Additional energy scale in SmB$_6$ at low temperature

TL;DR

This study uses low-temperature scanning tunneling microscopy and spectroscopy to reveal complex electronic states in SmB$_6$, shedding light on its topological surface states and the breakdown of the Kondo effect at low temperatures.

Contribution

It provides detailed spectroscopic evidence of surface and bulk states in SmB$_6$, clarifying its electronic structure and the temperature-dependent Kondo breakdown.

Findings

Surface states develop strongly below 7 K

Hybridization gap of about 20 meV observed

Evidence of Kondo effect breakdown at the surface

Abstract

Topological insulators give rise to exquisite electronic properties due to their spin-momentum locked Dirac-cone-like band structure. Recently, it has been suggested that the required opposite parities between valence and conduction band along with strong spin-orbit coupling can be realized in correlated materials. Particularly, SmB$_6$ has been proposed as candidate material for a topological Kondo insulator. By utilizing scanning tunneling microscopy and spectroscopy measurements down to 0.35 K, we observed several states within the hybridization gap of about $\pm$20 meV on well characterized (001) surfaces of SmB$_6$. The spectroscopic response to impurities and magnetic fields allows to distinguish between dominating bulk and surface contributions to these states. The surface contributions develop particularly strongly below about 7 K which can be understood in terms of a breakdown of the Kondo effect at the surface. Our high-resolution data provide detailed insight into the electronic structure of SmB$_6$, which will reconcile many current discrepancies on this compound.

Full text

Additional energy scale in SmB6 at low temperature

L. Jiao

Max-Planck-Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany

  

S. Rößler

Max-Planck-Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany

  

D. J. Kim

Department of Physics and Astronomy, University of California, Irvine, CA 92697

  

L. H. Tjeng

Max-Planck-Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany

  

Z. Fisk

Department of Physics and Astronomy, University of California, Irvine, CA 92697

  

F. Steglich

Max-Planck-Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany

  

S. Wirth

Max-Planck-Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany

Topological insulators give rise to exquisite electronic properties due to their spin-momentum locked Dirac-cone-like band structure. Recently, it has been suggested that the required opposite parities between valence and conduction band along with strong spin-orbit coupling can be realized in correlated materials. Particularly, SmB6 has been proposed as candidate material for a topological Kondo insulator. By utilizing scanning tunneling microscopy and spectroscopy measurements down to 0.35 K, we observed several states within the hybridization gap of about $\pm$20 meV on well characterized (001) surfaces of SmB6. The spectroscopic response to impurities and magnetic fields allows to distinguish between dominating bulk and surface contributions to these states. The surface contributions develop particularly strongly below about 7 K which can be understood in terms of a breakdown of the Kondo effect at the surface. Our high-resolution data provide detailed insight into the electronic structure of SmB6, which will reconcile many current discrepancies on this compound.

In the past few years, the concept of strong topological insulators which exhibit an odd number of surface Dirac modes characterized by a $\mathbb{Z}_{2}$ topological index, has attracted great interest. In this context, it was theoretically predicted that some Kondo insulators, such as SmB6, Ce3Bi4Pt3, CeNiSn, CeRu4Sn6, are candidates for strong three-dimensional (3D) topological insulators Dze2010 ; Ale2013 . In particular, SmB6 is intensively studied due to its simple crystal structure and clear signatures of a hybridization gap. Theoretically, a common picture of the multiplet $f$-states and the Kondo coupling is shared among different band structure calculations for bulk SmB6 Ale2013 ; Tak2011 ; Lu2013 ; Junwon ; Baruselli ; KangJPSJ , as sketched in Fig. 1. Due to strong spin-orbit coupling and crystal field effects, the $f$-states of Sm are split into several multiplets as presented in Fig. 1a. Considering the symmetry of the multiplets, only the $\Gamma_{7}$ and $\Gamma_{8}^{(1)}$ bands are allowed to hybridize with the Sm $d$-band via the Kondo effect Lu2013 ; Baruselli . As a result, two hybridization gaps ($\Delta_{1}$, $\Delta_{2}$) may open at different energies as sketched in Fig. 1b (in principle only $\Delta_{2}$ is a well-defined gap). Although topological surface states (TSS) are unambiguously predicted to reside within the hybridization gap Ale2013 ; Tak2011 ; Lu2013 ; Junwon ; Baruselli ; KangJPSJ , no consensus has been reached on the structure of the TSS around the Fermi energy ($E_{F}$).

Nonetheless, the prediction of TSS provides an attractive explanation for the four decades-old conundrum Men1969 of SmB6 which exhibits a plateau in the resistivity typically below about 5 K All1979 ; Kim2014 .

