KONDO 101. Ganpathy Murthy. November 21, University of Kentucky

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1 KONDO 101 Ganpathy Murthy University of Kentucky November 21, 2006

2 Outline The resistivity minimum. Why is this surprising? How to think about resistivity. Kondo's Hamiltonian and perturbative explanation. Beyond perturbation theory: Poor man's scaling. Wilson's NRG Large-N mean eld theory. Mesoscopic Stoner eect Stoner + Kondo in mean-eld Spin uctuations Results, conclusions and open questions

3 The Experiment The system is a nonmagnetic metal such as Au doped with a small concentration of a magnetic metal such as Fe. The resistivity data show a minimum, whose position varies with the concentration of the impurities c imp

4 Thinking about resistivity To know why this is surprising, let us see how to think about resistivity. The most direct way is in terms of the Boltzmann transport equation. In this approach, the electrons have the usual Fermi distribution function in equilibrium f ( k) = f (ɛ( k)) = e β(ɛ( k) µ) When a uniform electric eld E is applied the distribution function changes to f E ( k) which needs to be determined. The point is that collisions between the electrons and other electrons, phonons, static impurities, or magnetic impurities tend to push the distribution function back towards the equilibrium distribution.

5 Boltzmann transport equation df E dt = k t K f E ( k) = 2π d 3 k { T kk 2 f E ( k)(1 f E ( k )) k k } Now one notes that by Newton's second law k t = e E and assumes that the complicated collision term on the RHS of the BTE can be expressed as f E ( k) f ( k) τ(k)

6 This is the relaxation time approximation, which to leading order implies f E ( k) = f ( k) + τ(k)e E k f ( k) Substituting this into the BTE one gets an expression for the relaxation time 1 τ(k) = 2π d 3 k T kk 2 (1 cos θ kk )

7 Now, since we have found the correction to the distribution function, we can nd the current J = e d 3 k (f E ( k) f ( k)) k m = e2 d 3 k k m τ(k) E k f ( k) which leads to the following expression for the conductivity σ = e2 m dɛ τ(ɛ)d(ɛ) df dɛ The key point is the following: Since df dɛ is nonzero only in a small energy range of order kt around the Fermi surface, only the values of τ and D(ɛ) near the Fermi surface matter at low temperatures.

8 Phonon scattering Let us consider scattering by acoustic phonons, with a density of states which goes as ω 2 dω. At a temperature of T there are roughly T 3 phonons with energy less than kt. However, we must take into account the fact that the phonon wavevector q has a much smaller magnitude than k F.

9 Recalling the 1 cos θ kk in the expression for 1/τ we see that small-angle scattering introduces an additional factor of θ 2 q 2 T 2. So the phonon scattering rate, and thus the phonon contribution to the resistivity goes as ρ ph = C ph T 5 This is monotonic in T. Similarly the electron-electron scattering contribution to the resistivity goes as T 2, which is also monotonic. This is why the resistivity minimum is puzzling.

10 Kondo's perturbation theory Kondo was the rst to realize that this might be explained as the competition between phonon scattering and magnetic impurity scattering. He dened the Hamiltonian which has become famous as the Kondo model H = ks ɛ(k)c ks c ks + J K Si S e (0) Here S i is a spin- 1 magnetic impurity interacting with a band of 2 conduction electrons. The impurity sits at position r = 0, and only talks to the electron at that position. The conduction electron spin at r = 0 is Se (0) = 1 2V kk ss c ks σ ss c k s The idea now is to calculate the scattering matrix element T kk using this interaction. by

11 To leading order we just get J K /2V which will give a T -independent scattering rate. This is clearly not good enough. So we have to go to at least order JK 2 to get anything interesting.

12 The famous log The sum over the intermediate state p gives an integral dɛ f (ɛ) ɛ log T The total resistivity now looks like J. Kondo, Prog. Theor. Phys. 32, 37 (1964) ρ(t ) = C ph T 5 + c imp (A B log T ) Minimizing this gives a temperature for the minimum resistivity going as c 1/5 imp, which was conrmed by experiment. Of course, there are some problems, the main one being that the perturbation theory diverges as T 0.

