Lecture 22: The Kondo Problem: Singlet Ground State and Kondo ResonanceResidual Interaction Between a Local Moment and the Fermi SeaThe Kondo ProblemVariational Approach to the Kondo ProblemKondo Temperatured-level OccupationKondo ResonanceFano ResonanceFurther Consequence of the Kondo ResonanceMIT OpenCourseWare http://ocw.mit.edu 8.512 Theory of Solids IISpring 2009For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 22: The Kondo Problem: Singlet Ground State and Kondo Resonance In recent lectures, we have spent considerable time discussing the effect of a local magnetic moment on the electronic structure of materials. The question we turn to now it that of the ultimate fate of a local moment immersed in a sea of conduction electrons. In the large U limit, that is the limit in which the on-site repulsion energy U is much greater than the local moment resonance width Δ, we will examine the residual interaction between the local moment and the conduction electrons. 22.1 Residual Interaction Between a Local Moment and the Fermi Sea For simplicity, we assume that the local moment has spin S = 1/2, as would be the case for an element such as titanium which has only one d-level available for coupling. Consider a single electron with wave vector �k in a state just above the Fermi surface, and a singly occupied d-level well below the Fermi surface. If both states are occupied with the same spin, the only interesting low energy process that can happen is potential scattering of the conduction electron into a state with new momentum �k�. When the states are occupied by opposite spins, virtual excitations resulting in spin exchange between the two sites are possible. The first way this for this to happen is for the conduction electron to jump into the d-level to form a doubly occupied intermediate state. This state can then decay back to the original state, or a state in which the spins are flipped relative to the original configuration. Due to the double occupation of the intermediate state, however, the system must pay the on-site repulsion cost U ; the energy cost of the doubly occupied intermediate state relative to the initial configuration is U + �d − �� k. An alternative process through which spin exchange can take place without incurring the same-site repulsion cost is for the d-electron to be excited to a new state of momentum �k�, after which the excited electron of opposite spin can drop down to the now unoccupied d-level. The energy cost of this intermediate is simply ��d. k� − �We can write the Hamiltonian for this interaction via the Schrieffer-Wolf transformation � ˆ = � · �H J� �SkS� �k + �K��c†ck k(22.1)� �kk� ��k� σ kσ �� �kk� �kk� σ where �S� � = c†� �σ�αβ c�k k(22.2)k�α kβ The second sum accounts for simple potential scattering of the electron outside the Fermi surface. The σ+ and σ− components of �σ in the definition of �S� ount f k �kacc or the possibility of �1� � � ��� � †�2 The Kondo Problem S = �spin exchange in the Hamiltonian. For a local moment, �σ. Due to the rotational invariance of spin space, the only spin operators that can appear in the Hamiltonian are the identity and scalar product �k� �.S · S��kNext we must determine the coefficients J�k� . Recall the Anderson Model, in which hy-k�bridization between the d-level and conduction band was accomplished by including the terms c cdσ + Vkkσ k σ Thus V�k is the matrix element for going from the d-level to a state of momentum �k above †cdσ∗ �ˆHhybrid = V�k ckσ the Fermi surface, and the reverse for Vk processes we get ∗ �. Thus for the total matrix element of all exchange 1 1 J�k� = −V�Vk�kk� ∗ �+ U + �d − ����k k� − �d (22.3) where the first term corresponds to the process with the doubly occupied intermediate, and the second corresponds to the process involving only singly occupied states. From this point forward, all energies denoted by the symbol ε will be measured relative to the Fermi energy. Because we are focusing on states with momentum near the Fermi surface, k ≈ 0, and hence ε�εd|, U. Thus to get an estimate of J�k� we can leave off the ε�termsk�kε�k � |U k�(22.4)J�k� ≈ −|V |2 εd(U − |εd)| | |In the strong repulsion limit, U → ∞, J�k�k� → −||Vεd|2 | Because this coupling constant is nonzero, the Hamiltonian for this system causes spin flips; as a result, the ground state is no longer simply a local moment occupied by an up-spin or a down-spin. 22.2 The Kondo Problem We now will focus on the spin-exchange part of the Hamiltonian 22.1 H = −J�ˆS · S�(22.5) with J > 0. This is an antiferromagnetic exchange, and is sometimes referred to as “S-d ex-change.” A long standing problem in solid state physics at the time of Kondo’s work was the existence of a resistivity minimum at low but finite temperatures.1 Based on considerations of electron-phonon scattering and impurity scattering, one would expect the resistivity to decrease with temperature down to a limiting value at T = 0. Observations, however, showed that below a certain temperature TK, the resistivity would increase again, and finally saturate at a higher-than-expected value at T = 0. 1 TK varies drastically with host/impurity, and can be anywhere in the range 10−3K to 100K.� � � � � � �� �� �� � � � � �†�†�Variational Approach to the Kondo Problem 3 The basic understanding of this phenomenon is that, even though one would expect the material to become “dead” as phonons freeze out as T → 0, spin flips are still possible and provide a res idual scattering mechanism for c onduction electrons. In 1964, Kondo perturbatively calculated the scattering amplitude t�k� of conduction electrons by a local moment, in powers of k�the c oupling constant J. While most of us probably would have been satisfied to stop with the first order result, Kondo continued to second order and obtained t�k�= J + J2N(0) ln �F (22.6)k� T As T → 0, the scattering