Weyl semimetals: the next (next) graphene? Leon Balents, KITP, UCSB
Anton Burkov, U. Waterloo
Gabor Halasz, U. Cambridge Outline
• Weyl fermions • Where to find them • TR-breaking and Hall effects • I-breaking • Graphene-like physics Weyl Fermion
• Massless Dirac fermion with fixed handedness • described by a 2-component spinor unlike 4-component (spin+particle/hole) Dirac spinor
H = vσ k · Level repulsion
• von Neumann and Wigner, 1929 • In QMs, 3 parameters must be tuned to make 2 levels cross • led to a whole field of statistics of energy levels, quantum chaos,... ACCIDENTAL D EGEN ERAC Y 373
For crystals with an inversion center, contacts X+(k„,i), E (k„ i), of odd and' of even eigen- of equivalent manifolds M'(k), 3f'(k) may occur functions fs, ' be counted which have energies at all points k of an endless curve, or of a number E'(k,) ~& E'(k„). Now the quantity of such curves, in k-space. These contact curves cannot be destroyed or broken by any infini- tesimal change in the potential U which pre- serves the inversional symmetry. It is vanishingly is an integer, and according to whether this improbable for such curves to lie inWeylplanes of integerpointsis odd or eveninthe bandnumber of circuits of symmetry in the B-Z; however a contact curve the second type along which contact between the may pass through a symmetry axis at a point bands i and j occurs must be odd or even. Since where necessary degeneracy or contact of anytheorycrystal with an inversion center can be made inequivalent manifolds occurs. In 3d band bystructuresan infinitesimal withchange non-degeneratein the form of U into Suppose that for a crystal with• an inversion one whose space group is merely its translation center a contact of inequivalent bandsmanifolds - lackinggroup eitherplus theinversioninversion, or TRthis - implies this certain 3E'(k), M'(k) occurs at a point k happenson a sym- at restrictionsisolated pointson the numbers of contact curves metry axis, and suppose that m'(k) and m'(k) which may occur for crystals of higher sym- are each one-dimensional. Then if the thevector non-degeracyg metry. Prediction of courseof the requiresexistence of curves of • & (proportional in the Hartree case to (P„',breakingiVPq, ))' contactspin-rotationof equivalent symmetrymanifolds - may therefore does not vanish, a curve of contactACCIDENTAL DEGENERACYmust pass be365 possible from a knowledge merely of the basis vectors for a real representation of the is just like the theory of electronic wave functions space group of the crystal, and that the normal and their energies: frequency can be plotted as a through k. This curvemodes maybelonging to aberepresentationa curvewhich is functionoftypicallyof wavecontactvector, and sticking together byenergiesof SOCof the different M'(k, ) at the eight irreducible in the field of real numbers, even two or more of these frequency bands will occur though reducible in the complex field, must all at wave vectors k where G' has multidimensional of equivalent manifoldshave the sameoffrequency.the 7 Thustypemathematicallyjustrepresentationsdescribed,or where case (b) or case points(c), as k„. the theory of normal modes and their frequencies defined above, occurs. It is a pleasure for me to express my thanks to or it may be a curve ~ Cf.ofE. Wigner,contactGott. Nachr. (1930), p. 133.of inequivalentProfessor E. Wigner, who suggested this problem. For a crystal without an inversion center, the manifolds in a planeAUGUSTof15,symmetry.1937 PHYSICALNaturallyREVIEW if VOLUMenergyE 52 separation 8E(k+x) in the neigborhood there is no such symmetryAccidentalplaneDegeneracy ininthe EnergytheBands spaceof Crystals of a point k where contact of equivalent mani- CONYERS HERRING Princeton University, Princeton, Net Jersey group, the former alternative must(Received hold.June 16, 1937) folds occurs may be expected to be of the order The circumstances are investigated under which two wave functions occurring in the Hartree or I'ock solution for a crystal can have the same reduced wave vector and the same energy, It is found that coincidence of the energies of wave functions with the same — For a crystal whose space group consists symmetryonlyproperties, of ~ as ~ +0, for all directions of x. as well as those with different symmetries, is often to be expected. Some qualitative features are derived of the way in which energy varies with wave vector near wave vectors for which degeneracy occurs. All these results, like those of the preceding paper, should be applicable of its translation group alsoplusto the frequency anspectrum ofinversion,the normal modes of vibration of a crystal.three For a crystal with an inversion center, the "N previous papers, by Bouckaert, Smoluchow- The analysis necessary to answer these ques- k' types of contact curves- - ski, and Wigner,may' and by the author,occur,' certain tionswhichis rather tedious. Despitearethis and theenergyfact separation 8E(k') at a point near a properties of the wave functions and energy that it may not be of practical significance to values of an electron moving in the periodic field bother about too fine details in an approximate most easily describedof a crystalwhenwere derived.energyThese properties wereis theory,consideredthe discussion to be given below maycurvebe of contact of equivalent manifolds may be the properties necessitated by the symmetry of of value in forming pictures of the energy band the crystal and by the reality of the Hamiltonian. structures of metals, especially of multivalent k' as a trebly periodic Thefunctiontwo questions to be discussedofin thiswavepaper ones. Invectorparticular, it is hopedinthat the completeexpected to be of the order of the distance of are: determination of energy as a function of wave (1) In the solution of Hartree's or Fock's vector by interpolation from the results of cal- the infinite reciprocalequations forlatticea crystal to what extentspace.may one culationsTheof the Wigner-Seitz-Slaterfirst type willfrombe the curve. expect to encounter accidental coincidences in facilitated and made more reliable. The results energy between two one-electron wave functions of this paper also apply, as did those of I, to the type is a simple closedwith the samecircuitwave vector? Bywhich"accidental" frequencyis spectrumdistinctof the normal modes of All kinds of contacts of equivalent manifolds coincidences are to be understood coincidences vibration of a crystal; however numerical cal- not necessitated by the symmetry and reality of culation of these frequencies has not yet ad- from the circuit obtainedthe Hamiltonian.from it thevanced inversionas far as has the calculation of electronicexcept the ones described above are vanishingly (2) If the energies of two or morebybands bands. 3 coincide at wave vector k, whether accidentally The notation to be used is the same as in I. k~ —k. The second or for reasons isof symmetrya simpleand reality, how mayclosedIn addition, thecircuitsymbol LM', 3P] will be improbable.intro- In particular, the occurrence of typethe energies of these bands be expected to vary duced to represent the subspace of Hilbert space with wave vector in the neighborhood of k? spanned together by any two linear manifolds of wave functions M' and M'. which either coincides' Bouckaert,withSmoluchowski, and theWigner, Phys. inverseRev. 50, circuit isolated points of contact of equivalent manifolds 58 (1936), hereafter referred to as BSW. ' Calculations for a simple cubic lattice have been made ' Preceding paper, hereafter referred to as I. M. Blackman, Proc. Soc. A159, 416 or can be brought into coincidence withby it byRoy. 2x (1937).for crystals with an inversion center is vanish- times a translation of the reciprocal lattice. The ingly improbable. third type is a curve extending periodically to infinity. Now consider any energy band i, and I should like to express my gratitude to Pro- the band j next above it. For each of the eight fessor E. Wigner for his interest in this work, and distinct points k„(r= 1 to 8) of the B-Z whose to Dr. L. P. Bouckaert and Dr. R. Smoluchowski G~" contain the inversion let the numbers for some interesting discussions. Introduction Resistivity (polycrystalline samples) pyrochlore oxides Ln2Ir2O7 107 106 Ln3+: (4f)n Localized moment 6 5 10 10 Dy Magnetic frustration Ho
