently large values of the
inter-dot tunneling matrix, t,
2) the existence of a transition (crossover) between a quantum state characterized by a
single peak at zero bias in the dI/dV for small values of t, to the double- peaked
behavior at large t (item 1 above), where a depression is observed at zero bias in place
of the peak structure;
3) the width of the peaks on the double-peak side is comparable to that on the single-peaked
side, to within a factor of ∼ 2, indicating that they likely arising from the physical
origin, i.e. correlated Kondo physics;
4) a −lnT dependence in the zero bias conductance below the transition, t < tc , reminiscent
of the Kondo effect in a single QD, while just on the double-peaked side, the zero-bias
36
conductance is non-monotonic in temperature, first decreasing with a reduction in
temperature, then increasing, followed by a further decrease below ∼ 80mK, where
tc could either be determined by the DQD level broadening, Γ, or by the effective
exchange J = 4t2 /U compared to TK ; and
5) the peak separation on the double-peaked side increases rapidly beyond the crossover
region.
The first three observations provide key indications that by tuning t, the ground state
change character, while maintaining Kondo correlations through out. These points will be
discussed in detail below. The last two illustrate the behavior of the transition (cross-over)
in the vicinity of the cross-over point. In addition, the behavior of the split peak in the
presence of an in-plane magnetic field, B, which causes a Zeeman splitting in the spin states
but has minimal effect on the orbital motion, will be shown to be consistent with such an
interpretation. In particular, the peaks cross and then split again for one direction of B, while
it splits further apart for the opposite B direction. In what follows, we present experimental
data illustrating the above behaviors.
In Fig. 11(a) and (b), we show the double-peaked behavior in the differential conductance
versus bias, dI/dV versus Vsd (Vsd = Vbias ), at large values of the tunneling coupling, t, for
a parallel-DQD coupled by direct tunneling, and for a series-coupled DQD, respectively.
The value of t decreases from the top trace to the bottom trace. The peak separation is
observed to decrease as t is reduced, as the peaks gradually merge. In Fig. 11(c), we show
the presence of the single-peaked Kondo resonance at smaller values of t obtained in the
parallel-DQD only. Note that the peak-width here is narrower than that of the merged
peaks in Fig. 11(a). The single peak is always centered at zero bias, while the peak height is
observed to decrease with a reduction in t. Fig. 12 contains the transition from the doublepeaked to single-peaked behavior in one continuous tuning of the inter-dot coupling t, via
gate V 5 (see Fig. 7(b)). The data were obtained under the condition of symmetric couplings
of the two QD to the leads. Similar transition behavior was observed in RKKY-coupled
DQD, where the RKKY anti-ferromagnetic coupling takes place through a third, large QD,
as shown in Fig. 13. Here the respective coupling to the leads is slightly asymmetric for the
two side dots, causing the resonance peak to appear slightly offset from zero bias at weak
inter-dot coupling.
37
(b)
2
dI/dV(e /h)
(a)
0.3
0.09
0.08
0.07
-0.12
(c)
2
dI/dV(e /h)
2
dI/dV(e /h)
0.4
0.10
0.2
-0.2
0
0.12
V SD (mV)
0.09
0.08
0.07
-0.12
0.1
0
0
0.12
V SD (mV)
0.2
V SD (mV)
FIG. 12. The behavior of the Kondo resonance peak in one continuous sweep of t, from larger to
smaller
values:
(a)
From
top
to
bottom:
V5
=
−0.5940V + ∆V ,
∆V = 1.5, 1.2, 0.9, 0.6, 0.3, 0.0, −0.3, −0.6, −0.9, −1.2, −1.5, −1.8, −2.1, −2.4, −2.7, −3.0, −3.3, −3.6, −4.0,
−4.3, −4.6mV , third cool-down as measured in Kondo valley 2 of Fig. 9, offset by 0.02e2 /h for each trace.
