Kondo effect

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Alex C Hewson and Jun Kondo (2009), Scholarpedia, 4(3):7529. doi:10.4249/scholarpedia.7529 revision #91408 [link to/cite this article]
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The Kondo effect is an unusual scattering mechanism of conduction electrons in a metal due to magnetic impurities, which contributes a term to the electrical resistivity that increases logarithmically with temperature as the temperature T is lowered (as \(\log(T)\)). It is sometimes used more generally to describe many-body scattering processes from impurities or ions which have low energy quantum mechanical degrees of freedom. In this more general sense it has become a key concept in condensed matter physics in understanding the behavior of metallic systems with strongly interacting electrons.


Background to the Kondo Effect

The dominant contribution to the electrical resistivity in metals arises from the scattering of the conduction electrons by the nuclei as they vibrate about their equilibrium positions (lattice vibrations). This scattering increases rapidly with temperature as more and more lattice vibrations are excited. As a result the electrical resistivity increases monotonically with temperature in most metals; there is also a residual temperature-independent resistivity due to the scattering of the electrons with defects, impurities and vacancies in the very low temperature range where the lattice vibrations have almost died out. In 1934, however, a resistance minimum was observed in gold as a function of temperature (de Haas, de Boer and van den Berg 1934), indicating that there must be some additional scattering mechanism giving an anomalous contribution to the resistivity--- one which increases in strength as the temperature is lowered. Other examples of metals showing a resistance minimum were later observed, and its origin was a longstanding puzzle for about 30 years. In the early 1960s it was recognised that the resistance minima are associated with magnetic impurities in the metallic host --- a magnetic impurity being one which has a local magnetic moment due to the spin of unpaired electrons in its atomic-like d or f shell. A carefully studied example showing the correlation between the resistance minima and the number of magnetic impurities is that of iron impurities in gold (van den Berg, 1964). In 1964 Kondo showed in detail how certain scattering processes from magnetic impurities --- those in which the internal spin state of the impurity and scattered electron are exchanged--- could give rise to a resistivity contribution behaving as \({\rm log}(T)\ ,\) and hence provide a satisfactory explanation of the observed resistance minima --- a solution to the longstanding puzzle (see Figure 2).

Details of Kondo's Calculation

Consider a small amount of magnetic impurities in a metal. In order to calculate the electrical resistivity arising from these impurities one first calculates the scattering probability for an electron from a single impurity and then multiplies it by the number of impurities. Taking into account the spins of the electron and the impurity, we consider the case when the electron with wave number \( k\ ,\) and spin down \(\downarrow\ ,\) collides with the impurity in a state with its spin up \( \uparrow\) and is scattered into a state with wave number\( k'\) with spin down \(\downarrow,\) while the impurity remains in a state with spin up \(\uparrow\ .\) Let us write the matrix element for this process as

\[\tag{1} J(k\downarrow,\uparrow\to k'\downarrow,\uparrow) \]

This type of scattering process had already been taken into account. Kondo (1964) considered a higher order correction term where the electron is scattered into the state with wavenumber \( k''\) and spin up \( \uparrow\) leaving the impurity is a spin down state \(\downarrow\) ---- a scattering process involving a spin flip of the impurity. This is only an intermediate state, and we have to take into account a further scattering process to arrive at the same final state as in equation (1), in which the spin flip is reversed, so that the scattered electron is in the state \( k',\downarrow\) and the impurity is returned to the state with spin up \(\uparrow\) (for a diagrammatic representation of this scattering process see Figure 1). We sum \(k''\) over all possible intermediate states and so, according to quantum mechanics, the total matrix element for this process is given by

\[\tag{2} \sum _{k''}J(k\downarrow,\uparrow\to k''\uparrow,\downarrow). J(k''\uparrow,\downarrow\to k'\downarrow,\uparrow) {1-f_{k''}\over \epsilon_k-\epsilon_{k''}}. \]

Here \( \epsilon_k\) is the energy of the electron with wavenumber \( k\ ,\) \(f_{k}\) is unity if the state \(k \) is occupied and zero if it is empty. The factor \( 1-f_{k''} \) is to exclude an occupied state \( k'' \) from the sum.