104 Ir4+: 5d5 Conduction electrons 105 Tb 103 Ir[t2g]+O[2p] conduction band 104 Dy 60 80 100 300 Itinerant electron system cm) Ln2Ir2O7 Pyrochlores" Ho on the pyrochlore lattice (m 103
! Tb Gd Eu 2 IrOB6 10 Sm Nd 101 Pr Ln2Ir2O7 100 0 50 100 150 200 250 300 Ln A T(K) O! Metal Insulator Transition (Ln=Nd, Sm, Eu, Gd, Tb, Dy, Ho)
K. Matsuhira et al. : J. Phys. Soc. Jpn. 76 (2007) 043706. (Ln=Nd, Sm, Eu) • Series of materials shows systematic 1MITs D. Yanagashima, Y. Maeno, 2001 J. Phys. Soc. Jpn. 80 (2011) 094701 F•ULL PIrAPERS4+ has λ≈0.5eVK. MATSUHIRA et al. T-linear contribution in C T is known to be attributed to spin wave excitations for one-dimensionalð Þ antiferromagnets; it is difficult for the pyrochlore lattice to induce a T-linear contribution. As another possible origin for T-linear contribution in C T in the insulating state, Anderson localization may beð considered.Þ 35) Further investigation is required to reveal the origin of the T -linear contribution. Now we will discuss the Ln dependence of the entropy associated with the MIT (ÁS). To estimate ÁS, a smooth polynomial was fitted to the data outside the region of the anomaly; these fitting lines (broken line) for Ln = Nd, Sm, and Eu are shown in Figs. 5(a), 6(a), and 6(b), respectively. K. Matsuhira et al, 2011 The background contribution was subtracted from the raw data; the electronic portions of the C=T (ÁC=T ) for Ln = Sm and Eu are shown in the inset. By integrating ÁC=T , we obtained ÁS 0:47, 2.0, and 1.4 J/(K mole) for Fig. 7. (Color online) Phase diagram of Ln Ir O based on Ln3 ionic ¼ Á 2 2 7 þ Ln = Nd, Sm, and Eu, respectively. ÁS is much smaller radius dependence of TMI. than 2R ln 2. If we assume that a localized 5d electron from 4 Ir þ ions with S 1=2 causes a conventional magnetic transition, we can¼ expect a change in entropy of 2R ln 2 already been reported.38–40) Now, we point out the difference 11:5 J/(K mole). The reduction in the amount of change in¼ in the phase diagram between Ir and Mo pyrochlore oxides. entropy isÁ considered to be caused by a short-range ordering As is described in the introduction, as the ionic radius of 3 due to frustration or a reduction in magnetic moment due to Ln þ decreases, the electrical conductivity in Ln2Mo2O7 the itinerancy of 5d electrons. Next, recently, the Raman becomes semiconducting. Interestingly, the magnetic transi- scattering spectra of Ln2Ir2O7 for Ln = Nd, Sm, and Eu tion of Ln2Mo2O7 goes from the spin glass insulating state 36) have been measured. Below TMI, new peaks appear for (Ln = Gd, Tb, Dy, and Ho) to the ferromagnetic metallic 3 Ln = Sm and Eu, but no remarkable change is seen for state (Ln = Eu, Sm, and Nd) as the ionic radius of Ln þ Ln = Nd. The result indicates that Sm2Ir2O7 and Eu2Ir2O7 increases; the ferromagnetic transition comes from 4d accompany a structural change with MIT, but this does not electrons. Although the spin glass transition temperature Tg occur with Nd2Ir2O7. Therefore, the ÁS for Ln = Sm and is independent of Ln (Tg 20 K), the ferromagnetic 3 Eu involve the lattice contribution. Indeed, ÁS for Ln = Nd transition temperature increases as the ionic radius of Ln þ 4 is smaller than those for Ln = Sm and Eu. If we consider increases. In addition, semiconducting Ln2Ru2O7 [Ru þ: this ÁS in Ln = Nd to be caused by only the electronic (4d)4] shows the frustrated AFM transition originating from 41) contribution without the lattice contribution, we can estimate 4d electrons. The Ne´el temperature TN monotonically the electronic specific heat coefficient above TMI 14 mJ/ increases from TN 84 K for Ln = Yb to TN 160 K for (K2 mole) by the relation ÁS=T . As Sm Ir¼O and Ln = Pr as the ionic¼ radius of Ln3 increases.¼ The present Á ¼ MI 2 2 7 þ Eu2Ir2O7 are both semimetallic from the behaviors of their result shows that the magnetic transition (or MIT) in 3 T and S T , it is speculated that the for Ln = Sm and Ln2Ir2O7 decreases as the ionic radius of Ln þ increases. ð Þ ð Þ 3 Eu are smaller than that for Ln = Nd. Then, the opposite dependence of the ionic radius of Ln þ on the magnetic transition temperature is realized in Ln2Ir2O7. 3.5 Phase diagram It is speculated that the difference in their phase diagrams is Figure 7 shows the phase diagram of Ln2Ir2O7, which is due to the feature of the 5d electron system, which has a 3 based on the Ln þ ionic radius dependence of TMI; the ionic strong spin–orbit interaction and a reduced on-site Coulomb 3 21) radius of Ln þ is for an 8-coordination-number site. TMI repulsion in comparison with the 4d electron system. 3 monotonically increases as the ionic radius of Ln þ Further theoretical study is needed to understand this phase decreases. Obviously, TMI does not depend on the de Gennes diagram in Ln2Ir2O7. 2 3 factor gJ 1 J J 1 or the magnetism of Ln þ. This ð À Þ ð þ Þ 3 4. Conclusions MIT is not associated with the magnetic ordering of Ln þ. For T>TMI, Ln = Pr and Nd are metallic. Then, Ln = Sm, We report the physical properties (resistivity, thermo- Eu and Gd are semimetallic and Ln = Tb, Dy, and Ho are electric power, magnetization, and specific heat) of Ln2Ir2O7 semiconducting. Ln = Pr is a unique metal located near the for Ln = Nd, Sm, Eu, Gd, Tb, Dy, and Ho. Ln2Ir2O7 for critical point of MIT. In this figure, the extrapolation Ln = Nd, Sm, and Eu show MITs at 33, 117, and 120 K, between Ln = Nd and Pr is based on a recent result for respectively. In this study, we revealed that Ln2Ir2O7 for 37) resistivity in the solid solution (Pr1 xNdx)2Ir2O7. From Ln = Gd, Tb, Dy, and Ho exhibit MITs at 127, 132, 134, À the result, the substitution of Pr by 20% Nd leads to MIT and 141 K, respectively. These MITs in Ln2Ir2O7 has at around 3 K; below TMI, the increasing resistivity in this some common features: They are second-order transitions sample is suppressed, and resistivity reaches a finite value at since no thermal hysteresis or no discontinuous change lower temperatures. in their physical properties is observed at TMI. Under Next, we discuss the phase diagram of Ln2Ir2O7 in the FC condition, a weak ferromagnetic component 3 comparison with that of other rare-earth pyrochlore oxides. ( 10À B/f.u.) caused by 5d electrons from Ir is observed 4 2 The phase diagrams of Ln2Mo2O7 [Mo þ: (4d) ] have below TMI. The entropy associated with MIT supports the
094701-6 #2011 The Physical Society of Japan Exotic Possibilities
• Topological Mott Insulator?