(b), (c) Selected data from (a) without offset. (b) The second quantum state regime (from top to bottom:
V 5 = −0.5943, −0.5940, −0.5937, −0.5934V ). (c) The first quantum state regime (from top to bottom:
V 5 = −0.5961, −0.5964, −0.5970, −0.5986V ) [13].
Taken as a whole, these observations provide compelling evidence that a change in the
non-equilibrium transport characteristics takes place as t is increased beyond some value,
as the single-peaked behavior gives way to double-peaked behavior. The existence of the
change is in agreement with theoretical predictions. Two scenarios are possible. The change
may take place either when t exceeds the critical value tc = Γ (tc /Γ = 1), or otherwise if
t < Γ (t/Γ < 1), when the effective anti-ferromagnetic coupling, J = 4t2 /U exceeds 2.5TK
(J/TK exceeds 2.5, see Fig. 2). The theoretical predictions were deduced using approximate
methods, and the experimental observations in three different coupled-QD configuration
thus lend considerable support to the physical scenarios discussed in such analysis. Further confirmation of the relevance of Kondo physics to the data is obtained by tuning the
electron occupation number on one of the two QDs, while maintaining all other parameters
unchanged. As shown in Fig. 14(b), when this is done, the doubled-peaked structure appears
in panels 1 and 3, and disappears in panels 2, 4, and 6, as the number of electron is changed
one by one, from odd to even and to odd again. When the electron number is even (2, 4,
6), an ordinary Kondo still remains in the other QD. Thus, in a parallel-coupled geometry,
38
in which both QDs are simultaneously connected to the source and drain in a parallel manner, the dI/dV still will exhibit a Kondo resonance, arising from the contribution of the
remaining QD with an odd number of electron.
(a)
(b)
FIG. 13. (a) SEM micrograph of the RKKY-coupled parallel DQD devices used by Craig et al. to study
the 2IKE. The RKKY interaction between the left and right dots is mediated through the third, large dot in
the middle. The bias voltage is applied between the bottom lead, simultaneously to the left and right leads.
(b) Differential conductance, dI/dV , through the left dot, for strong and weak inter-dot RKKY coupling,
showing the evolution of the Kondo resonance from a double-peaked to a single-peaked behavior [From N.J.
Craig, J.M. Taylor, E.A. Lester, C.M. Marcus, M.P. Hanson, and A.C. Gossard, Science 304, 565 (2004).
Reprinted with permission from AAAS.]
39
(a) -625
V2(mV)
4
-635
-650
2
dI/dV (e /h)
-630
-620
-610
V4(mV)
2.0
4
2
1.6
1
6
1.2
0
1
2
1
3
1
0
-1
0
1
0.5
0
3
2
2
0.25
0.20
-0.5
2
-640
(b)
0.30
3
1
-645
0
-1
6
1
2
-1
0
1
0
-1
0
1
VSD(mV)
FIG. 14. (a) Charging diagram for the parallel-coupled DQD in the first cool-down: Valleys 1 and
3 contain the odd electron-odd electron configuration, while valleys 2, 4, and 6 contain the odd-even or
even-odd electron configuration. (b) For the odd-odd valleys 1 and 3, a double-peaked Kondo resonance is
observed in the dI/dV , as shown in panels 1 and 3. In contrast, increasing the electron number by one for
either dot (valleys 2, 4, and 6) causes the Kondo resonance to revert to the usual single-peaked behavior in
the absence of inter-dot Kondo coherence in panels 2, 4 and 6, respectively [96].
It is worthwhile to point out that the presence of the depression at zero source drain bias in
the double-peaked regime is a clear indication that the total spin of the two-impurity system
is going away, signaling the formation of a entangled, spin-singlet S = 0 state. Suppose the
two spin moments were to form a state of total spin different from zero, i.e. S = 1, then one
would have instead expected that the lead conduction electrons will attempt to screen this
non-zero spin moment, leading to a two-stage Kondo effect, and a peak in the differential
conductance at zero-bias rather than a depression.
