Formation of a Two-Dimensional Electron Gas in a Dense Lithium-Beryllium Alloy Richard G. Hennig, Ji Feng, Neil W. Ashcroft and Roald Hoffmann
Do Li and Be form alloys? What are their electronic structures? Can they have higher superconducting temperaturesThe most cthanomp pureetiti vLie andBe2 LBe?i structure
• Li and Be form intermetallic compounds under pressure • Larger core of Li and smaller core of Be push valence electron density into 2D electron gas
Possible enhancement of Tc • 1.98 Å through increased density of states
Supported by NSF Be2Li (80 GPa) Computational resources provided by OSC Nature 451,P6/mm m445 (2008)
MgB Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois The beryllium story
! Elemental beryllium: The Berylliumo Highest DStoryebye temperature among metallic elements o Superconducting transition TC = 0.026 K Elemental beryllium o Because Be is barely a metal • Highest Debye temperature of all metallic elements: ΘD = 1,100 K • Superconducting transition temperature of only Tc = 26 mK • BCS theory of supeconductivity 1 T = 1.13 θ exp c · D −g V ! 0 · " • Low density of states at Fermi level leads to low Tc
Beryllium is barely a metal.
Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Alloying to improve Tc
Alloying with light elements – Lithium • Light, metallic, electropositive • However, Li and Be do not mix or form any intermetallic compounds
Can pressure lead to compound formation?
Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Computational Structure Predictions
Computational structure prediction based on optimizationLocating the global minimum of a potential energy surface
• Stable crystal structure ⇒ Lowest free energy Energy • Minimize the free energy • Non-trivial for the following reasons: ‣ High-dimensional search space ‣ Rough free energy surface, i.e. sensitive to small changes Configuration coordinate ‣ Representation of structures by unit cells leads to redundancies ‣ Accurate ab-initio free energy calculations are computationally expensive Search method of choice depends on affordable number of energy evaluations • Only limited success of conventional optimization“Accurate” methods methods such as first principles DFT are required ‣ Simulated annealing, Metadynamics, Minima hopping • Recent advances in optimization methods: 2 ‣ Random search (Pickard & Needs) ⇒ Used in this work ‣ Evolutionary algorithms (Oganov)
Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Random Search Method
Generate a population of random structures and relax them: • Choose random unit cell translation vectors • Renormalize the volume to a reasonable range of values • Choose random atomic positions within the cell
May constrain the initial positions: • Fix the initial positions of some of the atoms (e.g., defect) • Insert molecules randomly (rather than atoms) • Choose a particular space group
Relax population of random structures • Use accurate density functional methods • Increase accuracy during optimization
Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois 718 C.W. Glass et al. / Computer Physics Communications 175 (2006) 713–720
Table 1 Standard parameter values for a 20-atom system Parameter Value Symbol Population size 30–60 PercentageEvolutionary heredity Algorithms 85 Pher Percentage mutation 10 Pmut Percentage permutation 5 Pperm Percentage shifting along a 100 !ch Percentage shifting along remaining vectors 5 Percentage of individuals with selection probability zero 40 In evolutionaryAverage number algorithms of two-atom-exchanges a duringpopulation permutation of candidate solutions 2–3is evolved overN perm Standard deviation of lattice strain matrix 0.7 σlattice Standard deviation of atomic position shifts 0.0 σatoms successive iterations of random variation and selection. Random1 variation Resolution for ki determination (smaller values required for metals) 0.12 Å− kresol provides theWeight mechanism for VUC adaptation from generationfor discovering to generation new solutions. Selection 0.5 determinesW adapt Number of (best) individuals to be averaged over for VUC adaptation 4 Nadapt which solutionsNumber of (best) to individualsmaintain of parental as population a basis to be considered for for further environmental exploration. selection 1–2 Ncons Evaluation2.8. function: Parameters Ab initio free energy For a system with 20 atoms in the unit cell with unknown lat- Variation tice,operators no starting structures and a reasonable guess at the unit cell volume, a reasonable setting for the parameters can be found in • Heredity Table 1. Hard constraints are system specific. For the angles ‣ Combining60 ◦a! fractionα, β, γ ! 120 of◦ eachmakes senseof two since structures for any structure there exists a unit cell with these constraints. The minimal lattice vec- ‣ Use spatiallytor length coherent should not slab be larger to retain than the structural diameter of themotifs largest atom. • Mutation A given parameter setting results in a certain behavior of the ‣ algorithm. Depending on the system chemistry and the pres- Random atomence/absence displacements of input (lattice and parameters, lattice startingstrains structures) • Permutationthe desired behavior will change. Therefore there is no univer- sal optimal set of parameters.