D. Pesin+LB, 2010 Exotic Possibilities
• Topological Mott Insulator?
D. Pesin+LB, 2010
Probably not: commensurate S. Zhao et al, 2011 magnetic order seen in μSR Weyl semimetal? X. Wan et al, 2011 • LDA+U calculationsTOPOLOGICAL SEMIMETAL find Weyl AND FERMI-ARC state! SURFACE ... PHYSICAL REVIEW B 83,205101(2011) WAN, TURNER, VISHWANATH, AND SAVRASOV PHYSICAL REVIEW B 83,205101(2011) along high-symmetry lines [see Fig. 3(b)]alsoappearstobe it was shown that the insulating ground states evolve from a high-temperature metallic phase via a magnetic transition.9,10 insulating, and at first sight one may conclude that this is The magnetism was shown to arise from the Ir sites, since it also occurs in A Y, Lu, where the A sites are nonmagnetic. an extension of the Mott insulator. However, a closer look While its precise= nature remains unknown, ferromagnetic ordering is considered unlikely, since magnetic hysteresis is using the parities reveals that a phase transition has occurred. not observed. At the L points, an occupied level and an unoccupied level We show that electronic structure calculations can naturally account for this evolution and point to a novel ground state. with opposite parities have switched places. It can readily First, we find that magnetic moments order on the Ir sites in a noncollinear pattern with moment on a tetrahedron be argued that only one of the two phases adjacent to the pointing all in or all out from the center. This structure retains inversion symmetry, a fact that greatly aids the electronic U where this crossing happens can be insulating (see the structure analysis. While the magnetic pattern remains fixed, the electronic properties evolve with correlation strength. For Appendix). Since the large U phase is found to be smoothly weak correlations, or in the absence of magnetic order, a connected to a gapped Mott phase, it is reasonable to assume metal is obtained, in contrast to the interesting topological insulator scenario of Ref. 8.Withstrongcorrelationswefind FIG. 1. (Color online) Sketch of the predicted phase diagram the smaller U phase is the noninsulating one. This is also aMottinsulatorwithall-in/all-outmagneticorder.However, for pyrochlore iridiates. The horizontal axis corresponds to the increasing interaction among Ir 5d electrons while the vertical axis for the case of intermediate correlations, relevant to Y2Ir2O7, borne out by the LSDA U SO band structure. A detailed the electronic ground state is found to be a Weyl semimetal, corresponds to external magnetic field, which can trigger a transition + + with linearly dispersing Dirac nodes at the chemical potential out of the noncollinear “all-in/all-out” ground state, which has several analysis perturbing about this transition point (also in the k p and other properties described above. electronic phases. · We also mention the possibility of an exotic insulating They also pointed out very unusual surface subsection) allows us to show that this phase is expected to be phase emerging when the Weyl points annihilate• in pairs full Hamiltonian to linear order. No assumptions are needed as the correlations are reduced; we call it the θ π axion aWeylsemimetalwith24Weylnodesinall. beyond the requirement that the two eigenvalues become insulator. Although our LSDA U SO calculations= find statesdegenerate at k .Thevelocityvectorsv are generically that a metallic phase intervenes+ before+ this possibility is 0 i Indeed, in the LSDA U SO band structure at U nonvanishing and linearly independent. The energy dispersion realized, we note that local-density approximation (LDA) + + = 3 2 1.5eV,wefindathree-dimensionalDiraccrossinglocated systematically underestimates gaps, so this scenario could well is conelike, $E v0 q i 1(vi q) .Onecanassigna = · ± = · occur in reality. Finally, we mention that modest magnetic chirality (or chiral charge) c" 1tothefermionsdefinedas =±# within the "-X-L plane of the Brillouin zone. This is illustrated fields could induce a reorientation of the magnetic moments, c sgn(v1 v2 v3). Note that, since the 2 2 Pauli matrices leading to a metallic phase. Previous studies include Ref. 18,an appear,= our· Weyl× particles are two-component× fermions. In in Fig. 4 and corresponds to the k vector (0.52,0.52,0.3)2π/a. ab initio study which considered ferromagnetism. In Ref. 19, contrast to regular four component Dirac fermions, it is not the tight-binding model of Ref. 8 was extended to include possible to introduce a mass gap. The only way for these modes There also are five additional Weyl points in the proximity of tetragonal crystal fields, but in the absence of magnetism. The to disappear is if they meet with another two-component Weyl topological Dirac metal and axion insulator discussed here do fermion in the Brillouin zone, but with opposite chiral charge. the point L related by symmetry (three are just inside each of not appear in those works, largely due to the difference of Thus, they are topological objects. By inversion symmetry, the the two opposite hexagonal faces of the Brillouin zone, which magnetic order from our study. band touchings come in pairs, at k0 andFIG.k0,andthesehave 4. (Color online) Semimetallic nature of the state at U We begin by giving a brief overview of the theoretical opposite chiralities (since the velocity1.5 vectors eV− are according reversed). to the LSDA U SO method. (a) Calculated= are identified with one another) When U increases, these points ideas that will be invoked in this work, before turning to our This semimetallic behavior would not occur (generically) + + LSDA U calculations of magnetic and electronic structure in a system without magnetic order.energy In materials bands such in as the plane Kz 0withbandparitiesshown;(b)energy move toward each other and annihilate all together at the L of the pyrochlore+ iridates. We then discuss the special surface bismuth, with both time reversal and inversion symmetry, = states that arise in the Weyl semimetal phase and close with Dirac fermions always contain bothbands left- and in right-handed the plane kz 0.6π/a, where a Weyl point is predicted to point close to U 1.8eV.ThisishowtheMottphaseisborn a comparison to existing experiments and conclusions. Our components and are thus typically gapped.20 = = results are summarized in the phase diagram Fig. 1. When the compound has stoichiometricexist. Thecomposition, lighter-shaded and plane is at the Fermi level. (c) Locations from the Weyl phase. Since we expect that for Ir 5d states the all the Weyl points are related by symmetry,of the Weylthe Fermi points energy in the three-dimensional Brillouin zone (Ref. 29) can generically line up with the energy of the touching points. actual value of the Coulomb repulsion should be somewhere I. WEYL SEMIMETALS AND INVERSION-SYMMETRIC Under these circumstances, the density(nine of are states shown, is equal to indicated by the circled or signs). INSULATORS zero and the behavior of the Weyl fermions controls the + − within the range 1 eV
205101-5 Fermi Arcs
