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11 The Kondo Effect Frithjof B. Anders Theoretische Physik II Technische Universität Dortmund Contents 1 Introduction Resistance minimum Anderson model

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11 The Kondo Effect Frithjof B. Anders Theoretische Physik II Technische Universität Dortmund Contents 1 Introduction Resistance minimum Anderson model Renormalization group Anderson s poor man s scaling Wilson s numerical renormalization group approach Exotic Kondo effects in metals Kondo effect in lattice systems Heavy Fermion materials Dynamical mean field theory (DMFT) Impurity solver Kondo effect in nano-devices Kondo effect in single-electron transistors Charge Kondo effect Conclusion 24 E. Pavarini, E. Koch, F. Anders, and M. Jarrell Correlated Electrons: From Models to Materials Modeling and Simulation Vol. 2 Forschungszentrum Jülich, 2012, ISBN 11.2 Frithjof B. Anders 1 Introduction Jun Kondo was intrigued [1, 2] by the puzzling experimental observation [3] that the resistance in noble or divalent metals typically shows a minimum at low temperatures when containing small concentrations of transition metals. It was expected that the inelastic scattering is reduced with decreasing temperature and, therefore, the resistance should be a monotonic function of T, which reaches a finite temperature-independent value for T 0 proportional to the remaining lattice imperfections. It had been noted that the increase of the residual resistance is proportional to the transition metal concentration [3,4] and only occurs when those impurities are magnetic. In 1961, Anderson [5] proposed a simple model for the understanding of the formation of stable magnetic moments in transition metals ions. Since the Coulomb interactions is only weakly screened on atomic length scales, valence fluctuations on unfilled d and f shells are suppressed at integer fillings, and a finite total angular momentum is formed according to Hund s rules. The Anderson model, which we will discuss in Sec. 1.2, provides a microscopic understanding of the Friedel sum rule [6] which relates the phase shifts of the conduction electrons scattered on the impurity to the number of displaced electrons. The overwhelming experimental evidence hints towards the generic nature of this effect: the details of the conduction bands actually do only enter into a single material-dependent low energy scale T K, the so-called Kondo scale. Kondo realized that the position of the resistance minimum remains unaltered when reducing the concentration of the magnetic impurities which rules out interaction induced correlation effects between different localized spins. From the first observation [3] in 1934, it took three decades until Kondo [1, 2] proposed his seminal Hamiltonian, which provides a simple physical picture and explains the experimental data. In the Kondo model, H = H b + H K, (1) the conduction electrons are described by a non-interacting electron gas H b = kσ ε kσ c kσ c kσ (2) and the interaction with a localized magnetic moment S is modelled by a simple Heisenberg term H K = J S s b. (3) c kσ (c kσ ) generates (destroys) a conduction electron with momentum k and spin σ, S represents the impurity spin, s b s b = 1 1 c 2 N kα σ αβ c k β (4) k k αβ is the spin of the conduction electrons at the impurity site and σ are the Pauli matrices. The lattice has a finite size of N sites which are sent to N in the thermodynamic limit. Over the period of the last 50 years we have learned that the Kondo problem is not restricted to magnetically doped noble or divalent metals: it has turned out to be one of the most fundamental The Kondo Effect 11.3 problems in solid state physics. It involves the change of ground states when going from highenergy to low energy physics indicated by the infrared divergent perturbation theory. 1.1 Resistance minimum Before we proceed to the Kondo problem itself, let us investigate the scattering of free conduction electrons on a finite number of impurities. The N imp identical impurities are located at positions { R i }, and each contributes a potential V ( r R i ) to H b generating the additional potential scattering term V = e i q Ri V ( q) c k+ q,σ c kσ, (5) i k qσ where V ( q) is the Fourier transform of V ( r). For a given configuration of impurities, { R i }, the single-particle Green function of the conduction electrons is determined by Dyson s equation [7], G k, k (z) = δ k, k z ε kσ + i e i q R i V ( q) G 0 k (z)g k q, k (z), (6) qσ where G 0 k (z) = [z ε kσ ] 1. After expanding this equation in powers of V ( q), we need to average over the different configurations { R i } in order to obtain the configuration averaged Green function G k, k (z) conf. In linear order, we obtain e i q R i V ( q) = N imp V (0) δ q,0, (7) i while in second order, two terms [4] arise e i q R i V ( q) e i q R j V ( q ) = N imp V (q)v (q )δ q+ q,0 + N imp(n imp 1)V 2 (0)δ q,0 δ q,0. i j (8) The first describes two scattering events on a single impurity and the other a single scattering of two different impurities. Summing up all these zero momentum transfers V (0) produces a uniform background which we absorb into the dispersion ε kσ. In higher order, there are two types of skeleton diagrams [8] generated: either the diagram describes multiple scattering on a single impurity, or several impurities are involved. The latter include interference effects which can be neglected if the mean free path is shorter than the average distance between two impurities. In the following, we assume such a small concentration of c imp that the condition c imp = N imp /N 1 is always fulfilled. Then G k (z) = [z ε kσ Σ k (z)] 1 acquires a self-energy Σ k (z) = c imp T k (z), (9) which is proportional to the impurity concentration. The scattering matrix T k (z) accounts for the sum of all multi-scattering processes on a single impurity. 11.4 Frithjof B. Anders The imaginary part of the self-energy is related to the single-particle life-time τ k : ImΣ k (ε k iδ) = 1/2τ( k), whose value close to the Fermi energy might be mistaken for a transport life-time entering the simple Drude model for the conductance, σ = ne2 τ Drude, (10) m where n is the concentration of electrons. According to Kubo s transport theory [7], however, the conductance is obtained from the current-current correlation function. A closer inspection reveals immediately that this correspondence of single-particle and transport life-time is incorrect in general: Clearly, forward scattering by the T -matrix contributes less to resistance than backward scattering. Hence, the average over the momentum transfer directions is required to connect τ Drude with T k (z). However, we can employ the optical theorem to connect the imaginary part of the forward scattering ImT k (z) to the angular integrated matrix elements ImT k (z) dω k ˆT k 2. Since we deal with isotropic s-wave scattering in the Kondo problem, the T -matrix becomes angular independent and the angular averaging yields τ Drude = τ kf. Just taking the contribution linear in J, the spin-diagonal scattering of conduction electrons reads JS z and the spin flip terms yields k ˆT k = JS, so that all three contributions from the scalar product JS s b add up to ImT = J 2 S(S + 1) + O(J 3 ) using the optical theorem. In second order in J, we just find a constant contribution similar to a residual potential scattering term. Its magnitude, however, is proportional to the square of the effective moment. In order to understand Kondo s theory of the resistance minimum, the second order contribution to the T -matrix [1] kσ ˆT k σ (2) = 1 kσ H K H K k σ (11) z Ĥ0 is needed. Since the details of the calculation can be found in Hewson s book [4], and a similar calculation is presented in the Sec. 2.1 below, we will only state the final result for the resistivity contribution of a single impurity: ρ imp = 3πmJ [ ( ) ] 2 S(S + 1) kb T 1 Jρ(ε 2e 2 F ) ln + O(J 3 ), (12) ε F D where m is the electron mass, ρ(ε F ) the conduction band density of state at the Fermi energy ε F and D the band width. The infrared divergent logarithm arises from the integration of the resolvent 1/(z Ĥ0) 1/(z ε) over all intermediate conduction electron states since the spinflipped local states are degenerate. In a magnetic field, however, the logarithmic divergency will be cut off on the energy scale given by the Zeeman energy. Typically, the bare coupling g = Jρ(ε F ) 1 is small. However, the logarithmic corrections causes an increase in ρ imp for decreasing temperatures and J 0, which diverges for T 0. The effective scattering rate becomes of the order O(1) at temperatures of an exponentially small energy scale T K T K De 1 ρj, (13) The Kondo Effect 11.5 which is non perturbative in the bare coupling constant ρj. This scale indicates the breakdown of the perturbation theory and is called the Kondo scale. At the heart of the problem are divergent spin-flip contributions, which occur in quantum impurity problems with degenerate local quantum states. 1.2 Anderson model As mentioned already in the introduction, Anderson proposed a model [5] for the understanding of local moment formation in Its simplest version comprises of a single localized spindegenerate level with energy ε d and a Coulomb repulsion U when the level is filled with two electrons of opposite spin. This local Hamiltonian H imp = σ ε d d σd σ + Un n (14) is then trivially diagonalized by the four atomic states 0, σ, 2. For the single particle energy we use the notation ε d which is identical to ε f in Bulla s lecture. This different notation roots historically in the modeling of either d-electron or f-electron systems. Such an atomic orbital [5] is then coupled to a single conduction band H b = kσ ε kσ c kσ c kσ (15) via a hybridization term H mix = kσ V k ( c kσ d σ + d σc kσ ). (16) The local dynamics of the single-impurity Anderson model (SIAM), defined by the