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Home , Nevill Francis Mott, Rudolf Peierls

Course Outline: Feb. 15-19, 2021 ZOOM: https://uzh.zoom.us/j/98416350390?pwd=WEhkaTBkMzdjL21BYjZSdGMyZUdNUT09 Monday Tuesday Wednesday Thursday Friday Room ZOOM ZOOM ZOOM ZOOM ZOOM 10-10h45 Lecture 1 Lecture 4 Lecture 7 Lecture 10 Lecture 13 Johan Johan Johan Marc Johan Fermi-liquids Strongly Supercond. Magnetism Anomalous Correlated Hall effect Insulators 11-11h45 Lecture 2 Lecture 5 Lecture 8 Lecture 11 Lecture 14 Marc Marc Marc Marc Johan Kondo-physics Quantum Supercond. Skyrmions Thermal Hall Phase Effect and Transitions Conductivity Lunch Break Lunch Break Lunch Break Lunch Break Lunch Break 13h30- Lecture 3 Lecture 6 Lecture 9 Lecture 12 Lecture 15 14h15 Marc Johan Johan Marc Johan Heavy Fermions Non-Fermi Nematicity Spin Liquids Charge Order liquids Exercise Class Exercise Class 14h30-16 14h30-16 Metal insulator transitions (MIT)

Scientific RepoRts | 6:23652 | DOI: 10.1038/srep23652 MIT applications: Smart windows Fathers of Insulators

Rudolf Peierls Nevill Francis Mott Jun Kondo

Nobel Prize 1977 Background information: Material classification

Band Peierls Mott Kondo …. Background information: Band structure

Band gap

Band insulator Metal Fathers of Insulators

Rudolf Peierls Nevill Francis Mott Jun Kondo Crystal structure -> Band structure Peierls Transition

Half-filling

1d Chains

http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/PeierlsTrans.htm Bechgaard Salts

CRYSTAL STRUCTURE

beta-co-(BEDT-TTF)2I3 Bechgaard Salts

RESISTIVITY

J. Phys.: Condens. Matter 25 (2013) 343201 Fathers of Insulators

Rudolf Peierls Nevill Francis Mott Jun Kondo Kinetic Energy Dominates

Potential Energy Dominates 1042 Imada, Fujimori, and Tokura: Metal-insulator transitions

half and so-called upper Hubbard band). However, if the ps level - filling becomes closer, the character of the minimum charge N[ nis , (1.1d) excitation gap changes to that of a gap between a singly (is M. Imada et al., occupied d band with fully occupied p band and a singly where the creation (annihilation) of the single-band occupied d band with a ps hole. This kind of insulator, Rev. Mod. Phys. 70, 1040 (1998) † electron at site i with spin s is denoted by cis(cis) with which was clarified by Zaanen, Sawatzky, and Allen † nis being the number operator nis[ciscis . In this sim- (1985), is now called a charge-transfer (CT) insulator as plification, various realistic complexities are ignored, as contrasted to the former case, the Mott-Hubbard (MH) we shall see in Sec. II.A. However, at the same time, insulator, which we discuss in detail in Secs. II.A and low-energy and low-temperature properties are often III.A, and the distinction is indeed observed in high- well described even after this simplification since only a energy spectroscopy. Correspondingly, compounds that small number of bands (sometimes just one band) are have the MH insulating phase are called MH com- crossing the Fermi level and have to do with low-energy pounds while those with the CT insulating phase are excitations. The parameters of the simplified models in called CT compounds. The term ‘‘Mott insulator’’ is this case should be taken as effective values derived used in this review in a broad sense which covers both from renormalized bands near the Fermi level. types. Recent achievements in the field of strongly cor- One of the most drastic simplification in the Hubbard related electrons, especially in d-electron systems, have model is to consider only electrons in a single orbit, say brought us closer to understanding more complicated the s orbit. In contrast, the experimental study of the situations in which there is an interplay between orbital

Rev. Mod. Phys., Vol. 70, No. 4, October 1998 1042 Imada, Fujimori, and Tokura: Metal-insulator transitions

the single-particle (carrier) number, as in the transition MIT in correlated metals has been most thorough and to a band insulator. For example, this takes place in the systematic in d-electron systems, namely, transition- transition from an antiferromagnetic metal to an antifer- metal compounds. Many examples will be reviewed in romagnetic Mott insulator, where the folding of the Bril- this article. In d-electron systems, orbital degeneracy is louin zone due to the superstructure of the magnetic an important and unavoidable source of complicated be- periodicity creates a completely filled lower band. Car- havior. For example, under the cubic crystal-field sym- riers are doped into small pockets of Fermi surface metry of the lattice, any of the threefold degenerate t2g whose Fermi volume vanishes at the MIT. bands, dxy , dyz , and dzx as well as twofold degenerate From this heuristic argument, one can see at least two eg bands, dx22y2 and d3z22r2, can be located near the distinct routes to the Mott insulator when one ap- Fermi level, depending on transition-metal ion, lattice proaches the MIT point from the metallic side, namely, structure, composition, dimensionality, and so on. In ad- the mass-diverging type and carrier-number-vanishing dition to strong spin fluctuations, effects of orbital fluc- type. The diversity of anomalous features of metallic tuations and orbital symmetry breaking play important phases near both types of MIT is a central subject of this roles in many d-electron systems, as discussed in Secs. review. Mass enhancement or carrier-number reduction II.H and IV. The orbital correlations are frequently as well as more complicated features have indeed been strongly coupled with spin correlations through the observed experimentally. The experiments were exam- usual relativistic spin-orbit coupling as well as through ined from various, more or less independently devel- orbital-dependent exchange interactions and quan- oped theoretical approaches, as detailed in Sec. II. In drupole interactions. An example of this orbital effect particular, anomalous features of correlated metals near known as the double-exchange mechanism is seen in Mn the Mott insulator appear more clearly when the MIT is oxides (Sec. IV.F), where strong Hund’s-rule coupling continuous. Theoretically, this continuous MIT has been between the eg and t2g orbitals triggers a transition be- a subject of recent intensive studies in which unusual tween the Mott insulator with antiferromagnetic order metallic properties are understood from various critical and the ferromagnetic metal. Colossal negative magne- fluctuations near the quantum critical point of the MIT. toresistance near the transition to this ferromagnetic A prototype of theoretical understanding for the tran- metal phase has been intensively studied recently. sition between the Mott insulator and metals was Another aspect of orbital degeneracy is the overlap or achieved by using simplified lattice fermion models, in the closeness of the d band and the p band of ligand particular, in the celebrated Hubbard model (Anderson, atoms which bridge the elements in transition-metal 1959; Hubbard, 1963, 1964a, 1964b; Kanamori, 1963). compounds. For example, as clarified in Secs. II.A and W=4t The Hubbard model considers only electrons in a single III.A, in the transition-metal oxides, the oxygen 2ps band.Hubbard Model Its Hamiltonian in a second-quantized form is level becomes close to that of the partially filled 3d band given by near the Fermi level for heavier transition-metal elements such as Ni and Cu. Then the charge gap of the 5 1 2mN, (1.1a) HH Ht HU Mott insulator cannot be accounted for solely with d KINETIC ENERGY † electrons, but p-electron degrees of freedom have also t52t( ~ciscjs1H.c.!, (1.1b) to be considered. In fact, when we could regard the H îj& Hubbard model as a description of a d band only, the POTENTIAL ENERGY 1 1 charge excitation gap is formed between a singly occu- U5U ~ni 2 2 !~ni 2 2 !, (1.1c) H (i " # pied d band (the so-called lower Hubbard band) and a doubly occupied (with spin up and down) d band (the and so-called upper Hubbard band). However, if the ps level Mott Transition becomes closer, the character of the minimum charge N[ nis , (1.1d) excitation gap changes to that of a gap between a singly U ~ comparable to Band width W(is occupied d band with fully occupied p band and a singly where the creation (annihilation) of the single-band occupied d band with a ps hole. This kind of insulator, Lower Upper † Hubbard Hubbard electron at site i with spin s is denoted by cis(cis) with which was clarified by Zaanen, Sawatzky, and Allen † Band Band nis being the number operator nis[ciscis . In this sim- (1985), is now called a charge-transfer (CT) insulator as plification, various realistic complexities are ignored, as contrasted to the former case, the Mott-Hubbard (MH) we shall see in Sec. II.A. However, at the same time, insulator, which we discuss in detail in Secs. II.A and low-energy and low-temperature properties are often III.A, and the distinction is indeed observed in high- well described even after this simplification since only a energy spectroscopy. Correspondingly, compounds that small number of bands (sometimes just one band) are have the MH insulating phase are called MH com- crossing the Fermi level and have to do with low-energy pounds while those with the CT insulating phase are excitations. The parameters of the simplified models in called CT compounds. The term ‘‘Mott insulator’’ is this case should be taken as effective values derived used in this review in a broad sense which covers both from renormalized bands near the Fermi level. types. Recent achievements in the field of strongly cor- One of the most drastic simplification in the Hubbard related electrons, especially in d-electron systems, have model is to consider only electrons in a single orbit, say brought us closer to understanding more complicated the s orbit. In contrast, the experimental study of the situations in which there is an interplay between orbital

