MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For 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 e ect 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 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 rst 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 ipped relative to the original con guration 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 con guration 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 k d We can write the Hamiltonian for this interaction via the Schrie er Wolf transformation S H J k k S K 22 1 k k k kk c c k k k k k where S k k c c k k 22 2 The second sum accounts for simple potential scattering of the electron outside the Fermi surface The and components of in the de nition of S k k account for the possibility of 1 The Kondo Problem 2 Due to the rotational invariance spin exchange in the Hamiltonian For a local moment S of spin space the only spin operators that can appear in the Hamiltonian are the identity and S scalar product S k k Next we must determine the coe cients J k k Recall the Anderson Model in which hy bridization between the d level and conduction band was accomplished by including the terms H hybrid V k c cd V k c d c k k k Thus V k is the matrix element for going from the d level to a state of momentum k above the Fermi surface and the reverse for V k Thus for the total matrix element of all exchange processes we get 1 1 22 3 J k k V k V k U d k k d where the rst 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 will be measured relative to the Fermi energy Because we are focusing on states with momentum near the Fermi surface k 0 and hence k d U Thus to get an estimate of J k k we can leave o the k terms J k k V 2 U d U d 22 4 In the strong repulsion limit U J k k V 2 d Because this coupling constant is nonzero the Hamiltonian for this system causes spin ips 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 J S S H 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 nite 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 nally 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 3 K 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 ips are still possible and provide a residual scattering mechanism for conduction electrons In 1964 Kondo perturbatively calculated the scattering amplitude t k k of conduction electrons by a local moment in powers of the coupling constant J While most of us probably would have been satis ed to stop with the rst order result Kondo continued to second order and obtained F t k k J J 2 N 0 ln 22 6 T As T 0 the scattering amplitude diverges logarithmically in the second order term Resistivity R is proportional to the scattering probability which is in turn proportional to t k k 2 Thus to lowest diverging order F R J 2 J 3 N 0 ln 22 7 T This result explains the resistivity minimum but does not explain the observed saturation as T approaches 0 The perturbative result breaks down in this regime but a full solution is possible using Renormalization Group methods Wilson Review of Modern Physics 1975 The conceptual answer to the question of the fate of a local moment in a conducting Fermi sea is that a doublet state will bind to another electron to give a singlet state leaving behind just a renormalized strength of potential scattering in the system 22 3 Variational Approach to the Kondo Problem Due to the breakdown of perturbation theory with the appearance of a logarithmic singularity in the electron local moment scattering amplitude we must seek a di erent approach to analyzing the Kondo S d model 2 The method we employ here is the variational method starting with the …