Hamiltonian is completely determined by the hybridization function H SIAM = H imp + H b + H mix (17) Γ σ (ω) = π k V k 2 δ(ω ε kσ ). (18) The SIAM and the Kondo model belong to the class of quantum impurity models (QIM) which are defined by a finite number of local degrees of freedom, which are coupled to one or more bath continua. In the regime ε d 0 and ε d + U 0, the energies of the empty and double occupied state, 0 and 2, lie above the σ states and can be neglected at low temperatures: a local moment represented by a spin 1/2 is formed. Although local charge fluctuations on the d-level are suppressed at low temperatures and odd integer fillings, virtual exchange of electrons with the conduction are still possible, leading to spin-flip processes. Using a unitary transformation, Schrieffer and Wolff have derived [9] an effective energy dependent Kondo coupling J eff between the local conduction electron and the two local moment states σ. First, the Fock space 11.6 Frithjof B. Anders is partitioned into a low energy sector, which contains σ, projected out by ˆP L and its complement ˆP H = (ˆ1 ˆP L ) which includes 0 and 2. The Hamiltonian can then be divided into a diagonal part H d = ˆP L H ˆP L + ˆP H H ˆP H and an off-diagonal part λv = ˆP L H ˆP H + ˆP H H ˆP L. The subsequent unitary transformation U = exp(λs) H = e λs He λs = H d + λ2 2 [S, V ] + n=2 λ n+1 n n + 1! [S, V ] n, (19) is defined by the requirement of eliminating V in first order. The generator S is determined by the condition [S, H d ] = V. Then, the effective Hamiltonian of the low energy subspace H LL = ˆP L H ˆPL = ˆP L H ˆP L + P L λ 2 2 [S, V ]P L + O(λ 3 ) (20) acquires renormalized parameters and additional interaction terms via virtual transitions between the low and the high energy sectors mediated by V up to second order in λ. By applying this transformation to the H SIAM, H K = ˆP λ 2 L [S, V ] ˆP 2 L takes the form of the Kondo interaction [9] H K = 1 J k 2 k c kα σ αβ c k βs k, k αβ ( ) 1 J k k = V k V k ε k (ε d + U) + 1 ε k (ε d + U) 1 ε k ε 1 d ε k ε d since the local low energy sector is only comprised of the two singly occupied spin states σ while 0, 2 P H. For a constant hybridization V k 2 = V 2 and conduction band energies close to the Fermi energy, ε k can be neglected, and J k k J = 2V 2 U/[ε d (ε d + U)] 0 for the local moment regime where ε d 0 and ε d + U 0. At the particle-hole symmetric point ε d = U/2 the dimensionless Kondo coupling ρj = 8Γ 0 /(πu) determines the charge fluctuation scale at, where Γ 0 = Γ (0). We have demonstrated that the Schrieffer-Wolff transformation generates an effective Kondo Hamiltonian for the low energy sector of the SIAM in second order in the hybridization. This clearly reveals the connection between the SIAM, which includes all orbital and spin degrees of freedom, and the Kondo model focusing solely on the local spin degrees of freedom. The numerical renormalization group approach [10, 11], discussed in the lecture of R. Bulla, is able to explicitly track the flow from a free-orbital fixed point for β = 1/T 0 to the Kondo model at intermediate temperatures T/Γ and odd integer fillings of the orbital by iteratively eliminating the high energy degrees of freedom, which involve charge fluctuations. More realistic descriptions of 3d and 4f-shell dynamics requires more than one orbital. H imp is easily generalized from a single to many orbitals: (21) H imp = iσ ε d i n d iσ + σσ mnpq U mnpq d nσd mσ d pσ d nσ. (22) The Kondo Effect 11.7 The direct and exchange Coulomb matrix elements U mnpq will differ but are related by symmetry in the absence of relativistic effects, such as the spin-orbit interaction. This is discussed in more detail in the lecture of R. Eder. The Coulomb interaction in H imp takes the rotational invariant form H U = U 2 J 2 iσ n d iσn d i σ + 2U J 4 m m σ m m σσ d nmσn d m σ J Sm Sm m m d mσd m σd m σd m σ (23) in spin-space by identifying U nnnn = U, U nmmn = U 2J = U, U nmnm = J, U nnmm = J. Clearly, neglecting the orbital pair-transfer term d mσd m σ d m σd m σ breaks this rotational invariance. Since J 0, the S msm term is responsible for the Hund s rules, which favor the maximizing of the local spin and of the angular momentum by a ferromagnetic alignment. 2 Renormalization group We have learned several important points in the previous sections: (i) The low energy physics of the Kondo effect shows universality and is characterized by a single energy scale T K. (ii) The universality suggests that the problem can be tackled by approaches which were developed in the context of phase transitions: the renormalization group approach. (iii) The perturbative analysis breaks down due to infrared divergencies. These divergencies indicate that the ground state of the starting point, a free conduction band coupled to a single spin, is orthogonal to the ground state of the strong-coupling fixed-point which governs the low energy physics. 