Experimentally, the existence of metallic surface states below about 5 K has been best illustrated by electrical transport measurements on SmB6 Kim ; Kim2014 ; Wol2013 . However, the origin of these surface states and their topological properties remain controversial in spite of intensive investigations. Several properties of SmB6 interfere with a straightforward interpretation. One major issue arises with respect to the size of the hybridization gap. Spectroscopic measurements observed a large hybridization gap of about 15–20 meV Zhang ; Frantzeskakis ; Den2014 ; Neupane ; Jiang ; Denlinger ; Xu2013 ; Xu2014 ; Min2014 ; Xu ; Zhu ; Hlawenka , which is normally understood by considering a single coherent $f$-band hybridizing with a conduction band (Supplementary Fig. 1). Typically, additional features within this energy scale are assumed to be in-gap states. In some cases, the in-gap states are further ascribed to TSS Neupane ; Jiang . On the other hand, analyses of thermal activation energies derive a small excitation energy of 2–5 meV, which shows bulk properties and is understood in terms of a small, likely indirect, bulk gap Coo1995 ; Chen ; Zhou16 or in-gap states Gab ; Gor ; Fla2001 . Obviously, different probes as well as different ranges in the measurement temperatures reveal only either the bigger or the smaller hybridization gap sketched in Fig. 1b. Nevertheless, these measurements provide essential constraints to the sizes of the two hybridization gaps. In terms of topology (i.e. trivial or non-trivial surface states), experimental results, even obtained by using the very same method, are conflicting among many reports Den2014 ; Frantzeskakis ; Neupane ; Jiang ; Denlinger ; Xu2013 ; Xu2014 ; Min2014 ; Xu ; Zhu ; Hlawenka ; Li2014 ; Tan2015 ; Tho2013 ; Nakajima . Considering the exotic phenomena which appear only within $\pm$20 meV and below 5 K, measurements with very high energy resolution and at very low temperature are highly desired.

Another severe difficulty, which contributes to such a wide discrepancy among the experimental results, is caused by the surface itself. Specifically, the (001) surface of SmB6 is polar Zhu . This can induce different types of band bendings Den2014 , quantum well confinements Kan2015 , charge puddles, and surface reconstructions Yee ; Sahana ; Ruan ; roe15 . Specifically the latter may give rise to conducting surface layers on its own Zhu . Frequently, different types of surfaces (B- and Sm-terminated, reconstructed and non-reconstructed) coexist at different length scales on one and the same cleaved surface which may complicate interpretation of spectroscopic results, e.g., by angle-resolved photoemission spectroscopy (ARPES).

We therefore conducted scanning tunneling microscopy/spectroscopy (STM/STS) down to the base temperature of 0.35 K with an energy resolution of about 0.5 meV. This allowed us to identify the fine structure of the hybridization gaps on large and non-reconstructed surfaces in the sub-meV scale. Moreover, by measuring the impurity, magnetic-field and temperature dependence of the STS spectra, we were able to attribute bulk and/or surface contributions to these states, and unveil a new energy scale of $\simeq$ 7 K, which provides an important piece of the puzzle for a unified picture of SmB6.

Results

Topography and STS spectra at base temperature. In Fig. 2b, the topography of a non-reconstructed surface with clear atomic resolution is presented. The distance of about 4.1 Å and arrangement of the corrugations is in good agreement with the lattice constant $a=$ 4.133 Å of the cubic crystal structure of SmB6 (Fig. 2a). The very small number of defects compared to the number of unit cells within the field of view (more than 5200) not only indicates high sample quality but also ensures that the measured spectrum is not influenced by defects.

The absence of any corrugation other than along the main crystallographic axes, as nicely seen in the inset to Fig. 2b, clearly indicates a B-terminated surface Sahana ; roe15 .

The tunneling conductance $g(V)$ $\equiv$ d$I(V)/$d$V$, measured at $T=$ 0.35 K and far away from any impurity, exhibits several anomalies close to $E_{\rm F}$, marked by i) – v) in Fig. 2c. A change in the slope of $g(V)$ around $\pm$20 meV suggests a pronounced loss of local density of states (LDOS) within this energy range. Around the same energy, the opening of a gap has been widely observed by a number of spectroscopic tools as mentioned above Frantzeskakis ; Jiang ; Denlinger ; Xu2013 ; Xu2014 ; Min2014 ; Xu ; Zhu ; Hlawenka , including STS Yee ; Sahana ; Ruan . Based on the band structure displayed in Fig. 1b, the kinks marked by i) can be ascribed to the Kondo coupling between the $f$-band and the conduction band, which results in a decreased conduction electron density inside the hybridization gap below the Kondo temperature $T_{K}$ ErnstNature2011 .