40
2
dI/dV(e /h)
Amp(e2 /h) W(µV) δV(µV) g(e 2/h)
0.10
0.09 (a)
0.08
0.07 -0.1
0.04
0
0.02
2G
VSD (mV) 0.1
(b)
0.00
1G
120 (c)
80
2G
40
0
60 (d)
40
20
0.03 (e)
0.02
0.01
0.00
-0.596
-0.598
2G Peak left
2G Peak right
1G
2G Peak left
2G Peak right
1G
-0.594
-0.592
V5(V)
FIG. 15. The linear conductance g ≡ dI/dV |Vbias =0 above background, peak splitting δV , width W ,
and peak amplitude(s) extracted by first subtracting a 2-Boltzmann simulated background signal followed
by a two or one-Gaussian fit [13]. The data are taken from Fig. 12(a) for the parallel-coupled DQD. The fit
and data are virtually indistinguishable. (a) Background subtraction. The fitting results are shown in (b)
- (e). The double peak feature visibly disappears for V 5 < −0.5943V , marked by the dashed line, where
δV ≤ W and a one-Gaussian fit is equally viable as a two-Gaussian fit.
Next we turn to the detailed characterization of the transition behavior. To do so,
we extract the linear conductance at zero bias g, the peak separation δV , peak width W ,
and peak amplitudes from the curves of Fig. 12. On both sides of the transition point,
at which V 5 ≈ −0.5943mV , we subtract a background (see Fig. 15 (a) solid curve) and
then fit to either two peaks or a single peak, depending on which side of the transition.
In the absence of precise theoretical expressions, several different functional forms were
used for the background and for the resonance line shape. The extracted parameters were
found to be fairly insensitive to the functional forms, and are presented in Fig. 15. Because
of the presence of the finite tunneling, t, which is a relevant perturbation to the critical
behavior as discussed in Sec. III B 1, the transition is not sharp and is in fact a rounded
crossover. Consequently, it is difficult to precisely locate the transition point. In Fig. 15
it is approximately indicated as the point at which the double-peaked splitting vanishes,
delineated by the vertical light dashed line. The extracted parameters, g, δV , W , and peak
41
amplitude (Amp), behave as follows (see Fig 15(b)-(e)). The amplitude of the single peaked
resonance reaches a maximum near the transition, while the peak separation increases rapidly
beyond the transition point, as t in further increased. The width of the resonances, whether
single- or double-peaked, are of similar value, varying of the order of a factor of 2 over
the entire range. This observation indicates that the energy scale is likely of similar origin,
namely due to Kondo-type correlations for both sides of the cross-over. Overall, the observed
behavior is similar to what is predicted for the series-coupled DQD. A full comparison is
somewhat problematic even if assuming the series case is relevant to the experiment situation
since theory assumes both T = 0 and the unitarity limit, a condition rarely realized. The
difference of parallel versus series geometries also further complicates matter. Nevertheless,
qualitatively, the experimental observations and theoretical predictions are largely similar.
Aside from the characterization above, the temperature dependence of the zero-bias conductance is distinct on the two sides of the crossover point. On the single-peaked side, the
peak height exhibits a dependence which is roughly logarithmic in temperature, as shown in
Fig. 16(a). This is what is expected from a 1CK Kondo behavior of a single QD. In contrast,
on the double-peaked side just beyond the cross-over point, the temperature dependence of
dI/dV at zero-bias is non-monotonic (Fig. 16(b),(c)). As temperature is reduced, the amplitude first decreases, then increases between ∼ 300mK down to ∼ 100mK, below which it
decreases again. This non-monotonic dependence is reminiscent of the singlet-triplet transition familiar in the even numbered electron valleys of a single QD. In that case, when an
unusually large exchange causes the triplet S = 1 state to be nearly degenerate with the
singlet S=0 state, this type of temperature dependence can arise due to two distinct Kondo
scales [97,98]. A larger scale, TK1, is associated with the screening of the S = 1 total spin
by one channel of electron sea. Above TK1 , the differential conductance dI/dV decreases
initially with decreasing temperature as a result of Coulomb blockade in the even valley.