13 Even if all leading logs are summed, the answer still diverges at a nite temperature A. A. Abrikosov, Physics 2, 5 (1965). T K W exp 1/DJ K where W is the bandwidth. How do we get a consistent theory at low T? Anderson introduced powerful renormalization group methods into this problem. The idea is to consider a physical quantity which can be measured at low energy. The Hamiltonian contains degrees of freedom at all energies up to the bandwidth W and some couplings, here J K.

14 ψ h = (E l H h ) 1 V ψ l Poor Man's Scaling Anderson and Yuval, PRL 23, 89 (69) Yuval and Anderson, PRB 1, 1522 (70) Anderson, Yuval and Hamann, PRB 1, 4464 (70) Consider a generic Hamiltonian which has a high-energy and a low-energy sector, and some coupling between the two H = H l V V H h Now consider a low-energy eigenvector of H, whose components are ψ l, ψ h. The eigenvalue equation is H l ψ l +V ψ h = E l ψ l V ψ l +H h ψ h = E l ψ h We can use the second equation to solve for ψ h

15 and put this into the rst equation to obtain (H l V (H h E l ) 1 V )ψ l = E l ψ l This is an equation for the low-energy wavefunction alone. This procedure is called by various names, including projection into the low-energy sector, or integrating out high-energy states. RG consists of doing this repeatedly. The eective Hamiltonian is H e H V H 1 h V

16 The matrix, or operator ˆV is the following part of the Kondo coupling ˆV = J K 2V S iz (c p c q c p c q ) + S i+ c p c q + S i c p c q + h.c. For the states q whose energy is between W and W dw away from µ, H h can be replaced by W. To second order we again get the same two types of diagrams

17 For a spin- 1 2 we have S 2 iz = 1 4 S iz S i+ = 1 2 S i+ S i+ S iz = 1 2 S i+ and so on. Using these relations, it can be seen that the second-order term V H 1 h V renormalizes the Kondo coupling. To this order the change can be written as dj z = 2D 0 dw W J 2 xy dj xy = 2D 0 dw W J zj xy

18 When one repeats the transformation it is convenient to let the running bandwidth be W (l) = W 0 e l, leading nally to the scaling equations dj z dl dj xy dl = 2D 0 J 2 xy = 2D 0 J z J xy J z J x,y

19 The main lesson is that if the Kondo coupling is ferromagnetic, then J xy ows to zero, and the impurity spin and the electrons are decoupled. If, on the other hand the Kondo coupling is antiferromagnetic, then both J xy and J z ow to strong coupling. So we need to investigate what happens at strong-coupling. Before we do that, let us see at what energy the couplings become strong. Taking J z = J xy = J K, we have dj K dl which has the solution = 2D 0 J 2 K dj K J 2 K = 2D 0 dl J K (l) = J K (0) 1 2D 0 J K (0)l

20 So J K (l) becomes large roughly when l 1/2D 0 J K. Recalling that l corresponds to an energy scale, the relevant energy scale at which the coupling becomes nonperturbative is K = kt K W e l = W e 1/2D 0JK

21 Strong coupling xed point To see what the system looks like at strong coupling, we take J K much larger than the bandwidth W. In this case the system consists of the impurity spin coupled with a very strong antiferromagnetic exchange to a single site of the conduction system. It will form a singlet with that site, and the singlet will be invisible to the conduction electron system. This is perfect spin screening, and is at the heart of the Kondo eect. Roughly speaking, above the scale K the impurity spin is free, while below K it is bound into a singlet with the conduction electrons.

22 Wilson's Numerical RG K. G. Wilson, Rev. Mod. Phys. 47, 773 (75). This procedure is a renement of Poor man's scaling, and keeps a nite number of low-energy states every time high-energy states are integrated out, and gives precise answers for the physical properties. T K K W exp χ imp (T = 0) = 0.1 (gµ B) 2 T K W = Bandwidth { 1 } J K D 0

23 There is also a peak in the density of states at the Fermi energy whose width is K. This is because the electronic density of states is now mixed up with that of the impurity spin, which has entered the Fermi sea.

