Kondo Effect in Coupled Quantum Dots. Abstract

Transcription

1 Kondo Effect in Coupled Quantum Dots A. M. Chang +, J. C. Chen + Department of Physics, Duke University, Durham, NC Institute of Physics, Academia Sinica, Taipei, Taiwan + Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan 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

2 Contents I INTRODUCTION 3 II THE KONDO EFFECT 6 A Single Impurity Kondo Effect B Connection to other Physical Systems and Models C Two Impurity and Two-Channel Kondo Effects The two-impurity Kondo Effect The two-channel Kondo effect III QUANTUM DOT MODEL SYSTEMS 14 A 1CK Model in a single Quantum Dot B 2IK Model in Coupled-Quantum-Dots Series-coupled Double-Quantum-Dots Parallel-coupled Double-Quantum-Dots C 2CK Model in Coupled-Quantum-Dots IV V VI ELECTROSTATIC MODELS OF SINGLE AND DOUBLE QUAN- TUM DOT SYSTEMS 23 A Electrostatic Model of a Single Quantum Dot B Electrostatic Model of a Double-Quantum Dot DQD CHARACTERIZATION AND EXPERIMENTAL STABILITY DIAGRAMS 31 A Quantum Dot Parameters Weak-coupling to leads Strong-coupling to leads-kondo limit DATA PRESENTATION-SEMICONDUCTOR COUPLED QUAN- TUM DOTS 36 A Experimental Results-2IKE B Experimental Results-2CKE VII METALLIC SYSTEMS 48 VIII SUMMARY 50 2

3 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

4 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

5 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

6 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 E K, or equivalently the Kondo temperature, T K = E K /k, where k is the Boltzmann constant. At temperatures much above T K, the impurity behaves much like a free spin an indication of asymptotic freedom at high energies, while below T K, 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 = kσ ǫ k c kσ c kσ + JS s(0). (1) where ǫ k 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) = ρ o + at 2 + bt 5 + c{j 2 + J 3 D(E F )ln(e BW /T)}, (2) where ρ o 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(E F ) the conduction electron density of states at the fermi energy E F, and E BW the conduction electron bandwidth. The logarithmic divergence in the third order term J 3 ln(e BW /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 E K = kt K = E BW exp[ 1/JD(E F )]. (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) = E BW 1 [1 JD(E F )ln(e/e BW )], (4) 6

7 and diverges at E K = kt K, 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 T K as indicated in Eq. 2 in this regime where perturbation theory is valid, attaining a saturated value as T approaches zero below T K. 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 IIIC. Aside from the contribution to the transport characteristics, the Kondo scattering also modify the thermodynamic quantities, such as the specific heat, c v, and the magnetic susceptibility, χ. In bulk metals doped with magnetic atoms, the impurity contributions are again fermi-liquid-like, with c v T, and χ constant. In addition, the Wilson ratio, R W = (δχ imp /χ)/(δc v,imp /c v ), 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

8 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 l K v F h kt K (a o, λ F ), (5) where a o is the lattice size and λ F 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

9 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 distinct from the fermi liquid. These nfl fixed points yield a spectral density, which behaves as ω at low energies hω 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 c v and χ, which differ from a fermi liquid. In particular, the low temperature correction to the transport coefficients behaves as T rather than T 2, 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/e BW ). 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

10 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: with H = H o + H imp, (6) H o = kσ ǫ k c kσ c kσ, (7) and H imp = J o [s(r 1 ) S 1 + s(r 2 ) S 2 ] + J 1 S 1 S 2. (8) Here H o refers to the Hamiltonian of the conduction electrons, with energy dispersion ǫ k k (after linearization about the fermi point), and electron creation operators, c kσ, with σ 10

11 labeling the spin. The interaction term H imp involving the impurity spin moments S i, i = {1, 2}, at locations r i, respectively, contains an exchange coupling to the conduction electron spin at their respectively positions, with coupling constant J o, as well as an impurity-impurity exchange with coupling J 1. 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]: H imp = J + (S 1 + S 2 ) (c k e σc ke + c k o σc ko) (9) k k + J (S 1 + S 2 ) (c k e σc ke c k o σc ko) k k + J m (S 1 S 2 ) (ic k e σc ko + h.c.) + J 1 S 1 S 2. k k 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 +, = (J e ± J o )/2, and J m = (J e J o ) 1/2, with the even and odd couplings given by J e,o = (J o /2)(1 ± sink F R/k F 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, k F, 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 J 1, a total effective impurity-impurity coupling J is present. By taking J +, J, J m, and J 1 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 j c = J/kT K 2.2. The staggered spin susceptibility and the heat 11

