The Kondo Effect and the Kondo Problem

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The Kondo Effect and The Kondo Problem Johar M. Ashfaque It is known that for certain metals with magnetic impurities, the resistivity decreases as temperature decreases but then reaches a minimum and after that it increases as − ln T as the temperature decreases further. Moreover, the temperature dependence does not increase indefinitely but disappears below a characteristic temperature called the Kondo temperature TK . Moreover, the magnetic and spin properties change in the neighbourhood of the Kondo temperature. The magnetic susceptibility follows a Curie 1/T law above TK and tends to a constant below TK . Furthermore, at TK the impurity and conduction electron spins condense into singlet states thus having a vanishing of the local magnetic moment, the opposite of the ferromagnetic behaviour. The Kondo model was introduced to describe the above behaviour in a perturbation theory to second order in J, an exchange interaction between the local impurity spin and the conduction electrons. It will obtain a minimum of resistivity as a function of temperature T but as T → 0 we have diverging resistivity (as − ln T ). The perturbation theory breaks down at TK so the the solution is only valid for T TK . Finding a solution to this divergence is known as the Kondo problem. Anderson in the 1970s introduced a scaling hypothesis showing that the exchange interaction increases in magnitude as one includes the effects of more high-energy excitations. I.

THE KONDO HAMILTONIAN AND IMPURITY-CONDUCTION SPIN INTERACTION

The Kondo model is based on an interaction between the spin of conduction electrons and the spin of the impurities HK =

X

~k n~kĎƒ −

~ k,Ďƒ

X J~k,~k0 ~2

~ k,~ k0

~ ~ ) ¡ (Ďˆ †SĎˆ ~ i ). (Ďˆ~†0 SĎˆ i k k

Here Ďˆ~k =

a~k↑ a~k↓

~ ~ ) the transition spin operator between conduction is a doublet for the conduction electrons with (Ďˆ~†0 SĎˆ k k states veck and ~k 0 and a~i↑ Ďˆi = a~i↓ ~ i ) the spin operator for impurity state i. As usual, is a doublet for the impurity electrons with (Ďˆi†SĎˆ ~ S = ~~Ďƒ /2 with ~Ďƒ being the Pauli spin matrices. We now show that J~k,~k0 < 0. The coefficient is related to the amplitude for spin flip between the conduction electron and an impurity electron 1 M(~k,Ďƒ)(i,âˆ’Ďƒ)→(~k0 ,âˆ’Ďƒ)(i,Ďƒ) = − J~k,~k0 2 which can be easily seen by considering Ďƒx =

0 1 1 0

0 −i i 0

and Ďƒy =

2 in the Kondo Hamiltonian HK and considering only the components a~k↓ , a~k0 ↑ , ai↑ and ai↓ . More precisely, the interacting part of the Hamiltonian H 0 can be written as H0 = −

X J~k,~k0 â€ â€ â€ ĎˆiĎƒ Siz ĎˆiĎƒ (a~†0 a~k↑ − a~†0 a~k↓ ) + ĎˆiĎƒ Si+ ĎˆiĎƒ (a~†0 a~k↑ ) + ĎˆiĎƒ Si− ĎˆiĎƒ (a~†0 a~k↓ ) k ↑ k ↓ k ↓ k ↑ 2~

~ k,~ k0 ,Ďƒ

where SiÂą = Six Âą iSiy . But this spin flip process is obtained in second order perturbation theory, through an intermediate state, in which one has generically M âˆź V~ki

1 V ~. Einitial − Eintermediate ik

We apply this to the case where the impurity energy i is situated in the middle of the conduction band, the conduction states ~k and ~k0 are at the upper side of it and the excited impurity state is above the conduction band, at i + U as can been seen in the diagram below.

FIG. 1. Impurity Electron at i , conduction electron at ~k and excited state at i + U . (a) Process where first a conduction electron jumps to an excited state. (b) Process where first the impurity electron jumps to a conduction state.