Rev. Mod. Phys., Vol. 70, No. 4, October 1998 Mott-insulator: Filling condition Simplest Mott insulator: Hydrogen under press Mott-insulator: Filling condition

Hydrogen Case

Electron interaction U is fixed

Lattice constant: a

t = probability for hopping M. Imada et al., Rev. Mod. Phys. 70, M. Imada et al., 1040 (1998) Rev. Mod. Phys. 70, 1040 (1998)Disclaimer: Not realized experimentally - yet Transition metal oxides

∆�

M. Sala et al., NJP 13, 043026 (2011) Transition metal oxides

La2CuO4 2x(+3) 4x(-2)

∆�

M. Sala et al., NJP 13, 043026 (2011) Transition metal oxides

Half filled

∆�

M. Sala et al., NJP 13, 043026 (2011) MottCuprate-insulator starting point for SCphase diagram

Mott-insulator: Filling condition

M. Imada et al., Rev. Mod. Phys. 70, 1040 (1998) Fathers of Insulators

Rudolf Peierls Nevill Francis Mott Jun Kondo Orbital Hybridization ORIGIN OF THE MATERIAL DEPENDENCE OF Tc IN ... PHYSICAL REVIEW B 85,064501(2012)

No hybridization

FIG. 4. (Color online) The top panel shows the main components Orbital Hybridization of the two Wannier orbitals (having different types of σ bonding) (mixing) Sakakibara et al., considered in the present two-orbital model. The bottom panel shows PRL 105, 057003 (2010) the schematic deﬁnition of the level offsets !E, !Ed ,and!Ep.

[with the ratio ( t2 t3 )/ t1 being 0.14 (0.37) for La (Hg)], resulting in the| smaller|+| curvature| | | of the Fermi surface in the former as mentioned. On the other hand, in the two-orbital model that considers the d 2 orbital explicitly, the ratio ( t2 z | |+ FIG. 3. (Color online) The band structure (with EF 0) in t3 )/ t1 within the dx2 y2 orbital changes to 0.35 (0.41) for the = | | | | − the two-orbital (dx2 y2 -dz2 ) model for La2CuO4 (left column) and La (Hg). The value is nearly the same between the single- and − HgBa2CuO4 (right). The top (middle) panels depict the weights of two-orbital modeling of Hg, while the value is significantly the d 2 2 (d 2 )characterswiththickenedlines,whilethebottom x y z increased in the two-orbital model for La. The reason why t2 panels− are the Fermi surface for the band filling of n 2.85. The = and t3 in the two-orbital model for La are large as compared to inset shows the band structure of the three-orbital model (see text) those in the single-orbital model can be understood from Fig. 5 for La system, where the 4s character is indicated by thick lines. as follows. Let us consider the diagonal hopping (t2). There is from Table I as follows. In the single-orbital model, the La adirect(dx2 y2 dx2 y2 )diagonalhopping,butthereisalso − − − cuprate has smaller t2 and t3 as compared to the Hg cuprate an indirect diagonal hopping that becomes effective when !E is small, that is, dx2 y2 dz2 dx2 y2 .Inthesingle-orbital TABLE I. Hopping integrals within the d 2 2 orbital for the − → → − x y model, where the dz2 component is effectively included in the single- and two-orbital models (upper half), interorbital− hopping dx2 y2 Wannierorbital,thecontributionofthe dx2 y2 dz2 (middle), and !E Ex2 y2 Ez2 (bottom). − − → → ≡ − − dx2 y2 path is effectively included in t2.Thelattercontribution has− a sign opposite that of the direct diagonal hopping (the One-orbital Two-orbital reason for which is clarified later), so that we end up with La Hg La Hg asmalleffectivet2 in the single-orbital model when !E is small as in the La cuprate. A similar argument applies to t3. t(dx2 y2 dx2 y2 ) Conversely, the Hg cuprate has a large !E so that the d 2 t (eV)− → − 0.444 0.453 0.471 0.456 z 1 − − − − contribution barely exists in the single-orbital model, and the t2 (eV) 0.0284 0.0874 0.0932 0.0993 ratio ( t2 t3 )/ t1 is similar to that in the two-orbital model. t3 (eV) 0.0357 0.0825 0.0734 0.0897 | |+| | | | In the La cuprate, the d 2 2 and d 2 orbitals strongly mix ( t t )/ t −0.14− 0.37− 0.35− 0.41 x y z | 2|+|3| | 1| around the N point, so that− the upper and lower bands repel t(dx2 y2 dz2 ) − → each other there, and the saddle point of the upper band that t1 (eV) 0.178 0.105 corresponds to the van Hove singularity is pushed up to nearly t2 (eV) Small Small touch the Fermi level for the band filling of n 2.85. Thus, the t3 (eV) 0.0258 0.0149 = !E (eV) 0.91 2.19 Fermi surface almost touches the wave vectors (π,0), (0,π). In the Hg cuprate, there is no such splitting of the two bands, and

064501-3 ORIGIN OF THE MATERIAL DEPENDENCE OF Tc IN ... PHYSICAL REVIEW B 85,064501(2012) Orbital Hybridization

X X

N Γ N Γ

Hybridization gap

No hybridization gap

FIG. 4. (Color online) The top panel shows the main components Sakakibara et al., of the two Wannier orbitals (having different types of σ bonding) PRL 105, 057003 (2010) considered in the present two-orbital model. The bottom panel shows the schematic deﬁnition of the level offsets !E, !Ed ,and!Ep.