• On most surfaces, metallic Fermi surfaces which are not closed - “arcs” - terminate at the projections of the Weyl points
TOPOLOGICAL SEMIMETAL AND FERMI-ARC SURFACE ... PHYSICAL REVIEW B 83,205101(2011) state for this subsystem [see Fig. 5(b)]. Hence, this surface state crosses zero energy somewhere on the surface Brillouin zone kλ0 .Suchastatecanbeobtainedforeverycurveenclosing the Weyl point. Thus, at zero energy, there is a Fermi line in the surface Brillouin zone, that terminates at the Weyl point momenta [see Fig. 5(c)]. An arc beginning on a Weyl point of chirality c has to terminate on a Weyl point of the opposite chirality. Clearly, the net chirality of the Weyl points within the (λ,kz)toruswasakeyinputindeterminingthenumberof these states. If Weyl points of opposite chirality line up along the kz direction, then there is a cancellation and no surface states are expected. In the calculations for Y2Ir2O7,atU 1.5eV,aDirac (or Weyl) node is found to occur at= the momentum (0.52,0.52,0.30)2π/a (in the coordinate system aligned with the cubic lattice of the crystal) and equivalent points (see Fig. 4). They can be thought of as occurring on the edges of a cube, with a pair of Dirac nodes of opposite chirality occupying each edge, as, for example, the points (0.52,0.52,0.30)2π/a and (0.52,0.52, 0.30)2π/a. For the case of U 1.5eV,the sides of this cube− have the length 0.52(4π/a). Thus,= the (111) and (110) surfaces would have surface states connecting the projected Weyl points [see Fig. 6 for the (110) surface states FIG. 6. (Color online) Surface states. The calculated surface and the theoretical expectation for the (111) surface]. If, on energy bands24 correspond Weyl to the points (110) surface of the pyrochlore the other hand, we consider the surface orthogonal to the (001) iridate Y2Ir2O7.Atight-bindingapproximationhasbeenusedto direction, Weyl points of opposite chirality are projected to the simulate the bulk band structure with three-dimensional Weyl points same surface momentum along the edges of the cube. Thus, as foundpredicted by our LSDA U SOin calculation. Y2Ir2 TheO plot7 corresponds + + no protected states are expected for this surface. to diagonalizing 128 atoms slab with two surfaces. The upper inset To verify these theoretical considerations, we have con- shows a sketch of the deduced Fermi arcs connecting projected structed a tight-binding model which has features seen in our bulk Weyl points of opposite chirality. The inset below sketches the theoretically expected surface states on the (111) surface at the Fermi electronic structure calculations for Y2Ir2O7.Thecalculated (110) surface band structure for the slab of 128 atoms together energy (surface band structure not shown for this case). with the sketch of the obtained Fermi arcs is shown in Fig. 6. This figure shows Fermi arcs from both the front and the back face of the slab, so there are twice as many arcs coming out of interconnection between insulating behavior and magnetism each Weyl point as predicted for a single surface. observed experimentally.9,10 It is also consistent with being The tight-binding model considers only t2g orbitals of Ir proximate to a metallic phase on lowering the correlation atoms in the global coordinate system. Since Ir atoms form strength, such as A Pr (Ref. 17). In the clean limit, a three- = atetrahedralnetwork(seeFig.2), each pair of nearest- dimensional Weyl semimetal is an electrical insulator and can neighboring atoms forms a corresponding σ -like bond whose potentially account for the observed electrical resistivity. The hopping integral is denoted as t and another two π-like noncollinear magnetic order proposed has Ising symmetry bonds whose hopping integrals are denoted as t".Tosim- and could undergo a continuous ordering transition. The ulate the appearance of the Weyl point it is essential to observed “spin-glass”-like magnetic signature could perhaps include next-nearest-neighbor interactions between t2g orbitals arise from defects like magnetic domain walls. A direct probe which are denoted as t . With the parameters t 0.2, t of magnetism is currently lacking and would shed light on this "" = " = 0.5t, t"" 0.2t,thevalueoftheon-sitespin-orbitcoupling key question. At lower values of U,thesystemmayrealize equal to= 2.5−t and the applied on-site “Zeeman” splitting of 0.1t an “axion insulator” phase with a magnetoelectric response between states parallel and antiparallel to the local quantization θ π,althoughwithinourcalculations(whichareknownto = axis of the all-in/all-out configuration we can roughly model underestimate stability of such gapped phases) a Fermi surface the bulk Weyl semimetal state; when this model is solved on a appears before this happens. lattice with a boundary, the surface states shown in the figure In summary, a theoretical phase diagram for the physical appear. system is shown in Fig. 1 as a function of U and applied magnetic field, which leads to a metallic state beyond a critical field. The precise nature of these phase transformations is not V. DISCUSSION addressed in the present study. We now discuss how the present theoretical description Note: An experimental paper35 appeared recently in which compares with experimental facts. We propose that the low- it is found that the spins in a related compound (Eu2Ir2O7) form temperature state of Y2Ir2O7 (and also possibly of A aregularlyorderedstateratherthanaspin-glass,consistent Eu, Sm, and Nd iridates) is a Weyl semimetal, with all-= with our results. It would be interesting to learn whether this in/all-out magnetic order. This is broadly consistent with the compound is a Weyl metal or not.
205101-7 Heterostructuring
Can we engineer Weyl points in a heterostructure?
• A: yes! And you can do it with topological insulators TI normal I TI to NI transition
TI NI
strong Tunneling tunneling across TI across the slabs kills the NI “heals” TI 3d TI TI to NI transition
TI NI in between is a (quantum) phase transition
strong Tunneling tunneling across TI across the slabs kills the NI “heals” TI 3d TI TI to NI transition
TI NI in between is a (quantum) phase transition
strong Tunneling tunneling We can turn this across TI across the critical point into the slabs10 kills the NI “heals” TI Weyl semimetal by 3d TI breaking I or TR S. Murakami, 2007 Figure 5. Phase diagram for the QSH and ordinary insulating (I) phases for (a) 2D and (b) in 3D. m is a control parameter which drives the phase transition, and δ represents a parameter which describes the breaking of I-symmetry. δ 0 is the case with I-symmetry. =
phase appears between the two phases which are gapful in the bulk. The difference of the Z2 topological numbers ν between the two sides of the phase transition can also be calculated as in the 2D case [19]. When the band crossing occurs at k 1 (n b + n b + n b ), the factor i = 2 1 1 2 2 3 3 δi (n1n2n3) in (4) changes sign, and some of the four Z2 topological numbers, ν0 (ν1ν2ν3) in (4=), change accordingly. This applies to the I-symmetric systems. The I-asymmetric; cases are similarly treated because they can be associated with the I-symmetric cases with perturbation.