Then consider the two possible processes. First, the conduction electron jumps to the opposite spin excited impurity state then the impurity electron jumps to the conduction electron state as can be seen in Fig. 1 (a). The other possibility is that first the impurity electron jumps to the opposite spin conduction electron state and then the conduction electron jumps to the excited impurity state as in Fig. 1 (b). In the first process, Eintermediate = 2 i + U and in the second Eintermediate = ~k + ~k0 . In both the cases Einitial = i + ~k . Also, in both cases, there is a minus sign in the amplitude due to the exchange in the final state of the two electrons (fermions) between the conduction and the impurity. Adding up the two processes, we obtain 1 1 M(~k,Ďƒ)(i,âˆ’Ďƒ)→(~k0 ,âˆ’Ďƒ)(i,Ďƒ) = −V~ki Vi~k0 + ~k − i − U i − ~k0 meaning J~k,~k0 = 2V~ki Vi~k0

1 1 + . ~k − i − U i − ~k0

Now consider that both conduction electron states are around the Fermi energy ~k ' ~k0 ' F

3 and the impurity states i and i + U are symmetric aroung F and since then V~ki = (Vi~k0 )∗ , we obtain J~k,~k0 = −8

|V~ki |2 < 0. U

From hereon in we will assume that it is momentum independent J~k,~k0 = J < 0.

II.

KONDO TEMPERATURE

To calculate the resistivity of the material, we compute the scattering of the conduction electrons off impurities. We are interested in the process where only the momentum of the conduction electron changes but not the spins or the state of the impurity i.e. the process (~k ↑)(i ↓) → (~k 0 ↑)(i ↓). Note that because of the antiferromagnetic interaction, we pair the conduction and impurity spins to be opposite since this configuration gives the smallest energy. There is also another process, where the spin of the conduction electrons is flipped, but will leads to the same temperature behaviour and so we will not consider that here. Consider first the zeroth order process. The amplitude is J − ms ≥ M 2 where ms = 12 is the spin projection for the impurity. The scattering rate is obtained by summing the amplitude squared over the final states to get (0) (k ↑)(i ↓)→(~ k0 ↑)(i ↓)

Γ~

=

2 2Ď€ X J δ( ~k − ~k0 ) m2s . ~ 2 ~ k

The sum over final states, for ~k close to the Fermi energy, gives Ď 0 V , where Ď 0 =

3ne mkF = 2Ď€ 2 ~2 4 F

is the single-spin electron density of states at the Fermi surface so that (0) (k ↑)(i ↓)→(~ k0 ↑)(i ↓)

Γ~

=

Ď€ Ď 0 V J 2 m2s . 2~

But we need to consider also the second-order process of the generic form M âˆź V~ki

1 V~ Einitial − Eintermediate ik

since it will lead to divergence. There are four possible processes. 1. An electron in ~k can scatter to unoccupied state with ~q and from there to ~k 0 without spin flip. 2. Reversely, first an electron is occupied state ~q scatters to ~k 0 then the initial electron ~k scatters in the vacated ~q. 3. The same as in case (1) but with a spin flip for the intermediate electron ~q and a corresponding opposite spin flip for the impurity. 4. The same as in case (2) but with a spin flip for the intermediate electron ~q and a corresponding opposite spin flip for the impurity.

4 A.

Process 1

Applying M âˆź V~ki

1 V~ Einitial − Eintermediate ik

for V~ki = M , we obtain the contribution |M |2 I1 where the sum over the intermediate states is I1 =

X q ~

1 − fq~ . ~k − q~ + iΡ

Here, 1 − fq~ with fq~ the Fermi-Dirac distribution guarantees that we sum only over states ~q that are empty at T = 0 and the infinitesimal iΡ is added to obtain only the outgoing waves. B.

Process 2

Similar in its entirety, apply M âˆź V~ki

1 V~ Einitial − Eintermediate ik

with V~ki = M obtaining the contribution |M |2 I2 where now the sum over the intermediate states is different X fq~ I2 = − . q~ − ~k0 + iΡ q ~

This time round, the factor fq~ guarantees that we sum only over states ~q that are occupied. We also have a minus sign because of the interchange in the order of the fermionic creation and annihilation with respect to the Process 1 since a~k a~†0 = −a~†0 a~k . k

k

In the sum of these two processes, the Fermi-Dirac distribution cancels for an energy-conserving process ~k = ~k0 obtaining a contribution 2 X Jms 1 2 |M | (I1 + I2 ) = − . 2 ~k − q~ q ~

Since the Fermi-Dirac distribution cancels, these terms will not lead to a temperature dependence. These two processes will actually be subleading and will not lead to the stated divergence. C.

