Emergence of Topological Electronic Phases in Elemental Lithium Under

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Li electronic elemental of phases pre- high-pressure some that dicted demonstrate density. we increasing calculations, with first-principles Using character semiconducting up even taking num- and pressure, coordination bers, low under symmetry, reduced diagram conditions, with phase structures ambient several rich 17, under surprisingly December review metal a for simple (received has 2019 prototypical 26, March a approved and Lithium, NJ, Piscataway, Jersey, New of University State The 2018) Rutgers, Vanderbilt, David by Edited 94720 CA Berkeley, www.pnas.org/cgi/doi/10.1073/pnas.1821533116 94720; a Mack A. 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Jeffrey and , hs,w n Li find we phase, b aeil cecsDvso,Lwec eklyNtoa aoaoy ekly CA Berkeley, Laboratory, National Berkeley Lawrence Division, Sciences Materials 1073/pnas.1821533116/-/DCSupplemental h atta ii h hr-iheteeeti h eidctable periodic the in element coupling. third-lightest spin–orbit negligible the with is Li despite which that features, nontrivial fact structure nonsymmorphic, of the band likelihood additional mostly hosting the and are increasing topology pressure four- sticking, becomes band high Li promotes at which predicted adopt after is Li GPa to groups space 450 low-symmetry The least coordinated. at fold of up pressures symmetries, of to nonsymmorphic sequence with a phases, in (20) structural persists coordinated. different experiments coordination threefold and threefold this 19) only shown (18, are have calculations atoms initio the ab which Subsequent in phase (17) measured been close-packed has from lithium transform GPa, to 40 and around measured At of phases. range coordi- diverse predicted a lower in pressure exhibits with lithium structures nated that confirmed have experiments velocities band 16). with (15, graphene behavior of fermionic those to Dirac comparable massless nonsymmorphic of a tive exhibited group, phases space semiconducting re- predicted zero-gap with initially the structures Interestingly, open numbers. coordination is to duced density transitions kinetic charge via The valence lowered (14). nonuniform subsequently energy increasingly of Fermi the character the of p-orbital near energy in (13) increase structure intersti- an band the to in the core leading reside (12), to 1s regions density between electron tial valence overlap 2s experimen- increasing forces but states densities, high achievable At (11). tally distortions symmetry-lowering of hsatcecnan uprigifrainoln at online information supporting contains article This 1 the under Published Submission.y Direct PNAS a is article This interest.y of conflict no declare authors The ...... n ...aaye aa n ...... n ...woetepaper. the wrote J.B.N. and S.M.G., S.A.M., research; and performed data; S.A.M. analyzed research; J.B.N. and designed S.M.G., J.B.N. S.A.M., and S.M.G. contributions: Author owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence whom To int narayrmral ope hs iga o an for diagram phase complex solid. remarkable elemental already dimen- an another to adding sion unexplored pressure, previously with that has properties indicate lithium results topological of Our isolated structure level. well electronic Fermi are the the at structure bands the band where other solid the from elemental of light a features is it topological fermions. that topologically in massless unique becomes as is behaving Lithium 80 structure electrons At electronic the with lithium. its nontrivial, structure of find electronic phases we the high-pressure GPa, predicted study several to use of We theory pressure. functional standard under structures density under symmetry tran- lower phase phase to of sitions sequence metallic unusual an lightest undergoes and conditions the forms Lithium Significance ic h rgnlpeitosnal w eae g (11), ago decades two nearly predictions original the Since a,b,c,d,1 d al nryNnsine nttt,Uiest fCalifornia, of University Institute, Nanosciences Energy Kavli ihlnal ipriebns(1 indica- (11) bands dispersive linearly with Cmca, NSlicense.y PNAS . y Fm 3m ¯ www.pnas.org/lookup/suppl/doi:10. oalower-symmetry a to NSLts Articles Latest PNAS | I f5 of 1 43d ¯ y