12 capacity, c v, were found to diverge at the fixed point as 1/(j j c ) 2. In terms of the low temperature dependence, however, there is no significant deviation from the fermi-liquid functional dependence. However, the Wilson ratio, R W, measuring the fractional change (due to the impurities over the overall metallic values) of c v 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

13 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 behaves as ω at low energies. The low temperature χ and c v are modified from the fermi liquid behavior by an addition logarithmic T factor. Here, the Wilson ration R W 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

14 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]: where H = H o + H imp + H I, (10) H o = kσ ǫ k c kσ c kσ, (11) is the Hamiltonian of the conduction electrons, with dispersion ǫ k, and electron creation operator c kσ. H imp = σ ǫ d d σd σ + Un n, (12) is the localized impurity Hamiltonian with energy level ǫ d 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 H I = k V k [c kσ d σ + h.c.] (13) arising from coupling between the conduction electrons and the local impurity site. When U, the on-site repulsion, is large, i.e. U (ǫ d, Γ), with Γ = k V k 2 δ(ǫ k E F ), being the impurity level broadening, a Schrieffer-Wolff transformation eliminates terms linear in V k. In the limit where a local moment exists on the impurity site, given by the condition [72]: ǫ d = [ǫ d + Γ/πln(E B 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: At low temperatures, below the Kondo scale: J 2 V k 2 U ǫ d (ǫ d + U). (15) T K = UΓexp[ π ǫ d (ǫ d + U) ], (16) 2UΓ 14

15 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 V U (V L ) to the upper (lower) lead, respectively, we have: where H = H U + H L + H QD + H I, (17) 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 ǫ k. H QD = σ ǫ d d σ d σ + Un n, (19) is the QD Hamiltonian with energy level ǫ d 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 H I = k V Ukσ [c Ukσ d σ + h.c.] + k V Lkσ [c Lkσ d σ + h.c.] (20) 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 Ukσ + sinθc Lkσ c kσ = sinθc Ukσ + cosθc Lkσ (21) where tanθ = V U /V L, to decouple the c kσ channel from any tunnel coupling to the QD. This yields an interaction: H I = VUkσ 2 + V Lkσ 2 [c +kσ d σ + 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

16 FIGURES (a) U (b) U QD1 QD2 Γ U t Γ L QD1 L QD2 (c) L U (d) U QD1 QD2 QD1 QD2 L 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 = V k 2 D(E F ). 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: where as in the single QD case, H = H U + H L + H UQD + H LQD + H UI + H LI + H DQD, (23) H (U,L) = kσ ǫ (U,L)k c (U,L)kσ c (U,L)kσ, (24) 16

17 are the Hamiltonians of the conduction electrons in the upper and lower leads, with electron creation operators c (U,L)kσ, and identical dispersion ǫ k. H (U,L)QD = 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 ǫ Ud (ǫ Ld ) measured from the equilibrium chemical potential of the conduction electron fermi sea in each lead, impurity electron creation operator d Uσ (d Lσ ), and on-site repulsion U U (U L ) for double occupancy of an up and a down spin electron, and H (U,L)I = k V (U,L)k [c (U,L)kσ d (U,L)σ + h.c.] (26) 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: H DQD = t σ [d Uσd Lσ + h.c.)]. (27) Defining the level broadening for each dot due to coupling to the respective leads, Γ U and Γ L, as: Γ (U,L) = k V (U,L)k 2 δ(ǫ (U,L)k E F ), (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: T K = 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 U U = U L = U, Γ U = Γ L = 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 = 4t 2 /U plays a role analogous to the AF coupling between impurity moments in the 2IK model in a metal. Thus for J/T K 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/T K > 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

