A long-standing and multi-faceted problem in condensed matter physics is posed by the introduction of a magnetic impurity in a metal. Considering the very successfully adopted non-interacting or “free-electron” approach of our understanding of ordinary metals, it appears to be rather counter-intuitive that a magnetic moment may survive on the impurity at all when the impurity is immersed into the environment of a sea of degenerate conduction electrons. This question was the subject of much debate [[1,2], see, e.g., M. B. Maple in Part I] that accompanied the development of the related subject of Kondo physics from the late 1960s. When a magnetic ion with spin and charge residing in an unfilled d − or f − electron shell is inserted into a metal, charge neutrality and spin parity result in conduction electrons nearby to the impurity spin screening or compensating the spin in an antiferromagnetic coupling, with characteristic energy k _B T _K, where k _B is Boltzmann’s constant and T _K is the Kondo temperature. The formation of the Kondo cloud leads essentially to a gain in energy per magnetic impurity which is customarily written as follows: k B T K ∼ E F e − 1 / N E F J ex ∝ D * e − 1 / N E F J ex , (1) where J _ex measures the exchange coupling between the f − electrons and conducting electrons that gives rise to a hybridized band between these two types of electrons, of width D*. This expression then connects the Sommerfeld coefficient γ (see discussion below and Eq. 2), the electronic density of states at the Fermi energy N E F , and the electron mass m in a straightforward but very consequential manner. The heat capacity attributable to the electrons in a metal follows C el T = γ T , with an enhancement in the heat capacity that directly tracks the electronic density of states at the Fermi energy.

The screening of a magnetic moment by the “Kondo cloud” is a many-body problem; it takes place gradually as a function of temperature and in the T → 0 limit with complete screening when a non-magnetic singlet state is achieved. The theoretical treatment of the spin-only case was a convenient and adequate yet simplified treatment of the problem dealing with, for example, Fe impurities dissolved in a metal such as Cu or Au, but a more embracive approach that includes spin-orbit effects and crystal-electric field splitting is demanded when dealing with orbitally active ℓ > 0 rare-earth magnetism [3]. It is insightful to consider the two main theoretical approaches with which moment formation in solids was treated in the years leading up to the era of strongly correlated electron physics in 1979 [4]. The first of these was the Friedel-Anderson model [5,6]. Here, the proximity of the unfilled magnetic d − or f − shell to the metallic Fermi energy is considered to lead to quantum-mechanical hybridization of the magnetic orbital with the conduction electrons and, in effect, the washing out of the magnetic moment. In the Kondo model [2, see Part II], on the other hand, the formation of a singlet is the outcome of screening of the impurity spin by conduction electrons. The full or partial demise of a magnetic moment on an atom in a metal is, thus, a matter of comparing energies: the prevailing thermal energy, the bandwidth and location in energy with respect to the Fermi energy, and the magnitude of the exchange between the magnetic moment and its surroundings.

The study of Kondo physics has a strong foundation in the magnetism of rare-earth-based compounds, and especially so for the elements Ce, Sm, and Yb with orbital angular momentum L = 3 when their predisposition toward the trivalent ionic state leads to the spin S = 1 2 and total angular momentum J = 5 2 quantum-mechanical states. Having an odd number of electrons in their unfilled f − shells means that these elements are known as Kramers ions in which the ground multiplet retains at least a two-fold degeneracy (Kramer’s rule [7] states that the time-degeneracy of the ( J + 1 2 ) components of the crystal-field multiplet cannot be lifted by an electric field). The exchange between the f − electron shell and conduction electrons is not negligible; however, this results for the most part in polarization of the conduction electrons, while the magnetic moment of the ion remains the same as the Hund’s rule magnetic moment in free space or in insulators, for example. On the one hand, the localized magnetic moments of these elements provide exemplary cases of the moment being conserved in the solid state, but at the same time, they are known for many examples where the f − electron shell lies below but close to the metallic Fermi energy. This means that the conduction electrons, −and by extension the material’s physical properties, acquire a measure of f − electron character.