3. Towards materials search for QSH systems
3.1. Bismuth thin film and 2D QSH system In [15], the bilayer bismuth is studied as a candidate for the 2D QSH phase. From the 2D bilayer tight-binding model, truncated from the 3D tight-binding model [27], the bilayer bismuth is proposed to be a 2D QSH system [15]. Bilayer antimony is studied in a similar way, and it is an ordinary insulator. It is predicted from the calculation of Z2 topological number and from a band structure calculation for the geometry with edges (i.e. the strip geometry). The Z2 topological number is calculated in [15] by the Pfaffian of the matrix for the time-reversal operator proposed in [8]. The calculation of the Pfaffian involves fixing of phases of the wavefunction as an analytic function of the wavenumber k, which is numerically a challenging problem, even for a simplified model presented in [15]. It can be tackled by discretizing the k space and counting the vortex of the Pfaffian matrix [28, 29]. Instead, for I-symmetric systems, the method of calculating parity eigenvalues (3) proposed in [18] is much easier. For bilayer bismuth, we checked that this method leads us also to the same conclusion that the Z2 topological number is odd and nontrivial, and it is in the QSH phase. For the system to be in the QSH phase, it should have a gap in the bulk. The 3D bulk bismuth is semimetallic, and has a small band overlap between the conduction and the valence band, while the direct gap is finite for all wavenumbers. By making it into a thin film, the perpendicular motion is quantized and tends to open a band gap. Earlier theoretical estimates [30, 31] and experiments [32, 33] (see also [34]) show that thin-film
New Journal of Physics 9 (2007) 356 (http://www.njp.org/) TR breaking
Bandstructure HgTe
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) Dope with magnetic impurities (alreadyE 4.0nm 6.2 nm 7.0 nm
• k achieved in Bi-based TIs) E1 H1 normal inverted Bandstructuregap HgTe gap H1 E1
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) Δs TI E 4.0nm 6.2 nm 7.0 nm k
TI E1 H1 normal inverted gap gap H1 E1 Δd
model just in terms of surface states m = exchange energy TR breaking
Bandstructure HgTe
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) Dope with magnetic impurities (alreadyE 4.0nm 6.2 nm 7.0 nm
• k achieved in Bi-based TIs) E1 H1 normal inverted Bandstructuregap HgTe gap H1 E1
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) Δs TI E 4.0nm 6.2 nm 7.0 nm k
TI E1 H1 normal inverted gap gap H1 E1 Δd
z z x H = [vF τ (ˆz σ) k δi,j + mσ δi,j +∆Sτ δi,j × · ⊥ k ,ij ⊥ 1 + 1 + ∆dτ δj,i+1 + ∆dτ −δj,i 1 ck† ick j 2 2 − ⊥ ⊥ m = exchange energy TR breaking
Bandstructure HgTe
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) Dope with magnetic impurities (alreadyE 4.0nm 6.2 nm 7.0 nm
• k achieved in Bi-based TIs) E1 H1 normal inverted Bandstructuregap HgTe gap H1 E1
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) Δs TI E 4.0nm 6.2 nm 7.0 nm k
TI E1 H1 normal inverted gap gap H1 E1 Δd
2 2 2 2 k = vF k +[m ∆(kz)] ± ⊥ ±
∆(k )= ∆2 +∆2 + 2∆ ∆ cos(k d) m = exchange energy z s d s d z TR breaking
• Dope with magnetic impurities (already achieved in Bi-based TIs)
2 Δs TI Δ Δ TI D D Ins TI TI Δd Weyl m semimetal d TI Ins QAH Ins 0 0 FIG. 1. Schematic drawing of the proposed multilayer struc- 0 ΔS 0 m ΔS ture. Unhashed layers are the TI layers, while hashed layers are the ordinary insulator spacers. Arrow in each TI layer (a) m=0 (b) m≠0 shows the magnetization direction. Only three periods of the superlattice are shown in the figure, 20-30 unit cells can per- FIG. 2. (Color online) Phase diagrams for (a) m =0and haps be grown realistically.m = exchange energy (b) m = 0. In (a), the red line represents the phase bound- ary between topological insulator (TI) and ordinary insulator (Ins). In (b), due to TR symmetry breaking, the distinction bottom surface of each layer. k is the momentum in ⊥ between topological and ordinary insulators is moot, so the TI the 2D surface BZ (we use ¯h =1units),σ is the triplet in (a) has been converted to Ins. QAH denotes the quantum of Pauli matrices acting on the real spin degree of free- anomalous Hall phase. dom and τ are Pauli matrices acting on the which sur- face pseudospin degree of freedom. The indices i, j label distinct TI layers. The second term describes exchange which can be brought to a block-diagonal form, explic- spin splitting of the surface states, which can be induced, itly revealing a pair of two-component Weyl fermions for example, by doping each TI layer with magnetic im- with opposite chirality, by a π/2 rotation around the x- purities, as has been recently demonstrated experimen- axis in the pseudospin space. Alternatively, in total this tally [10]. The remaining terms in Eq.(2) describe tun- is a conventional 4-component massless Dirac fermion. neling between top and bottom surfaces within the same As discussed above, since the two Weyl fermions are lo- cated at the same point in momentum space, they are TI layer (the term, proportional to ∆S), and between top and bottom surfaces of neighboring TI layers (terms, topologically unstable. Any perturbation, for example any deviation of the ratio ∆ /∆ from unity, immedi- proportional to ∆D). Longer-range tunneling is assumed S D ately eliminates the degenerate Dirac node and produces to be negligible. We will regard m and ∆S,D as tunable parameters and study the phase diagram of Eq.(2) as a a fully gapped spectrum. With m = 0, the massless function of these parameters. Dirac point can be understood [5] as a critical point be- Let us initially assume that the spin splitting is ab- tween topological (∆D > ∆S) and ordinary (∆D < ∆S) sent, i.e. set m = 0. Diagonalizing Eq.(2) one finds the insulators with both inversion and time-reversal symme- following band dispersion: try preserved (see Fig. 2a). To produce a topologically stable phase with 3D Dirac nodes, the nodes have to be 2 2 2 2 2 k = vF (kx + ky)+∆ (kz), (3) separated in momentum space. As mentioned above, this ± can generally be accomplished by breaking either TR or 2 2 where ∆(kz)= ∆S + ∆D +2∆S∆D cos(kzd), and d is I symmetries and there are in principle many ways to do the superlattice period (i.e. TI layer plus spacer layer this. Here we will focus on one particular way, which is thickness) in the growth (z) direction. This bandstruc- perhaps the simplest from the point of view of a practi- ture is fully gapped when ∆S = ∆D , but contains cal realization. Namely, as already mentioned above, we Dirac nodes when ∆ /∆ |= | 1. | The| nodes are lo- S D ± will assume that each TI layer is doped with magnetic cated at kz = π/d when ∆S/∆D = 1 and at kz =0 impurities, producing a ferromagnetically-ordered state when ∆ /∆ = 1(k = k = 0 always). While both S D − x y within each layer, with magnetization along the growth cases are possible, we will assume the former for concrete- direction of the heterostructure. This leads to spin split- ness and will take both tunneling matrix elements to be ting of the surface states of magnitude m,describedby positive (this choice does not affect any of our results). the second term in Eq.(2). The band dispersion is now Expanding the band dispersion near the Dirac point at given by: kx = ky =0,kz = π/d to leading order in the momentum 2 2 2 2 2 one obtains: k = vF (kx + ky)+[m ∆(kz)] . (6) ± ± 2 2 2 2 2 2 k = vF (kx + ky)+˜vF kz , (4) This dispersion has two nondegenerate Dirac nodes, sepa- ± rated along the z-axis in momentum space, with locations wherev ˜F = d√∆S∆D. The momentum-space Hamilto- given by kz = π/d k0,where: nian near the Dirac node has the form: ± 1 2 2 (k)=v τ z(ˆz σ) k +˜v τ yk , (5) k0 = arccos 1 (m (∆S ∆D) )/2∆S∆D . (7) H F × · F z d − − − TR breaking
