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THÈSE présentée par Pavel Vitushinsky pour obtenir le grade de Docteur de l Université Joseph Fourier - Grenoble 1 Spécialité : Physique Etude théorique de l effet Kondo dans les boîtes quantiques Soutenue

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THÈSE présentée par Pavel Vitushinsky pour obtenir le grade de Docteur de l Université Joseph Fourier - Grenoble 1 Spécialité : Physique Etude théorique de l effet Kondo dans les boîtes quantiques Soutenue le 3 Novembre 2005 devant la Commission d Examen : M. Gilles Montambaux Rapporteur M. Peter Wölfle Rapporteur M. Frank Hekking Examinateur M. Philippe Nozières Président Mme. Mireille Lavagna Directrice de thèse M. Andrés Jerez Codirecteur de thèse Mémoire Préparé au sein du Service de Physique Statistique, Magnétisme et Supraconductivité, DRFMC, CEA Grenoble Contents Remerciements 7 1 General introduction Kondo effect Metals with impurities Scattering by impurities Kondo effect at low temperatures a Perturbation expansion b Scaling c Numerical renormalization group approaches d Local Fermi liquid e Bethe-Ansatz f Large-N approach Universality and crossover Phase shift Measuring the electron transmission phase Quantum dots Transport properties of the quantum dots a Coulomb Blockade b Kondo effect in quantum dots Two-terminal Interferometers. Experiment of Yacoby et al. Phase locking Open interferometers a Experiment of Schuster. Coulomb blockade regime b Experiment of Heiblum. Kondo correlation and unitary limit regimes Theoretical works a Phase shift b Phase lapses Theoretical analysis of the transmission phase shift of a quantum dot at zero temperature in the presence of Kondo correlations Introduction Scattering Phase Shift Two-reservoir Anderson model 4 CONTENTS Scattering theory for the Anderson model a S-matrix b The Friedel sum rule c The partial Friedel sum rule d Levinson s theorem Landauer conductance and Aharonov-Bohm effect a Landauer approach b Aharonov-Bohm effect in an open interferometer c Experimental check of the dependence of the conductance with the phase shift Scattering phase shift a Diagonalization of the hamiltonian of the Anderson model with two reservoirs b Solution of the Anderson model Conclusion Phase Lapses Introduction Phase lapse at T = 0 in the unitary limit and Kondo correlation regimes Magnetic and non-magnetic regimes. Phase diagram at T = Net current through an Aharonov-Bohm ring a Unitary limit regime b Kondo correlation regime Dependence of the source-drain current with the transmission amplitude of the reference arm t ref Conductance evolution in the low-temperature (T T K ) and hightemperature (T T K ) regimes Phase diagram at finite T a T T K b T T K c T Γ d Experimental situation Ring current in the Coulomb blockade regime Conclusions Transmission phase shift at finite temperature in the out of equilibrium situation Introduction e Quantum dots vs impurity atoms f Nonequilibrium g Anderson model h Calculation of the current i Calculation of the phase Noncrossing approximation (NCA) CONTENTS Large-N expansion NCA Slave bosons Keldysh formalism NCA for the Anderson model in the slave-boson representation Application to the out-of-equilibrium regime Numerical solution of the NCA equations Transmission phase shift of a quantum dot out of equilibrium Phase shift Modelling Results Results for the occupation number Results for the transmission phase shift a Effect of the temperature at equilibrium b Asymptotic behavior c Intermediate regime d Large bias voltage regime Comparison with experiments Conclusions a What is good? b What is bad? General conclusion 130 List of figures 132 Bibliography 141 6 CONTENTS empty page Remerciements J adresse en premier lieu mes profonds remerciements à Mireille Lavagna pour m avoir accueilli dans le cadre d un stage pré-doctoral et pour m avoir proposé ensuite un projet de thèse sous son encadrement. Je la remercie pour sa participation active au déroulement de la thèse, pour ses conseils, ses encouragements, sa disponibilité permanente, sa patience, pour m avoir donné la possibilité de bénéficier de ses compétences et de m avoir éclairé sur les subtilités relatives à la théorie de la matière condensée. Je remercie Andrés Jerez, le co-directeur de ma thèse, pour les nombreuses discussions, ses conseils, son aide, sa contribution au projet de thèse, pour m avoir fait bénéficier de sa compréhension des phénomènes complexes et des méthodes adoptées à la théorie de la matière condensée. Je remercie Peter Woelfle et Gilles Montambaux pour avoir accepté une lourde charge en étant rapporteurs du manuscrit, ainsi que pour les commentaires et suggestions qu ils ont apportés. J exprime ma profonde reconnaissance au professeur Philippe Nozières, qui a accepté de lire mon manuscrit et je le remercie pour l honneur qu il m a fait en présidant le jury chargé d examiner cette thèse. Je remercie Frank Hekking pour avoir accepté d examiner ma thèse, pour son soutien et pour sa contribution à ma