APPLIED PHYSICAL SCIENCES Metal Insulator Dirac semimetal insulating (Fig. 1). Using Z2Pack, we compute Aba2 to be topologically trivial with a Z2 index of zero. At 80 GPa, a transition to the semimetallic Pbca phase is predicted; the crystal structure bcc fcc I43d Aba2 Pbca Cmca-24 Cmca-56 of Pbca at 80 GPa is shown in Fig. 2A. Our calculated DFT band Pbca SI Appendix 0 100 200 structure for at 80 GPa (see for lattice parame- Pressure ters) features two fourfold degenerate Dirac points at the Fermi (GPa) energy located along the Γ–X and Γ–Y directions as shown in Fig. 1. Predicted phase diagram of Li below 250 GPa from ref. 18 at zero Fig. 2B. Interestingly, these Dirac points are isolated from other temperature. Note that from 100 GPa to 165 GPa lithium assumes a Cmca bands. At the Dirac points (and away from the Fermi level at X unit cell with 24 atoms per unit cell and a 56-atom unit cell from 165 GPa to and Y) we observe “band sticking” (27), or band degeneracies, 220 GPa, after which P42/mbc is the preferred space group. enforced by the nonsymmorphic symmetries in the Pbca space group. Similar band degeneracies arise in the electronic structure of lithium at lower pressure in the I 43¯ d phase (SI Appendix, Here, we use first-principles density functional theory (DFT) Fig. S1), which is also nonsymmorphic. However, the eightfold calculations to examine the most current and accepted predicted degeneracies at the H point, predicted in the band structure of structures in the pressure phase diagram of Li. We compute any crystal with I 43¯ d symmetry (28), are far below the Fermi and analyze their electronic structure and demonstrate that, in energy and topologically trivial nearly free-electron–like bands the proposed high-pressure structural phases of elemental Li, dominate the electronic structure at the Fermi level. topologically nontrivial electronic structures emerge under pres- Our computed Fermi surface of the Pbca phase at 80 GPa sure. We calculate a change in the topological properties of the appears in Fig. 2C. The fourfold degenerate Dirac points form near–Fermi-level band structure coinciding with the predicted part of a nodal ring located at the Fermi energy in the kz = 0 structural phase transition at 80 GPa from the trivially insulat- plane of the Brillouin zone, enforced by the glide plane {2001 ing Aba2 phase to the Pbca phase which forms a Dirac nodal line |1/2 0 1/2} (in Seitz notation). Our calculations predict that the semimetal. The nodal ring in the Pbca phase at the Fermi energy nodal ring is well isolated from the nearest bands over a broad is well separated from trivial bands by 0.8 eV in our DFT calcula- energy range of 0.8 eV. We verify that the nodal ring is protected tions. We further show that the structural transition between two by a nonzero Berry phase (winding number = −1), and there- predicted higher-pressure Cmca phases is also accompanied by a fore it is topologically nontrivial (see SI Appendix for details). The Lifshitz transition, wherein the Fermi surface topology changes Fermi velocities, computed with DFT-LDA, range from 2.8 to from a single Dirac nodal ring at the Fermi level in the Cmca- 6.6 × 105 m/s, comparable to measured values for other verified 24 phase to two nodal rings in the higher-pressure Cmca-56. The Dirac semimetals, such as Na3Bi (29) and Cd3As2 (30). Topolog- higher-pressure P42/mbc phase is predicted to have a nodal ring ically nontrivial electronic bands will lead to unique surface states just below the Fermi energy. Finally we compute that the highest- arising from the bulk topological features. Although such surface pressure predicted phase Fd3¯m features a distorted hexagonal states would ostensibly be challenging to probe experimentally, honeycomb network in strong analogy with graphene and a Dirac our DFT calculations of the Pbca (001) surface-projected band crossing 1 eV below the Fermi energy. structure (Fig. 2D) verify the existence of the expected drumhead surface state bands connecting the bulk Dirac points (31, 32). Results That high-pressure phases of solid Li possess topological band Experimental determination of the high-pressure structures of structures with pressure is notable, given its low atomic number. lithium is challenging for diffraction experiments due to its small atomic number (Z = 3). In fact, only four lower-pressure structural phases (Im3¯m, Fm3¯m, I4¯3d, Aba2) have been experimentally confirmed. Beyond 70 GPa, although the Pearson class A B has been determined experimentally, the full crystallographic symmetry of the high-pressure phases is not yet known (20). Based on the Pearson class and using first-principles calculations, two prior studies (18, 19) used structure searching algorithms to determine the low-enthalpy space group symmetries at a range of pressures and predicted the zero-temperature phase diagram of lithium from 0 GPa to 500 GPa. We adopt structures from one of these studies (18), which is in good agreement with an earlier independent study by Pickard and Needs (19); the dif- C D ference between the two studies is that ref. 18 predicts two intermediary structures at 71 GPa and 227 GPa, not reported in ref. 19. Starting with atomic coordinates from ref. 18, we use structures at a representative pressure in the predicted phase diagram (SI Appendix) and then use DFT to compute and analyze their electronic structure. All DFT calculations are performed within the local density approximation (LDA) with projector augmented-wave (PAW) potentials, treating all three electrons Fig. 2. Solid Li at 80 GPa in its predicted Pbca phase. (A) Pbca unit cell. in Li as valence using the Vienna ab initio simulation pack- Coloring indicates different atomic layers along the b direction. (B) Band age (VASP) code (21, 22). We use the postprocessing software structure at 80 GPa, where the colors indicate the dominant orbital contri- Z2Pack (23, 24) and WannierTools (25) to compute topological bution to the band, with the s, px , py , and pz orbitals represented by orange, invariants and surface states and to determine whether a given light blue, medium blue, and dark blue, respectively. The Fermi energy is set to 0 eV and marked by a black line. (C) The nodal ring is shown in the kz = 0 phase is topologically trivial or nontrivial. Further details of these plane where the color gradient represents the size of the band gap in elec- calculations can be found in SI Appendix. tronvolts. The projection of the Fermi surface on the 2D k plane indicates Although solid lithium is metallic (26) and considered nearly there is a nodal ring in the kz = 0 plane. (D) The projected band structure free-electron–like at low pressures, between 70 GPa and 80 GPa along the [001] direction. See SI Appendix for details of our surface state it is predicted to adopt the Aba2 phase, which is predicted to be calculations.