064501-3 NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03266-0 ARTICLE

kz = 0 - plane # 2 ab e d d 0 xy xz/yz d 2 d 2 2 1 z x – y " g [eV] [eV]

F 0 –1 F E E – – Z E

E –1 R A –2 $/$' –2

Γ MMXXΓ MXX Γ M X X Γ Min Max dz 2 -weight TB

kz = ! - plane M k 2 z k 0 cd f y 1 kx [eV] F

[eV] 0 E F –1 E – – E

E –1

–2 –2

XARAARRZ RZ A R

Fig. 2 Comparison of observedORIGIN and calculated OF band THE structure. MATERIALa–d Background DEPENDENCE subtracted (see Methods section) OF T soft-X-rayc IN ... ARPES EDMs recorded on PHYSICAL REVIEW B 85,064501(2012) La2−xSrxCuO4, x = 0.23 along in-plane high-symmetry directions for kz = 0 and kz = π/c′ as indicated in g. White lines represent the two-orbital (dz2 and dx2 y2 ) tight-binding model as described in the text. The line width in b, d indicates the orbital weight of the dz2 orbital. e, f Corresponding in-plane DFT À band structure at kz = 0 and kz = π/c′, calculated for La2CuO4 (see Methods section). The colour code indicates the orbital character of the bands. Around the anti-nodal points (X or R), strong hybridisation of dz2 and dx2 y2 orbitals is found. In contrast, symmetry preventsC. Matt et al., any hybridisation along the nodal lines À Orbital (Γ–M or Z–A). g SketchHybridization of the 3D BZ of LSCO with high symmetry lines and points as indicated Nat. Comm. 9, 972 (2018) Results Min Max –4t 4t Material choices. Different dopings of LSCO spanning from #" #" x = 0.12 to 0.23 in addition to an overdoped compound of 0.0 !– - pol La1.8−xEu0.2SrxCuO4 with x = 0.21 have been studied. These compounds represent different crystal structures: low- –0.5 X temperature orthorhombic, low-temperature tetragonal and the Γ t#" = 0 eV high-temperature tetragonal. Our results are very similar across all crystal structures and dopings (Supplementary Fig. 1). To keep t#" = –0.21 eV [eV] –1.0 M the comparison to band structure calculations simple, this paper F focuses on results obtained in the tetragonal phase of overdoped LSCO with x = 0.23. E – E –1.5

Electronic band structure. A raw ARPES energy distribution map (EDM), along the nodal direction, is displayed in Fig. 1a. –2.0 ~2t#" Near EF, the widely studied nodal quasiparticle dispersion with 12 predominately dx2 y2 character is observed . This band reveals XXΓ the previously reportedÀ electron-like Fermi surface of LSCO, x = Fig. 3 Avoided band crossing. Left panel: ultraviolet ARPES data recorded 0.2324,25 (Fig. 1b), the universal nodal Fermi velocity v ≈ 1.5 F along the ant-inodal direction using 160 eV linear horizontal polarised eVÅ26 and a band dispersion kink around 70 meV26. The main photons. Solid white lines are the same tight-binding model as shown in observation reported here is the second band dispersion at ~1 eV Fig. 2. Right panel: tight-binding model of the dx2 y2 and dz2 bands along the below the Fermi level E (Figs. 1 and 2) and a hybridisation gap Hybridization gap À F anti-nodal direction. Grey lines are the model prediction in absence of inter- splitting the two (Fig. 3). This second band—visible in both raw orbital hopping (tαβ = 0) between dx2 y2 and dz2 . In this case, the bands are momentum distribution curves (MDC) and constant energy À crossing near the -point. This degeneracy is lifted once a finite inter-orbital maps—disperses downwards away from the BZ corners. Since a Γ hopping parameter is considered. For solid black lines t = −210 meV and pronounced k dependence is observed for this band structure αβ z other hopping parameters have been adjusted accordingly. Inset indicates (Figs. 2 and 4) a trivial surface state can be excluded. Subtracting the Fermi surface (green line) and the − X cut directions. Coloured a background intensity profile (Supplementary Fig. 2) is a stan- Γ background displays the amplitude of the hybridisation term (k) that dard method that enhances visualisation of this second band Ψ vanishes on the nodal lines No hybridization structure. In fact, usinggap soft X-rays (160–600 eV), at least two additional bands (β and γ) are found below the dx2 y2 dominated band crossing the Fermi level. Here, focus is set entirelyÀ on the β calculation (see Methods section) of La2CuO4 is shown in Fig. 2. band dispersion closest to the dx2 y2 dominated band. This band À The eg states (dx2 y2 and dz2 ) are generally found above the t2g is clearly observed at the BZ corners (Figs. 1–3). The complete in- À FIG. 4. (Color online) The top panel shows the main components bands (dxy, dxz and dyz). The overall agreement between the plane (kx, ky) and out-of-planeSakakibara (kz) band dispersionet al., is presented experiment and the DFT calculation (Supplementary Fig. 3)of thus the two Wannier orbitals (having different types of σ bonding) in Fig. 4. suggests that the two bands nearest to the Fermi level areconsidered com- in the present two-orbital model. The bottom panel shows PRL 105, 057003 (2010posed predominately) of dx2 y2 and dz2 orbitals. This conclusion Orbital band characters. To gain insight into the orbital char- can also be reached by pureÀ experimental arguments. Photo-the schematic definition of the level offsets !E, !Ed ,and!Ep. acter of these bands, a comparison with a DFT band structure emission matrix element selection rules contain information