18 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/T K < 2.5. The different phases in the parameter space is summarized in the phase diagram in Fig. 2. t / Γ 1.0 Coherent Bonding Kondo Singlet Kondo Regime Atomic-like Kondo 2.5 Stot = 0 No Local Moment j = J/TK U3 U2 U1 < U2 < U3 j(t) for U1 2 J=4t /U 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/T K are not independent, with J = 4t 2 /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, T K the Kondo temperature (of each dot), and Γ the level-broadening of each dot due to its coupling to the respective lead. T K 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 S tot = 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/T K approaches the critical value of j c 2.5. Instead, the divergence is cutoff as: χ o χ (j j c ) 2 + (1.5t/23T K ) 2. (30) 18

19 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 T K 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 ev bias 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 IIIC 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 = Γ U + Γ L. 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 2e 2 /h near the critical point where j c 2.5. On the other 19

20 hand, if a singlet is formed for j = J/T K > 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 4e 2 /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 SD Γ U Γ L L Γ LD LD 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 Γ LD, from coupling to the large dot (LD), respectively, with Γ (U,L) = V (U,L) 2 D(E F ). In Section IIIB1 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

21 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 u LD 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 u LD, but still sufficient large to far exceed the level spacing of the large dot, u LD kt LD, the near continuum of state in the large dot act as a fermi sea for screening, but the charging energy u LD 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(N SD, N LD ), with N SD electrons on the small dot (SD), and N LD electrons on the large dot (LD): E(N SD, N LD ) = 1 2 E SD(N SD C g SD V gsd e +E DQD (N SD C g SD V gsd e ) E LD(N LD C g LD V gld ) 2 (31) e )(N LD C g LD V gld ), e ( ) ( e where E (SD,LD) = 2 1 e C (SD,LD) 1 CDQD 2 /C SDC LD, E DQD = 2 1 C DQD C SD C LD ), and /CDQD 2 1 C (SD,LD) is the total capacitance of the small dot (SD) (large dot (LD)) to its surroundings, C g(sd,ld) the capacitance of the respective dot to its control gate, V g(sd,ld) the applied gate voltage, and C DQD the capacitance between the SD and LD. From this expression, one obtains: u LD = 1 2 E DQD (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 > u LD, the 1CKE is recovered, as charge fluctuations 21

22 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 J SD and J LD, given by [70]: and J SD = Γ SD ( [E(2, 0) E(0, 1)] 1 + [E(0, 0) E(0, 1)] 1), (33) J LD = Γ LD ( [E(1, 0) E(0, 1)] 1 + [E(2, 1) E(0, 1)] 1). (34) 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 J SD = J LD, 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 correction is related to the spectral density near the fermi energy E F, 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 E F. 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]: { G 2CK (0, T) G 2CK (V sd, T) 3 = π x 1 for x 1 T x 2 for x 1 (35) where x = ev sd kt in the form: (note V sd = V bias. In the low bias region, x 1, the scaling may be rewritten G 2CK (0, T) G 2CK (V sd, T) = f 2CK x 2, (36) with f 2CK (T) = T for an individual Kondo impurity. The above can be contrasted with the Fermi-liquid scaling for the 1CK problem: G 1CK (0, T) G 1CK (V sd, T) T 2 with γ a model dependent numerical factor of order unity. 22 = γ x 2, (37) TK 2

23 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

24 (a) U (c) C gv g = n e (n+1) e (n+1/2) e (b) Q,V L U R U, CU C g R L, C L R C V g E (d) E (e) Ec N-1 N N+1 N+2 N N+1 Ec C gv g e n-1 n n+1 n+2 N N+1 Q,V L C U C g C L V g E Ec n-1 n n+1 n+2 C gv g 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, V g. 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 C g V g / 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 C g V g / e = (n + 1/2). (d) An alternative representation of the energy parabolas plotted versus V g, 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

25 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. R U, R L. (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 2C C g C QV g, (38) where C = C U +C L +C g 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, R U and R L 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 V g to favor an integral number (n, n+1, etc.) of electrons, i.e. favoring C g V g / 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 = ( N e + C gv g ) 2 2C C2 g 2C V 2 g, (39) where we have Q = N e, i.e. charge is quantized in units of e, yielding a Coulomb charging energy of 1E 2 C = e2 for the addition or removal of an electron. Here, the second 2C term quadratic in V g 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 < E c /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: C g V g = (n + 1 ) e. (40) 2 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 V g at fixed number of electrons as shown in Fig. 4(d). At a V g 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, E C, 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