In calculating eq.(2) let us assume that \( J \) is a constant. The sum over \(k'' \) is replaced by an integral using the density of states \( \rho(\epsilon_{k''}) \ ,\) which we assume is also a constant. Then eq. (2) becomes \[\tag{3} J^2\rho \int {1-f_{k''}\over \epsilon_k-\epsilon_{k''}}\,d\epsilon_{k''}=J^2\rho\int_{\epsilon_{\rm F}}^D {1\over \epsilon_k-\epsilon_{k''}}\,d\epsilon_{k''}. \]

Here we assume that the electron energy takes a value between \( 0\) and \( D\) and that the states below the Fermi level \( \epsilon_{\rm F}\) are occupied. The integral in eq.(3) is easily calculated and we find eq. (3) can be expressed as

\[\tag{4} J^2\rho\,{\rm log}\left(\left|{\epsilon_k-\epsilon_{\rm F} \over\epsilon_k- D}\right|\right). \]

The spin flip contributions do not cancel due to the To this correction term to the matrix element we must add the first term \(J \ .\) The scattering probability \( W_k \ ,\) in which the electron \( k \) is scattered to any state, is proportional to the square of this total matrix element, giving \[\tag{5} W_k\propto J^2+ 2J^3\rho\,{\rm log}\left(\left|{\epsilon_k-\epsilon_{\rm F} \over\epsilon_k- D}\right|\right) +{\rm O}(J^4). \]

Earlier calculations of the resistivity used only the leading term in eq. (5), but \(J\rho\) is typically of order \(0.1 \) so the second term is not so small when the electron energy approaches the Fermi energy as it increases logarithmically. This singularity arises from the factor \( 1-f_{k''} \) in the integral of eq.(3), and implies that, when one considers the scattering of an electron, one must take into account the influence of all the other electrons. Logarithmic terms can also arise from scattering with the \( S_z \) component of the spin of the local magnetic moment, which do involve a spin flip. These logarithmic terms, however, cancel. The fact that the logarithmic terms arising from the spin flip scattering do not cancel is due to the fact that the operators involved in the spin flip processes, \( S^+=S_x+iS_y \) and \( S^-=S_x-iS_y \ ,\) do not commute \( S^+ S^- - S^- S^+=2S_z \ .\)

When calculating the resistivity, one just considers the electrons whose energy lies within a window of about \( k_{\rm B}T \) about the Fermi energy. This means \(|\epsilon_k-\epsilon_{\rm F}|\approx k_{\rm B}T \) and we replace \(\epsilon_k-\epsilon_{\rm F} \) in eq. (5) by \( k_{\rm B}T \) and find a contribution to the resistivity of the form,

\[\tag{6} R(T)=R_0\left[ 1+2J\rho {\rm log}\left(\left|{k_{\rm B}T \over D-\epsilon_F}\right|\right)\right] , \]

where \( R_0 \) is the resistivity obtained by considering only the first term of eq.(1). The sign of the exchange interaction \( J \) between the conduction electrons and the impurity is important. If \( J>0 \ ,\) then this interaction tends to align the magnetic moments of the conduction electron and impurity magnetic moments in the same direction (ferromagnetic case). If \(J<0 \ ,\) then this interaction tends to align the magnetic moments of the conduction electron and impurity magnetic moments in the opposite direction (antiferromagnetic case). Only in the antiferromagnetic case does the extra scattering term give a contribution to the resistivity that increases as the temperature is lowered. Such an antiferromagnetic exchange coupling can be shown to arise when a degenerate 3d or 4f state of a magnetic impurity hybridizes with the conduction electrons (see Schrieffer and Wolff (1966)).

Combining the contribution in the antiferromagnetic case with that from the scattering with lattice vibrations, Kondo was able to make a detailed comparison with the experiments for iron impurities in gold, demonstrating that this extra scattering mechanism could provide a very satisfactory explanation of the observed resistance minima, as is shown in Figure 2.

Figure 1: A diagrammatic representation of the spin-flip scattering process in which a down-spin conduction electron (thick line) is scattered by the impurity (dotted line) into an intermediate spin-up state.
Figure 2: A comparison of the experimental results (points) for the resistivity of iron impurities in gold at very low temperatures with the predictions (full curves) that include logarithmic term due to the Kondo effect (taken from the paper of Kondo (1964))

The Kondo Problem

Although the extra contribution to the resistivity explains the resistance minimum very well, the term \(J\rho{\rm log}\left(T/(D-\epsilon_{\rm F}\right)) \) diverges at low temperatures as \( T\to 0 \ ,\) and higher order scattering gives terms proportional to \([J\rho{\rm log}\left(T/D-\epsilon_{\rm F}\right)]^m \) with \( m>1 \ ,\) which diverge even more rapidly. The perturbation result is clearly unreliable at a temperature \( T \)such that \(J\rho\,{\rm log}\left(T/(D-\epsilon_{\rm F}\right))\sim 1 \ .\) Extensions of the perturbation approach (Abrikosov, 1965), which involved summing up the leading logarithmic contributions from the higher order scattering processes, gave a result which diverged in the case of an antiferromagnetic coupling \( J=-|J| \) at the temperature \(T_{\rm K}\) given by \[ T_{\rm K}\sim (D-\epsilon_{\rm F})e^{-1/|J|\rho}. \] The temperature \(T\)K has become known as the Kondo temperature.