formation, spécialement dans le domaine de la physique mésoscopique. J exprime ma reconnaissance à Louis Jansen, chef de SPSMS, pour son soutien tout au long de mon projet de thèse, pour sa disponibilité permanente qui a permis que mon séjour dans son laboratoire se passe dans les meilleures conditions. Je remercie Vladimir Mineev, chef du Groupe Théorie au SPSMS, pour l intérêt qu il a exprimé à mon travail, ainsi que pour les nombreuses discussions et suggestions. Je suis très sincèrement reconnaissant à Stephan Roche, Jacques Villain, Jacques Schwiezer, Michel Bonnet, membres du SPSMS, qui ont toujours manifesté une grande curiosité à mon travail, pour leurs conseils sur le fond ainsi que sur la forme de mes exposés scientifiques. Je les remercie pour la chaleur de leur accueil qui m a permis de m adapter à la société française, ce qui a rendu mon travail durant la thèse plus efficace. Je tiens à exprimer ma gratitude à Ramon Aguado pour les discussions que nous avons eues et ses conseils qui m ont aidé à bien avancer la partie de la thèse reliée au déphasage hors équilibre à température finie. Je remercie Denis Feinberg, Pascal Simon, Laurent Saminadayar, Christopher Bauerle, Karyn Le Hur pour leur intérêt à mon travail, les nombreuses discussions et suggestions. J ai aussi profité de nombreuses discussions avec Thierry Champel, Damien Bensimon, Vu Hung Dao, François Triozon, étudiants et post-docs dans le Groupe Théorie du SPSMS. Qu ils en soient remerciés. Chapter 1 General introduction 10 Chapter 1. General introduction next page 1.1 Kondo effect Kondo effect Metals with impurities Conduction electrons in normal metals behave as weakly coupled (interacting) quasiparticles. A good description is provided by the theory of Fermi liquid developed by L. Landau. Considering normal metals containing impurities it was shown that in the framework of the Fermi liquid description the electric resistivity of the metal sample drops when temperature decreases. The resistance starts saturating as temperature is lowered below about 10 K due to static defects in the material. Generally the temperature dependence of the resistance is given by ρ(t) = ρ 0 ( 1 + at 2 ) (1.1) where ρ 0 is the zero-temperature conductance. ρ 0 comes essentially from the scattering of conduction electrons by static impurities while the quadratic term is due to other types of scattering processes, like the scattering of electrons by electrons or by lattice vibrations, which becomes weaker and weaker as temperature is lowered. The value of the low-temperature resistance depends on the number of defects in the metallic sample but the character of the temperature dependence remains the same. Some metals, however, can have their electric resistance dropped at low temperature and become superconducting. In our work we will not consider the case of superconducting metals and focus instead on normal metals. However the low-temperature behavior changes drastically when magnetic atoms are added. The electronic shells of these atoms correspond to only partially filled outer d or f shells and may have nonzero net magnetic moment, such as for example cobalt, 27 Co, iron, 26 Fe, or manganese 25 Mn. The resistance of this alloys first decreases and then increases as temperature is lowered. The origin of this increase of the resistance has been the subject of many theoretical studies. It was established experimentally that the minimum appears when and only when the alloy contains magnetic impurities, the resistance minimum is thus a universal phenomenon of dilute magnetic alloys. Another important point was clarified later on through the measurements of the resistance of diluted alloys with an impurity concentration less than 0.1 at.%. This result showed that the residual resistance is proportional to the impurity concentration and increases as temperature is lowered. Thus it was established that the phenomenon is a single impurity effect rather than due to the interaction between impurities. To summarize, any theoretical analysis is confronted with three main obstacles. The first one is the resistance increase when temperature is lowered. Any source of electron scattering should vanish as temperature is lowered and the scattering probability should decrease in metals except in very special cases. The second one is the fact that residual resistance is not a constant but varies at very low temperatures. The origin of the corresponding energy scale was not clear. The third difficulty is the universality of the phenomenon. A large number of alloys were tested experimentally and all lead to the similar results. From the latter reason one expect the model to be relatively simple and very general, universal for all magnetic alloys. In fact, the standard model introduced 12 Chapter 1. General introduction Figure 1.1: Temperature dependence of the resistivity of pure metal (Cu) and dilute magnetic alloy [1]. to describe localized moments in a metal is the so-called s d model, which was treated extensively and found to give a