26 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, ev bias ) = e2 h Γ L Γ R de[f(e + ev bias ) f(e)] Γ 2 + (E E o ) 2. (41) where f(x) is the Fermi-Dirac distribution function, Γ L,R denote the level broadening due to coupling to the left and right leads, respectively, and Γ = 1 2 (Γ L + Γ R ), and E o the energy of the resonant level which may be tuned by the plunger gate, V g. (a) U (b) U R U, C U C = C U l Q1,V1 C g1 R t, C t V g1 Q1,V1 V g1 C g1= C g C t Q2,V2 L C g2 R L, C L V g2 Q2,V2 L V g2 C = C g2 C = C L l g 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 V g1 and V g2, 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

27 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 Q i, is given by: Q i = j q ij = j c ij (V i V j ). (42) Casting this into vector form with Q = (Q 1,..., Q i,...q N ) T, V = (V 1,..., V i,...v N ) T, and defining the capacitance matrix C, where C ij = ( k c jk )δ ij c ij, yields: and an electrostatic energy for the system, E, of: Q = CV, (43) E = 1 2 VT CV = 1 2 QT C 1 Q. (44) Inclusion of the charge, Q v, and voltage, V v, of voltage sources, the voltage, V c, on the conductors may conveniently be related to its charge, Q c, and the voltage settings on the sources, and the capacitance submatrices between the conductors, C cc, and between the conductors and the voltage sources, C cv [91 93,85]: V c = C cc 1 (Q c C cv V v ). (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: E(N 1, N 2 ) = 1 2 E C1N E C2N E Ct N 1 N 2 (46) 1 e [C g1v g1 (N 1 E C1 + N 2 E Ct ) + C g2 V g2 (N 2 E C2 + N 1 E Ct )] + 1 e 2[1 2 C2 g1 V 2 g1 E C C2 g2 V 2 g2 E C2 + C g1 V g1 C g2 V g2 E Ct ] = 1 2 E C1(N 1 C g1v g1 e +E Ct (N 1 C g1v g1 e ) E C2(N 2 C g2v g2 ) 2 e )(N 2 C g2v g2 ), e ) ) where E Ci = e2 1 Ct 2/C 1C 2, ECt = e2 C 1 C 2 /C, and t 2 Ci C 1 U,L + C gi + C t is the total 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 C i ( 1 C t ( 1 27

28 with the same capacitive coupling to its respective plunger gate, i.e. C U = C L = C l and C g1 = C g2 = C g. Eq. [46] then reduces to the form: (b) C =0, E =0 t ct (c) C >0, E >0 t ct (a) (1,0) (1,0) (0,0) (0,1) (0,1) (1,1) V g2 V g2 V g1 V g1 E splitting for (ii) (i) (ii) (d) C >> C +C t l g E ~ E ct c (iii) V g2 C gv g / e V g1 FIG. 6. (a) The energy diagram for a fully symmetric series-coupled, double quantum dot, for different values of the inter-dot capacitance, C t. The energy at fixed occupancy, (N 1, N 2 ), is plotted versus the gate voltage, V g = V g1 = V g2. (i) For C t = 0, E Ct = 0, and the two dots are isolated (short dashed curve). (ii) For C t > 0 but small compared to (C l +C g ), E Ct < E C 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 C t = 0. (iii) When C t dominates, E Ct E c, 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 V g (long dashed curve). (b) Charge stability, honeycomb diagrams in the V g1 versus V g2 plane, for the three cases depicted in (a), where (b) C t = 0, (c) C t > 0 but small, and (d) C t (C l + C g ), respectively. The situations depicted in (a) correspond to a cut along the diagonal in these three diagrams, respectively. E(N 1, N 2 ) = 1 2 E C(N 1 C g1v g1 e +E Ct (N 1 C g1v g1 e ) E C(N 2 C g2v g2 ) 2 (47) e )(N 2 C g2v g2 ), e with C = C 1 = C 2 and E C1 = E C2 = E C. In the limit of zero inter-dot-coupling, for which C t = 0 and E Ct = 0, the system behaves as two isolated but identical dots, with energy: E(N 1, N 2 ) = E C 2 (N 1 C g1v g1 ) 2 + E C e 2 (N 2 C g2v g2 ) 2, (48) e where E C = e 2 /C o and C o = C l + C g. In the opposite limit of large inter-dot coupling and dominant C t (C l + C g ), so that C C t, we have E Ct E C = e 2 /2C o and: 28