The problem of how to extend Kondo's calculations to obtain a satisfactory solution in the low temperature regime, \(T< T_{\rm K}\ ,\) became known as the Kondo Problem, and attracted the attention of many theorists to the field in the late 1960s and early 1970s. The physical picture that emerged from this concerted theoretical effort, in the simplest case where the magnetic impurity has an unpaired spin \(S=1/2\) (2-fold degenerate), is that this spin is gradually screened out by the conduction electrons as the temperature is lowered, such that as \(T\to 0\) it behaves effectively as a non-magnetic impurity giving a temperature independent contribution to the resistivity in this regime. Furthermore it was concluded that the impurity contributions to the magnetic susceptibility, specific heat, and other thermodynamic properties, could all be expressed as universal functions of\( T/T_{\rm K}\ .\)

Definitive results confirming this picture were obtained by Wilson (1975) using a non-perturbative renormalization group method, which built upon the earlier scaling approach of Anderson (1970). Further confirmation came in the form of exact results for the thermodynamics of the Kondo model by Andrei (1980) and Wiegmann (1980), by applying the Bethe Ansatz method, which was developed by Bethe in 1931 to solve the one dimensional Heisenberg model (interacting local spins coupled by an exchange interaction \( J\)). Shortly after Wilson's work, Nozieres (1974) showed how, in the very low temperature regime, the results could be derived from a Fermi liquid interpretation of the low energy fixed point. In the Landau Fermi liquid theory, the low energy excitations of a system of interacting electrons can be interpreted in terms of quasiparticles. The quasiparticles correspond to the original electrons, but have a modified effective mass \(m^*\) due to the interaction with the other electrons. There is also a residual effective interaction between the quasiparticles which can be treated asymptotically exactly (\(T\to 0\)) in a self-consistent mean field theory. In the Kondo problem, the inverse effective mass of the quasiparticles \( 1/m^*\) and their effective interaction are both proportional to the single renormalized energy scale \(T_{\rm K}\ .\) The density of states corresponding to these quasiparticles takes the form of a narrow peak or resonance at the Fermi level with a width proportional to \(T_{\rm K}\ .\) This peak, which is a many-body effect, is commonly known as a Kondo resonance. It provides an explanation why the anomalous scattering from magnetic impurities leads to an enhanced contribution to the specific heat coefficient and magnetic susceptibility at low temperatures \(T<<T_{\rm K}\) with leading correction terms behaving as \((T/T_{\rm K})^2\ .\) At high temperatures such that \(T>>T_{\rm K}\ ,\) when the magnetic impurities have shed off the screening cloud of conduction electrons, the magnetic susceptibility then reverts to the Curie law form (ie. proportional to \( 1/T\) ) of an isolated magnetic moment but with logarithmic corrections (\({\rm log}(T/T_{\rm K})\)).

Direct observation of the Kondo resonance in quantum dots

Direct experimental confirmation of the presence of a narrow Kondo resonance at the Fermi level at low temperatures \( T<<T_{\rm K}\) has been obtained in experiments on quantum dots. Quantum dots are isolated islands of electrons created in nanostructures that behave as artificial magnetic atoms. These islands or dots are connected by leads to two electron baths. Electrons can only pass easily through the dots if there are states available on the dot in the vicinity of the Fermi level, which then act like stepping stones. In the situation where there is an unpaired electron on the dot, spin \(S=1/2\ ,\) in a level well below the Fermi level, and an empty state well above the Fermi level, there is little chance of the electron passing through the dot, when a small bias voltage is introduced between the two reservoirs--- this is known as the Coulomb blockade regime (for a schematic representation of this regime see Figure 3). However, at very low temperatures when a Kondo resonance develops at the Fermi level, arising from the interaction of the unpaired dot electron with the electrons in the lead and reservoirs, the states in the resonance allow the electron to pass through freely (see Figure 4). The observation of an electron current passing through a dot at very low temperatures, in the Coulomb blockade regime on the application of a small bias voltage, was first made in 1998 (Goldhaber-Gordon et al 1998). It provides a direct way of investigating and probing the Kondo resonance. Experimental results of the current through a dot spanning the temperature range to \( T>>T_{\rm K}\) to \( T<<T_{\rm K}\) are shown in Figure 5. Other related many-body effects have been investigated by using different configurations of dots and various applied voltages, and this is currently a very active research field.

