Kondo Breakdown in Topological Kondo Insulators

Kondo insulators are a class of strongly correlated electron material in which the screening of local moments by conduction electrons gives rise to an insulating gap at low temperatures Fisk et al. (1995); Coleman (2007). The first Kondo insulator, SmB₆, discovered more than 40 years agoMenth et al. (1969), has attracted renewed interest due to its unusual surface transport properties: while its insulating gap develops around $T_{K}\simeq 50$K, the resistivity saturates below a few KelvinAllen et al. (1979). Although this excess conductivity was originally ascribed to mid-gap impurity statesNickerson et al. (1971), renewed interest derives from the proposal that SmB₆ is a topological Kondo insulator, developing protected conducting surface states at low temperatures Dzero et al. (2010, 2012); Alexandrov et al. (2013); Lu et al. (2013). ExperimentsWolgast et al. (2013); Zhang et al. (2013); Kim et al. (2013, 2014) have since confirmed that the plateau conductivity derives from surface states, and these states have been resolved by angle-resolved photoemission spectroscopy (ARPES) Jiang et al. (2013); Neupane et al. (2013); Xu et al. (2013); Frantzeskakis et al. (2013). The most recent spin-ARPES experiments have also resolved the helicoidal spin polarization of the surface quasiparticles expected from topologically protected Dirac conesXu et al. (2014).

Yet, despite this success, certain aspects of these materials remain unexplained. A particularly notable problem, is that both quantum oscillationLi et al. (2013) and ARPESNeupane et al. (2013); Jiang et al. (2013); Xu et al. (2013); Frantzeskakis et al. (2013); Kim et al. (2014); Xu et al. (2014) studies show that the surface quasiparticles are light, with Fermi velocities $v_{s}$, ranging from $220{\rm meV\AA }$Jiang et al. (2013) to $300~{\rm meV\AA }$Neupane et al. (2013); yet current theoriesDzero et al. (2010, 2012); Alexandrov et al. (2013); Lu et al. (2013) predict heavy Dirac quasiparticles with velocities $v_{s}\sim 30-50~{\rm meV\AA }$, an order of magnitude slower.

Here we propose a resolution to this problem, by taking into account of the breakdown of the Kondo effect at the surface. The essence of our theory is based on the observation that the reduced co-ordination of the Sm³⁺ ions at the surface causes a marked reduction in the surface Kondo temperature $T_{K}^{s}\sim T_{K}/10$, so that the screening of local moments at the surface is either suppressed to much lower temperatures, or fails completely due to an intervention of surface magnetic order. “Kondo breakdown”Si et al. (2001); Coleman et al. (2001); Senthil et al. (2003); Pépin (2007); Paul et al. (2007) liberates unquenched moments at the surface, and has the effect of shifting the mixed valence of the surface Sm ions towards the higher entropy, $4f^{5}$ (3+) configuration of the unquenched moments, as observed in X-ray absorption spectroscopy Phelan et al. (2014). Most importantly, Kondo breakdown will liberate a large number of carriers previously bound inside Kondo singlets at the surface (see Fig1(b) and (c)): this process localizes $4f$-holes, driving up the electron count in the surface states to form large spin-polarized Fermi surfaces determined by the Luttinger sum rule

where $\mathcal{A}_{FS}$ is the total Fermi surface area of the singly-degenerate surface states, $\Delta n_{f}$ is the change the $f$ valence, i.e. for Sm $4f^{5.6+}\rightarrow 4f^{6+}$, $\Delta n_{f}=0.4$, in units where the lattice constant $a=1$. A detailed analysis presented later, shows that the dispersion of these highly doped surface states are dominated by a quadratic inter site hopping term

where $t_{f}$ is the effective $f$ hopping, while $r(\delta)$ and $p(\delta)$ are functions that depend on the surface scattering phase shift $\delta$, as defined later. The quadratic term, absent in a conventional topological surface state, appears when the perfect compensation between heavy and light electrons is disrupted at the surface by Kondo breakdown: this term increases the velocity of the surface states by about an order of magnitude, providing a natural explanation to the light surface states observed in experiments. Moreover, the heavy doping of the surface states shifts the Dirac point into the valence band, leading to an unusual protection of surface states against decreasing thickness and surface magnetism. We will show that the interaction of the local moments and the light surface states is described by a new kind of “chiral Kondo lattice” with the potential for a rich phase diagram of competing interacting surface states.

To describe the large $U$ limit we adopt a slave-boson mean-field theory and carry out a saddle point approximation in the bulk, giving rise to the mean-field Hamiltonian Coleman (1987).