29 E(N 1, N 2 ) = E C 2 [ (N 1 + N 2 ) e + C g (V g1 + V g2 )] 2, (49) and the system behaves as a single large dot, albeit with a charging energy, E C, which is one half the isolated case. If we were to tie V g1 and V g2 together so that V g1 = V g2 = V g, and plot the electrostatic energy at fixed N 1, N 2 : E(N 1, N 2 ) = E C 2 [( N 1 + C gv g e ) 2 + ( N 2 + C gv g e ) 2 ] + E Ct ( N 1 + C gv g e )( N 2 + C gv g ), e versus V g, for the three cases of: (a) C t = 0 with E Ct = 0, (b) C t < (C l + C g ) with E Ct > 0, and (c) C t (C l + C g ) with E Ct E C, we see behaviors at the charge degeneracy point(s) corresponding to two identical isolated dots, the development of a splitting due to the finite E Ct, 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 (C t = 0), when V g1 and V g2 are tied together. Here we have plotted the energy for fixed electron numbers (N 1, N 2 ), which denote the excess occupancy above up-down spin paired occupied quantum levels, with N i =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, C t 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, C t, 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 V g1 V g2, it is useful to examine the stability diagram in the V g1 versus V g2 plane. Such a stability diagram can be obtained by first defining a chemical potential for each dot, µ i : (50) µ i E(N 1, N 2 ) E(N 1 δ i,1, N 2 δ i,2 ) (51) = E C [(N i C gv gi e ) 1] + E Ct (N j C gv gj ), e and the addition energy for adding an electron on either dot, E add : E add µ i (N 1 + δ i,1, N 2 + δ i,2 ) µ i (N 1, N 2 ) = E C, (52) 29

30 where i={1,2} and i j. With the definition of µ = 0 for each lead at zero bias, stability for occupation (N 1, N 2 ) is given by the requirement of µ 1 < 0 and µ 2 < 0. Such stability diagrams in this idealized situation are shown in Fig. 6(b)-(d) for different values of C t and corresponding E Ct. 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 E Ct 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 E C and E Ct, 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 (N 1, N 2 ) is controlled by V gi. A plot in the (V g1,v g2 ) 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, E Ct 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

31 V. DQD CHARACTERIZATION AND EXPERIMENTAL STABILITY DIAGRAMS A. Quantum Dot Parameters (a) (b) V5 V1 V3 V2 V4 V1 V5 1mm V3 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/Al x Ga 1 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 = cm 2 and µ = cm 2 /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 C Σ 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 r e 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

32 1. Weak-coupling to leads V2(mV) G( e /h ) V4(mV) 0 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

33 will contain 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 V2(mV) x 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 V bias. 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

34 (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 V bias for each dot for the parallel-coupled case during the separate tuning process. Fig. 10 depicts a similar stability 34

35 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) (c) 0.20 di/dv(e 2 /h) V SD (mv) V (mv) SD V (mv) SD FIG. 11. The differential conductance, di/dv, versus V sd, 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 = V + V, V = 3, 2.5, 2, 1.7, 1.5, 1.3, 0.8, 0.3, 0mV, first cool-down, offset by 0.005e 2 /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 = V + V, V = 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4.0, 4.5mV, second cool-down [13]. 35

36 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 V bias 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 < t c, reminiscent of the Kondo effect in a single QD, while just on the double-peaked side, the zero-bias 36

37 conductance is non-monotonic in temperature, first decreasing with a reduction in temperature, then increasing, followed by a further decrease below 80mK, where t c could either be determined by the DQD level broadening, Γ, or by the effective exchange J = 4t 2 /U compared to T K ; 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 V sd (V sd = V bias ), 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