‣ Swap pairs If,of foratoms example, a set of good starting structures is available, Fig. 4. MgSiO3 C.at 120W. GPa.Glass, Enthalpy A. R. ofOganov, the best individualN. Hansen, versus generation. the proposed parameter values change significantly. Most im- Population size: 30. Comp. Phys. Comm. (2006) Iterate untilportantly low-energyNcons, Pmut and P permstructureincrease, while isNperm found, Pher and σlattice decrease. Thus the search would be more localized and by keeping more individuals, be more restrained to the currently parameters where this is specified. For non-molecular systems best region—both enhancing exploitation of the information with up to 80 atoms/cell, we have observed a success rate of present in the starting structures. close to 100%. Usually the correct prediction was achieved in the first run. With up to approximatelyElectronic one Structure dozen atoms/cell, Workshop the [email protected] June, 2008 • Urbana-Champaign, Illinois 2.9. Parallelization global minimum can be found with reasonable effort by random search. Molecular systems are generally harder to predict. The computationally expensive part of the algorithm is Furthermore USPEX yields numerous metastable struc- the local optimization. Locally optimizing different candidates tures, some of which highly competitive, and is extremely within one generation is independent and can thus be processed efficient: e.g. for structure prediction of MgSiO3 at 120 GPa (20 in parallel. However only calculations within the same popula- atoms/cell) with minimal input, only between 150 and 40018 in- tion can be parallelized.17 dividuals were calculated before the structure of post-perovskite was found.19 An example, where both perovskite and post- 3. Results perovskite were identified, can be found in Fig. 4.
The method has been successfully tested on various systems with known structure. An overview of the systems can be found 18 Exact timing differed between runs, depending on parameter setting and in Table 2. For all these systems calculations were performed random factors. 19 with minimal input (see Section 2.6) or providing the lattice High-pressure behavior of MgSiO3 was thoroughly studied by standard computational methods over the last 20–30 years, but the post-perovskite phase was found [17,18] only after an analogous phase was identified for Fe2O3 [19]. 17 Since in order to generate a new population, all fitness values of the old This discovery has significantly changed models of the Earth’s internal struc- population need to be known (see Section 2). ture and evolution. USPEX finds this structure straightforwardly. Comparison of Search Methods
Random search Evolutionary algorithm • Simple to program • More complex rules • Successful for small unit cells • Successful for structures with • Increasingly more difficult for large unit cells and structural large structures motifs Locating the global minimum of a potential energy surface • SiH4, LiBe, H2O, H, N • CaCO3, MgSiO3, CO2, O, H
Energy
Configuration coordinate
Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Search method of choice depends on affordable number of energy evaluations “Accurate” methods such as first principles DFT are required
2 Structure Maps Structure maps
• Pettifor structure map of A-B ordered alloy • Adapted from Villars et al.
• Advantage: Fast, simple • Disadvantage: Structures have to be known
Pettifor structure map of AB ordered alloys
Adapted from Villars et al
Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Structural prediction
! Do we understand the structures of intermetallic compounds? Structure Prediction and Search Algorithm! Can we “predict” the structure of a compound given the stoichiometry? • Do we understand the structures of intermetallics? • Can we predict the structureStru coftu compounds?ral search aSlgtoruricthtmural prediction
Be4Li, Be3Li, Be2Li, ! Dpo we understand the structures of intermetallic Be3Li2, BeLi, Be2Li3, BemLin BeLi2, BeLi3, BeLi4 compounds? ! Can we “predict” the structure of a compound given the
stoichiometry? MMgBgB2 2 CrB GaLi2
Guess structure ! Now what about MgB , or Cr B ? Random starting 4 0.08 3.10 from chemical Structure structure design ! Predict high-pressure structures?! Search
Algorithm Structural Structural optimization optimization
GaLi2 MgBm2in{ H } CrB GaLi2
! Now what about MgB4, or Cr0.08B3.10?