This decrease is followed by an increase in dI/dV for T < TK1 as the S = 1 spin is partially
screened, leaving a residual spin of 1/2. Below a second Kondo scale, TK2 < TK1 , a second
channel of conduction electrons screens the remaining 1/2 spin, forming a singlet state. As
this occurs, conduction is suppressed at T → 0. This scenario is qualitatively similar to the
data in Fig. 16(b) and (c).
In a DQD, as a first approximation one may start with a situation in the absence of any
Kondo correlation. Then, there is a singlet- triplet splitting of the DQD levels, associated
with the symmetric and anti-symmetric linear combinations of the QD orbital levels. We
only consider the mixing of levels where double occupancy on either QD is forbidden due to
the presence of a sizable on-site repulsion energy U. If the orbital levels, one on each QD,
are tuned to degeneracy, then in the the presence of the tunneling coupling t, the splitting is
given by 4t. If t is small, or if the exchange energy is large, the situation is analogous to the
single QD case close to singlet-triplet degeneracy. This scenario is only fully correct when
the Kondo energy scale is small compared to t. The situation here is even more intriguing in
our coupled DQD since TK is not small. As a consequence, the singlet-triplet transition is of
many-body origin, taking place near the degeneracy point of the coherent bonding (singlet)
and anti-bonding (triplet) Kondo states corresponding roughly to the symmetric and anti42
symmetric combinations of the correlated Kondo states of each QD. The resultant splitting,
as measured by δV and shown in Fig. 15(c), is strongly renormalized from the bare value of
2t [50].
2
(e /h)
(b)
0.09
V=0
0.100
g
dI/dV(e 2/h)
(a)
270
0.068
0.08
0.064
0.07
100
T(mK)
100
T(mK)
280
0.075
0.06
47 mK
-0.12
0
0.12
-0.12
VSD(mV)
(c)
50 mK
0
0.12
VSD(mV)
0.16
0.11
dI/dV(e 2/h)
0.14
0.09
600
0.12
0.10
100
T(mK)
500
400
300
100
0.08
40
0.06
-1.0
65
200
-0.5
0.0
0.5
1.0
VSD(mV)
FIG. 16. Temperature dependence of dI/dV versus Vsd = Vbias for the parallel-coupled DQD within
the Kondo valley of Fig. 12(a) at slightly displaced voltage settings (due to temporal drift over a 2 day
period): (a) First quantum state regime with single-peaked resonance. (b) Second regime near the transition
point. Insets: dI/dV |Vsd =0 versus T (large dots are from curves shown). (c) Second regime exhibiting the
double-peaked resonance near the transition point, within the Kondo valley of Fig. 11(a). Inset: dI/dV |Vsd =0
versus T .
43
Additional information regarding the double-peaked quantum state can be obtained from
the dependence of the splitting of the peaks as a function of the in-plane magnetic field. The
application of the in-plane B field adds a Zeeman term to the spin states, but does little
to alter the orbit behavior. Thus, the splitting merges and splits apart again as shown in
Fig. 17. However, it is surprising that there are only two peaks. Since the triplet excited state
is expected to split by the Zeeman field into three levels, naively one might have expected
three peaks for each polarity of the voltage bias, producing 6 peaks in total. Instead, likely
due to increased decoherence at large Vsd , only two peaks are observable.
FIG. 17. In plane magnetic field dependence of a symmetric Kondo resonance peak in the series-coupled
DQD (Fig. 7(a) and 11(b)). Traces are for B = 0, 0.25, 0.5, 0.75, 1.0, and 1.25 T. The curves are offset by
0.02e2/h for clarity. The Zeeman splitting from two split peaks enhance the conductance at zero bias as the
field increases because of the overlap of the density of states from two peaks. At higher fields, they separate
again in a manner similar to the single-peaked resonance [12].