38 2 di/dv(e /h) (a) (b) 2 di/dv(e /h) (c) 2 di/dv(e /h) V (mv) SD V (mv) SD 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: V 5 = V + 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 0.02e 2 /h for each trace. (b), (c) Selected data from (a) without offset. (b) The second quantum state regime (from top to bottom: V 5 = , , , V). (c) The first quantum state regime (from top to bottom: V 5 = , , , V) [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 t c = Γ (t c /Γ = 1), or otherwise if t < Γ (t/γ < 1), when the effective anti-ferromagnetic coupling, J = 4t 2 /U exceeds 2.5T K (J/T K 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

39 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

40 (a) V2(mV) (b) V4(mV) di/dv (e /h) 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

41 2 di/dv(e /h) 2 g(e /h) δv(µv) W(µV) 2 Amp(e /h) (a) V SD (mv) 0.1 (b) (c) (d) (e) G 2G 2G V5(V) G Peak left 2G Peak right 1G 2G Peak left 2G Peak right 1G 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 < V, 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 mV, 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. IIIB1, 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

42 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, T K1, is associated with the screening of the S = 1 total spin by one channel of electron sea. Above T K1, 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 < T K1 as the S = 1 spin is partially screened, leaving a residual spin of 1/2. Below a second Kondo scale, T K2 < T K1, 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 T K 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 anti- 42

43 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]. di/dv(e 2 /h) (a) g 2 (e /h) V= (b) T(mK) T(mK) 270 (c) 47 mk di/dv(e 2 /h) V (mv) SD T(mK) 50 mk V SD(mV) V SD(mV) FIG. 16. Temperature dependence of di/dv versus V sd = V bias 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

44 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 V sd, 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 0.02e 2 /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 coupled- DQDs. 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

45 or 2J = 8t 2 /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 V bias is increased. At finite t, when V bias exceeds T K, 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 indicate any evidence of a non-fermi-liquid type V bias correction away from V bias = 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

46 V ex V ds (a) (b) (c) I sp g (e 2 /h) c bp c = 282 mv 260 mv 256 mv 244 mv B 1 µm n 1 G g(0, T ) g(v ds, T ) (K 2 ) T mk 24 mk 28 mk 38 mk (d) mk (e) (f) 2CK scaling V ds (mv) g(0, T ) g(v ds, T ) (K 0.5 ) T mk 22 mk 28 mk 39 mk 49 mk Theory g(0, T ) g(v ds, T ) (K 2 ) T mk 21 mk 27 mk 39 mk 45 mk ( ds) ev 2 ( ds) ev 2 kt kt g(0, T ) g(v ds, T ) (K 2 ) T CK scaling 12 mk 22 mk 28 mk 39 mk 49 mk ( ev ) 0.5 kt ( 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 E LD kt u LD. (b) 1CK scaling to the universal curve in Eq. 37 when the device is tuned to asymmetry with J LD < J Leads with Kondo coupling between the small dot and leads. Conductance is peaked at zero bias. (c) 1CK scaling with J LD > J Leads 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 J LD J Leads occurs near c = 260mV. (See (a) for the c-gate). (d) Scaling of data at different drain-source bias, V ds, 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 density is different from the usual ω 2 form familiar in the context of the fermi-liquid to a nfl Here ω is measured relative to the fermi-energy. This nfl ω dependence. ω form translates to a T ev bias form, depending on which energy scale functional form at low temperatures, or a 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 u LD and even 46

47 smaller level spacing LD, there is a range of temperatures, LD T u LD 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 ev bias /kt, as pointed out in Section IIIC. Thus, there are two independent knobs to turn, namely T and V bias. Secondly, from a technical standpoint, one must deal with the delicate fine-tuning of the controlling electrostatic gates in the coupled- QD device, in order to achieve symmetry between the AF couplings of the two independent channels, J LD and J Leads 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 mK 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 J LD and J Leads 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 (V bias = V ds ) 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

48 VII. METALLIC SYSTEMS (c) (d) 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/T K ). (b) Differential conductance, G = di/dv, as a function of bias, V b, and gate voltage, V g. The split zero-bias anomaly vanishes when an extra electron is added to the quantum dot (V g = 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/dv b as a function of bias V b, and (d) non-monotonic temperature dependence of the conductance at V b = 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 (ev bias ). For example, single-dot Kondo resonance is routinely 48