NATURE COMMUNICATIONS | (2018)9:972 | DOI: 10.1038/s41467-018-03266-0 | www.nature.com/naturecommunications 3 [with the ratio ( t2 t3 )/ t1 being 0.14 (0.37) for La (Hg)], resulting in the| smaller|+| curvature| | | of the Fermi surface in the former as mentioned. On the other hand, in the two-orbital model that considers the d 2 orbital explicitly, the ratio ( t2 z | |+ FIG. 3. (Color online) The band structure (with EF 0) in t3 )/ t1 within the dx2 y2 orbital changes to 0.35 (0.41) for the = | | | | − the two-orbital (dx2 y2 -dz2 ) model for La2CuO4 (left column) and La (Hg). The value is nearly the same between the single- and − HgBa2CuO4 (right). The top (middle) panels depict the weights of two-orbital modeling of Hg, while the value is significantly the d 2 2 (d 2 )characterswiththickenedlines,whilethebottom x y z increased in the two-orbital model for La. The reason why t2 panels− are the Fermi surface for the band filling of n 2.85. The = and t3 in the two-orbital model for La are large as compared to inset shows the band structure of the three-orbital model (see text) those in the single-orbital model can be understood from Fig. 5 for La system, where the 4s character is indicated by thick lines. as follows. Let us consider the diagonal hopping (t2). There is from Table I as follows. In the single-orbital model, the La adirect(dx2 y2 dx2 y2 )diagonalhopping,butthereisalso − − − cuprate has smaller t2 and t3 as compared to the Hg cuprate an indirect diagonal hopping that becomes effective when !E is small, that is, dx2 y2 dz2 dx2 y2 .Inthesingle-orbital TABLE I. Hopping integrals within the d 2 2 orbital for the − → → − x y model, where the dz2 component is effectively included in the single- and two-orbital models (upper half), interorbital− hopping dx2 y2 Wannierorbital,thecontributionofthe dx2 y2 dz2 (middle), and !E Ex2 y2 Ez2 (bottom). − − → → ≡ − − dx2 y2 path is effectively included in t2.Thelattercontribution has− a sign opposite that of the direct diagonal hopping (the One-orbital Two-orbital reason for which is clarified later), so that we end up with La Hg La Hg asmalleffectivet2 in the single-orbital model when !E is small as in the La cuprate. A similar argument applies to t3. t(dx2 y2 dx2 y2 ) Conversely, the Hg cuprate has a large !E so that the d 2 t (eV)− → − 0.444 0.453 0.471 0.456 z 1 − − − − contribution barely exists in the single-orbital model, and the t2 (eV) 0.0284 0.0874 0.0932 0.0993 ratio ( t2 t3 )/ t1 is similar to that in the two-orbital model. t3 (eV) 0.0357 0.0825 0.0734 0.0897 | |+| | | | In the La cuprate, the d 2 2 and d 2 orbitals strongly mix ( t t )/ t −0.14− 0.37− 0.35− 0.41 x y z | 2|+|3| | 1| around the N point, so that− the upper and lower bands repel t(dx2 y2 dz2 ) − → each other there, and the saddle point of the upper band that t1 (eV) 0.178 0.105 corresponds to the van Hove singularity is pushed up to nearly t2 (eV) Small Small touch the Fermi level for the band filling of n 2.85. Thus, the t3 (eV) 0.0258 0.0149 = !E (eV) 0.91 2.19 Fermi surface almost touches the wave vectors (π,0), (0,π). In the Hg cuprate, there is no such splitting of the two bands, and

064501-3 Kondo PhysicsObservation of Kondo Lattice Coherence (Charge Channel)

ARPES: CeCoGe1.2Si0.8

From Lecture 3

Image courtesy of Shin-ichi KIMURA (Osaka Univ.) & Takahiro Ito (Nagoyo Univ.) Kondo Insulator

Hybridization gap LETTERS PUBLISHED ONLINE: 23 MARCH 2014 | DOI: 10.1038/NMAT3913

Topological surface state in the Kondo insulator samarium hexaboride

D. J. Kim*, J. Xia and Z. Fisk

Topological invariants of electron wavefunctions in condensed a 1,2 matter reveal many intriguing phenomena . A notable exam- 100 ple is provided by topological insulators, which are character- SmB6 ized by an insulating bulk coexisting with a metallic boundary state3,4. Although there has been intense interest in Bi-based 10

5,6 ) Sm

topological insulators , their behaviour is complicated by Ω the presence of a considerable residual bulk conductivity7–10. 11,12 1 B Theories predict that the Kondo insulator system SmB6, which is known to undergo a transition from a Kondo lattice metal to a small-gap insulator state with decreasing tem- Resistance ( 0.1 LETTERS perature, could be a topological insulator. Although the PUBLISHED ONLINE: 23 MARCHinsulating 2014 | bulkDOI: and10.1038/NMAT3913 metallic surface separation has been demonstrated in recent transport measurements13–15, these 0.01 have not demonstrated the topologically protected nature of the metallic surface state. Here we report thickness-dependent 1 10 100 transport measurements on doped SmB6, and show that T (K) magnetic and non-magnetic doping results in contrasting b 3.5 Kondo Insulator: SmBbehaviour that supports the conclusion that SmB6 shows virtually no residual bulk conductivity. 6 f-electron Systems as Prototypes For Strong Electronic Correlations3.0 The Kondo insulator SmB6 is a dense lattice of Sm magnetic SmB 320 µm/110 µm Topological surface state in the Kondo insulator6 − moments, which transforms from a poor metal at room temperature SmB 320 µm/170 µm 2.5 10 K 6 − 5 K BaB 330 µm/210 µm to an insulator with some residual conductance at low tempera- 6 − 16 CeAuSb 470 µm/310 µm ture . This transition is one of the most remarkable phenomena 2.0 2 − samarium hexaborideof Kondo lattices, which also exhibit unconventional super- 17 18 19 conductivity , hidden order transition and quantum criticality 1.5

3 Resistance

as a result of immersion of magnetic moments into a conduction Resistance ratio ratio band. The resistance of SmB6 increases exponentially as the 1.0 2 D. J. Kim*, J. Xia and Z.temperature Fisk decreases (Fig. 1a), with a non-universal ratio of low to high temperature resistance. Usually, the higher quality sample 0.5 1 2345 exhibits the higher ratio (Supplementary Information). T (K) From Ohm’s law, the electrical resistivity of a rectangular 0.0 0 50 100 150 200 250 300 Topological invariants of electronparallelepiped wavefunctions shaped bulk conductor in condensed is defined by thea product 1,2 T (K) matter reveal many intriguingof measured phenomena resistance .R Aand notable geometrical exam- factor A/L, where100L and A are the length and cross-sectional area. As ideal topological Figure 1 | Failure of Ohm’s law. a, Resistance versusSmB temperature of a ple is provided by topologicalinsulators insulators, (TIs) do which not have are bulk character- conductance, they cannot 6 rectangular parallelopiped shaped SmB6 sample; the inset shows the ized by an insulating bulk coexistingsatisfy the basic with Ohm’s a metallic law. Thus, boundary if a three-dimensional (3D) crystal structure. b, Thickness dependences of SmB6, BaB6 and CeAuSb2 3,4 TI transforms from a conventional bulk conductor at high10 state . Although there has been intense interest in Bi-based resistance. SmB6 has very clear thickness independence and its resistance temperature to an insulator with only surface conduction at low 5,6 ) ratios (deﬁned by R_thin/R_thick) for three diferent thicknesses converge Sm