• Dope with magnetic impurities (already achieved in Bi-based TIs)
Δs TI kz +
Δd k0 -
m = exchange energy TR breaking
• Dope with magnetic impurities (already achieved in Bi-based TIs) 1 B (k)= dˆ ∂ dˆ ∂ dˆ Δs TI kz µ 8π µνλ · ν × λ +
Δd k0 - ∂ B (k)= q δ(k k ) µ µ i − i i m = exchange energy “monopoles” of Berry curvature Quantum Hall effect
c.f. Haldane 1988
x y z kz H = vkxσ + vkyσ + m(kz)σ m(kz) Quantum Hall effect
c.f. Haldane 1988
x y z kz H = vkxσ + vkyσ + m(kz)σ m(kz)
σxy =0 Quantum Hall effect
c.f. Haldane 1988
x y z kz H = vkxσ + vkyσ + m(kz)σ m(kz) e2 σ = xy h TR breaking
• Dope with magnetic impurities (already achieved in Bi-based TIs) c.f. Volovik, 2005 Δs TI kz +
Δd k0 -
2 e k0 σxy = m = exchange energy h 2π semi-quantum AHE TR breaking
• Dope with magnetic impurities (already achieved in Bi-based TIs)
Δs TI kz +
Δd k0 - e2 σµν = µνλQλ in 2πh m = exchange energy general Q = kiqi + Q RLV i TR breaking
• Dope with magnetic impurities (already achieved in Bi-based TIs)
TI Δs Σxy 1.0 0.8 Δd 0.6 0.4 0.2 m
0.5 1.0 1.5 S QAHE in finite multilayer m = exchange energy I breaking
• Asymmetric heterostructure, or intrinsic I breaking +V electrostatic potential asymmetry Δs TI -V
2 2 2 2 Δd k = vF ( k V ) + ∆(kz) ± | ⊥|± | |
m = exchange energy I breaking
• Asymmetric heterostructure, or intrinsic I breaking +V electrostatic potential asymmetry Δs TI -V
2 2 2 2 Δd k = vF ( k V ) + ∆(kz) ± | ⊥|± | |
Naively gives nodal ring at critical point with Δs=Δd m = exchange energy I breaking
• Asymmetric heterostructure, or intrinsic I breaking +V electrostatic potential asymmetry Δs TI -V
2 2 2 2 Δd k = vF ( k V ) + ∆(kz) ± | ⊥|± | |
Need to include k- dependence of Δs,Δd m = exchange energy I breaking
• Asymmetric heterostructure, or intrinsic I breaking
Δs TI ky - +
Δd kx
+ -
no AHE m = exchange energy I breaking
• Asymmetric heterostructure, or intrinsic I breaking ky - + Δs TI
kx Δd +- σxx σyy
m = exchange energy TI NI Δs-Δd Hg1-xCdxTe structures
• Checked this with semi- realistic 10-orbital tight HgTe binding model for CdTe (Hg,Cd)Te superlattices with asymmetry • Advantage: can be grown with very high quality • Disadvantage: strain must be controlled 5
the choice presented in Table III, the most general contribu- complexity of the full model, its understanding requires a nu- tion to the Hamiltonian takes the form merical treatment. In perspective of this, we numerically in- vestigate the phenomenon of robust band touching in the func- x x y z D = τ [βxky σ + βykx σ + βzkzσ ] , (14) tion of the layer thicknesses N , the strain (HgTe) in H 1,2 0 ≡ the multilayer structure, the amplitude U0 of the superlattice (l) where the coefficients βx,y,z are again to be determined from potential, and the asymmetric displacement δ0. a comparison with the full model. The reduced Hamiltonian We first consider the dependence on . If U =0and 0 0 finally reads = 0 + S + D. It is a considerable simpli- δ0 =0, there is a range in 0 close to zero where band touch- fication withH respectH to HH, andH it only contains seven parame- ing is observed. This band touching is robust because it re- ters that need to be extracted from the full model. mains intact for an infinitesimal change in any of the external parameters 0, U0, and δ0. The upper and lower limits of the range are functions of U0 and δ0 as illustrated in Fig. 2, and HgC. Conditions1-x forCd robust band touchingxTe structureswe verify the expectation from Section II C that the range in- creases with both U0 and δ0. For the reasonable values of Band touching between the HOB and the LUB occurs in U0 0.1 eV and δ0 a/2, this range is ∆0 0.002. ∼ ∼ ∼ the full model when the two middle eigenvalues are equal in 6 Ε the simplified model. It can be shown that for a 4 4 ma- 0 Ε0 0.03 0.02 TI trix of the form , this is only possible if the direction× of the TI W H qualitative0.02 level. Nevertheless, it provides useful guidelines Σ ! Σ0 vector BChecked=(β k , β k ,thisβ k ) lieswith halfway semi- between the di- 0.015 • x y y x z z for the realization of the Weyl semimetal phase in this multi- 1.0 Σyy Σxx (1,2) (1,2) (1,2) NI NI W rections of the vectors A =(αx kx , αy ky , 0). The layer0.01 structure. The strain 0 has to be positive to avoid band realistic 10-orbital tight 0.01 two bands then cross each other as k is increased without overlap but not too large becauseU !eV that would be hard to achieve∆0!a | | 0.00 0 changingbinding the direction model of k, whereas for anti-crossing happens experimentally.strain 0.0 0.1 This gives0.2 a restriction0.3 on0.0 the0.2 thickness0.4 0.6 of the0.8 1.0 HgTe layers: the ideal dimensionless thickness of 4 N 5 NIW TI otherwise. Since the above condition requires the three vec- ≤ 1 ≤ (Hg,Cd)Te superlattices corresponds to an actual thickness of 2 nm
It was shown in Ref. [1] that the Weyl semimetal phase is B. Surface states metallic in the sense that its conductivity is finite in the limit of zero temperature. This is a characteristic experimental fea- ture, especially in contrast with the neighboring NI and TI Since Weyl semimetals are topological phases of matter, phases. In this subsection, we demonstrate that the finite con- they are characterized by topological surface states [1]. In ductivity at T 0 becomes highly anisotropic when the Weyl this subsection, we consider the model in Section I D, and semimetal phase→ occurs due to broken inversion symmetry. demonstrate the existence of these surface states. Although To achieve this, we consider the model in Section I D, and we choose a specific situation and also make a couple of sim- derive an expression for the conductivity tensor in the limit of plifying assumptions in the following, the topological nature of the surface states ensures that they exist under more generic small δT δN . When the condition of the Weyl semimetal − circumstances as well. phase is satisfied, there are 4 Weyl points at angles θ1 = θ, In our specific situation, the interface is in the x, z plane, θ2 = θ, θ3 = π + θ, and θ4 = π θ. Due to the convention { } 0 θ− π/2 we find that θ gradually− decreases from π/2 to therefore any spatial variation is in the y direction only. This ≤ ≤ Graphene-like Physics
• 2d graphene physics can already be achieved in HgTe quantum wells Bandstructure HgTe Dirac peak at B=0
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
E 4.0nm 6.2 nm 7.0 nm
k
E1 H1 normal inverted gap gap H1 E1
Peak width and mobilities comparable with/better than free standing graphene Scattering mechanisms: probably mass fluctuations + Coulomb (fit is Kubo model) L. Molenkamp: HgTe QWs are “better graphene” “3d graphene”
• The transport behavior of 3d Dirac/Weyl fermions is subtle and interesting! • Naive argument (no disorder or interactions):
Re σ(ω,T = 0) ω ∝ • insulating? With impurities