The evidence presented above enables to identify the experimentally observed transition
(crossover) to be that discussed in the theoretical analysis of the 2IK problem in coupledDQDs. Although to date no theoretical work has addressed the parallel-coupled DQD
with inter-dot coupling under non-equilibrium conditions. Nevertheless, because the Kondo
anomaly occurs under slightly non-equilibrium conditions it is likely that we may identify
the observed transition with the quantum critical phenomenon discussed in the two-impurity
Kondo problem in series-coupled DQDs under non-equilibrium conditions [73,50,74–79]. Beyond the identification of the transition between quantum states, semi-quantitatively, it is
informative to roughly estimate key parameters associated with the transition behavior.
Within the theoretical scenarios, the peak splitting δV must be compared to 4t ≈ 600µeV
44
or 2J = 8t2 /U ≈ 180µeV . (Note that in the open dot regime U is expected to be reduced
from its closed dot value by roughly 1/3 [5], yielding U ≈ 1meV .) The reduction of δV
compared to 4t and 2J are in agreement with theory and lends further credibility to our
identification of the quantum transition (cross-over).
Despite the considerable amount of information obtained thus far from these detailed
experimental studies, it is clear that these experiments are yet unable to address two important aspects. Firstly, theory predicts an interesting phase transition under non-equilibrium
conditions as the source drain bias Vbias is increased. At finite t, when Vbias exceeds TK ,
the Kondo state in both sides of the transition was predicted to be destroyed by decoherence effects. A dramatic consequence is a sudden reduction of the differential conductance,
leading to the phenomenon of the negative-differential-conductance (NDC) [50]. Thus far,
no experimental evidence for such an intriguing behavior has been found. This could be a
result of thermal smearing and residual asymmetry in the DQD; in any event it has proven
to be difficult to reach the unitarity limit of maximal conductance in semiconductor QDs,
and this maybe symptomatic of the relative ease of non-idealities to smear out more subtle
effects and to reduce their visibility. Secondly, it is evident from the data in Fig. 12 that
the differential conductance dI/dV deviates from the zero-bias value in a roughly quadratic
manner. This is the expected behavior in a fermi-liquid
system. Thus the data does not inq
dicate any evidence of a non-fermi-liquid type |Vbias | correction away from Vbias = 0. Thus,
it has not proven possible to attain a close-approach to the intriguing nFL critical point due
to rounding in the cross-over region. Nevertheless, despite the inability to approach the nFL
quantum critical point, the results presented here go a significant ways to helping unravel the
behavior in an 2IK system in the context of the coupling of two Anderson-type impurities. It
thus may shed light on the relative ease, or difficulty, in reaching this novel nFL fixed point
in strongly correlated systems, such as heavy fermion systems. At the minimum, it points
to the fact that in order for the nFL behavior to emerge, it will require an AF coupling
between impurities, but very likely, this must occur in the absence of tunnel-coupling, so as
to prevent particle exchange between reservoir (fermi seas) through the entire DQD (through
both dots), a process which proceeds through a connection between the QDs.
45
Vds
n
1G
I
250
0
(K–2)
12 mK
24 mK
28 mK
38 mK
200
150
T2
g(0, T ) – g(Vds , T )
Vex
sp
(c)
300
B
100
50
–40
–60
–100
–60 –40 –20 0 20 40 60
eVds 2
kT
–60 –40 –20 0 20 40 60
eVds 2
kT
( )
( )
0.2
1CK scaling
12 mK
22 mK
28 mK
39 mK
49 mK
Theory
1.2
0.8
(K–2)
(K–0.5)
(f)
2CK scaling
0.4
0.0
0
–10 0 10
Vds (mV)
–4
–2
0
2
4
()
eV 0.5
kT
200
T2
0.4
c=
–282 mV
–260 mV
–256 mV
–244 mV
(e)
g(0, T ) – g(Vds , T )
g (e2/h)
0.6
12 mK
T 0.5
0.8
g(0, T ) – g(Vds , T )
(d)
12 mK
21 mK
27 mK
39 mK
45 mK
–80
–120
0
1 µm
–20
T2
(b)
bp
(K–2)
c
g(0, T ) – g(Vds , T )
(a)
100
0
–100 –50
12 mK
22 mK
28 mK
39 mK
49 mK
0
50
100
()
eV 2
kT
FIG. 18. (a) Electron micrograph of the type of device used by Potok et al. to study the 2CKE. The
large dotfermi seaprovides a second channel to screen the impurity moment on the small dot under the
condition ∆ELD ≪ kT ≪ uLD . (b) 1CK scaling to the universal curve in Eq. 37 when the device is tuned to
asymmetry with JLD < JLeads with Kondo coupling between the small dot and leads. Conductance is peaked
at zero bias. (c) 1CK scaling with JLD > JLeads with Kondo coupling between the small dot and large dot.