topological insulators , their behaviour is complicated by Ω temperature, the sample thickness should not aﬀect the measured to unity, indicating bulk (insulator) and surface (conductor) separation as 7–10 the presence of a considerablelow temperature residual limiting bulk conductivity resistance but be independent. of it. the temperature is lowered below 10 K. In contrast, BaB and CeAuSb 1 6 2 11,12 Figure 1b shows this most unusual behaviour of SmB6, manifesting show conventional bulk conducting behaviour. B Theories predict that thetheKondo thickness independence insulator systemof sample resistance SmB6, caused by the which is known to undergobulk a transition and surface separation, from a which Kondo is a necessary lattice condition for an metal to a small-gap insulatorideal TI. state The samples with are decreasing mirror-polished tem-to follow the previousResistance ( Quantum Design PPMS with an LR700 a.c. bridge and automation scanning tunnelling microscopy result20 for metal terminated non-0.1 software21. From room temperature down to 10 K the comparative perature, could be a topologicalreconstructed insulator. surface, and the Although electrical leads the are spot-welded resistance ratio (RR, which is defined by R_thin)/R_thick) of three insulating bulk and metallicon one surface side. Then separation the thickness of has the samples been is adjusted by diﬀerent thicknesses of SmB6 follows the geometric ratios as with polishing the other side with the original13–15 leads remaining in0.01 place usual bulk metallic systems, but below 10 K RR starts dropping and demonstrated in recent transport(Supplementary measurements Information). The resistance, these is measured in a rapidly converges to unity below 5 K. This convergence indicates have not demonstrated the topologically protected nature of the metallic surface state. Here we report thickness-dependent 1 10 100 Department of Physics and Astronomy, University of California, Irvine, Irvine, California 92697, USA. *e-mail: [email protected] transport measurements on doped SmB6, and show that T (K) magnetic and non-magnetic doping results in contrasting 466 b 3.5 NATURE MATERIALS | VOL 13 | MAY 2014 | www.nature.com/naturematerials behaviour that supports the conclusion that SmB6 shows© 2014 Macmillan Publishers Limited. All rights reserved virtually no residual bulk conductivity. 3.0 The Kondo insulator SmB6 is a dense lattice of Sm magnetic SmB 320 µm/110 µm 6 − moments, which transforms from a poor metal at room temperature SmB 320 µm/170 µm 2.5 10 K 6 − to an insulator with some residual conductance at low tempera- 5 K BaB 330 µm/210 µm ➔ 4f and 5f valence electrons of rare-earth and actinide elements are typically well- 6 − From Lecture 316 CeAuSb 470 µm/310 µm ture . This transition is one of the most remarkable phenomena 2.0 2 − localized around theirof Kondo nucleus lattices, ( whichi. e. electrons also exhibit unconventional do not contribute super- to conduction). 17 18 19 conductivity , hidden order transition and quantum criticality 1.5

3 Resistance

as a result of immersion of magnetic moments into a conduction Resistance ratio ratio ➔ The localized unpairedband. The electrons resistance ofform SmB6 aincreases magnetic exponentially moment. as the 1.0 2 temperature decreases (Fig. 1a), with a non-universal ratio of low to high temperature resistance. Usually, the higher quality sample 0.5 1 2345 exhibits the higher ratio (Supplementary Information). T (K) From Ohm’s law, the electrical resistivity of a rectangular 0.0 0 50 100 150 200 250 300 parallelepiped shaped bulk conductor is defined by the product T (K) of measured resistance R and geometrical factor A/L, where L and A are the length and cross-sectional area. As ideal topological Figure 1 | Failure of Ohm’s law. a, Resistance versus temperature of a insulators (TIs) do not have bulk conductance, they cannot rectangular parallelopiped shaped SmB6 sample; the inset shows the satisfy the basic Ohm’s law. Thus, if a three-dimensional (3D) crystal structure. b, Thickness dependences of SmB6, BaB6 and CeAuSb2 TI transforms from a conventional bulk conductor at high resistance. SmB6 has very clear thickness independence and its resistance temperature to an insulator with only surface conduction at low ratios (defined by R_thin/R_thick) for three diferent thicknesses converge temperature, the sample thickness should not affect the measured to unity, indicating bulk (insulator) and surface (conductor) separation as low temperature limiting resistance but be independent of it. the temperature is lowered below 10 K. In contrast, BaB6 and CeAuSb2 Figure 1b shows this most unusual behaviour of SmB6, manifesting show conventional bulk conducting behaviour. the thickness independence of sample resistance caused by the bulk and surface separation, which is a necessary condition for an ideal TI. The samples are mirror-polished to follow the previous Quantum Design PPMS with an LR700 a.c. bridge and automation scanning tunnelling microscopy result20 for metal terminated non- software21. From room temperature down to 10 K the comparative reconstructed surface, and the electrical leads are spot-welded resistance ratio (RR, which is defined by R_thin)/R_thick) of three on one side. Then the thickness of the samples is adjusted by different thicknesses of SmB6 follows the geometric ratios as with polishing the other side with the original leads remaining in place usual bulk metallic systems, but below 10 K RR starts dropping and (Supplementary Information). The resistance is measured in a rapidly converges to unity below 5 K. This convergence indicates

Department of Physics and Astronomy, University of California, Irvine, Irvine, California 92697, USA. *e-mail: [email protected]

466 NATURE MATERIALS | VOL 13 | MAY 2014 | www.nature.com/naturematerials

Rudolf Peierls Nevill Francis Mott Jun Kondo

Vanilla Mott insulator Spin-orbit Mott insulator Band-Mott insulator LETTERS PUBLISHED ONLINE: 23 MARCH 2014 | DOI: 10.1038/NMAT3913

week ending PRL 101, 076402Topological (2008) PHYSICAL surface REVIEW state LETTERS in the Kondo insulator15 AUGUST 2008