• Usually impurities induce elastic scattering that dominates at low T • Here, Born approximation is valid (disorder is irrelevant in RG sense) 1/τ u ω2 ∼ imp • Contrast graphene: higher order corrections induce non-zero scattering c rate at zero frequency (SCBA) 1/τ e− uimp ∼ With impuritiesBandstructure HgTe
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
• Neutral impurities w/oE interactions4.0nm leads6.2 nm 7.0 nm to non-zero DC conductivity
2 k Re σ(ω,T) σ0f(ω/T ) ∝ 3 σ surface, not normal to the z-axis. For more details on Σxx this we refer the reader to Ref. [9]. 1.0 In the rest of this section we will focus on diagonal σ0 E1 H1 transport characteristics of the Weyl semimetal, namely 0.8 normal excited inverted its optical conductivity. Some of the results, presented gap gap here, were quoted in [9], but not derived in detail. As we 0.6 carriers will demonstrate, the frequency dependence of the opti- H1 E1 cal conductivity of the Weyl semimetal is very unusual, 0.4 and can be used for experimental characterization of this T 2 phase of matter. 0.2 ∝ To calculate the optical conductivity, we assume a Ω 0.0 model with short-range impurity scattering potential of 0.0 0.5 1.0 1.5 2.0 2.5 3.0 2 the form: ω T 2 1/τ uimpω V (r)=u0 δ(r ra), (8) FIG. 1. (Color online). Optical conductivity of the Weyl − 2 ∼ a semimetal, in units of the DC conductivity σDC.Theω/T 2 3 3 ratio on the horizontal axis is in units of 32π γ/h vF . where ra label the impurity positions. For simplicity we will assume that the impurity potential is diagonal in both the spin and the pseudospin indices. We will con- This gives a DC conductivity: sider a single Weyl fermion in the 3D BZ for simplicity: 2 2 e vF generalization to any number of distinct Weyl fermions is σDC = , (14) 3γh trivial, as they contribute additively to transport (we will assume that the impurity potential does not mix Weyl and a Drude-like peak in the optical conductivity, but fermions at different points in the BZ). In the first Born with a temperature-dependent width, scaling as T 2.This approximation, the impurity scattering rate is given by: is a very unusual property of the optical conductivity in a metal and can be used to characterize the Weyl 1 d3k semimetal phase experimentally. = γ Im GR(, k)=2πγg(), (9) τ() − (2π)3 λ The Drude peak also has a highly unusual shape, with λ a divergent first derivative, as can be seen from Fig. 1. where This can be obtained explicitly from Eq. (13). The first derivative of the optical conductivity with respect to the R 1 frequency is given by: Gλ = , (10) λvF k + iη − dRe σ (ω) e2v2 ω h3v3 2 xx = F F is the retarded Green’s function of the Weyl fermion, dω − 3γh 32π2γT 2 λ = labels the helicity of the positive and negative 4 2 ∞ x sech (x) energy± Dirac cones, γ = u2n ,wheren is the impurity 0 i i dx 4 3 3 2 2 2 2 . (15) × [x +(h vF ω/32π γT ) ] concentration, and the density of states g() is given by: −∞ The integral above diverges when ω 0 and at small 2 frequencies is dominated by the contribution→ near x = 0. g()= . (11) 2π2v3 Then we can set sech(x) 1 and obtain: F ≈ 2 2 2 3 3 Thus 1/τ() , which means that the conduc- e vF 1 ω vF h ∼ Re σxx(ω) 1 . (16) tivity can be calculated semiclassically, using Boltzmann ≈ 3γh − 8 2 γT 2 equation. Solving linearized Boltzmann equation with the energy-dependent momentum relaxation rate (9) in At high frequencies, on the other hand: the standard way, we obtain: 4 2 2 2 2 2 7π e vF 32π γT 2 2 Re σxx(ω) . (17) e vF ∞ dnF () 1/τ() 3 3 Re σ (ω)= g() , ≈ 720γh h vF ω xx − 3 d ω2 +1/τ()2 −∞ (12) where nF is the Fermi distribution function at temper- III. LINE NODE SEMIMETALS ature T . Introducing dimensionless integration variable x = /2T and restoring explicith ¯, we obtain: In this section we will describe a realization, in the same physical system of a TI multilayer, of a line-node 2 2 4 2 e vF ∞ x sech (x) semimetal: a distinct topological semimetal phase, with Re σxx(ω)= dx 4 3 3 2 2 2 , 6γh x +(h vF ω/32π γT ) zeros in the spectrum, forming continuous lines in mo- −∞ (13) mentum space. With interactions
• Coulomb interactions are marginal - characterized by dimensionless fine 2 structure constant α=e /εvF • Leads to strong scattering 1/τ α2max(ω,T) u ω2 ∼ imp • Then expect 2 k T power law σ e2 v2 τ B dc ∼ v3 F ∼ α insulator F Experiment?
• Experiments on Eu2Ir2O7 find “weak” insulator
2 2
2000 2000 2.06 GPa 2.06 GPa 2.88 GPa 40 7.88 GPa 7.88 GPa 2.88 GPa 3.49 GPa 40 3.49 GPa cm) 4.61 GPa cm) 4.61 GPa " 1500 10.01 GPa 1500 " 10.01 GPa m 20 6.06 GPa m 20 6.06 GPa ( ( 7.88 GPa 7.88 GPa ! ! 11.34 GPa 10.01 GPa 11.34 GPa 10.01 GPa cm) cm) 0 11.34 GPa 0 11.34 GPa 0 50 100 150 200 250 300 0 50 100 150 200 250 300 12.15 GPa
12.15 GPa " " 1000 T (K) 1000 T (K) (m (m
! T ! TMI MI 500 500
0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 T (K) T (K)
FIG. 1. Resistivity as a function of temperature at P = 2.06 to FIG. 1. Resistivity as a function of temperature at P = 2.06 to FIG. 2. The phase diagram for Eu2Ir2O7 constructed from FIG. 2. The phase diagram for Eu2Ir2O7 constructed from 12.15 GPa. The approximate location of TMI,corresponding 12.15 GPa. The approximate location of TMI,corresponding our resistivity data. At low pressures, P<6GPa, the fi- our resistivity data. At low pressures, P<6GPa, the fi- to a peak in the second derivative of ρ(T ), is indicated by to a peak in the second derivative of ρ(T ), is indicated by nite temperature MIT is indicated by red squares. At high nite temperature MIT is indicated by red squares. At high arrows for the three lowest pressure curves. The inset is an arrows for the three lowest pressure curves. The inset is an pressures, P>6GPa, the transition between conventional pressures, P>6GPa, the transition between conventional expanded view of the higher pressure curves (P ≥ 7.88 GPa). expanded view of the higher pressure curves (P ≥ 7.88 GPa). Tafti et al, 2011 and private communications and diffusive metallic states is indicated by blue circles. The and diffusive metallic states is indicated by blue circles. The ∗ ∗ T cross-over is shown by orange triangles. All the lines are T cross-over is shown by orange triangles. All the lines are guides to the eye. The quantum critical point (QCP) lies on guides to the eye. The quantum critical point (QCP) lies on ical pressure [9], however there are important differences the P axis at Pc =6.06 ± 0.60 GPa. Notice the weak temper- ical pressure [9], however there are important differences the P axis at Pc =6.06 ± 0.60 GPa. Notice the weak temper- in detail. The main features of our data are summarized ature dependence of both TMI and Tmin. ature dependence of both TMI and Tmin. in detail. The main features of our data are summarized in Fig. 2. Our phase diagram can be viewed as four quad- in Fig. 2. Our phase diagram can be viewed as four quad- rants, corresponding to four distinct regimes of electronic rants, corresponding to four distinct regimes of electronic transport, at high and low pressure and temperature. To discuss our results, we start with the incoher- transport, at high and low pressure and temperature. To discuss our results, we start with the incoher- In the top left quadrant (P<6GPa, T ! 100 K), the ent metallic phase whose negative ρ(T ) slope is sup- ent metallic phase whose negative ρ(T ) slope is sup- In the top left quadrant (P<6GPa, T ! 