P.Blaha (fromOleg Rubel, McMaster Univ, Canada) - Boston College

Wannier functionsMacroscopic polarization(Berry phase) and related propertiesEffective band structure of alloysP.Blaha(from Oleg Rubel, McMaster Univ, Canada)

Wannier functions Wannier90: A Tool for Obtaining Maximally-LocalisedWannier FunctionsA. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D.Vanderbilt and N. MarzariComput. Phys. Commun. 178, 685 (2008)[http://wannier.org]

Bloch vs Wannier functionsBloch functionsWannier functions (localized orbitalsIndexed bythe wavevectorΓ-pointIndexed bythe latticevector in realspaceBoth sets: complete and orthonormalWannier:PRB 52, 191 (1937)Marzari et al.:PRB 56, 12847(1997)Rev. Mod. Phys.(2012)

Max. localized Wannier functions (MLWF)Bloch functions (moreprecisely):gauge freedom ambiguityrandom phase non-unique WF!optimized ɸnkWF spreadchoose U(k) to minimize spread

maximally localized wannier functions

Two flavours of Wannier functionsGaAtom centered sp3-like includes bonding andantibonding statesbuilding effectivehamiltonianAsBond centered s-like Marzari et al.:Rev. Mod. Phys.(2012)includes valence statescharge transfer andpolarization

MLWF: applications analysis of chemical bondingbonding and antibonding states electric polarization and orbital magnetizationBerryPi (O.Rubel et al.) Wannier interpolationH(k) KH(R) K-1(eg. Woptic, transport, .)H(k) G building effective hamiltoniantight binding parametersinput for dynamical mean field theory

Wannier functions as a tight-binding basis (atom centered FW)Matrix element (eV)〈s1 H s1〉 Es1〈s1 s1〉 less GaAs-WANN hr.dat.Homeunit cell0000000000000000010203040506〈s2 0000.Neighbourunit cell1WF are well localized nearest-neighbour suffice(Im part 0)Matrix element (eV)〈s2 H s1〉 Vssσ〈p2 H s1〉 Vsp

Band structureEnergy (eV)GaAsEFWave vector original Wien2k band structure- Band structure computed from Wannier hamiltonian

DisentanglementEnergy (eV)GaNEF} Ga-3d & N-2sWave vector original Wien2k band structure- Band structure computed from WannierhamiltonianWannier functionswithout Ga-3dSouza et al.:PRB 65, 035109(2001)

Relation to polarization (bond centered WF)Bond-centered WFSiGaAs symmetric(non-polar)Ionic part ZAsWanniercenter qeZGanon-symmetric(polar)Electronic partKing-Smith & Vanderbilt,Phys. Rev. B 47, 1651 (1993)

Workflow Regular SCF calculation Band structure plot Initialize wien2wannier (init w2w):- select bands, init. projections, # of WF (case.inwf file)- projected band structure “bands plot project” (case.win file)- additional options related to entanglement (case.win file)Compute overlap matrix element Smn and projections Mmn (xw2w) Perform Wannierization (x wannier90):- position of Wannier centers and spreads (case.wout file)- Wannier hamiltonian (case hr.dat file) Initialize plotting, select plotting range, r-mesh (write inwplot)Evaluate WF on the r-mesh selected (x wplot)Convert the output of wplot into xcrysden format for plotting(wplot2xsf) Plot WF

Useful resources Jan Kuneš et al. “Wien2wannier: From linearizedaugmented plane waves to maximally localizedWannier functions”, Comp. Phys. Commun. 181,1888 (2010).Wien2Wannier home and user r90 home and user guide:http://www.wannier.org/Nicola Marzari et al. “Maximally localized Wannierfunctions: Theory and applications”, Rev. Mod.Phys. 84, 1419 (2012)

Macroscopic polarization BerryPI

Material properties related to polarizationPiezo- and FerroelectricityDielectric screeningE Effective chargePyroelectricity

What is polarization? 2q-O qdHH q PP 0PPolarization for periodic solids is undefined

Modern theory of polarizationPioneered by King-Smith, David Vanderbilt and Raffaele RestaAll measurable physical quantities arerelated to the change in polarization!E

Components of polarizationionicelectronic In Wien2k Zsion is the core charge0King-Smith and David Vanderbilt, Phys. Rev. B 47, 1651 (199

Berry phaseWIEN2WANNIERKing-Smith and David Vanderbilt, Phys. Rev. B 47, 1651 (1993)

Uncertainties it is challenging todetermine largepolarization difference 1 C/m2Solution: λ0 λ1/2 λ1