Figure 3: A schematic representation of the discrete energy levels of a quantum dot with an odd number of electrons which is coupled to two reservoirs of electrons. The quantum dot is in the Coulomb blockade regime with \( T>>T_{\rm K} \ .\) There are no states on the dot near the Fermi level \( E_{\rm F} \) to facilitate the transfer of an electron through the dot when a small bias voltage is applied between the reservoirs. The levels on the dot can be shifted up or down by changing the gate voltage \( V_{g} \) which is applied to the dot.
Figure 4: A schematic representation of a quantum dot in the low temperature regime such that \( T<<T_{\rm K} \ .\) There is a build up of states at the Fermi level, as the spin of the odd electron on the dot is screened by the coupling through leads to the electrons in the reservoirs. These states form of a narrow resonance (Kondo resonance) at the Fermi level \( E_{\rm F} \) which facilitates the transfer of an electron through the dot when a bias voltage between the reservoirs is applied.
Figure 5: Experimental results for the rate of change of the current with bias voltage (G in units of \( e^2/h\)) for various temperatures as a function of the gate voltage \( V_g \ ,\) taken from the paper of van der Wiel et al. (2000), reprinted with permission from AAAS. The red curve shows the results at the highest temperature \( T>>T_{\rm K} \ :\) there is a peak when one of the discrete levels on the dot passes through the region of the Fermi level \( E_{\rm F} \ ,\) and a dip when the Fermi level falls between the levels as in Figure 3 (Coulomb blockade regime). The black curve shows the results at the lowest temperature \( T<<T_{\rm K} \ :\) when there is an odd number of electrons on the dot the current is significantly enhanced due to the Kondo effect. When there is an even number of electrons on the dot, there is no net magnetic moment on the dot and hence no Kondo effect. The response in this case decreases as the Coulomb blockade becomes more effective at low temperatures. The right inset shows the response as a function of temperature for a case with an odd number of electrons, and the red line indicates that in the intermediate temperature regime the current varies logarithmically with temperature as predicted by the Kondo effect.

Related developments

Strictly speaking the Kondo scattering mechanism only applies to metallic systems with very small amounts of magnetic impurities (dilute magnetic alloys). This is because the impurities can interact indirectly through the conduction electrons (RKKY interaction), and these interactions can clearly be expected to become important as the number of magnetic impurities is increased. These interactions are ignored in the Kondo calculation, which treats the impurities as isolated. Nevertheless, certain non-dilute alloys with magnetic impurities, particularly those containing the rare earth ions, such as Cerium (Ce) and Ytterbium (Yb), show a resistance minimum. Resistance minima can also be observed in some compounds containing the same type of rare earth magnetic ions. In many cases the Kondo mechanism provides a very satisfactory quantitative explanation of the observations. Good examples are the cerium compounds La1-xCexCu6 (see Figure 6) and Ce1-xLaxPb3 where \( 0<x\le 1\ .\) In these systems the inter-impurity interactions are relatively small, and at intermediate and higher temperatures the magnetic ions act as independent scatterers. As a result, in this temperature regime, the original Kondo calculation is applicable. At lower temperatures, in the compounds (where \( x=1\)), which display a resistance minimum but are completely ordered, the interactions between the magnetic ions becomes important, and the scattering of the conduction electrons becomes coherent, in contrast to the incoherent scattering from independent scatterers. Hence, in these systems, the resistivity decreases rapidly below a coherence temperature T coh to a residual value due to non-magnetic impurities and defects. The resistivity curve then displays a maximum as well as a mininum as a function of temperature. See for example the resistivity curve shown in Figure 6 for the compound CeCu6 (curve x=1). Other examples of compounds displaying such a resistivity maximum can be seen in Figure 7. The most dramatic effects of this type occur in rare earth and actinide compounds, which have ions carrying magnetic moments but do not magnetically order, or only do so at very low temperatures. These types of compounds are generally known as heavy fermion or heavy electron systems because the scattering of the conduction electrons with the magnetic ions results in a strongly enhanced (renormalized) effective mass, as in the Kondo systems. The effective mass can be of the order 1000 times that of the real mass of the electrons. The low temperature behavior of many of these compounds can be understood in terms of a Fermi liquid of heavy quasiparticles, with induced narrow band-like states (renormalized bands) in the region of the Fermi level. Due to the variety and complex structures of many of these materials, there is no complete theory of their behavior, and it is currently a very active field of research both experimentally and theoretically.

Figure 6: Resistivity vs Temperature (note the log scale) for \(\mathrm{Ce}_x\mathrm{La}_{1-x}\mathrm{Cu}_6\) (from Sumiyama et al (1986)). The resistance minimum is seen even for \( x=1\)
Figure 7: The resistivity \( R(T) \) for the compounds CeAl3, UBe13, CeCu2Si2, and U2Zn11, which display a maximum as a function of temperature from Fisk et al., (1986), reprinted by permission from Macmillan Publishers Ltd. The increase as the temperature is lowered can be attributed to the incoherent scattering from the Ce and U magnetic ions. As the scattering becomes coherent at low temperature the resistivity decreases rapidly producing a maximum. The \( T^2 \) behavior at low temperatures, typical of a Fermi liquid, is shown for the compound CeAl3 in the inset.


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Further reading

See also

renormalization group

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