The conductance is a minimum at zero bias. (d) (e) (f)– 2CKE: (d) Tuning of the 2CK conductance from
a maximum to a minimum. The symmetry point JLD ≈ JLeads occurs near c = −260mV . (See (a) for the
c-gate). (d) Scaling of data at different drain-source bias, Vds , and temperature, to the 2CK scaling form,
Eq. 35, showing the collapse of the data to the universal curve (green). (e) Attempt to scale to the 1CK
form shows much poor quality [16].
B. Experimental Results-2CKE
The Hallmark of nFL behavior in a 2CK system is in the low temperature behaviors of
the system, whether in the transport characteristics or in thermodynamic quantities. The
behaviors should provide evidence that in this system the spectral densityqis different from
the usual ω 2 form familiar in the context of the fermi-liquid to a nFL |ω| dependence.
q
√
Here ω is measured relative to the fermi-energy. This nFL |ω| form translates to a T
q
functional form at low temperatures, or a |eVbias | form, depending on which energy scale
dominates. As discussed in Section III C, in the coupled-QD implementation proposed by
Oreg and Goldhaber-Gordon, where a second independent screening channel fermi sea is
provided by a large quantum dot characterized by a small charging energy uLD and even
46
smaller level spacing ∆LD , there is a range of temperatures, ∆LD ≪ T ≪ uLD in which
2CK behavior should be present. This experiment is extremely challenging from several
perspectives. Firstly, it is necessary to have a significant range in the temperature or bias
voltage, in order to fully distinguish between the fermi-liquid and non-fermi-liquid behaviors.
To this end, the situation is greatly aided by the availability of predicted universal scaling
behaviors in the variable eVbias /kT , as pointed out in Section III C. Thus, there are two
independent knobs to turn, namely T and Vbias . Secondly, from a technical standpoint, one
must deal with the delicate fine-tuning of the controlling electrostatic gates in the coupledQD device, in order to achieve symmetry between the AF couplings of the two independent
channels, JLD and JLeads to the impurity moment on the small QD. Any residual asymmetry
tends to drive the system away from the 2CK behavior towards the 1CK behavior. This fine
tuning is complicated again, by the complicated mutual capacitances between all metallic
islands/leads. Thirdly, to achieve temporal stability from charge fluctuations in nearby traps,
it is essential to work at very low temperatures. To reach such extremely low temperatures
(∼ 12mK) in the electronic system, it is imperative to adequately filter out any residual
electromagnetic noise in the system. All these considerations together restrict the useful
temperature and bias range between 10 − 100mK or equivalent in source-drain voltage.
Potok et al. [16] succeeded in performing this difficult experiment, providing the first clear
evidence for a marked difference in the differential conductance dI/dV between the 2CK
and 1CK Kondo regime. By tuning JLD and JLeads to a symmetric situation or asymmetric
situation, respectively, the 2CK and 1CK cases can be accessed and contrasted. Fig. 18(a)
shows an electron micrograph of the type of device they studied, with a coupled-small QD
and large QD. The small dot is connected to two electrical leads. Fig. 18(b) shows the
1CK behavior of the system, when the local spin moment of the small dot forms a Kondo
resonance with the electrons in the leads. A conductance maximum is observed at zero
voltage bias. The differential conductance at various values of temperature and bias voltage
can be fairly nicely collapsed into a single universal curve of the fermi-liquid form (Eq. 37).