! Predict Hhf i Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Computational Details Density functional theory (VASP) • Generalized gradient approximation (PBE) • Plane-wave basis and PAW potentials • Optimization of all parameters (atom positions and lattice vectors) at given pressure Random structural search • Use 20 – 50 starting structures for each selected pressure, composition and cell size ‣ Pressure range: 0 – 200 GPa ‣ Compositions: Be1-xLix x = 0, 20, 25, 33, 40, 50, 60, 66, 75, 100 % ‣ Cell size: Up to 15 atoms per primitive cell • Symmetry identification using ISOTROPY (Stokes & Hatch, BYU) • Check energy of higher symmetry structures • Phonon dispersion calculation to confirm mechanical stability Use of petascale computing for structure searches [email protected] Electronic Structure Workshop June, 2008 • Urbana-Champaign, Illinois Enthalpy of Formation of Li-Be Compounds 0.2 Computed enthalpies of formation Li4Be Cmmm LiBe P21/m Li4Be P-1 Li2Be3 C2/m Li Be C2/m LiBe P6/mmm 0.1 3 2 Li2Be P21/m LiBe3 R-3m Li Be C2/m LiBe R-3m ) ) 3 2 4 x x i i L L x x - - 1 1 e e 0 B B / / V V e e m m ( ( f f h h -0.1 Enthalpy of mixing [eV/atom] Be Li Be Li -0.2 0 50 100 150 200 Pressure [GPa] Stability increases with pressure dramatically at low pressures Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Li-Be Phasediagram under Pressure The most competitive BeLi3 structure The most competitive BeLi3 structure The most competitive BeLi structure Stability ranges Most stable Li3Be phase 3 Li Li3Be Be LiBe LiBe2 LiBe4 BeLi3 (80 GPa) BeLi3 (80 GPa) Li3Be C2/m(80 GPa) C2/m GaLi C2/m GaLi2 2 GaLi2 BeLi3 (80 GPa) C2/m GaLi2 Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Li-Be Phasediagram under Pressure The most competitive BeLi3 structure The most competitive BeLi structure The most competitive BeLi structure Stability ranges Most stable LiBe phase 3 Li Li3Be Be LiBe LiBe2 LiBe4 BeLi3 (80 GPa) LiBeBeLi (80 (82 GPa)GPa) P2 /m C2/m P211/m CrB GaLi CrB 2 BeLi3 (80 GPa) C2/m GaLi2 Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Li-Be Phasediagram under Pressure The most competitive BeLi3 structure The most competitive Be2Li structure The most competitive BeLi3 structure Stability ranges Most stable LiBe2 phase The most competitive Be2Li structure Li Li3Be Be LiBe 1.98 Å LiBe2 1.98 Å LiBe4 Be2Li (80 GPa) P6/mmm Be Li (80 GPa) 2 BeLi3 (80 GPa) LiBeP6/mmm2 (80 GPa) C2/m MgB P6/mmm 2 GaLi MgB2 2 BeLi3 (80 GPa) C2/m MgB2 GaLi2 Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Li-Be Phasediagram under Pressure The most competitive BeLi3 structure The most competitive Be4Li structure The most competitive BeLi3 structure Stability ranges Most stable LiBe4 phase The most competitive Be2Li structure Li Li3Be Be LiBe 1.98 Å LiBe2 LiBe4 Be2Li (80 GPa) P6/mmm BeLi3 (80 GPa) LiBe4 (80 GPa) Be4Li (80 GPa) C2/m MgB R3mR3m 2 GaLi MgB2 2 BeLi3 (80 GPa) C2/m MgB2 GaLi2 Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Li-Be Phasediagram under Pressure Stability ranges LiBe4 LiBe2 Li3Be LiBe LiBe Li3Be LiBe2 LiBe4 Four novel Li-Be phases become stable under pressure Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Fermi DOS’s Electronic Structure of Li-Be phases ! g(!F) for beryllium is relatively constant over the entire • Beryllium’spressu DOSre r aatn thege :Fermi~ 0.0 level4 eV is-1 nearlyper e constantlectron over entire -1 pressure range: g(εF) = 0.04 eV per valence electron Be4Li Be2Li BeLi 80 GPa LiBe4 LiBe2 LiBe 80 GPa R-3m P6/mmm P21/m R-3m P6/mmm P21/m g(! ) (eV-1 per g(εF)F in eV-1 per 0.060.06 0.060.06 0.120.12 vavalencelence electron electron) LiBeBe4L4i LiBeBe2L2i LiBeBeLi For a comparable e-ph coupling Tc would be about 32 K Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois NATURE | Vol 451 | 24 January 2008 LETTERS To understand how such effective electronic two-dimensionality Figure 4 gives the computed electron density on a cross-section could arise in a material that by structural criteria is patently three- (6.4 A˚ 3 16.2 A˚ ) of the stable LiBe phase at 82 GPa. Outside the ionic dimensional, we propose the following model hamiltonian in the core regions, a sharp separation is apparent between