24 Large-N approach Start by generalizing the twofold degeneracy of spin to an N-fold degeneracy, labelled by m H = km ɛ k c km c km + J K c k N m c km f m f m kk mm f m f m = n f = 1 m Coqblin and Scrieer, PR 185, 847 (69) Chakravarty BAPS (82) Read and Newns, J. Phys. C 16, 3273 (83) Decouple the interaction in the partition function (path integral) by a Hubbard-Stratanovich transformation, introducing the auxiliary elds σand σ

25 Z = DσDσ DλDψD ψ exp S S = β dτ km ψ km ( τ + ɛ k ) ψ km + m f m ( τ + ɛ f + iλ) f m 0 + km ( σ ckm f m + σ f m c km ) + 2N σσ J K iλn f Now look for a time-independent saddle point σ = σ ɛ f + iλ = ɛ f H MF = ɛ k c km c km + ɛ f f m f m + σ km m km ( ) c km f m + f m c km

26 The mean-eld ground state energy is 2 K ( K log W K πj K D 0 π 2 K k = πσ 2 D 0 2 ) We minimize with respect to K to { nd K = D exp 1 } J K D 0 Which reproduces the nonperturbative Kondo scale at a mean-eld level.

27 There are also phase uctuations of σ = σ e iθ, which have the eective action 2N π 2 dτdτ sin2 [θ (τ) θ (τ )] (τ τ ) 2 Fluctuations in θdisorder the mean-eld ground state at long times, but the mean-eld results for fermionic properties are robust. Read, J. Phys. C 18, 2651 (85) Chakravarty and Nayak, Int. J. Mod. Phys. B 14, 1421 (2000)

28 Summary for Kondo 101 The Kondo problem is the simplest strongly correlated problem in condensed matter physics. For a ferromagnetic J K the impurity spin decouples from the conduction electrons at low energies. For an antiferromagnetic J K the coupling becomes very strong at low energies, leading to perfect screening of the impurity spin. The excitations of the impurity spin enter the Fermi sea, and lead to an extra density of states at the Fermi energy. If there is a lattice of impurity spins, one has the heavy fermion phenomenon.

29 The Mesoscopic Stoner Eect Single-particle states within E T of E F are controlled by Random Matrix Theory E T = { vf L D L 2 Ballistic/chaotic Diusive δ = Mean level spacing g = E T δ = Thouless number How about interactions? For the time-reversal invariant GOE H = ɛc αs c αs V αβγδ c αs c βs c γs c δs

30 Using Random Matrix Theory we can calculate all statistical properties of V αβγδ to get < Vαβγδ ss > = 0 all indices dierent < Vαββα ss = U < V s, s αββα > = J < V s, s ααββ > = λ < (V ss αβγδ )2 > δ2 g 2 Since the typical variance of the matrix elements vanishes as g it seems natural to ignore all matrix elements which have a vanishing ensemble average.

31 This leads to the Universal Hamiltonian H = ɛc αs c αs + U 2 N 2 J S 2 + λ( c α c α )( c β c β Andreev and Kamenev, PRL 81, 3199 (98) Brouwer, Oreg, and Halperin, PRB 60, R13977 (99) Baranger, Ullmo, and Glazman, PRB 61, R2425 (00) Kurland, Aleiner, and Altshuler, PRB 62, (00) But the number of neglected interactions (g 4 ) is much larger than the number of those kept (g 2 )! Fortunately, H U is the infrared xed point of Fermionic RG for weak coupling while more interesting things happen for strong coupling. Murthy and Mathur, PRL 89, (02) Murthy and Shankar, PRL 90, (03) Murthy, Shankar, Herman, and Mathur, PRB 69, (04)

32 E(0) = 0 E(1) = δ 2J E(2) = 4δ 6J by denition E(S) = S 2 δ JS(S + 1) This leads to submacroscopic magnetization long before the bulk Stoner instability, which occurs at J = δ. This is the mesoscopic stoner eect

33 If the exchange is not spin-rotation invariant, but is Ising, then E(S) = δs 2 JS 2 implying that there is no mesoscopic Stoner eect in the equal spacing model. Of course, in a real dot, the randomness of the levels will produce some distribution of ground state spin even in this case. Moral: For spin-rotation-invariant exchange, as J δ the system can acquire a large but submacroscopic spin, but this will not happen in the equal spacing model for Ising interactions.