However, when the system is tuned so that the Kondo state is formed between the local
moment and the large QD ”fermi sea” instead, a suppression of the conductance is observed.
The different curves at various values of temperature and bias voltage (Vbias = Vds ) are now
collapsible into an inverted V-shaped curve as shown in Fig. 18(c), but still of the fermi liquid
form. In contrast, when the system is tuned to the 2CK regime (Fig. 18(d), (e), and (f)),
the differential conductance curves fail miserably to produce a single scaling curve (f), when
plotted in the fermi-liquid form. Instead, if plotted in the 2CK scaling form, all the curves
come quite close to collapsing into a single universal curve (e). This set of data indicates that
very likely, nFL behavior is indeed present in this regime, thus providing the first evidence
of nFL behavior in a Kondo system. These data provide the first compelling evidence for
such novel behavior in a Kondo system.
47
VII. METALLIC SYSTEMS
(d)
(c)
FIG. 19. Split zero-bias peak in a gold nano-particle device, sprinkled with magnetic cobalt atoms [23].
(a) Schematic of the device. Bottom: Expected dependence of the zero- bias conductance on the scaled
RKKY coupling (I/TK ). (b) Differential conductance, G = dI/dV , as a function of bias, Vb , and gate
voltage, Vg . The split zero-bias anomaly vanishes when an extra electron is added to the quantum dot
(Vg = −1). Dashed lines (diamond edges) are a guide to the eye. Color scale from 28µS (black, dark blue
online) to 55µS (dark gray, dark red online). T = 2.3K. (c) and (d) Temperature dependence of the split
zero-bias peak: (c) G ≡ dI/dVb as a function of bias Vb , and (d) non-monotonic temperature dependence of
the conductance at Vb = 0V . Line is a guide to the eye.
Thus far we have focused on fully tunable semiconductor systems. Metallic quantum
dots and tunnel junctions have also exhibited similar Kondo physics. In metallic systems,
the tunnel coupling is difficult to tune. However, if a large number of devices are fabricated,
by chance some devices possessed the appropriate coupling to the leads to exhibit Kondo
behavior. Because of their small size, typically below 10nm down to atomic [20–23], the
relevant energy scales tend to be large, thus giving the advantage of larger dynamic range
in temperature and energy (eVbias ). For example, single-dot Kondo resonance is routinely
48
observed in gold nanoparticles coupled to leads, fabricated by a break junction technique
using electro-migration [23]. The addition of magnetic impurities to such devices can lead to
physics similar to the 2IK model. In Fig. 19(c) and (d), we show the temperature evolution of
the differential conductance for a gold grain positioned between a break junction (Fig. 19(a)),
with cobalt atoms randomly sprinkled onto the gold nanoparticle. The gold nanoparticle
itself forms a Kondo resonance with the leads, with a Kondo temperature of ∼ 60K. The
cobalt spin (S = 1/2) and the gold nanoparticle spin (S = 1/2) moment interact via an
RKKY interaction, I. The coupling I can be either anti-ferromagnetic (AF) or ferromagnetic
(FM). In either case, a split peak results. In the first case, the two spins lock into a singlet,
giving rise to a zero-bias depression. This is the 2IK case. In the other case, the two spins
lock into a spin S = 1 state. However, the Kondo scale for the triplet state is greatly reduced
compared to the scale for individual spins, to much below 60 K. Thus, at high temperatures
above the triplet scale but below the indvidual Kondo scale, a split peak is also observed.
FIG. 20. (a) 2CK scaling for the differential conductance in Al/Al-oxide/Sc planar tunnel junctions
(black curves in color version). The Kondo effect is believed to originating from a few localized spin-1/2 Sc
atoms situated slightly inside the Al-Oxide/Sc interface [24]. Solid curves stand for high T data, and dotted
√
curves for low T data. (b) f2CK , the scaling function as a function of T for three different junctions. (c)
Expanded view of the 2CK scaling for Samples A and B of panel (a).