BerryPI workflowNeed wien2k, wien2wannier, python 2.7.x and numpy[command line] berrypi -k 6:6:6 [-s] [-j] [-o]completed SCF cyclegenerate k-mesh in the full BZ(kgen)Spin-polarizedOrbital potential(e.g., LDA U)Spin-orbitprepare nearest-neighbour k-pointlistcalculate wavefunctions(lapw1)calculate overlap matrix Smn (w2w)determine electron. and ion.phasesPolarization vectorComput. Phys. Commun. 184, 647 (2013)

Choice of a reference structureλ0 structure file must preserve the symmetrybegin with the lowest symmetry (λ1) casecopy case λ1 to case λ0edit structure file for case λ0do not initialize calculation (init lapw)update density (x dstart)run SCF cycle (run[sp] lapw [-so -orb])run BerryPIλ1

Demonstration: Effective charge of GaNGeneral definitionNΔuzGa“Shortcut” (i j, novolume change)

Reality check

Useful resources Sheikh J. Ahmed et al. “BerryPI: A software forstudying polarization of crystalline solids withWIEN2k density functional all-electron package”,Comp. Phys. Commun. 184, 647 (2013).BerryPI home and iRaffaele Resta “Macroscopic polarization incrystalline dielectrics: the geometric phaseapproach” Rev. Mod. Phys. 66, 899 (1994)Raffaele Resta and David Vanderbilt “Theory ofPolarization: A Modern Approach” in Physics ofFerroelectrics: a Modern Perspective (Springer,2007)

Effective band structure ofalloys fold2Bloch

Semiconductor alloysThermoelectric:Si1-xGex(InGaAl)P(InGa)NEg 1 eV junction:(InGa)(NAs)IR detector:(HgCd)Te1.55 μm lasers:(InGa)As(InGa)(NAsSb)Ga(AsBi)

Band structureSilicon2-atom basisSilicon250-atom supercellEffective massEnergy gapEFXForbidden opticaltransitionBloch wave vector

Zone foldingThe character of changes between – Xfrom bonding to anti-bondingDoubling the unit cell halfs the BZ backfolding of X to the wavefunction can still tell you if an eigenvalue was or X

Unfolding the first-principle band structurePlane waveexpansionBlochspectralweightPopescu & Zunger:Phys. Rev. Lett. 104, 236403 (2010)Rubel et al.Phys. Rev. B 90, 115202 (2014)

Workflow Construct primitive unit cell Make supercell (supercell) Run SCF calculation Create k-path (case.klist band file) Compute wave functions (case.vector[so] file) for theselected k-path:- x lapw1 [-p]- x lapwso [-p] (in the case of spin-orbit coupling)fold2Bloch Unfold band structure (fold2Bloch) Plot effective band structure (ubs dots*.m)

Demonstration: Band structure of Si1-xGex alloy (x 0.2)

Thermoelectric material: Si0.7Ge0.3Landoldt-Bornstein (1982, 1987);Sasaki et al. (1984).Busch & Vogt. (1960)

(Hg,Cd)Te band structure evolutionHg22Cd5Te27

Impact of alloying disorder on charge transportCdTe (HgCd)TeGaAs Ga(AsBi)LΓWave vectorLμh 200 10 cm2V-1s-1μe 1,100 1,000,000 cm2V-1s-1μe 4,000 2,500 cm2V-1s-1

Useful resources V. Popescu and A. Zunger, Phys. Rev. Lett. 104,236403 (2010).O. Rubel, A. Bokhanchuk, S. J. Ahmed, and E.Assmann “Unfolding the band structure ofdisordered solids: from bound states to highmobility Kane fermions” Phys. Rev. B 90, 115202(2014)fold2Bloch home and tutorials:https://github.com/rubel75/fold2Bloch

AcknowledgementBerryPI contributors: Jon Kivinen Sheikh J. Ahmed Ben Zaporzhan Sam Pichardo Laura Curiel David Hassan Victor XiaoWIEN2WANNIER: Elias Assmann Jan Kunes Philipp Wissgottfold2Bloch: Anton Bokhanchuk Derek Nievchas Elias Assmann Sheikh J. Ahmed

Berry phase King-Smithand David Vanderbilt, Phys. Rev. B 47, 1651 (1993) WIEN2WANNIER. Uncertainties . "BerryPI: A software for studying polarization of crystalline solids with WIEN2k density functional all-electron package", Comp. Phys. Commun. 184, 647 (2013).