! was obtained by using Kramers-Kronig (KK) trans- one remaining electron in the Jeff 1=2 band, the system formation.ð Þ The validitysamarium of KK analysis hexaboride was checked by is effectively reduced to a half-filled¼J 1=2 single band eff ¼ independent ellipsometry measurements between 0.6 and system [Fig. 1(c)]. The Jeff 1=2 spin-orbit integratedweek ending PRL6.4 eV.101, XAS076402 spectraD. (2008) J. were Kim obtained*, J. Xia andat 80PHYSICAL Z. K Fisk under vacuum REVIEWstates form LETTERS a narrow band so¼ that even small15U AUGUSTopens a 2008 10 of 5 10À Torr at the Beamline 2A of the Pohang Light Mott gap, making it a Jeff 1=2 Mott insulator [Fig. 1(d)]. ! Âwas obtainedTopological by using invariants Kramers-Kronig of electron wavefunctions (KK) trans- in condensedone remaininga electron¼ in the J 1=2 band, the system Source with Áh 0:1 eV. 1,2 The narrow band width is due to reducedeff hopping elements ð Þ ¼matter reveal many intriguing phenomena . A notable exam- 100 ¼ formation.Here we The propose validityple a is schematic provided of KK by model topological analysis for insulators, wasemergence checked which of are a by character-ofis the effectivelyJ 1= reduced2 states to with a half-filled isotropicSmBJ orbitaleff6 1 and=2 single mixed band eff ¼ ¼ independentnovel Mott ground ellipsometryized state byby an insulatingmeasurements a large SO bulk coupling coexisting between energy with a 0.6 metallic and boundaryspinsystem characters. [Fig. The1(c) formation]. The Jeff of the1=J2 spin-orbitbands due integratedto the state3,4. Although there has been intense interestSO in Bi-based 10 ¼ eff 6.4as shown eV. XAS in Fig. spectra1. Under wereobtained the O symmetry5, at6 80 K under the 5d vacuumstates largestatesSO formexplains) a narrow why Sr band2IrO4 so(SO that even0:4 eV small) is insulat-USm opens a topological insulatorsh , their behaviour is complicated by Ω 10 7–10 ofare5 split10 intoÀ Torrt andtheat thepresencee Beamlineorbital of a states considerable 2A by of the residual Pohang crystal bulk field Light conductivityingMott while. gap,Sr2 makingRhO4 ( itSO a Jeff0:151 eV=2)Mott is metallic. insulator [Fig. 1(d)]. 2g g 11,12 1 B Â Theories predict that the Kondo insulator system SmB6, ¼ Sourceenergy with 10Dq.Áh In general,0:1 eV4.d and 5d TMOs have suffi- TheThe narrowJeff band band formation width is is due well to justified reduced in hopping the LDA elements and which¼ is known to undergo a transition from a Kondo lattice Here we propose a schematic model5 for emergence of a LDAof theUJ Resistance ( calculations1=2 states on Sr withIrO isotropicwith and orbital without and in- mixed ciently large 10Dqmetal to to yield a small-gap a t2g insulatorlow-spin state state with decreasing for tem- eff0.1 2 4 perature, could be a topological insulator. Although theþ ¼ novel Mott ground state by a large SO coupling energy cludingspin characters. the SO coupling The formation presented of in the Fig.Jeff 2bands. The due LDA to the Sr2IrO4, and thusinsulating the system bulk would and metallic become surface a metal separation withSO has been 13–15 resultlarge [Fig. 2(a)explains] yields why a metalSr IrO with( a wide0t:24g eVband) is as insulat- in aspartially shown filled in Fig. wide1demonstrated.t Under2g band the [Fig. inO recenth1(a)symmetry]. transport An unrealistically the measurements5d states , these SO0.01 2 4 SO have not demonstrated the topologically protected natureFig. of1(a), and the Fermi surface (FS) is nearly identical to arelarge splitU intoWt2couldg and leadeg orbital to a typical states spin by theS crystal1=2 Mott field ing while Sr2RhO4 (SO 0:15 eV) is metallic. the metallic surface state. Here we report thickness-dependent energy 10Dq. In general, 4d and 5d TMOs¼ have suffi- thatThe of SrJeff2RhOband1 4 [ formation12,13]. The is10 well FS, justified composed in100 the of LDA one- and insulator [Fig. 1(b)transport]. However, measurements a reasonable on dopedU cannot SmB6, lead and show that T (K) 5 dimensional yz and zx bands, is represented by holelike cientlyto an insulating large 10Dq statemagnetic toas seen yield and from non-magnetic a t thelow-spin fact doping that Sr results stateRhO in for contrastingLDA U calculations on Sr2IrO4 with and without in- LETTERS 2g 2 4 þb 3.5 behaviour that supports the conclusion that SmB6 showscludingand X sheets the SO and coupling an electronlike presented M insheet Fig. centered2. The at LDA Sris2IrO a normal4, andPUBLISHED thus metal.virtually the ONLINE: As system no the 23 residual MARCH SO would coupling bulk 2014 become conductivity. | DOI: 10.1038/NMAT3913 is a taken metal into with À,resultX, and [Fig.M3.0points,2(a)] yields respectively a metal [12 with]. As a the wide SOt couplingband as in account, the t2g statesThe effectively Kondo insulator correspond SmB6 is ato dense the lattice orbital of Sm magnetic SmB 320 µm/110 µm 2g partially filled wide t2g band [Fig. 1(a)]. An unrealistically 6 − moments, which transforms from a poor metal at room temperatureis included [Fig. 2(b)], the FS becomesSmB 320 µm/170 rounded weekµm endingbut retains angular momentum L 1 states with zx Fig. 1(a),2.5 and the Fermi10 K surface6 − (FS) is nearly identical to largePRLU 101,W076402couldto an (2008) lead insulator to a with typical some residualPHYSICALm spinl 1S conductance1=2 Mott at REVIEW low tempera- LETTERS5 K BaB 330 µm/21015 µ AUGUSTm 2008 ¼Æ the overall topology. Despite small6 − variations in the FS 16 ¼ ¼¼ Çðj i Æ that of Sr RhO [12,13]. The FS, composed of one- i yz =p2 and ture . Thisxy transition. In the is one strong of the SO most coupling remarkable phenomena 2.02 4 CeAuSb −470 µm/310 µm insulator [Fig. 1(b)ml].0 However, a reasonable U cannot lead topology, the band structure changes2 remarkably: Two j iÞ of¼ Kondo¼ j lattices,i which also exhibit unconventional super-dimensional yz and zx bands, is represented by holelike tolimit, an insulating the t band state splits as seen into17 effective from thetotal fact angular that18 Sr RhO mo- 19 ffiffiffi 2g conductivity , hidden order transition and quantum2 4 criticalitynarrow bands1.5 crossing EF are split off from the rest due

TopologicalNovel Jeff 1=2 Mott surface State Induced state by Relativistic inand the Spin-OrbitX sheets Kondo and Coupling an electronlike insulator in Sr3 2 Resistance IrOM sheet4 centered at