Here, $V=\tilde{V}b$ is the renormalized hybridization, where $b$ is the slave boson projection amplitude. The f-hopping now becomes $t^{f}=b^{2}\tilde{t}^{f}$. We consider a general nearest neighbor and next nearest neighbor dispersion, $\epsilon_{k}=-2t\sum_{i}\cos k_{i}-4t^{\prime}\sum_{i\neq j}\cos k_{i}\cos k_{j}$ while $\epsilon_{f{\bf k}}=-\alpha\epsilon_{k}$, where, for simplicity, we have taken the ratio between the $f-$- and $d-$ electron hoppings to be a single fixed constant $\alpha=t_{f}/t=t_{f}^{\prime}/t^{\prime}$ for both nearest neighbor and second-neartest neighbor hoppings. The quantity $\lambda$ is the constraint field that imposes the mean-field constraint $Q=n_{f}+b^{2}$, where $Q$ is the local conserved charge associated with the slave boson treatment of the infinite $U$ limit, here taken to be $Q=1$; ${\cal N}_{s}$ is the total number of sites. In momentum space the form factor of the hybridization now takes the form $\Phi({\bf k})=\vec{s}_{{\bf k}}\cdot\vec{\sigma}$, where $\vec{s}_{{\bf k}}=(\sin{k_{x}},\sin{k_{y}},\sin{k_{z}})$, which reduces to ${\bf k}\cdot\sigma$ at small k. This odd-parity hybridization is reminiscent of the gap function in topological superfluid ³He-3B. This model can be regarded as an adiabatic continuation from small to large $U$ at infinite spin-orbit coupling. Band crossing between the odd and even parity bands generates the topological band structureHasan and Kane (2010); Qi and Zhang (2011) with protected surface states.

In order to obtain the surface state spectrum, we adopt Volovik’s approach, mapping the reflection at the boundary onto transmission through an interface where the hybridization changes signVolovik (2009). This method enables us to solve for the surface states using a linearized Hamiltonian. For pedagogical purposes, we take $t^{\prime}=t_{f}^{\prime}=0$. We can treat the surface eigenstate at $k_{x,y}=0$ as a one-dimensional problem, and since the main surface scattering effect takes place close to the Fermi surface, we can linearize the dispersion(5) normal to the surface and obtain the surface eigenstates at $k_{x,y}=0$. The two interface eigenstates at $k_{x,y}=0$ are given by

where $k_{z}=\lambda/[t(\alpha+1)]$ and $\kappa=V/(\sqrt{\alpha}t)$. $\tau$ and $\sigma$ denote the orbital and the spin component of the wave function. The corresponding energy of a state at the Dirac point is $E_{0}=\lambda/(\alpha+1)$. The transverse dispersion that develops at finite $k_{x,y}$ is then treated using first order degenerate perturbation theory, projecting the full Hamiltonian $H_{MF}$ onto the surface bound-states, $H_{eff}=\langle\psi_{\pm}|H_{MF}|\psi_{\pm}\rangle$. This leads to the dispersion

where $k_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}$. Since $\alpha\ll 1$, the surface states are composed of heavy quasiparticles. Note that at this stage, the quadratic $k^{2}$ term is absent.

Surface Kondo breakdown localizes mobile $f$-quasiparticles at the surface, removing them from the Fermi sea. In SmB₆, this is equivalent to a shift in the effective f-valence from $4f^{5.6}\rightarrow 4f^{6}$ at the surface. This is a reduction in the number of f-holes, that corresponds to an increase $\Delta n_{f}=0.4$ in the density of electrons on the surface. From Luttinger’s sum rule (eq. 1), the area enclosed by the Fermi surface is equal to the total electron density. The Fermi surface area observed in ARPES is 0.3-0.35Neupane et al. (2013); Jiang et al. (2013); Xu et al. (2013); Frantzeskakis et al. (2013); Kim et al. (2014); Xu et al. (2014), in agreement with Luttinger sum rule.

Next we examine the effect of Kondo breakdown on the dispersion of the surface states. The effect of Kondo breakdown can be incorporated by introducing a scattering phase shift $\delta=k_{z}a$ into the $f$ component of the surface state wave function, which ensures that surface node of the $f$ part of the wave function is shifted one lattice unit in towards the bulk, depleting the surface of mobile f-electrons (see supplementary information for detailssupplementary material (2014)). In Volovik’s approach, $\delta$ becomes a transmission phase shift. The unperturbed energy of the surface bound-state is now modified $E_{0}^{\prime}=(\lambda+2V\sqrt{\alpha}\sin\delta)/(1+\alpha)$ and the dispersion becomes