More importantly, we were able to disentangle several anomalies which were hitherto not resolved individually by STS at higher temperature Yee ; Sahana ; Ruan . Benefitting from this improvement, we can investigate the fine structure of bulk/surface bands and go beyond a simple Kondo hybridization analysis of the bulk states, which is based on only one $f$-band and one conduction band Den2014 . Around $-13.5$ meV, there is a small peak marked by ii). Excitations with similar energy have been reported before, e.g. by ARPES ($-15$ meV) Neupane , X-ray photoelectron spectroscopy (XPS) ($-15$ meV) Den2014 and inelastic neutron scattering (14 meV) Fuhrman ; ale95 , yet with differing explanations as to its origin. As discussed below, this small peak is most likely related to the indirect tunneling into the localized $\Gamma^{(2)}_{8}$ states. Compared to delocalized $f$-states, such localized $f$-states may give rise to only small anomalies in spectroscopy measurements Ramankutty .

Compared to peak ii), peak iii) (at around $-6.5$ meV) is very sharp and pronounced. Such a peak has been observed on different types of surfaces, including reconstructed ones Yee ; Sahana ; Ruan , which clearly indicates that there are significant bulk contributions to this state. Very likely, the weakly dispersive structure of the hybridized $\Gamma^{(1)}_{8}$ band around the $X$-point along with the Fano effect can induce a peak in the conductance spectra at this energy level. In a Kondo system, the Fano effect is due to a quantum mechanical interference of electrons tunneling into the localized states and the conduction bands mal09 ; Figgins . Either a sharp drop (like feature i)) or a pronounced peak will show up around the gap edge, depending on the tunneling ratio between the two channels as well as the particle-hole asymmetry of the conduction band. However, as has been reported previously, the spectrum deviates from a simple Fano model at low temperature Ruan ; Yee , indicating additional components to peak iii) (see also discussion below). This is consistent with our inference that the hybridized $\Gamma^{(1)}_{8}$ band resides within the big gap $\Delta_{1}$ and also contributes to the intensity of this peak. Hence, the position of peak iii) can provide an indication with respect to the energy level of the $\Gamma^{(1)}_{8}$ band and therefore the size of the small gap $\Delta_{2}$. Note that its energy level is also comparable with the size of the small bulk gap observed by transport measurements Coo1995 ; Chen ; Zhou16 . Therefore, peak i) to iii) can directly be compared to the band structure in Fig. 1b. To verify the bulk/surface origins of these peaks at low temperature, impurity, magnetic field, and temperature dependence of STS have been conducted. As we will show below, besides bulk components, peak iii) also contains components from the surface layer below 7 K.

Crucially, we also observe small anomalies iv) and v) at $\pm$3 meV, which reside just inside the bulk gap $\Delta_{2}$ (cf. also Fig. 4c and d). The shoulder-like shape of these small anomalies indicates the existence of two weakly dispersive bands or localized states near $E_{F}$. It is noted that both features at about $\pm$3 meV also reveal spatial inhomogeneity (see Supplementary Fig. 2), which—given the electronic inhomogeneity of even atomically flat surfaces roe15 —hints at a surface origin of these states.

Spatial dependence of the STS spectra. For STM measurements, one possible way to distinguish bulk and surface states is to carefully investigate the tunneling spectra at/near impurities or other defects, because the surface states are more vulnerable to such defects.

Therefore, $g(V)$ was measured across two impurity sites at 0.35 K, shown in Figs. 3a and b. The bigger impurity at $\#$A with an apparent height of $\approx$160 pm is probably located on top of the surface, while the smaller one at $\#$E (apparent height $\approx$50 pm) is likely incorporated into the crystal. According to Fig. 3c, the $g(V)$-curves are all very similar for positions $\#$B to $\#$F. Even at position $\#$A, i.e. on top of the big impurity, the spectrum exhibits similarities; in particular all anomalies i)–v) can be recognized. In addition, a new peak occurs at $-10$ meV, which may be assigned to an impurity bound state. In Fig. 3d, we plot the height of the peaks ii) to iv) at different positions. A combined analysis of Figs. 3c and d reveals spatial stability of peak ii), being consistent with the expectation for bulk states as discussed above. On the other hand, peaks iii) and iv) are not as stable as peak ii); their heights are suppressed by both the big and the small impurity, which implies that at this temperature both peaks contain contributions from the states pertained to the surface.