100 K), the system is an “incoherent metal” characterized by a neg- pressed by increasing pressure. Eu2Ir2O7 is located pressed by increasing pressure. Eu Ir O is located system is an “incoherent metal” characterized by a neg- 2 2 7 ative ρ(T ) slope (Fig. 1). A broad MIT at TMI separates right at the metal-insulator boundary in the R2Ir2O7 se- ative ρ(T ) slope (Fig. 1). A broad MIT at TMI separates right at the metal-insulator boundary in the R2Ir2O7 se- the incoherent metallic phase from the “insulating” phase ries. Gd2Ir2O7 is an insulator at all temperatures, while the incoherent metallic phase from the “insulating” phase ries. Gd2Ir2O7 is an insulator at all temperatures, while (P<6GPa, T " 100 K), characterized by a tempera- Eu2Ir2O7 is the first compound in the series to support a (P<6GPa, T " 100 K), characterized by a tempera- Eu2Ir2O7 is the first compound in the series to support a ture dependent gap. Note that the resistivity does not metallic phase at high temperatures. Metallic phases in ture dependent gap. Note that the resistivity does not metallic phase at high temperatures. Metallic phases in diverge at T → 0, however, ρ(0) shows a strong sample the vicinity of localization transitions are usually subject diverge at T → 0, however, ρ(0) shows a strong sample the vicinity of localization transitions are usually subject dependence, so this behaviour may imply the existence of to strong fluctuations in the spin, charge and orbital de- dependence, so this behaviour may imply the existence of to strong fluctuations in the spin, charge and orbital de- conducting surface channels, impurity states in the bulk, grees of freedom, resulting in unconventional transport conducting surface channels, impurity states in the bulk, grees of freedom, resulting in unconventional transport or small residual Fermi pockets. properties. The negative ρ(T ) slope in the incoherent or small residual Fermi pockets. properties. The negative ρ(T ) slope in the incoherent In the top-right quadrant (P>6GPa,T ! 100 K), the metallic regime of Eu2Ir2O7 is likely to be the result of In the top-right quadrant (P>6GPa,T ! 100 K), the metallic regime of Eu2Ir2O7 is likely to be the result of system is a “conventional metal” characterized by a posi- incoherent scattering of electrons off spin and/or charge system is a “conventional metal” characterized by a posi- incoherent scattering of electrons off spin and/or charge tive ρ(T ) slope (Fig. 1, inset). A cross-over at Tmin sepa- fluctuations. tive ρ(T ) slope (Fig. 1, inset). A cross-over at Tmin sepa- fluctuations. rates the conventional metal and the “diffusive metal” Since the residual resistivity of the incoherent metallic rates the conventional metal and the “diffusive metal” Since the residual resistivity of the incoherent metallic (P>6GPa, T " 100 K), whose resistivity increases phase is two orders of magnitude higher than the Ioffe- (P>6GPa, T " 100 K), whose resistivity increases phase is two orders of magnitude higher than the Ioffe- with decreasing temperature in a non-Fermi liquid (NFL) Regel limit (ρIR =1.3mΩ cm), the MIT at TMI cannot power law fashion (Fig. 1, inset). be a simple disorder driven Anderson localization. In with decreasing temperature in a non-Fermi liquid (NFL) Regel limit (ρIR =1.3mΩ cm), the MIT at TMI cannot power law fashion (Fig. 1, inset). be a simple disorder driven Anderson localization. In At intermediate pressures, P = 6.06 and 7.88 GPa, a fact, disorder wipes out the insulating phase and leaves cross-over appears between the incoherent and the con- the system metallic at all temperatures [9]. Frustration At intermediate pressures, P = 6.06 and 7.88 GPa, a fact, disorder wipes out the insulating phase and leaves ∗ cross-over appears between the incoherent and the con- the system metallic at all temperatures [9]. Frustration ventional metallic phases at a temperature T > 100 K at induced localization [13] may be relevant to the MIT in ∗ which the negative ρ(T ) slope becomes positive (Fig. 3). the pyrochlore lattice of Eu2Ir2O7. The recent revelation ventional metallic phases at a temperature T > 100 K at induced localization [13] may be relevant to the MIT in ∗ T =180and270KatP =6.06 and 7.88 GPa respec- of a commensurate AFM order in Eu2Ir2O7 from µSR which the negative ρ(T ) slope becomes positive (Fig. 3). the pyrochlore lattice of Eu2Ir2O7. The recent revelation ∗ ∗ tively. At higher pressures, T seems to be pushed above measurements raises the possibility of a Slater transition T =180and270KatP =6.06 and 7.88 GPa respec- of a commensurate AFM order in Eu2Ir2O7 from µSR ∗ room temperature (Fig. 2). [14]. tively. At higher pressures, T seems to be pushed above measurements raises the possibility of a Slater transition ∗ room temperature (Fig. 2). [14]. Our phase diagram suggests a strong connection be- The broad peaks at T (Fig.3) mark a coherent- tween the high and low temperature phases of Eu Ir O : incoherent cross-over of the quasiparticle dynamics Our phase diagram suggests a strong connection be- The broad peaks at T ∗ (Fig.3) mark a coherent- 2 2 7 the incoherent metal becomes insulating below TMI;the generic to correlated oxides in proximity to a Mott transi- tween the high and low temperature phases of Eu Ir O : incoherent cross-over of the quasiparticle dynamics ∗ 2 2 7 conventional metal crosses over to the diffusive metal tion [15, 16]. The T cross-over has not been observed in the incoherent metal becomes insulating below TMI;the generic to correlated oxides in proximity to a Mott transi- below Tmin; and the transition between the insulat- the previous chemical pressure measurements by replac- conventional metal crosses over to the diffusive metal tion [15, 16]. The T ∗ cross-over has not been observed in ing and the diffusive metallic ground states at Pc = ing the R site with larger atoms. Moreover, it probably below Tmin; and the transition between the insulat- the previous chemical pressure measurements by replac- 6.06 ± 0.60 GPa coincides with the appearance of the cannot be realized by alloying the Eu site with larger ing and the diffusive metallic ground states at Pc = ing the R site with larger atoms. Moreover, it probably coherent-incoherent cross-over at T ∗ > 100 K. lanthanides either, because of the extreme sensitivity of 6.06 ± 0.60 GPa coincides with the appearance of the cannot be realized by alloying the Eu site with larger coherent-incoherent cross-over at T ∗ > 100 K. lanthanides either, because of the extreme sensitivity of Donors
• This is likely related to combined effect of small carrier density and Coulomb scattering from donors - O vacancies • Follow ideas of calculation for graphene but for 3d
c.f. Nomura+MacDonald, 2007 Donors
• Screening 2 e 2 2 V (q) 2 2 ξ− αkF ∼ q + ξ− ∼ • Scattering
1 3 2 τ − n d qδ( ) V (k + q) v(k q) ∼ q − F | | · 2 1 cos2 θ e k α d cos θ − 2 ∼ F [2(1+cos θ)+α] c.f. Nomura+MacDonald, 2007 Donors
• Conductivity k2 σ e2 F v2 τ ∼ v F F 1 f(α)e2n1/3 f(α) 1+ ∼ ∼ α2 ln α • Mean free path 2 σ e kF kF k f(α) ∼ · F ∼ c.f. Nomura+MacDonald, 2007 Conclusions
• Weyl semimetals occur in the same sorts of materials as topological insulators (and others!), if inversion or time reversal are broken • They can be designed as intermediate states between certain TIs and NIs • They have unique transport properties and surface states, and in some respects are 3d analogs of graphene, with interactions and defects playing crucial roles