Alternatively, 2CK behavior has recently been reported in Al/Al-oxide/Sc planar tunnel
junctions. The Kondo effect is believed to originating from a few localized spin-1/2 Sc atoms
situated slightly inside the Al-Oxide/Sc interface. Thus just as the above, in this system since
the active impurity is atomic, the energy scale is considerably larger than the corresponding
49
semiconductor case, with a TK ∼ 10 − 20K! Yeh and Lin found that after removal of a
background and symmetrizing the Kondo resonance in the differential conductance versus
source-drain bias, the curves in the 2CK temperature regime can be plotted in the 2CK
scaling form, producing a very good collapse of √the data into one single universal curve,
as shown in Fig. 20(a). Fig. 20(b) depicts the T behavior of the function f2CK in the
universal scaling curve (Eq. 36) for the intermediate temperature range, where 2CK scaling
is observable. Thus, it appears that it is possible to produce nFL Kondo behavior in this
atomic system. It is not entirely clear whether the identification of 2CK behavior is accurate.
It is also possible that it arises from a 2IK scenario, with AF coupling between two nearby
Sc spin moments, each of which couplings to the fermi sea at its own location. In either case,
it is remarkable that the scaling behavior works so nicely; from a theoretically standpoint,
it is not obvious what key ingredients have enabled the system to approach the nFL fixed
point with such closeness. Nevertheless, if confirmed this may indicate that Kondo physics
is potentially relevant to the heavy fermion system. There, a debate has arisen as to whether
coupled Kondo spins is responsible for the observed nFL behavior, or instead, the proximity
to a quantum critical point of some other variety is at play. The key issue, which has made
it less favorable for the Kondo scenario is the perceived difficulty to access the nFL critical
point. As discussed in Section III C, the critical behavior is quite delicate, and can easily
be rounded into a cross-over by asymmetries in e-h channels, or in the Kondo coupling of
different fermi sea channels. The observation of nFL scaling, and the elucidation of the
conditions for achieving nFL behavior would go a ways to bringing a fuller understanding of
these strongly correlated systems.
VIII. SUMMARY
In this review, we have presented the latest experimental results and theoretical understanding of the Kondo effect in the coupled-quantum dot systems. Several significant predictions for the 2-impurity Kondo model and for the 2-channel Kondo model were observed in
the experimental systems. In particular, in the 2IK case, the transition (cross-over) between
a ground state exhibiting a double-peaked behavior in the dI/dV , with a suppressed conductance at zero bias voltage, to a different ground state with a single-peaked Kondo resonance
exhibiting a conductance maximum was observed [12–14,23]. In the 2CK case, recent work
provided the first evidence for non-fermi-liquid behavior [16,24]. These pioneering studies
have now opened an avenue for more detailed studies of the complex and intriguing quantum
phase transitions, which have been predicted to exist in these model Kondo systems.
One important direction for the next generation of investigations would be to gear toward elucidating the relevance of the coupled-Kondo systems to heavy fermion physics, by
uncovering the conditions under which nFL behavior can be observed. For example, more
sophisticated inter-dot coupling [51,52], or tunable capacitive-coupling, may allow the 2IK
nFL fixed point to be closely approached. Other interesting directions include: (i) the study
of multi-channel systems beyond two channels and to observe the evolution of the power-law
exponent in the temperature scaling, or bias voltage scaling, with increasing channel number,
50
(ii) under-screened Kondo effect, when one-channel of fermi-sea attempts to screen two or
more impurity spins, and (iii) study of SU(4) Kondo systems, such as in carbon nanotubes.
Such studies will undoubted further deepen our understanding of Kondo physics, and its
relevance to the strong-correlated systems such as the heavy fermion systems, and other
related systems.
Acknowledgement We wish to acknowledge valuable contributions of our collaborators, M.R.
Melloch, and J. Jeong. We also would like to acknowledge helpful discussions with S. N. Li, J.
J. Lin, and C. H. Chung. This work was supported in part by NSF grant No. DMR-9801760,
DMR-0135931, and DMR-0401648.
51
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