34 Mesoscopic Stoner + Kondo H = ks ɛ k c ks c ks J S 2 d + JK δ kk ss Sf c ks Sd = c σ ss ks 2 c ks kss σ ss 2 c k s We assume the levels equally spaced ɛ k = ( k + 1 2) δ and consider an even number of electrons in the dot. By spin-rotation invariance the total spin S = S d + S f and S z are conserved. We will work with the state S z = S. Treat the Kondo interaction in large-n and decouple the Stoner interaction by a eld h to get the lagrangian 2δ σσ J K + c ks {( τ + ɛ k ) δ ss 1 2 h σss } c ks + h 2 4J + fs ( τ + ɛ f + iλ) f s iλn f + δ ( σ cks f s + σ f s c ks )

35 As usual, look for a static mean-eld solution σ (τ) = σ (τ) = σ ɛ f + iλ (τ) = ɛ f h z (τ) = 2b h x (τ) = h y (τ) = 0 The mean eld single-particle energies are roots of ω j = jδ b δ ( ) jδ b π arctan K ω j = jδ + b δ ( ) jδ + b π arctan K They are to be lled consistent with N tot = N e + 1 and S z = S

36 The ground-state energy as a function of the auxiliary parameters is ( E gs = b2 J + δ S { + K π log ( [b Sδ] K 2 K 0 ) 2bS 2 π (b Sδ) arctan ( b Sδ ) 2 } K ) Minimizing w.r.t b and K and gives E MF gs b = JS = K 0 = D exp ( = δ S 2 1 ) JS ( 1 J K ) Looks Ising, so there seems to be no mesoscopic Stoner eect!

37 Spin uctuations To recover spin-rotation invariance, which is broken by the mean-eld solution, we need to integrate over uctuations of h x,y whose action is given by the diagrams shown. For large S we need to only consider the quadratic uctuations. This gives a ground state energy ( E gs = δ S 2 1 ) ( 1 + b 2 4 J + 4JS 3 2 K { } + K π log ( ) [b Sδ] K 2 2 K 0 ) 2bS JS 2 (b Sδ) π arctan ( ) b Sδ K

38 Now minimize. In the limit K one gets ( E gs = δ S 2 1 ) JS (S + 1) 4 which shows the restored spin-rotation invariance. More generically, for K 0 > ( S 0 δ we ) nd ( ) S gs J, JK < S gs (J, 0) = S 0 K J, JK > K 0 Counterintuitively, a larger S increases the Kondo scale. In contrast to the bulk case, a Zeeman coupling to an external magnetic eld also increases S and increases K. This is very dierent from the noninteracting Kondo!

39 The competing state There is no Kondo eect, and the dot is (almost) fully polarized and the impurity spin lies (almost) completely opposite to the dot spin 2p + 1 2p + 2 { p + 1, p , p+2 p + 1, p 1 2, 1 2 } The lowest collective excitation involves an impurity spin ip, and should appear in nite bias tunnelling.

40 Conclusions There are two regimes, a Stoner-enhanced Kondo regime with S gs (J, J K ) < S gs (J, 0) and K (J, J K ) > K (0, J K ), and a polarized dot regime. In the Stoner-enhanced Kondo regime, a small Zeeman coupling enhances the Kondo scale, but a strong Zeeman coupling pushes the system over into the polarized dot regime. In the polarized dot regime, there is no Kondo eect, but there should be a collective excitation at an energy scale of J K.

41 Future directions Mesoscopic Fluctuations Open dots Finite temperature Smaller S Two impurities + RKKY + Stoner Simon, Lopez, and Oreg PRL 94, (2005) Vavilov and Glazman, PRL 94, (2005) Two-channel Kondo + Stoner Oreg and Goldhaber-Gordon, PRL 90, (2003) Connection to bulk Kondo near the ferromagnetic transition Larkin and Melnikov, Zh. Eksp. Teor. Fiz. 61, 1231 (1971) Maebachi, Miyake, and Varma, PRL 68, (2002) Loh, Tripathi, and Turkalov, PRB 71, (2005)