Magnetic field dependence of the STS spectra. In Fig. 4a and b, $g(V)$-curves measured at sites $\#$A and $\#$C of Fig. 3a for different applied magnetic fields are presented. There is no distinct change detected up to the maximum field of 12 T for features i) to v), except an enhanced peak amplitude for the impurity state at $-10$ meV, see Fig. 4b. The magnetic-field independence of these states is consistent with the observation of metallic surface conductance up to 100 T by transport Chen ; Biswas ; Coo95b ; Coo99 and spectroscopic measurements Fla2001 ; Yee . This observation can be understood by considering a very small $g$-factor (0.1–0.2) of the $f$-electrons Ert2015 .

Temperature dependence of the STS spectra. We now turn to the temperature dependence of the features i) to v). The temperature evolution of the STS spectra was measured continuously on the same unreconstructed, B-terminated surfaces away from any defect between 0.35 and 20 K, see Fig. 4c. Above 15 K, the spectra show a typical asymmetric lineshape which arises from the Fano effect mal09 ; Figgins , being in good agreement with previous work Sahana (see Supplementary Fig. 4). Upon cooling, the amplitude of peak iii) increases sharply, accompanied by a sudden appearance of peaks iv) and v) below 7 K, with the latter effect being beyond thermal smearing (see Supplementary Fig. 3). The low-temperature evolution of the spectra is clearly seen after the measured $g(V,T)$-curves were subtracted by the data at 20 K, see Fig. 4d. In an effort to quantitatively investigate the evolution of the spectra with temperature, we describe the low-temperature $g(V)$-curves by a superposition of four Gaussian peaks on top of a co-tunneling model (see Supplementary Fig. 4). However, fits to data obtained at higher temperature ($T$ $>$ 10 K) turned out to be less reliable (Supplementary Figs. 5 and 6).

To further analyze the temperature evolution of peak iii), we normalized the spectra by its size at $V_{b}$ = $\pm$30 mV. The resulting $g(T)$-values of peak iii) are plotted in Fig. 4e. Clearly, a change in the temperature dependence is observed around 7 K. This is further supported by a comparison to data obtained by Yee et al. Yee (blue circles and blue dashed line) in a similar fashion but on a ($2\!\times\!1$) reconstructed surface (which may explain the scaling factor, right axis). Also, the spectral weights of the $-10$ meV peak by Ruan et al. Ruan (green squares) indicate a similar trend at $T$ $\geq$ 5 K. Note that even the temperature evolution above about 7 K cannot be explained by a mere thermal broadening effect Yee ; Ruan . By tracing the temperature evolution of the d$I$/d$V$-spectra between about 7–50 K Yee ; Sahana ; Ruan a characteristic energy scale of about 50 K was derived. This can be accounted for by the Kondo effect of the bulk states, with an additional contribution from a resonance mode Ruan which is likely (as discussed above) related to the $\Gamma_{8}^{(1)}$ state. The same energy scale of $\gtrsim 50$ K has also been observed by transport All1979 ; Wol2013 and other spectroscopic measurements Zhang ; Xu2014 ; Caldwell ; CHMin . However, below 7 K, the intensity of peak iii) shows a sudden enhancement in Fig. 4e, indicating the emergence of an additional energy scale. Considering the fact that this new energy scale as well as many other exotic transport phenomena related to the formation of a metallic surface Kim ; Kim2014 ; Wol2013 set in simultaneously, the increase in intensity of peak iii) (as well as the appearance of peaks iv) and v)) is expected to rely on the same mechanism that is responsible for the formation of the metallic surface. Both observations appear to evolve out of the bulk phenomena associated with the primary hybridization gap at elevated temperatures. In the following section, we will argue that this new energy scale is related to the breakdown of the Kondo effect at the surface.

Discussion

In this study, the topographic capabilities of the STM allow us to distinguish features i) to v) on non-reconstructed (001) surfaces of a single termination and without apparent defects. Therefore, we can simply exclude the possibility that they are driven by surface reconstructions or defects. Especially the observation of new states on clean surfaces below about 7 K indicates that the exotic properties of SmB6 are intrinsic rather than due to impurities. The observation of well-resolved

features in our tunneling spectra (discussed above) enables the direct comparison with results of bulk band structure calculations Lu2013 ; Denlinger ; Junwon ; Baruselli ; Peters ; KangJPSJ . This not only reveals the energy levels of the multiplet $f$-states, but can also reconcile the long-standing debate of ‘small’ versus ‘large’ bulk gap in SmB6 Den2014 . Consequently, our data shows that a dedicated hybridization model with two—instead of one—multiplet $f$-states is necessary to interpret the low-energy properties of SmB6. In particular, peak iii) has multiple components including bulk and surface states, the ratio of which changes dramatically with temperature.