A particularly useful property with which we inspect how the electronic density of states at the Fermi energy ∂ N ϵ / ∂ ϵ ϵ = E F = N ( E F ) affects the behavior of electrons in a metal is the heat capacity C(T). To understand this, let us introduce a few salient concepts. We first assume that in a metallic solid, the electronic and lattice degrees of freedom under a prevailing finite thermal energy are decoupled to a good approximation. By virtue of the Pauli principle, we also assume that low-energy orbitals occupied by the electrons may not be multiply occupied, and thus, also electron–electron interactions are non-existent for practical purposes. This is just the fundamental picture of the non-interacting Fermi liquid, which provides a basis to treat the electrons in a metal as a free-electron gas. In momentum space, the Fermi momentum p _F = ℏ k _F separates the occupied |k| ≤ |k _F| states from the unoccupied |k| > |k _F| ones. The Fermi energy is related to the Fermi wavevector k _F through the dispersion relation ϵ k = k F ≡ E F = ℏ k F 2 / 2 m . In momentum space, the occupied states lie within a sphere of radius |p _F| with the associated Fermi energy chosen by the principle that summation over all the accessible states of particles within the Fermi sphere must yield precisely the number of particles in the system.

The matter of adiabaticity and interactions of particles across the boundary p _F is important. In the Landau Fermi liquid phenomenological treatment of electrons in metals, the following fundamental principles hold: spin and momentum remain good quantum numbers with which we describe non-interacting particles and interacting (quasi)particles, the latter which derives from excitations across p _F at T > 0, and the interacting system may be assimilated by adiabatically turning on interactions over time t, which measures the lifetime of the resulting excitations or quasiparticles. Through the theorem of adiabaticity, the ground state of the non-interacting Fermi gas transforms into the ground state of the interacting system which is known as a Fermi liquid. In this process, the dynamical properties such as mass and magnetic moment become renormalized to a new set of numbers, while spin, charge, and momentum of the fermionic electrons remain invariant [8]. We write for the particle density of N number of non-interacting particles per unit volume V, n = N/V, and it may be shown that the density of states N ( ϵ ) = 3 / 2 n / E F ϵ / E F , or at the Fermi energy, N E F = 3 / 2 n / E F . For electron momenta at |p _F|, we have for the electron mass m of the electron; m = p _F/v (p = p _F) in terms of the Fermi velocity |v _F|. At the Fermi surface, the quasiparticle density of states may be written as N E F = m | p F | / π 2 ℏ 3 . We now turn to the interacting case. We consider a filled Fermi sphere plus a single particle of momentum |p| such that p > p _F, but p − p _F ≪ p _F. For the energy of this particle, ϵ − E F = p 2 / 2 m − E F ≈ p F / m * p − p F where the effective mass m* now accounts for the interaction-dressed fermion instead of the bare electron mass m for the non-interacting case.

The aforementioned introduced concepts are brought together by the Sommerfeld coefficient γ [[9] and references within]. γ = π 2 3 k F 2 N E F = k B 2 3 ℏ 2 k F m * . (2)

Here, we see the implication that a high effective mass has on the heat capacity of a metal, since the Sommerfeld coefficient describes the electronic heat capacity through C el T = γ T . The problem may also be described by approaching Eq. 2 from the density of states. A so-called f − electron many-body Kondo resonance (also referred to as the Abrikosov–Suhl resonance) forms in the electronic density of states near the Fermi energy as a result of the virtual bound state consisting of conduction electrons that screen the local moment. The existence of this resonance has been amply illustrated by using model calculations and by a variety of spectroscopic experiments, [1, 10]

An important consequence of the Kondo resonance is the prediction that the conduction-electron/f − electron (c − f) hybridization leads to the formation of an (indirect or pseudo-) energy gap at the Fermi energy [[11], and references within], of width ϵ _i ∼ T _K and which, in turn, is governed by the effective mass enhancement factor through m * / m ≃ ϵ i / T K 2 [11]. This is a common feature of Kondo systems, with correspondence between the single-ion Kondo effect in a dilute impurity system and the Kondo lattice case, on the other hand, that has a Kondo ion in every unit cell of the lattice [12]. In a small number of known Kondo materials, however, −all of which are characterized by a strong c − f hybridization, gapping, which we refer to as ϵ _d, takes places at the Fermi energy, which produces a marked loss of charge carriers. The hybridization is temperature driven, and so is the gapping. In panel (a) of Figure 1, the unperturbed case of a |k|² dispersed wavevector is compared with the situation where strong band hybridization produces a direct energy gap ϵ _d and a small, indirect energy gap ϵ _i. The shape of the gap within the renormalized density of states, − see panel (b) of Figure 1, is a matter of the strength of the hybridization.