2.1 Anderson s poor man s scaling This section covers the simplest perturbative renormalization group (RG) approach to the Kondo model developed by Anderson [12] Although it does not solve the problem, it sets the stage for the deeper understanding of the physics provided by Wilson s numerical renormalization group approach. We begin with the definition of s-wave conduction band annihilation operators c εσ 1 c εσ = δ(ε ε k )c kσ, (24) Nρ(ε) k which are obtained by angular } integrating on a shell of constant energy ε. Starting from the anti-commutator {c kσ, c k σ = δ σσ δ k, k of a discretized system, the prefactor [ Nρ(ε)] 1 ensures the proper normalization of } {c εσ, c ε σ = δ σσ δ(ε ε ) (25) 11.8 Frithjof B. Anders in the continuum limit since the density of state ρ(ε) is defined as ρ(ε) = 1 N k δ(ε ε k ). (26) By supplementing the δ-function with some suitable symmetry adapted form factor B( k), δ(ε ε k ) δ(ε ε k )B( k), such as a spherical harmonics Y lm (Ω) or a Fermi surface harmonics [13], we could generalize these operators to the appropriate symmetry beyond simple s-wave scattering considered here. With those operators, the Hamiltonian (1) takes on the continuous form H = dε εc εσc εσ + dε dε J(ε, ε ) c εα σc ε βs imp (27) σ αβ where we have defined J(ε, ε ) = 1J ρ(ε)ρ(ε 2 ). This formulation is still very general. It turns out, however, that the occurrence of the infrared divergence is linked to finite density of states at the Fermi energy. For simplicity, we assume a constant density of states restricted to the interval ε [ D, D]. ρ 0 = 1/(2D) on this energy interval, and the Fermionic operator c ε has the dimension of 1/ E. After introducing the dimensionless coupling constant g = ρ 0 J/2, the dimensionless energy x = ε/d and the dimensionless operators c xσ = Dc εσ, we obtain the dimensionless isotropic Kondo Hamiltonian H = H D = σ 1 dx xc xσc xσ + g dx 1 1 dx αβ c xα σc x β S imp (28) which will be subject to a perturbative renormalization group treatment. Since δ(ε ε ) = δ([x x ]D) = δ(x x )(1/D), the rescaled operators also obey a normalized anti-commutator relation {c xσ, c x σ } = ( D) 2 {c εσ, c ε σ } = Dδ σσ δ(ε ε ) = δ σσ δ(x x ). (29) The key ingredients to any renormalization group (RG) transformation are 1. separation of energy scale 2. eliminating high energy contributions by renormalizing low energy coupling constants 3. rescaling of all parameters and quantum fields In the first step we define the appropriate low and high energy sector ˆP L and ˆP H = 1 ˆP L by partitioning the Fock-space appropriately. In the second step we perform the same transformations as outlined in Eq. (19). By eliminating the coupling between these sectors up to quadratic order, the effective Hamiltonian of the low energy subspace H LL = ˆP L H ˆPL = ˆP L H ˆP L + ˆP L λ 2 2 [S, V ] ˆP L + O(λ 3 ) = H LL + H (2) LL + O(λ3 ) (30) acquires renormalized parameters via virtual transitions between the low and the high energy sectors mediated by V up to the second order in λ. The Kondo Effect 11.9 k k k k k k (a) (b) Fig. 1: The particle (a) and the hole (b) spin-flip processes in second order in J contributing to the renormalization of J. Since the procedure is rather trivial for the free electron gas, we illustrate the steps on this part of the Hamiltonian. We introduce a dimensionless parameter s 1 and split H b into two contributions: H b D = σ 1/s 1/s dx xc xσc xσ + σ ( 1/s dx /s dx ) x c xσc xσ. (31) One contribution contains all low energy modes x 1/s and the other all high energy modes 1/s x 1. Defining ˆP L as the operator which projects onto all modes x 1/s, the Hamiltonian is written as H b = ˆP L H b ˆPL + ˆP H H b ˆPH and, therefore, V = 0. Focusing on the low energy part H b = H LL = ˆP L H b ˆPL = σ 1/s 1/s dx x c xσc xσ (32) we have to rescale the energy modes x to x = sx in order to restore the original mode distribution x 1 and obtain H b = 1 s 2 dx x c x(x )σ c x(x )σ. (33) σ 1 Since original Fermionic operators have the dimension 1/ E, they must also be scaled as c x = sc x(x )σ on expansion of the scale from 1/s 1, which leads to H b = 1 s σ 1 1 dx x c x σ c x σ. (34) This completes the third and last step of the RG procedure. The dimensionful Hamiltonian H b remains invariant under the mode elimination procedure if H b /D = H b /D. Comparing the rescaling of the integrals and fields after the mode elimination, Eq. (34) yields the scaling equation of the band width: D = D/s. Such an invariance is called a fixed point under the RG transformation, and the Hamiltonian of the free electron gas is obviously such a fixed point Hamiltonian. 11.10 Frithjof B. Anders Before we come back to the Kondo interaction, we briefly review the scaling of an additional local Coulomb interaction H U c x 1 σc x 2 σ c x3 σ c x 4 σ under the RG transformation. Performing the same RG steps in linear order, we accumulate [s 1/2 ] 4 for the rescaling of the four fields and

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