ismentum aKondo Insulator: SmB normalJ metal.1=as2 adoublet resultAs the of immersion and SOJ coupling of magnetic3=2 quartet is moments taken bands into into a conduction Resistance ratio eff eff ¼ ¼ ¼ À, X, and M points,6 respectively [12].ratio As the SO coupling account,[Fig. 1(c) the][17t ].states Noteband. effectivelythat The theresistanceJ correspond of1 SmB=2 6 isincreases energetically to the exponentially orbital as the 1.0 2 B. J.samarium Kim,21gHosubtemperature Jin,1 S. decreases J. Moon, hexaborideeff ¼ (Fig.2 J.-Y.1a), with Kim, a non-universal3 B.-G. Park, ratio4 C. of(a)is lowS. LDA included Leem,5 Jaejun [Fig. 2(b) Yu,1],T. the W. FS Noh, becomes2 C. Kim, rounded5 S.-J. Oh, but1 retains f-electron Systemsangularhigher than momentum as the PrototypesJefftoL high3 temperature=12,states seemingly with resistance.For againstStrong Usually,3,4, the the Hund’s higherElectroniczx quality6 sample 6 Correlations0.5 7 1 ¼ J.-H. Park,ml 1* V. Durairaj, G. Cao,the overalland E. Rotenberg topology. Despite2345 small variations in the FS rule, since the J exhibits¼1= the2 higheris branched ratio (Supplementary off¼Æ from¼ Information). theÇðjJ i Æ T (K) E-E i yz =p2 and eff 1 xy . In the strong SO coupling5=2 M ml ¼From0 School Ohm’s oflaw, Physics the electrical and Astronomy, resistivity Seoulof a rectangular Nationaltopology, University,0.0 the Seoul band 151-747, structure Korea changes remarkably: Two ¼2 0 50 100 150 200 250 300F (j5di5Þ=2) manifoldD. J. Kim due*parallelepiped, J. to¼ Xia thej large andi shaped Z.crystal Fiskbulk conductor field as depicted is defined inby the product (eV) limit, the t2g bandReCOE splits into & Schooleffective of Physicstotal and angular Astronomy, mo- SeoulnarrowX NationalΓ bands University, crossing Seoul 151-747,E areT Korea split(K) off from the rest due Fig. 1(e)ffiffiffi. As a3 result,of measured with the resistance filledRJand geometrical3=2 band factor andA/L, where L F mentum J 1Pohang=2 doublet Accelerator and Laboratory,J eff 3=2 Pohangquartet University bands of Science and Technology, Pohang 790-784, Korea Topologicaleff 4eSSCand invariants &A are Department the of length electron and ofeff Physics, cross-sectional wavefunctions¼ Pohang area. University in As condensed ideal topological of Sciencea Figure and Technology, 1 | Failure of Ohm’s Pohang law. a,790-784, Resistance versus Korea temperature of a ¼ insulators (TIs) do not¼ have bulk conductance, they cannot [Fig. 1(c)][17]. Note that the5 Jeff 1=2 is1, energetically2 rectangular parallelopiped shaped SmB6 sample; the inset shows the matter revealsatisfy many the intriguing basicInstitute Ohm’s phenomena of law. Physics Thus, if and. a A three-dimensional Applied notable Physics, exam- (3D) Yonsei(a) LDA100 University, Seoul, Korea 6 ¼ crystal structure. b, ThicknessM dependences ofX SmB6, BaB6 and CeAuSbΓ 2 higher thanple is the providedJeffTIDepartment transforms by3 topological=2, seemingly of from Physics insulators, a conventional and against Astronomy, which bulkthe are Hund’s conductor University character- at of high Kentucky, Lexington,Γ Kentucky 40506, USASmB6 ized by an7 insulating¼ bulk coexisting with a metallic boundary(b) LDA+SOresistance. SmB6 has very clear thickness independence and its resistance rule, since the J temperatureAdvanced1=2 Light tois an branched Source, insulator Lawrence with off only from surface Berkeley the conductionJ National at Laboratory, low ratios (defined Berkeley, by R_thin/ CaliforniaR_thick) for 96720, three di USAferent thicknesses converge E-E 3,4 eff 5=2 M10 state . Althoughtemperature,¼ there the has sample been(Received thickness intense should23 interest January not a inffect Bi-based2008; the measured published 15to unity, August indicating 2008) bulk (insulator) and surface (conductor) separation as F E-E 5,6 ) (5d ) manifold due to the large crystal field as depicted in (eV) 5=2 low temperature limiting resistance but be independent of it.M βM Sm topological insulators , their behaviour is complicated by X Ω the temperatureΓ is lowered below 10 K. In contrast, BaB6 and CeAuSb2 7–10 α β WeFigure investigated1b shows this the most electronic unusual behaviour structure of ofSmB56d, manifestingtransition-metal oxide XSr2IrO4 using angle-resolved F Fig. 1(e)the. As presence a result, of witha considerable the filled residualJeff bulk3=2 conductivityband and . show conventional bulk conducting behaviour. (eV) the thickness11,12 independence of sample resistance caused byX the Γ1 B Theoriesphotoemission, predict that optical the Kondo conductivity, insulator¼ x-ray system absorption SmB6, measurements, and first-principles band calcula- bulk and surface separation, which is a necessary condition for an which istions. known The to system undergo was a found transition to be from well described a Kondo by lattice novel effective total angular momentum Jeff states, in metal to aideal small-gap TI. The samplesinsulator are state mirror-polished with decreasing to follow the tem- previous Resistance ( Quantum Design PPMS with an LR700 a.c. bridge and automation whichscanning the relativistic tunnelling microscopy spin-orbit result coupling20 for metal is fully terminated taken into non- accountsoftware0.1 under21. From a large roomΓ crystal temperatureM field. down Despite toX 10 K the comparativeΓ perature, could be a topological insulator. Although the (b) LDA+SO delocalizedreconstructed Ir 5d surface,states, andthe J theeff states electrical form leads such are narrow spot-welded bands thatresistance even a smallratioΓ (RR, correlation whichM is definedenergy by leadsXR_thin)/ to R_thick)Γ of three insulatingon bulk one and side. metallic Then the surface thickness separation of the samples has isadjusted been(c) byLDA+SO+Udifferent thicknesses of SmB6 follows the geometric ratios as with the Jeff 1=2 Mott ground state with unique13 electronic–15 and magnetic0.01 behaviors, suggesting a new class of demonstratedpolishing in recent the other transport side with the measurements original leads remaining, these in place usual bulk metallic systems, but below 10 K RR starts dropping and E-E ¼ M βM have notJ demonstratedeff(Supplementaryquantum spin the Information). driven topologically correlated-electron The protected resistance isphenomena. nature measured of in a rapidly converges to unity below 5 K. This convergence indicates

α β E- M X F β (eV) the metallic surface state. Here we report thickness-dependent X αΓ X µ DOI: 10.1103/PhysRevLett.101.076402 PACS numbers:1 71.30.+h, 71.20. b, 78.70.Dm,10 79.60. i 100 (eV) transport measurements on doped SmB6, and show that X Γ À T (K) À magnetic and non-magnetic doping results in contrasting Department of Physics and Astronomy, University of California, Irvine, Irvine, California3.5 92697, USA. *e-mail: [email protected] behaviour that supports the conclusion that SmB shows b Mott physics based on the Hubbard Hamiltonian, which6 Sr2IrO4, however, is unexpectedly an insulator with weak virtually no466 residual bulk conductivity. NATURE MATERIALSΓ | VOL 13 |M MAY 2014 | www.nature.com/naturematerialsX Γ is at the root of various noble physical phenomena such as ferromagnetism(c) LDA+SO+U3.0 [8].Γ At this point,M it is naturalX to considerΓ The Kondo insulator SmB6 is a dense lattice of© Sm 2014 magnetic Macmillan Publishers Limited. All rights reserved SmB 320 µm/110 µm metal-insulator transitions, magnetic spin orders, high T the(d) spin-orbitLDA+U (SO) coupling as a candidate6 − responsible for moments, which transforms from a poor metal at room temperatureC SmB 320 µm/170 µm superconductivity, colossal magneto-resistance, and quan- the insulating2.5 nature10 since K its energy6 − is much larger than 5 K BaB 330 µm/210 µm E- to an insulator with some residual conductance at low tempera- M 6 − β E-E tum criticality,16 has been adopted to explain electrical and that inM 3d andα 4dX systems.CeAuSb Recent470 band µm/310 calculations µm µ ture . This transition is one of the most remarkable phenomena 2.0 2 − (eV) F

X Γ (eV) magneticof Kondo properties lattices, of various which also materials exhibit in unconventional the last several super-showedX that the electronic states near EF can be modiﬁed 17 18 19 Γ decadesconductivity [1–5]. Great, hidden success order transition has beenand achieved quantum in criticality3d considerably1.5 by the SO coupling in 5d systems, and sug-