high-electron- limit of non-interacting electrons; density zones associated with the Be layers and low-electron-density 2 2 zones associated with the Li layers. There are three density extrema B k { H~ ck ckz eon2iz eozW n2iz1z outside the nuclear region: a, b and c. Extremum a is a maximum 2m ðÞ 3 k à 2i 2iz1 ( 0.059 e/a , where e is the charge on an electron and a is the Bohr X X X 1 , o o NATURE | Vol 451 | 24 January 2008 ð Þ radius), from which the electronLETTERS density drops slowly toward the {tc{c zc{ c i iz1 iz1 i nearest Be atom, but then very rapidly toward the nearest Li atom. i X Extrema b and c (,0.015 e/a 3) are in the middle of the shortest Li–Li where the firstTo understand term describes how such the effective in-plane electronic (x two-dimensionality–y) levels of a two-Figure 4 gives the computed electron densityo on a cross-section could arise in a material that by structural criteria is patently three- (6.4 A˚ 3 16.2separations A˚ ) of the stable in LiBe the phase crystal at 82 (the GPa. Li Outside atoms the near ionic b are above and below the dimensionaldimensional, free-electron we propose gas the (where following the modelks have hamiltonian no components in the core regions,plane a sharp shown). separation These is apparent findings between are high-electron- entirely consistent with the two- in the z-direction),limit of non-interacting and the second electrons; and third terms embed a potentialdensity zonesdimensional associated with electron the Be layers gas and model low-electron-density described by our model hamiltonian, 2 2 zones associated with the Li layers. There are three density extrema energy difference, W . 0,B betweenk { Be and Li layers with site indices ni H~ ck ckz eon2iz eozW n2iz1z outside theand nuclear confirm region: thata, b and the c. two-dimensional Extremum a is a maximum electron gas states are clearly 2m ðÞ 3 (odd-numbered sites havek à higher2i energy).2iz1 In equation (1), m* is( the0.059 e/a , where e is the charge on an electron and a is the Bohr X X X 1 , associatedo with Be layers. o { ð Þ radius), from which the electron density drops slowly toward the electronic effective mass,{ andtc{c c andzc{ cc are annihilation and creation i iz1 iz1 i nearest Be atom,In principle, but then very the rapidly approximate toward the nearest segregation Li atom. of valence electrons into operators. The last termi describes the hopping perturbation between 3 X Extrema bhigh- and c (, and0.015 low-densitye/ao ) are in the middle regions of the could shortest arise Li–Li from the potential energy neighbouringwhere Be the and first Li term layers, describes characterized the in-plane (x– byy) levels a hopping of a two- energy,separationsdifference in the crystal (the between Li atoms Li near and b are Be above layers, and below as the characterized by W in the t . 0. Thedimensional above hamiltonian free-electron gas can (where be thesolvedks have exactly; no components in the limitplane shown). These findings are entirely consistent with the two- in the z-direction), and the second and third terms embed a potential model hamiltonian (equation (1)). The hopping energy t for Li W ? t, it yields a DOS as shown in Fig.3b top inset, where t ; t2dimensional/W. electron gas model described by our model hamiltonian, energy difference, W . 0, between Be and Li layers with site indices ni and confirmand that Be the has two-dimensional a typical electron value gasof states,1–2 are eV clearly (Methods). The very small In spite(odd-numbered of the simplicity sites have of higher this energy). hamiltonian, In equation it (1), reproducesm* is the associated the with Be layers. { value of t (,0.05 eV) in the computed DOS indicates that W is at essential featureelectronic of effective the lower mass, and valencec and c DOS,are annihilation computed and creation using a DFTIn principle, the approximate segregation of valence electrons into operators. The last term describes the hopping perturbation between least ,20 eV, and hence much large than t. More importantly, a large method that in principle includes full atomic potentials and elec-high- and low-density regions could arise from the potential energy