While theoretical development of our understanding of the formation of the Kondo insulating state remains an ongoing pursuit [13–15], the subject has recently gained attention [[16,17], and references within] on account of the contemporary physics of topology having relation to the formation of the heavy-fermion Kondo insulating state. We consider an electron situated at the edge of the valence band. A combination of an appropriate change in k together with an excitation in energy k _B T ≳ ϵ _i would accomplish a transfer into the conduction band. The viability of this process has an obvious temperature dependence and, thus, produces electrical conduction that is dependent on temperature by the rate at which the conduction band is populated. At high temperature, T ≫ T _K, the electrical resistivity of a Kondo insulator may resemble that of a metal, and the magnetic susceptibility would pick up a Pauli-like spin response of the free electrons above the Fermi energy, in the conduction band. As the temperature is lowered, bridging the energy gap becomes less and less feasible, and the conduction band slowly becomes depleted. The magnetic susceptibility diminishes or, at most, becomes constant in temperature in the event of spin pairing in the lower band, and the electrical conductivity likewise diminishes, contrary to what happens in an ordinary metal. Whether and to what extent the electrical resistivity ρ(T) of a Kondo insulator remains finite or not at low temperatures has been a matter of considerable debate. We discuss salient points in what follows. We refer to the so-called “impurity Kondo” effect as the single-ion Kondo problem, which for most practical purposes is relevant for the incoherent temperature region of a Kondo lattice where the Kondo ions act independently (or incoherent) from each other. Starting from the single-ion Kondo problem, the progressive screening or formation of the virtual bound state as temperature is lowered enhances the effectiveness of the Kondo ion as a scattering center of charge carriers. The interaction strength J _ex grows logarithmically, which was first demonstrated by Kondo [18] in a perturbation calculation beyond first order: J → J T = J + 2 J 2 N F ⁡ ln D * / T . The electrical resistivity in turn corresponds with a logarithmic correction [19]. ρ imp T = 3 π m J 2 S S + 1 2 e 2 ℏ E F 1 − J ex N F ⁡ ln k B T D * + O J ex 3 . (3)

Here, D* is the hybridized bandwidth, e is the electron charge, and S represents the spin on the Kondo ion.

As it stands, Eq. 3 presents a resistivity that increases without bound as T → 0. For this question, one has to be mindful of the limit of validity of the aforementioned equation. The temperature scaling of the resistivity posed by Eq. 3 breaks down when the hybridized bandwidth approaches D* ≃ T _K from the T > T _K side. Instead of continuing to diverge, experimentally the resistivity of a single-ion Kondo system is frequently observed to saturate at low temperatures. This behavior augurs well with the prediction when describing the scattering process of a conduction electron by the static potential of an isolated Kondo ion from a phase shift point of view [20], namely, that a complete phase shift π/2 is accomplished in the T → 0 limit; the scattering can be no greater and it, therefore, saturates at the unitarity limit. Apart from these considerations, it is instructive to look into the nature of the energy gap in real materials a little closer. From the shape of the gap in the normalized density of states N * ϵ posed in Figure 1, if the Fermi energy lies within the gap, or when charge carriers at low temperatures become unable to get excited into the conduction band, then the material would be an insulator. However, it was recognized early on [21] that spin degeneracy alone cannot account for real types of f − electron systems that have orbital degrees of freedom in addition. Furthermore, a particular behavior in the specific heat and its magnetic field dependence of the well-studied Kondo insulator CeNiSn [22] was modeled not by an energy gap structure as simple as is portrayed in Figure 1 but by a V− shape gap instead, having a residual density of states within the gap. This picture was born out by band-structure calculations [23], as well as by nuclear magnetic resonance studies on various 4f − and 5f − electron Kondo insulators [24–27]. It is worthwhile to note that the physics of materials having residual densities of states need to be treated with caution. With the procurement of higher and higher purity sample material of CeNiSn, the divergence in resistivity at low temperatures was found to be an artifact of impurities, and CeNiSn is possibly best described as a “failed Kondo insulator” [28].