monotonic decrease of the resistance below the Neel temperature [2, 3] Scattering by impurities In the case of the scattering by impurities which do not own any internal degree of freedom, as for example for the scattering by a potential v(r R i ) (1.2) i the scattering can be analyzed in terms of a one-particle problem since each electron sees the same scattering potential. The situation is very different when the scatterer owns internal degrees of freedom as, for example, spin or orbital degeneracy. After each scattering event the ground state of the scatterer and therefore the potential seen by the electrons may change. Correlations between electrons are introduced and the problem, thus, is no longer a one-particle but rather a many-body problem. In addition to the potential expressed in Eq.1.2 the interaction energy of an electron with a magnetic atom contains a coupling term between the spins σ and S of the conduction electron and the impurity. It can be written in the framework of the s-d model V s = (J/n) i σ S i δ(r R i ) (1.3) 1.1 Kondo effect 13 Figure 1.2: Temperature dependence of the resistivity for different concentrations of impurity atoms - comparison between experiment (circles) and theory (curves) [4]. The coefficient J has the dimension of energy. J is several times lower than the electron energies, which are of the order of µ, whereas the usual spin-independent interaction of an electron with an impurity atom is of the order of µ. The latter interactions do not interfere in the scattering probability and one may consider them separately. The spin part of the interaction gives a contribution of the order of J 2 S 2 /µ 2 to the scattering probability with respect to the usual interaction. It seems that this is a small effect which can be neglected. It is known that usually the interaction is not weak and one has to take into account higher order terms in the calculation of the scattering probability. However, more detailed calculations in this case show that the result remains similar. The situation is rather different for the spin-dependent part of the interaction. Although it is smaller than the spinless part there is nevertheless a significant difference. The importance of the higher order corrections was first shown by Jun Kondo [4]. The point is that the interaction described in Eq.1.3 corresponds to the physical process in which the electron spin can flip together with a simultaneous spin-flip of the impurity. When an electron is scattered off an usual atom its spin keeps its orientation. The correction appears to be dependent on the energy of electron leading to a temperature dependence of the resistance. Jun Kondo showed that the minimum of the resistance is due to the process of spin exchange between free electrons and the localized impurity. 14 Chapter 1. General introduction The scattering probability calculated at the third order in J is proportional to log T as a result of the non-commutativity of the spin operators. The sign of this log T term depends on the sign of J. The scattering amplitude decreases for positive values of J and increases if J 0. This effect has a physical explanation. If J 0, then the interaction is of ferromagnetic type and favors the parallel alignment of spins σ and S. In this case the spin-flip scattering is suppressed and the scattering amplitude (and thus the resistance) decreases with temperature. Conversely, for negative J the interaction is antiferromagnetic and the processes with a flip of electron spins are now possible. This leads to an increase of the resistance when the temperature is lowered. At high temperatures the main contribution to the resistance is due to electronphonon scattering. The thermal dependence of the resistance is given by ρ = ρ v + c m a ln µ T + bt 5 (1.4) where c m is the atomic concentration of magnetic impurities, a and b are constants and ρ v is the contribution to the resistivity brought by the potential scattering. The latter term, resulting from the electron-phonon interaction, increases with temperature. The interplay between the two last contributions gives rise to a minimum in the resistance as a function of temperature at T min given by T min = [ cm a ] 1/5 (1.5) 5b As can be seen T min is proportional to c 1/5 m and hence weakly depends on c m. J. Kondo have also shown that the scattering cross-section obtained by perturbation expansion diverges as a given temperature. The temperature at which the logarithmic term in the perturbation expansion becomes large is given by [ T K (Jµ) 1/2 exp n ] (1.6) J ν(µ) where ν(µ) is the electron density of states at the Fermi level. This temperature is called the Kondo temperature. The physical phenomena related to this process are known as the Kondo effect whereas the class of theoretical models proposed to describe this effect are called - the Kondo models. Since then the properties of Kondo systems have intensively been studied. A complete and detailed overview of this question can be found in the book of A.Hewson [5]. Earlier, in