neighbouring Be and Li layers, characterized by a hopping energy, differencevalue between of LiW andindicates Be layers, as that characterized there is by aW veryin the large effective electronega- . tron–electront 0. interactions. The above hamiltonian Moreover, can be in solved Fig. 3b exactly; left inset in the we limit see thatmodel hamiltonian (equation (1)). The hopping energy t for Li W ? t, it yields a DOS as shown in Fig.3b top inset, where t t2/W. tivity difference between Li and Be. But the first ionization potential ; and Be has a typical value of ,1–2 eV (Methods). The very small the Be layersIn of spite the of metastable the simplicity LiBe of this2 are hamiltonian, twice as it thick reproduces as in the the LiBe value of t of(, Be0.05 is eV) only in the about computed 4 eV DOS greater indicates than that thatW is of at Li, which is insufficient to phase, andessential the step feature in of DOSthe lower of valence LiBe DOS,occurs computed at using a lower a DFT energy 2 least ,20 eV,produce and hence the much narrow large than stepst. More at importantly, the bottom a large of the valence electron DOS method that in principle includes full atomic potentials and elec- with roughly half the height of the LiBe case. This indicates thatvalue of Wobtainedindicates that with there density is a very functional large effective theory, electronega- and it does not explain the the two-dimensionaltron–electron interactions. states are Moreover, associated in Fig. 3b left with inset the we see Be that layers.tivity difference between Li and Be. But the first ionization potential the Be layers of the metastable LiBe2 are twice as thick as in the LiBe of Be is onlysecond about 4 eV step greater at about than that 4 of eV Li, above which is the insufficient bottom to of valence bands in both Hence, thephase, high-potential-energy and the step in DOS of sites LiBe2 areoccurs associated at a lower with energy the Li produce thePmma narrowLiBe steps at2 and the bottomP21/m of theLiBe valence (Fig. electron 3b). DOS layers, andwith correspondingly, roughly half the height the of Be the layers LiBe case. can This be indicates pictured that as two-obtained with density functional theory, and it does not explain the the two-dimensional states are associated with the Be layers. second step atWe about suggest 4 eV above that the bottom the enhanced of valence bands electronegativity in both differential arises dimensionalHence, potential the high-potential-energy wells. sites are associated with the Li Pmma LiBebecauseand P2 / them LiBe potential (Fig. 3b). energy difference between Be and Li increases layers, and correspondingly, the Be layers can be pictured as two- 2 1 We suggest that the enhanced electronegativity differential arises dimensional potential wells. under compression as a result of differential core overlap between Be a LiBe R-3m LiBe P2 /m because the potential energy difference between Be and Li increases 1 1 4 1 under compressionand Li layers. as a result Be of differentialand Li have core nuclear overlap between charges Be of 4e and 3e, respec- a 80 Gpa LiBe4 R-3m 82LiBe Gpa P21/m and Li layers.tively, Be and so Li have the nuclear 1s core charges of Be of 1 has4e and a1 significantly3e, respec- smaller spatial extent 80 Gpa 82 Gpa tively, so thethan 1s core the of Li Be 1s hasorbital. a significantly At 80 smaller GPa in spatial BeLi, extent both Be–Be and Li–Li bonds LiBe P6/mmm Li Be C2/m than the Li 1s orbital. At 80 GPa in BeLi, both˚ Be–Be and Li–Li bonds 1 2 LiBe P6/mmm Li3 Be C2/m are between˚ 1.9 and 2.0 A. In fact, the electron1 density of a bare Li at Two-dimensional2 3 Electronicare between Gas 1.9 and 2.0in A. In LiBe fact, the electron density of a bare Li at 21 83 Gpa 83 Gpa 8080 GpaGpa 0.95 A˚ away0.95 from A˚ theaway nucleus from is 15 the times nucleus that of a is bare 15 Be times21 (for that of a bare Be (for hydrogenic wavefunctions). The difference in the spatial extent of 2323454523452345 hydrogenic wavefunctions). The difference in the spatial extent of k (Å–1) k (Å–1) core density is significantCrB The and CrB most even more competitive pronounced for the BeLi com- structure k (Å–1) k (Å–1) mon cationscore of these density elements, is significant whose ionic radii and are even 0.27 A more˚ (Be21) pronounced3 for the com- 0.30 1 21 b and 0.76 A˚mon(Li ). cations In LiBe, the of Li these atomic elements, cores have thus whose started ionic to radii are 0.27 A˚ (Be ) 2 LiBe (P2 /m) 0.30 τ 1 overlap significantly while˚ the1 Be cores have not. Indeed, in the DFT b ) ε LiBe (Pmma) and 0.76 A (Li ). In LiBe, the Li atomic cores have thus started to 0.25 ( 2 The most competitive BeLi structure g 2 LiBe (P2 /m) derived bandLiBe structure, the Li 1s bands show a dispersion of over 3 τ 1 overlap significantly while the Be cores have not. Indeed, in the DFT ) 1 eV, while those of Be remain within 0.1 eV. Substantial core overlap ε LiBe (Pmma) 0.25 ( 2 P21/m 9,10,17 0.20 g ε between Liderived atoms bandresults structure, in an even higherthe Li potential 1s bands energy show a dispersion of over /m /m for valence electrons,1 which1 evade the Li layers and move into the 1 eV,P2 while thoseP2 of Be remain within 0.1 eV. Substantial core overlap 0.15 neighbourhood of Be atoms where9,10,17 they form almost ideal two- 0.20 ε dimensionalbetween free-electron-likeLi GPa) (82 BeLi Li atoms GPa) (82 BeLi states. Withresults the enhanced in an electron even higher potential energy per valence electron) transfer betweenfor valence Be and Li, electrons, our high-pressure which LiBe evade alloys the resemble Li layers and move into the –1 0.10 Be neighbourhood of Be atoms where they form almost ideal two- 0.15 ) (eV ε ( g 0.05 dimensionala b free-electron-likec structure BeLi competitive most The states. structure BeLi competitive most The With the enhanced electron per valence electron) transferBe betweenLi Be and Li,Be ourLi high-pressure0.5 LiBe alloys resemble –1 0.10 –12 –10 –8 –6 –4 –2 0 2 0.4 Li Be Li Be e Å ) (eV ε – εF (eV) ε –3 ( 0.3 g Figure0.05 3 | Simulated X-ray diffraction patterns and electronic structures of Be a Li b Be Lic stable Li12xBex high-pressure phases. a, Approximate X-ray diffraction 0.2 patterns simulated for the four stable Li-Be phases at around 82 GPa. Shown Li Be Li BeBe Li Be Li 0.5 are the relative heights expected for the diffraction peaks, each associated 0.1 with–12 a reciprocal –10 lattice vector–8 belonging –6 to the selected –4 structure.–2 The 0 dash- 2 0.4 dotted lines indicate where twice the free-electron Fermi wavevector lies. Li Be BeLi3 (80Li GPa)Be e ε – εF (eV) Å b C2/m , The electronic density of states (g or DOS) of P21/m LiBe (82 GPa) and of Figure 4 | Electron density of the stable LiBe phase at 82 GPa. The 0.3 –3 the structurally related metastable Pmma (83 GPa) phase. Note that the DOS structure is slightly symmetrized so that we can show a plane with many GaLi Figure 3 | Simulated X-ray diffraction patterns and electronic structures of Be Li Be Li 2 is given as an intensive property, that is, in units per eV per valence electron. atoms. The structural change in symmetrization isBeLi3 very small; (80 theGPa) energy and stable Li Be high-pressure phases. a 0.2 12x OnlyElectronx the valence densities density of states are, shown. Approximateshows The dotted a line X-ray two-dimensional, indicates diffraction the electronic structure are essentiallylayered unchanged. structure Note theC2/m horizontal direction patterns simulatedFermi level. for Left the inset, four structure stable of Li-BePmma LiBe phases2. Top at inset, around the DOS 82 deduced GPa. Shownhere is the directionLi in whichBe Li and Be layers alternate, whichLi correspondsBe GaLi2 are the relativefrom heights a model hamiltonian expected (see for text). the diffraction peaks, each associatedto the vertical direction in Fig. 2d. a, b and c are density extrema (see text). 