In the last part of this discussion, we refer to the recent work carried out to develop the concept of topological Kondo insulators. Topological insulators as such represent a new state of matter, and the field is barely more than a decade in the making [29]. Topological insulators are similar to more traditional electrical insulators in that an energy gap separates the top of the occupied electronic band from the lowest energy of the empty band. In a non-interacting system such as a non-ordered magnetic material, surface states are protected by symmetry (importantly, time-reversal symmetry) which means that at the boundary or edge of a topological insulator with the surrounding vacuum, metallic surface states are formed by virtue of being gapless. In practice, this means that a bulk insulator is in effect short-circuited by the finite electrical conductivity on its surface(s). The reader is referred to the insightful demonstration of the involvement of surface states in the electrical conduction of the archetypal Kondo insulator SmB₆ [30]. Resistivity saturation at low temperatures in Kondo insulators is ubiquitous, but the theoretical foundation for this phenomenon in correlated semiconductors remains the subject of debate. A very recent study [15] searched for the source of this behavior in the charge carrier scattering rate itself operating as the relevant energy scale, with the finite lifetime of carriers being responsible for resistivity that saturates in the low-temperature quantum regime. Topological surface conductivity [31] and a non-trivial metallic state [32] were likewise found in the face-centered cubic Kondo insulator YbB₁₂.

In the paragraphs that follow, we review the physical properties of a selection of Kondo insulators. We highlight certain experimental commonalities among these systems but also focus on those aspects of these selected ones which challenge our understanding beyond the standard pattern of behavior of this materials class.

In the broad and well-studied subject of metal physics, strongly correlated electron systems form one of the most important developments in condensed matter physics in recent times. When the mixing of local-moment electron orbitals with the conduction band becomes very strong, the Kondo semimetal or insulating state forms at the boundaries between metal/semimetal and magnetic/non-magnetic phases. These phase boundaries are not crossed in the usual manner of a phase transition; instead, they are connected adiabatically and accessed by varying the material temperature. In this review, we have highlighted the foundation features of Kondo insulators but with emphasis on what makes the selected three compounds stand out from their otherwise more conventional counterparts.

The compounds discussed here deviate from the rule that Kondo insulators have a cubic crystal structure. This is shown to produce magnetocrystalline anisotropy in all three compounds, an attribute for which they occupy a niche in this materials class. CeRu₄Sn₆ has been shown to possess non-trivial topology [16] and is, thus, a candidate for the topological insulating state [17,29,38]. Initially identified with signatures of quantum criticality [34] by its electronic specific heat that diverges in a logarithmic fashion at low temperatures, a recent multi-parameter study [48] provided compelling evidence that CeRu₄Sn₆ is in close proximity to a quantum critical point. This finding places the compound CeRu₄Sn₆ at the confluence of quantum criticality and the topological Kondo insulating state, a situation that is probably unique.

U₂Ru₂Sn is a tetragonal Kondo insulator with pronounced anisotropy among several of its physical properties [26]. Anisotropy might intuitively be associated with the symmetry of a tetragonal crystal structure, but it should be remembered that the magnetic (5f) electrons in uranium compounds are known to be much more delocalized than what is the case in 4f-electron compounds of rare earths. Despite the dual nature of 5f − electrons having been confirmed by the electronic band-structure calculations and x-ray photoemission studies in U₂Ru₂Sn [87], the anisotropy in this Kondo insulator, nevertheless, points to magnetism that derives from a sufficient measure of non-spherical electronic orbitals.

In the case of CeRu₂Al₁₀, the semi-metallicity that is characterized by anisotropic gapping of electron states on the Fermi surface occurs already toward higher temperatures in the paramagnetic state far above the ground state, in the range where the first excited crystal-field doublet becomes depopulated upon cooling down. At the unusually high temperature (among cerium compounds) of 30 K, CeRu₂Al₁₀ orders in a long-range antiferromagnet state where competing crystal fields and anisotropic hybridization achieves to turn the ordered magnetic moment away from the crystal-electric field determined direction or “easy” axis.

The CeT₂Al₁₀ (T = Fe, Ru, and Os) series of compounds offers the rare opportunity to study the systematics of a trio of Kondo insulators.