It is widely accepted that the electronic properties of SmB6 can be divided into several temperature regions, which are based on transport measurements Denlinger ; Chen as well as other probes, like ARPES Denlinger ; Xu2014 . Apparently, 5-7 K is a crucial regime where the temperature-dependent properties undergo significant changes. Above this range, the electronic states in SmB6 are governed by the Kondo effect of the bulk Den2014 ; Jiang ; Frantzeskakis . At lower temperatures, several interesting observations—in addition to that of the saturated resistance—were made. For example, the Hall voltage becomes sample-thickness independent Kim ; the angular-dependent magnetoresistance pattern changes from fourfold to twofold symmetry Chen ; and the development of a heavy fermion surface state is found by magnetothermoelectric measurements Luo . These experimental facts provide convincing evidence for the formation of (heavy) surface states just around 5-7 K, which is in line with the appearance of a new energy scale.

Recently, a surface Kondo breakdown scenario was proposed based on the reduced screening of the local moments at the surface. As a result, the Kondo temperature of the outmost layer ($T_{K}^{s}$) can be strongly suppressed, resulting in a modified band structure Alexandrov . Slab calculations further show that below $T_{K}^{s}$ $f$-electrons gradually become coherent and form weakly dispersive band close to $E_{F}$ Ert2015 ; Peters . Remarkably, very narrow peaks with strongly temperature-dependent STS spectra near $E_{F}$ are regarded as a smoking gun evidence for a surface Kondo breakdown scenario Peters . Based on our experimental results, $T_{K}^{s}$ is inferred to be around 7 K, being about an order of magnitude smaller than $T_{K}$. The evolution of our tunneling spectra below about 7 K also fit excellently to the theoretical prediction and the related calculations for STS. In such a scenario, the additional component at $-6.5$ meV and shoulders at $\pm$3 meV are related to the heavy quasiparticle surface states, the formation of which supplies an additional tunneling channel in particular into the $f$-states. This provides a highly possible origin for the metallic surface states and a reasonable explanation to the various experimental observations listed above.

We note that theoretically a surface Kondo breakdown effect does not change the topological invariance of SmB6, which is determined by the topology of the bulk wave functions. Therefore, the surface-derived heavy quasiparticle states could still be topologically protected. Experimentally, for such topologically protected surface states backscattering is forbidden in quasiparticle interference (QPI) patterns as measured by STM Xu2016 . In line with this prediction and as shown in the Supplementary Fig. 7, no clear QPI pattern could be detected so far from our results, which is similar to the observation by Ruan et al., Ruan .

Methods

All samples were grown by the Al-flux method. A cryogenic (base temperature $T\approx 0.35$ K) STM with magnetic field capability of $\mu_{0}H\leq 12$ T was utilized. Three SmB6 single crystals were cleaved a total of five times in situ at $\approx$20 K to expose a (001) surface. Cleaved surfaces were constantly kept in ultra-high vacuum, $p<3\cdot 10^{-9}$ Pa. Tunneling was conducted using tungsten tips Ernst and the differential conductance ($g(V)$-curve) is acquired by the standard lock-in technique with a small modulation voltage. In our best cleaved sample, the size of non-reconstructed surface can reach to $100\,\times\,100$ nm2.

In principle, the low-temperature $g(V)$-curves can be well described by a superposition of four Gaussian peaks on top of a Fano model (see example of $g(V,T\!=\!0.35\,$K$)$ in Supplementary Fig. 4) or more elaborate hybridization models mal09 ; Figgins (Supplementary Fig. 6). A similar procedure with only one Gaussian was employed in Ruan . However, fits are less reliable at elevated temperature. Instead, our spectra measured at different $T$ in zero field overlap nicely for $V_{b}<-25$ mV and $V_{b}>10$ mV such that they can be normalized by using very similar factors. Consequently, we can directly trace the temperature dependence of the peak height (at least for peak iii)) by measuring the normalized peak intensity as shown in Fig. 4e. Note that the intensities of peak iii) as obtained from Fig. 4d, *i.e. *after subtracting the 20 K-data, yield very similar values as those shown in Fig. 4e from normalized spectra.

acknowledgments

We acknowledge valuable discussion with J. W. Allen, P. Coleman, X. Dai, I. Eremin, O. Erten, C.-L. Huang, Deepa Kasinathan, B. I. Min, C. J. Kang, G. Sawatzky, Q. Si and P. Thalmeier. This work was supported by the Deutsche Forschungsgemeinschaft through SPP 1666 and by the Defense Advanced Research Agency (DARPA) under agreement number FA8650-13-1-7374. L.J. acknowledges support by the Alexander-von-Humboldt foundation.