0.1 with a reciprocal lattice vector belonging to the selected structure. The dash- 447 © [email protected] Nature Publishing Group Electronic Structure Workshop dotted lines indicate where twice the free-electron Fermi wavevector lies. June, 2008 • Urbana-Champaign, Illinois b, The electronic density of states (g or DOS) of P21/m LiBe (82 GPa) and of Figure 4 | Electron density of the stable LiBe phase at 82 GPa. The the structurally related metastable Pmma (83 GPa) phase. Note that the DOS structure is slightly symmetrized so that we can show a plane with many is given as an intensive property, that is, in units per eV per valence electron. atoms. The structural change in symmetrization is very small; the energy and Only the valence densities of states are shown. The dotted line indicates the electronic structure are essentially unchanged. Note the horizontal direction Fermi level. Left inset, structure of Pmma LiBe2. Top inset, the DOS deduced here is the direction in which Li and Be layers alternate, which corresponds from a model hamiltonian (see text). to the vertical direction in Fig. 2d. a, b and c are density extrema (see text). 447 © 2008 Nature Publishing Group ElectronicThe most competitiveStructure BeLi of3 structure Li-Be phases CrB CrB The most competitive BeLi structure LiBe Analytically3 solvable model !2k2 P21/m /m /m 1 1 xy = c† ck P2 P2 k BeLi GPa) (82 BeLi GPa) (82 BeLi H 2m∗ Li !k Electronic structure of BeLi (P2 /m) Be 1A modezl =HaWminl2ti otniaci†nci+1 + h.c. h otcmeiieBL structure BeLi competitive most The structure BeLi competitive most The H − i i ! ! "2t2/W # BeLi • In the limit W >> t P21/m it yields the ) shown DOS E ( g BeLi3 (80 GPa) C2/m GaLi2 0 1 2 3 4 BeLi35 (806 GPa) 7 C2/m E Reduction of degrees GaLi2 Density of statesof freed ofromm model Hamiltonian matches calculation Be2Li Pmcm Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Electronic Structure of Li-Be phases • Extract W from step-shape of density of states: 2t2/W = 0.05 eV • Assuming a typical value of t = 1...2 eV yields a value of W > 20eV ‣ Observe large effective electronegativity difference between Li and Be ‣ However, first ionization potential is only 4 eV higher in Li than in Be ‣ Insufficient to produce narrow step at bottom of valence band What is the origin of the large potential difference between Li and Be layers? • Electronegativity difference arises because potential energy difference between Li and Be layers increases as a result of core overlap ‣ At 80 GPa: dLi-Li and dBe-Be = 1.9–2.0 Å ‣ Ionic radii: rLi+ = 0.76 Å and rBe2+ = 0.27 Å ‣ Li core electrons show 1 eV dispersion Substantial core overlap of the large Li ions pushes valence electrons into Be layers resulting in a quasi-2D electron gas. Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois Formation of a Two-Dimensional Electron Gas in a Dense Lithium-Beryllium Alloy Richard G. Hennig, Ji Feng, Neil W. Ashcroft and Roald Hoffmann Electronic Structure Workshop Nature 451, 445 (2008) June, 2008 • Urbana-Champaign, Illinois Formation of a Two-Dimensional Electron Gas in a Dense Lithium-Beryllium Alloy Richard G. Hennig, Ji Feng, Neil W. Ashcroft and Roald Hoffmann Do Li and Be form alloys? What are their electronic structures? Can they have higher superconducting temperatures than pure Li and Be? The most competitive Be2Li structure • Li and Be form intermetallic compounds under pressure • Possible enhancement of Tc through increased density of states • Larger core of Li and smaller core of Be push valence electron density into 2D electron gas 1.98 Å • Fascinating high-pressure chemistry of alloys from simple elements Be2Li (80 GPa) P6/mmm They used to be called the simple elements Nature 451, 445 (2008) MgB2 [email protected] Electronic Structure Workshop June, 2008 • Urbana-Champaign, Illinois Two Open Postdoc Positions Computational Materials Science – Cornell University 1. Modeling and design of strongly correlated transition metal oxides • Joined between RGH (Materials Science and Engineering) and Craig Fennie (Applied and Engineering Physics) • DFT, GW, and QMC computations to understand the physics and materials science of complex oxide bulk solids and nanostructures • MRSEC program “Controlling ComplexElectronic Materials” 2. Quantum Monte Carlo Petascale Algorithms and Applications • Joined between RGH (Materials Science and Engineering) and Cyrus Umrigar (Physics) • DOE program “Quantum Monte Carlo Endstation for Petascale Computing” If you are interested, talk to me or Cyrus or e-mail us. Electronic Structure Workshop [email protected] June, 2008 • Urbana-Champaign, Illinois