the late 50s the concept of virtual bound state was introduced by J.Friedel. These are states which are almost localized due to the resonant scattering by the impurity. This idea was developed later on by P.W. Anderson (1963) in a paper in which he proposed the Anderson model [6]. This model has then played a very important role. The model contains, in addition to the resonance at the impurity site, a strong on-site interaction arising from the Coulomb repulsion between impurity electrons. This interaction is responsible for the formation of a localized magnetic moment at the impurity site. For the calculation of the resistance, Kondo assumed 1.1 Kondo effect 15 (it was the intrinsic property of the model considered) a local magnetic moment to be associated with the spin S. The Anderson model is more general than the s-d model the latter can be deduced from the former in a given limit. The transformation mapping the Anderson Hamiltonian to the s d (or Kondo) hamiltonian can be performed using unitary operators. It is known as the Schrieffer-Wolff (canonical) transformation (1966) Kondo effect at low temperatures a Perturbation expansion Perturbation theory expansion in J leads to unphysical predictions such as a divergence of the resistivity at T = 0. The summation of the infinite series of leading order logarithmically divergent terms was performed by A.A.Abrikosov [7]. His result extends the calculation of J. Kondo. The expression that he gets for the resistivity leads to reliable results for the ferromagnetic case J 0. For the antiferromagnetic case, J 0, the resistance becomes infinite at the Kondo temperature T K. The divergence of the resistance results from the formation of a singlet state (Kondo singlet) between the spins of the conduction electrons and the spin of the impurity [8]. In other words, it corresponds to a complete screening of the impurity by the conduction electrons surrounding it, forming what is called the Kondo cloud. Perturbation theory provides a good description of the magnetic impurity systems for T T K but the expansion breaks down at T T K. The perturbation expansion predicts a log T-form for the resistance at low temperatures while experimental studies show that both thermodynamic and transport quantities give power laws in T. The resistivity, for example, deviates from its T = 0 value by T 2 terms. Non-perturbative techniques are required to investigate the low-t regime b Scaling In the late 60s Anderson and coworkers introduced a new theoretical framework based on the ideas of scaling [9, 10]. They proposed to perturbatively eliminate higher-order excitations allowing then to derive an effective model valid at low energy scales. The width of the conduction band in this method is gradually reduced so that finally only low-energy excitations are allowed. This procedure generates a set of effective models, each one characterized by its own coupling J. Additional couplings are also generated in this procedure but most of them are small and can be neglected. Thus the coupling J varies with the width of the conduction band: J = J(D). The scaling approach leads to the concept of the existence of fixed point when J(D) becomes scaling-invariant. The system is then described by the fixed point corresponding to the value of J = J(D ) where D is defined from [ ] dj(d) = 0. (1.7) dd D=D The scaling approach leads to an increase of the effective coupling strength between the spins of conduction electron and of the impurity when D decreases, allowing one 16 Chapter 1. General introduction Figure 1.3: Formation of the spin singlet complex at low temperatures. Magnetic impurity virtually traps one conduction electron in order to compensate its magnetic moment. to describe the low temperature behavior of the system. The perturbation approach breaks down when J becomes too large but Anderson assumed that the approach is still valid at very low temperatures, T 0. If the scaling is pursued further, the coupling J increases to infinity as D is reduced. Such a scaling behavior implies the formation of a ground state with an infinite coupling in which the impurity is bound to the conduction electrons in a spin singlet state. The behavior at low temperatures is similar to that of a non-magnetic impurity, Fig.1.3. An outstanding result of the scaling approach is that each set of effective models is characterized by a single energy scale, T K, which is a scaling invariant. Systems with different values of the parameters J and D but with the same T K = T K (J,D) exhibit the same low energy behavior. Hence, for example, the low-temperature dependence of the resistance is given by ( ) T ρ(t) = F (1.8) where F(x) is a universal function c T K Numerical renormalization group approaches An important contribution was made by K.G.Wilson. He developed a non-perturbative method - the numerical renormalization group approach (NRG). Wilson combined the renormalization group ideas coming from field theory with scalin

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