Process 3

We obtain the same sum I1 but the matrix element is modified with respect to Process 1. D.

Process 4

We obtain the same sum I1 but the matrix element is modified with respect to Process 2. Summing the contributions from the Processes 3 and 4, the Fermi-Dirac distributions do not cancel and there is a temperature dependence this time round. The temperature dependence only cancels in the I1 + I2 terms.

5 E.

Overall

All in all, at second-order in perturbation we have 2 2 J J (2) M~ [2m I +(s(s+1)−m )(I +I )] = [−2ms I1 +(s(s+1)+ms )(I1 +I2 )]. = s 2 s 1 2 (k ↑)(i ↓)→(~ k0 ↑)(i ↓) 2 2 We can ignore the temperature independent terms and write 2 J (2) M~ 'Âą 2ms I2/1 (k ↑)(i ↓)→(~ k0 ↑)(i ↓) 2 with indices 2/1 corresponding to the Âą signs, respectively. We now convert the sums I1 and I2 into integrals using the total density of states at momentum k, Ď = mk/(Ď€ 2 ~3 ) and ~k = k 2 /(2m) to obtain Z D Z kD Ď q 2 dq m d I1 = = 2 3 2 ~k − q~ + iΡ Ď€ ~ kF k − q 2 + iΡ F F

Z

Ď m d = 2 3 ~k − q~ − iΡ Ď€ ~

I2 = 0

Z

kF

0

q 2 dq k 2 − q 2 − iΡ

where D is the maximum energy in the conduction band for the electrons. We can do the k and ~k integrals obtaining

Z kF q 2 dq k

k − kF

k

k − kF

= −k − ln ' − ln F k 2 − q 2 − iΡ 2 k + kF

2 k + kF

0 Z

D

F

Ď kF

k − F

d ' ln

. ~k − q~ + iΡ 2 k − D

Finally, we obtain approximately (2) M~ (k ↑)(i ↓)→(~ k0 ↑)(i ↓)

2 TF J Ď 0 ln ' 2ms 2 T

in one way and in the other way we replace TF by (D − F )/KB . Adding this expression to the zeroth order one, we obtain Jms TF (2) M~ ' − 1 − JĎ ln + ... 0 (k ↑)(i ↓)→(~ k0 ↑)(i ↓) 2 T leading to the scattering rate of Γ'Γ

(0)

TF 1 − 2JĎ 0 ln + ... . T

This would lead to resistivity TF Ď R (T ) = Ď R (0) 1 − 2JĎ 0 ln + ... T where J < 0 so the resistivity decreases with temperature. However, note that we have a leading contribution to the resistivity that comes from phonons and scales at T 5 so the correct dependence of the total resistivity is of the form Ď R, total (T ) = aT 5 − b ln where a, b > 0 and it has a minimum at a temperature 15 b Tmin ' . 5a

T TF

6 III.

STRONG COUPLING AND THE KONDO PROBLEM

The Kondo problem is the fact that we have a diverging resistivity at small temperatures which is clearly unphysical. The method of Anderson of “poor man’s scalingâ€? based on a second-order perturbation in the effective spin-spin coupling, nevertheless shows how we can avoid this Kondo problem. Here we only explain the results. Consider an electronic band −D < ~k < D and we want to see how the second-order transition amplitudes change when we vary it D → D + δD. The second-order analysis in the isotropic limit gives the equation for the coupling J dJ = J 2 Ď 0 . d ln D The solution of this equation is J(D) =

J(D0 ) . 1 − J(D0 )Ď 0 ln(D/D0 )

Since we are interested in the effective bandwidth explored at finite T of width kB T around the Fermi surface, we can set D/D0 = kB T / F = T /TF valid to logarithmic accuracy. To determine the Kondo temperature, note that TK must be scale invariant since it is a physical quantity and independent of our choice of D ∂TK = 0. ∂D