3 Resistance

as a result of immersion of magnetic moments into a conduction Resistance ratio

transition-metal oxides (TMOs), in which the localized gested a new possibility of the Mott instabilityratio [14]. It indi- ➔ 4f and 5f valence electronsband. The of resistance rare-earth of SmB6 andincreases actinide exponentially elements as the are1.0 typically well- 3d states yield strongly correlated narrow bands with a cates that the correlation effectsM can be important2 X even in From Lecture 3temperature decreases (Fig. 1a), with a non-universal ratio of low Γ Γ M X Γ Γ (d) LDA+U localized aroundFIG. theirlarge 1.to on-siteSchematic highnucleus temperature Coulomb energy (i. repulsione. resistance. diagrams electronsU Usually,forand the a 5 thedo smalld5 higher( tnot5 band) conﬁgu- qualitycontribute width sample5d TMOs to 0.5conduction). when combined with strong SO1 coupling. W. As predicted, most stoichiometric 3d TMOs2g are anti- In this Letter, we show2345 formation of new quantum state ration (a)exhibits without the SO higher and ratioU, (b) (Supplementary with an unrealistically Information). large U FIG. 2 (color online). TheoreticalT (K) Fermi surfaces and band ferromagneticFrom Ohm’s (AFM) law, Mott the electrical insulators resistivity [5]. On of the a other rectangularbands with0.0 effective total angular momentum J in 5d E-E but no SO, (c) with SO but no U, and (d) with SO and U. dispersionsM 0 in (a) LDA,50 (b)100LDA 150SO, (c)200LDAeff 250SO U 300

parallelepiped shaped bulk conductor is defined by the product F ➔ The localized unpairedhand, 4 delectronsand 5d TMOs form were a magnetic considered as moment. weakly- electron systems under a large crystalþ field, in whichþ theþ (eV) Possible optical transitions A and B are indicated by arrows. (2 eV),X and (d)Γ LDA U. In (c), theT (K) left panel shows topology correlatedof measured wide band resistance systemsR and with geometrical largely reduced factor AU/L,due where LSO coupling is fullyþ taken into account, and also report for (e) 5d level splittings by the crystal field and SO coupling. of valence band maxima (EB 0:2 eV) instead of the FS. to delocalizedand A are the4d lengthand 5 andd states cross-sectional [6]. Anomalous area. As insulating ideal topologicaltheFigure first time1 | Failure manifestation of Ohm’s law. a¼, of Resistance a novel versusJ temperature1=2 Mott of a insulators (TIs) do not have bulk conductance, they cannot eff 076402-2 rectangular parallelopiped shaped SmB6 sample; the inset¼ shows the behaviorssatisfy were the basic recently Ohm’s reported law. Thus, in some if a4 three-dimensionald and 5d TMOs (3D)ground state realized in Sr2IrO4 by using ARPES, optical [7–10], and the importance of correlation effects was rec- conductivity,crystal structure. andb, x-ray ThicknessΓ absorption dependencesM spectroscopy of SmBX6, BaB6 and (XAS) CeAuSbΓ2 TI transforms from a conventional bulk5 conductor5 at high resistance. SmB has very clear thickness independence and its resistance FIG. 1. Schematic energy diagrams for the 5d (t2g) configu- 6 ognizedtemperature in 4d TMOs to an insulator such as withCa only2RuO surface4 and conductionY2Ru2O7, at lowand first-principles band calculations. This new Mott ration (a) without SO and U, (b) with an unrealistically large U FIG.ratios 2 (defined (color by online).R_thin/R_thick) Theoretical for three dif Fermierent thicknesses surfaces converge and band whichtemperature, were interpreted the sample as Mottthickness insulators should not near aff theect the border measuredgroundto unity, state indicating exhibits bulk novel (insulator) electronic and surface and (conductor) magnetic separation behav- as but no SO,low (c) temperature with SO limiting but no resistanceU, and (d) but with be independent SO and U of. it. dispersions in (a) LDA, (b) LDA SO, (c) LDA SO U line of the Mott criteria, i.e., U W [7]. However, as 5d ior,the for temperature example, isspin-orbit lowered below integrated 10 K. In contrast, narrowþ BaB bands6 and CeAuSb andþ an2 þ PossiblestatesFigure optical are spatially1b transitions shows more this most A extended and unusual B areand behaviour indicatedU is expected of SmB by6, arrows. manifesting to be exotic(2show eV), orbital conventional and dominated (d) LDA bulk conducting localU. In magnetic behaviour. (c), the left moment, panel suggest- shows topology (e) 5d levelthe splittings thickness independence by the crystal of field sample and resistance SO coupling. caused by the þ further reduced, insulating behaviors in 5d TMOs such as ingof a valence new class band of themaximaJ quantum (EB 0 spin:2 eV driven) instead correlatedof the FS. bulk and surface separation, which is a necessary condition for an eff ¼ Sr2IrOideal4 and TI.Cd The2Os samples2O7 have are mirror-polished been puzzling to [8, follow9]. the previous076402-2electronQuantum phenomena. Design PPMS with an LR700 a.c. bridge and automation 20 21 Srscanning2IrO4 crystallizes tunnelling in microscopy the K2NiF result4 structurefor metal as La terminated2CuO4 non- Singlesoftware crystals. From of roomSr2IrO temperature4 were grown down to by 10 flux K the method comparative and itsreconstructed4d counterpart surface,Sr2RhO and4 the[8, electrical11]. Considering leads are its spot-welded odd [15].resistance ARPES ratio spectra (RR, were which obtained is defined at by 100R_thin)/ K fromR_thick) cleaved of three 5 11 numberon one of electrons side. Then per the unit thickness formula of ( the5d ), samples one expects is adjusted a bysurfacesdifferentin thicknesses situ under of vacuum SmB6 follows of 1 the10 geometricÀ Torr ratiosat the as with polishing the other side with the original leads remaining in place usual bulk metallic systems, but belowÂ 10 K RR starts dropping and metallic state in a naı¨ve band picture. Indeed Sr2RhO4 beamline 7.0.1 of the Advanced Light Source with h# (4d5)(Supplementary is a Fermi liquid Information). metal. Its The Fermi resistance surface is (FS) measured mea- in85 a eVrapidlyand convergesÁE 30 to meV unity. The below chemical 5 K. This potential convergence" was indicates¼ ¼ sured by the angle-resolved photoemission spectroscopy referred to EF of electrically connected Au. The band (ARPES) agrees well with the band calculation results calculations were performed by using first-principles [12,13Department]. Since bothof Physics systems and Astronomy, have identical University atomic of California, arrange- Irvine, Irvine,density-functional-theory California 92697, USA. *e-mail: [email protected] with LDA and LDA U ments with nearly the same lattice constants and bond methods [16]. The optical reflectivity R ! was measuredþ 466 NATURE MATERIALS | VOL 13 | MAY 2014 | www.nature.com/naturematerials angles [8,11], one expects almost the same FS topology. at 100 K between 5 meV and 30 eV andð theÞ conductivity © 2014 Macmillan Publishers Limited. All rights reserved

0031-9007=08=101(7)=076402(4) 076402-1 Ó 2008 The American Physical Society Example: Ca2RuO4

4d electrons

Filling = 2/3 Band Structure Calculation Ingredient: Crystal Lattice Denys Sutter & Fabio Cossalter et al., Nature Communications 8, 15176 (2017) Example: Ca2RuO4

Physical Review Letters 93, 146401 (2004) Combine band – Mott insulator

½ filled band Crystal field splitting

Ca dxy filled band

dxz dyz Compression Elongation Apical oxygen