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D E − EF (eV) −2 −1 ntefia w hsso h rdce rsuephase pressure predicted the of phases two final the In the in persist Li solid for features topological nontrivial The 0 1 2 Pbca 0 P ma20Ga-Cc 350GPa -P4 220GPa -Cmca 100 GPa -Cmca adsrcue o ihrpesr ukpae o (A) for phases bulk higher-pressure for structures Band oa ig (E ring. nodal )(D) .) P nthe In Cmca-24. 4 ) 2 E P /mbc YadYTa hw nFg 4F Fig. in shown as Y–T and Γ–Y sdrvdfo h w rsig eni h electronic the in seen crossings two the from derived is 4 2 YadYX ( F Y–X. and Γ–Y EFG h eaieobtlcnrbto otebands the to contribution orbital relative The /mbc. ) hr r w oa ig.Tefis nodal first The rings. nodal two are There Cmca-56. thge rsue f30Ga the GPa, 350 of pressures higher At Pbca. hs.Cmuigteeetoi structure electronic the Computing phase. BC −2 −1 D–G. 0 1 2 Γ on n smc mle nksaeta for than space k in smaller much is and point k z Tefl DFrisrae r hw in shown are surfaces Fermi 3D full (The ln,tendllo slreyderived largely is loop nodal the plane, 0 = P 4 YadYTse tteFrieeg in energy Fermi the at seen Y–T and Γ–Y Γ 2 h eodndllo ssandby spanned is loop nodal second The ) ta nry20mVblwE below meV 250 energy an At /mbc. oadul opcnee tthe at centered loop double a to Pbca NSLts Articles Latest PNAS hs t8 GPa. 80 at phase −2 −1 0 1 2 D–F R hsLifshitz This . 3m ¯ si evolves it as (B) Cmca-24, Pbca hs is phase 2 /mbc | tthe at ) f5 of 3 case. SI F

APPLIED PHYSICAL SCIENCES A 450 GPa - R3mBC 500 GPa - Fd3m symmetries which enforce band stickings at high-symmetry 20 20 points, thus increasing the likelihood for crossings along high- symmetry directions. The distorted hexagonal honeycomb structural motifs are reminiscent of graphene and are consistent 10 10 with the nonsymmorphic symmetries that guarantee the pres-

(eV) ence of band stickings. Furthermore, the pressures at which these F structures are favored not only reorder the bands and lead to E