D Define the dimensionless coupling

g = Ď 0 J and its beta function, the same one defined in relativistic quantum field theory dg ≥ β(g) d ln D defining renormalization. Then the scaling equation is simply the statement that, up to second-order in perturbation, the beta function is β(g) = g 2 . Dimensional analysis tells us that kB TK = Df (g) where f (g) is a dimensionless function. From the scale invariance condition, D∂TK /∂D = 0, we obtain f (g) +

∂f ∂g =0 ∂g ∂ ln D

with solution 1

f (g) âˆź e g . Then the Kondo temperature is given by 1

kB TK = De JĎ 0 . The complete analysis was due to Wilson, using the numerical renormalization group approach. He found that, as the energy scale goes to zero, the effective coupling diverges, J → −∞. The divergence in the resistivity is then found to occur at T = 0 instead of at TK which is an artifact of perturbation theory. Note however that experiments show a finite resistivity as T → 0 or D → ∞. In Wilson’s analysis, one considers spherical shells in position space and discretizes energy bands. This is the key to avoiding divergences, since the continuum of energies and the presence of all energy scales, leads to divergence, which is somewhat similar to what one obtains in QCD at large distances (color confinement). A correct

7 renormalization group procedure gets rid of the divergence. The discretization of the Kondo Hamiltonian results in a chain Hamiltonian HK =

∞ X

† (fn+1 fn + fn† fn+1 ) + Hint .

n=0

Renormalization amounts to a discretization only up to an N followed by the introduction of a further point N + 1 modifying the Hamiltonian as X † √ † HN +1 = ΛHN + (fN +1,σ fN,σ + fN,σ fN +1,σ ). σ

Thus the renormalization group transformation is increasing the number of points and rescaling the √ energy by Λ : HN +1 = R(HN ). In conclusion, the Kondo problem is a strong coupling problem for which we still need a good resolution. IV.

HEAVY FERMIONS AND THE KONDO LATTICE

Heavy fermion materials are materials in which the effective mass of the electrons is much larger than the mass of the free electron. One defines the effective mass in correlation to the relaxation time τ τ∗ m∗ = m τ and one has conductivity σ=

ne2 τ∗ . m∗ 1 + ω 2 τ∗2

Therefore, one can use, at least in some regime, the Fermi liquid theory on which the above discussion is based. A heavy fermion material can be described by the presence of many local impurity moments forming a lattice called the Kondo lattice. Because the lattice preserves translational invariance, elastic scattering conserves momentum, leading to coherent scattering off the impurities. This in turn leads to, in the simplest heavy fermion materials, a dramatic drop in resistivity below the Kondo temperature TK . Generically, at high enough temperatures, the heavy fermion materials can be described by a Fermi liquid picture. Heavy fermion materials have • Curie-Weiss susceptibility at high temperature χ∼

1 T +θ

• Paramagnetic susceptibility at low temperatures χ ∼ const. • Much enhanced linear specific heat at small temperatures CV = γT • Quadratic low-temperature resistivity ρ = ρ0 + γT 2 . V.

SUMMARY

• The resistivity in some metals with magnetic impurities has a minimum as a function of T , below which it increases logarithmically ∼ − ln T until a Kondo temperature TK . Its explanation is the Kondo effect.

8 • The Kondo effect is the interaction of conduction electron spins with the impurity spins, described by the Kondo Hamiltonian. It contains an antiferromagnetic interaction. • A second order scattering effect of the electrons off impurities leads to logarithmic divergence − ln T in the scattering rate, thus in the resistivity. When compared with the leading effects, it leads to a minimum in the resistivity. • Since this is a perturbative effect comparing leading and subleading terms, it cannot be trusted except at large values. A resumming of leading logs gives an effective coupling that diverges at the Kondo temperature. • The Kondo problem is how to get rid of this divergence. A second order scaling relation based on a quadratic beta function reproduces TK . • Wilson’s renormalization group method shows that there is no divergence but we still need an analytical strong coupling resolution of the Kondo problem. • Heavy fermion materials have the effective electron mass m∗ up to thousands of times larger than the free me and obey the Fermi liquid theory at least at high temperatures and large couplings. • Most heavy fermion materials are described by a Kondo lattice model, are antiferromagnetic at small coupling and are a Fermi liquid at large coupling.

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