Note the appearance of a $k_{\perp}^{2}$ term, driven by an increase the conduction electron content of the surface states. The subleading linear term is also modified, containing the renormalization $r(\delta)$ of order one: $r(\delta)=2(\cos\delta+\frac{\kappa}{k_{z}}\sin\delta)/(1+\alpha\exp(\frac{2\kappa\delta}{k_{z}}))$. Substituting $\alpha t=t_{f}$ and $V\sim\sqrt{T_{K}t}$ in eq. 10 leads to eq. 2. This analytic result provides a natural explanation to the light quasiparticles in SmB₆. For a parameter choice of $t=500~{\rm meV}$, $t_{f}=5~{\rm meV}$, consistent with band structure calculationsLu et al. (2013), and the value of $V=100~{\rm meV}$ which leads to the experimentally observed gap $\Delta\simeq 20~{\rm meV}$ we obtain surface state velocity $v_{ss}\sim 58~{\rm meV\AA }$ without surface Kondo breakdown and $v_{ss}\sim 340~{\rm meV\AA }$ with surface Kondo breakdown in agreement with the $v_{ss}\sim 300~{\rm meV\AA }$ measured by ARPES experimentsNeupane et al. (2013). Here the quoted surface velocities were calculated at the mid-gap energy of the Dirac point in the absence of Kondo breakdown. We can not however, account for the much larger surface velocity $v_{ss}\sim 4000-7000$meV$\AA$ reported in quantum oscillations Li et al. (2013).

To confirm the results of our simplified analytical calculations, we have carried out a series of slab calculations with and without Kondo breakdown. As a first check, we simulated Kondo breakdown by setting the amplitude of the surface slave boson field to zero. Without surface Kondo breakdown (Fig 2(a)), the surface states are heavy, but as expected, when the slave boson amplitude is suppressed to zero on the surface, the topological states become much lighter as shown in Fig. 2(b). Moreover, the area of the light Fermi surface is significantly enhanced, corresponding to the extra density of carriers on the surface.

Next, to confirm that the assumption of surface Kondo breakdown we performed a fully self-consistent mean field calculation in which both $b(z)$ and $\lambda(z)$ were allowed to vary along the $z$ direction. In this self-consistent calculation, we chose parameters where the heavy and light bands cross at the three X-points, as seen SmB₆ Jiang et al. (2013); Neupane et al. (2013); Xu et al. (2013); Frantzeskakis et al. (2013); from a technical stand-point, the crossing at the three $X$ points is required to enhance the f-character, achieving the Kondo limit $n_{f}\sim 1$ while still in the strong topological insulator phase. We achieve this crossing in our model by taking $t=-t^{\prime}/2$ , such that the heavy and light bands cross at the 3 different $X$ points. We note that for the simpler choice $t^{\prime}=0$, bands cross at the $\Gamma$ point, restricting the strong topological insulating behavior to the region to $n_{f}\ll 1$. Fig 2 (c) presents the results of these calculations. We find as expected, that the self-consistently determined slave boson amplitude $b$ is depressed at the surface, corresponding to a reduction of the surface Kondo temperature. In our model calculation, for which $T_{K}/D\sim 0.025$ found that $b^{2}$ is depressed by $1/2$ at the surface, corresponding to $T_{K}^{s}/T_{K}\approx 0.5\sim(T_{K}/D)^{1/5}$, a result consistent with equation (8). We also found that as the temperature is raised above the the surface Kondo temperature $T_{K}^{s}$, the surface $b$ collapses to zero, even though it is still finite in the bulk. The dispersion displays a corresponding transition between heavy and light surface states as the temperature is raised through $T_{K}^{s}$. In practice, the possible intervention of magnetism or other instabilities may permanently prevent the low-temperature re-establishment of the Kondo effect at the surface.

Even the magnetic phase diagram of this lattice is expected to be rich. The RKKY interaction in a chiral Kondo lattice gives rise to Heisenberg, Dzyaloshniskii-Moriya and compass anisotropy terms Zhu et al. (2011) of strength $J$, $D$ and $A$ respectively. For $AJ/D^{2}>1$ the ground state is an in-plane ferromagnet whereas for $AJ/D^{2}<1$, it is a spiral of variable pitch set by $D/J$Banerjee et al. (2013). Such a spiral state evolves into a skyrmion crystal in applied magnetic fieldBanerjee et al. (2014). Experimentally, in SmB₆, indications of magnetism have been observed as hysteresis in magnetoresistance experimentsNakajima et al. (2013); Eo et al. (2014). The absence of ordered ferromagnetic moment in XMCD experimentsPhelan et al. (2014) suggests that the ground state might be a spiral with no net moment. Magnetic order breaks the time reversal symmetry and can in principle gap the surface states. We note that of the most recent single crystals show no plateau in resistivityPhelan et al. (2014) indicating that magnetism may play a crucial role in the topological behavior. Better sample quality and tuning the chiral Kondo lattice through a quantum critical point opens up the possibility of exotic superconductivity on the surface of topological Kondo insulators.

Supplementary material for ‘Kondo Breakdown In Topological Kondo Insulators’

This supplementary material presents the details of the surface state calculations. We first describe the Volovik methodVolovik (2009) and then apply it to our model, with and without Kondo breakdown.