− 0 0 dominant p-orbital character near the Fermi level, but also shift E many of the trivial electronic bands away from the Fermi energy as they become more significantly broadened. These two effects −10 −10 in combination, the dominant p character and the phase transitions to structures with nonsymmorphic symmetries, result in Fig. 5. (A and B) Band structures for higher-pressure bulk phases for (A) ¯ ¯ well-isolated Dirac nodes at the Fermi energy in high-pressure R3m and (B) Fd3m, where the coloring follows the same scheme as in Figs. lithium. When lithium assumes the Cmca symmetry, it under- 2B and 4. We note the graphene-like Dirac crossing at the W point 1 eV ¯ goes a Lifshitz transition as seen by the change in Fermi surface below EF in B and the bands have p character. (C) Fd3m unit cell. Coloring indicates different atomic layers which have the same buckling distortion topology between the two predicted structures. It then evolves to perpendicular to the atomic plane in the hexagonal motifs similar to the a more metallic P42/mbc phase where the nodal line is slightly Pbca structure; here the atoms are fourfold coordinated with their nearest below the Fermi energy as the s- and p-like bands become neighbors and all bond lengths are 1.23 A.˚ reordered with pressure. At 500 GPa, lithium’s structure consists of fourfold coordinated atoms in buckled hexagonal honeycomb layers, giving rise to a Dirac crossing 1 eV below the Fermi level ¯ free-electron–like. But as lithium adopts the Fd3m structure, as predicted by our DFT calculations. Using first-principles cal- the similarity to graphene becomes more pronounced in both culations, we show here that lithium’s complex structural phase its geometry and its electronic structure: Its structure consists diagram also features topological electronic structure, suggesting of offset, hexagonal layers with all bond lengths approximately similar features may be observed in other light elements under 1.23 A˚ (both within each hexagonal layer and between layers) pressure. Indeed, more generally, pressure can be used to realize and has the same “buckling” motif, as described earlier for the topological features in electronic structures in broad classes of Pbca phase. We note that at the W point on the Brillouin zone materials. edge, where the bands are dominated by p-like character, there is a Dirac point, similar to graphene’s Dirac point at its K point Materials and Methods (which also lies on the Brillouin zone edge); albeit, here in Density functional theory calculations are carried out with VASP (21, 22, 59) lithium, it is 1 eV below the Fermi energy whereas in graphene it within the LDA. We use a plane-wave basis set and projector augmented- is at the Fermi level. Lithium can thus be seen to form a 3D ana- wave pseudopotentials which, for Li, treat both 1s and 2s electrons explicitly logue to graphene at this extremely high pressure, where again as valence. Our plane-wave energy cutoff is 1,000 eV (to achieve energy con- the crystallographic symmetry and dominant p character at the vergence across all of the high-pressure structures) and a k-grid density of Fermi level facilitate the formation of the Dirac crossing. 0.01 A˚ −1 is used. We find that including spin–orbit coupling has a negligi- We note that at low temperatures, topological properties are ble effect on the structural and electronic properties of the different phases predicted to appear at about 80 GPa and above, a pressure and therefore all calculations are performed without spin–orbit coupling. range recently probed for lithium (20); pressures up to 400 GPa Z2Pack is used to calculate the winding number around the nodal ring using are experimentally achievable using diamond anvil cells (49), the evolution of Wannier charge center positions along some periodic direction of a surface in the Brillouin zone (23, 24), in this case a torus constructed and 500 GPa appears to be within reach in the near future around the nodal ring. To examine surface states, tight-binding models are (50–52). The closed loop in the Fermi surfaces would lead to constructed using the hopping parameters obtained from the Wannier func- characteristic quantum oscillations in de Haas–van Alphen mea- tions calculated using Wannier90 (60). As implemented in the open-source surements for the Pbca, Cmca, and P42/mbc structures (shown package WannierTools (25), the Fermi surface is calculated via our tight- in SI Appendix). Given the small lithium mass, nuclear motion binding analysis and the surface density of states is calculated using an associated with zero-point and finite temperature effects would iterative Green’s function technique (61). be expected to alter the structural phase diagram (53–55); such effects may shift the zero-temperature structural phase transition Note Added in Proof. Since submission of our manuscript, we became aware pressures predicted in refs. 18 and 19 and used here. Future cal- of recent unpublished calculations with conclusions about the topological properties of high-pressure Li phases broadly consistent with ours (62). We culations, for example including anharmonic effects in calcula- thank R. Hoffmann and S. Bonev for communicating their work to us. tions of structural energetics, as has recently been demonstrated for hydrogen (56), would be desirable. Additionally, apprecia- ACKNOWLEDGMENTS. This work was supported by the Theory Field Work ble electron–phonon interactions expected for lithium at high Proposal at the Lawrence Berkeley National Laboratory, which is funded by densities could potentially modify band dispersion, introducing the US Department of Energy (DOE), Office of Science, Basic Energy Sci- ences, Materials Sciences and Engineering Division under Contract DE-AC02- satellites and kinks as has been reported in angle-resolved pho- 05CH11231. Computational work performed at the Molecular Foundry was toemission studies of graphene (57, 58). Further theoretical and also supported by the Office of Science, Office of Basic Energy Sciences, experimental studies will be important to explore these issues in of the US DOE under the same contract number. This research also used detail and their consequences for the structural and electronic resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science phase behavior of lithium at these high densities. of the US DOE, under the contract number listed above. S.M.G. acknowl- In summary, as lithium is subjected to higher and higher pres- edges financial support by the Swiss National Science Foundation Early sures, it favors lower coordinated phases with nonsymmorphic Postdoctoral Mobility Program.

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