Periodic systems: Concepts

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1 Periodic systems: Cocepts Sergey V. Levcheo FHI Theory departmet

2 Remider: electroic-structure problem h = h = uow fuctios (oe-electro wave fuctios) ( ) = ( ) - basis set expasio ( ) - ow fuctios (basis fuctios) Geeralized eigevalue problem: h[ ] =

Exteded (periodic) systems There are 10 20 electros per 1 mm 3

(Bravais lattice): 3 3 2 2 1 1 3 2 1 ) (0,0,0 ),, ( a a a r r

3 Exteded (periodic) systems There are electros per 1 mm 3 of bul Cu x y z a 1 a 2 a 3 Positio of every atom i the crystal (Bravais lattice): ) (0,0,0 ),, ( a a a r r lattice vector: 2, 1, 0,,, ),, ( a a a R

4 Example: two-dimesioal Bravais lattice a 1 a 2 primitive uit cells 3a a The form of the primitive uit cell is ot uique

5 From molecules to solids Electroic bads as limit of bodig ad ati-bodig combiatios of atomic orbitals: electroic bad bad gap Adapted from: Roald Hoffma, Agew. Chem. It. Ed. Egl. 26, 846 (1987)

6 Bloch s theorem U ( r R) U ( r) Periodic potetial (traslatioal symmetry) R 1a 1 2a2 3a3 I a ifiite periodic solid, the solutios of the oe-particle Schrödiger equatios must behave lie ( r R) exp( ir) ( r) 0 r R r R Idex is a vector i reciprocal space g 1 2 x1g 1 x2g2 x3g3 i j ij a m a g l 2 reciprocal lattice vectors Cosequetly: g a ( r) exp( ir) u( r), u( r R) u( r) g 2

7 The meaig of a a a a exp( ix j ) 1s ( j a) j chai of hydroge atoms shows the phase with which the orbitals are combied: = 0: exp( 0) 1s ( j a) 1s ( a) 1s (2 ) 0 a j π = : 0 exp( i j) 1s ( j a) 1s ( a) 1s (2a) 1s (3a) a j is a symmetry label ad a ode couter, ad also represets electro mometum Adapted from: Roald Hoffma, Agew. Chem. It. Ed. Egl. 26, 846 (1987)

8 Bloch s theorem: cosequeces I a periodic system, the solutios of the Schrödiger equatios are characterized by a iteger umber (called bad idex) ad a vector : hˆ ( r) exp( ir) u ( r), u ( r R) u ( r) g 1 For ay reciprocal lattice vector G 1g1 2g2 3g3 G g 2 g 1 g 2 exp( ir )[ u exp( igr )] exp( ir ) u~ G G a Bloch state at +G with idex a lattice-periodic fuctio u ~ a Bloch state at with a differet idex Ca choose to cosider oly withi sigle primitive uit cell i reciprocal space

9 Brilloui zoes A covetioal choice for the reciprocal lattice uit cell For a square lattice I three dimesios: Face-cetered cubic (fcc) lattice Body-cetered cubic (bcc) lattice Wiger-Seitz cell For a hexagoal lattice Wiger-Seitz cell

10 Time-reversal symmetry For Hermitia, ca be chose to be real ĥ ) ( ) ( ˆ ) ( ) ( ˆ * * r r r r h h From Bloch s theorem: ) ( ) exp( ) ( ) ( ) exp( ) ( * * r R R r r R R r i i ) ( * ( ) Electroic states at ad are at least doubly degeerate (i the absece of magetic field)

11 Commoly used basis sets: Basis sets i ( r) C ip p ( r) p plae waves exp( i r) (delocalized, aalytic itegrals) gaussias x i y j z exp( r 2 ) (localized, aalytic itegrals) Slater-type x i y j z exp( r) (localized, uclear cusp) Numeric atomic orbitals (, ) (localized, flexible) grid-based ( r ri ) (localized) Core electros are ofte treated separately (pseudopotetials, plae-wave + localized basis)

12 Localized basis sets ad periodic systems = ( + ) New basis fuctios satisfyig Bloch s theorem: + = ( ) =, ( ) + = ( )

13 Localized basis sets ad periodic systems h + = + Multiply by ad itegrate over all space: h + = + h I practice, all itegratio poits ad pieces of bac to the origial uit cell: are mapped = 0

14 Electroic bad structure () -π/a 0 π/a Bad structure represets depedece of o For a periodic (ifiite) crystal, there is a ifiite umber of states for each bad idex, differig by the value of

15 Electroic bad structure i three dimesios Brilloui zoe of the fcc lattice z Al bad structure (DFT-PBE) x Γ Λ L U Σ Δ X K W y ε (), ev By covetio, are measured (agular-resolved photoemissio spectroscopy, ARPES) ad calculated alog lies i -space coectig poits of high symmetry

16 Fiite -poit mesh Charge desities ad other quatities are represeted by Brilloui zoe itegrals: ( r) occ j ( r) BZ j 2 d 3 BZ 8x4 Mohorst-Pac grid, smooth fuctios of ca use a fiite mesh, ad the iterpolate ad/or use perturbatio theory to calculate itegrals ( r) occ pt j N m1 w m j m ( r) 2 1 y H.J. Mohorst ad J.D. Pac, Phys. Rev.B 13, 5188 (1976); Phys. Rev. B 16, 1748 (1977) b 2 x b

17 Bad gap ad bad width (dispersio) 0.8 Å hydroge molecule chai a a a (DFT-PBE) a = 10.0 Å a = 5.0 Å ε() ε() 0 0 a a = 2.0 Å a a a = 1.6 Å a ε() ε() Overlap betwee iteractig orbitals determies bad gap ad bad width

18 Bad structure test example Orbital eergies are smooth fuctios of Example: chai of Pt-L 4 complexes (K 2 [Pt(CN) 4 ]) a x 2 -y 2 z y x z z = π/a ε() xz,yz y x z xy xz yz xy z 2 z 2 0 π/a Adapted from: Roald Hoffma, Agew. Chem. It. Ed. Egl. 26, 846 (1987)

19 Isulators, semicoductors, ad metals E g >> B T E g ~ B T ε F Isulators (MgO, NaCl, ZO, ) Semicoductors (Si, Ge, ) E g =0 Metals (Cu, Al, Fe, ) I a metal, some (at least oe) eergy bads are oly partially occupied The Fermi eergy ε F separates the highest occupied states from lowest uoccupied

m e 2 F K Cu Al Periodic table of Fermi surfaces: http://www.phys.ufl.

20 Fermi surface Plottig the relatio ( ) F i reciprocal space for differet yields differet parts of the Fermi surface For free electros, Fermi surface is a sphere 2 2 m e 2 F K Cu Al Periodic table of Fermi surfaces: The grid used i -space must be sufficietly fie to accurately sample the Fermi surface

21 Desity Of States (DOS) N j j j w d g pt BZ 1 3 )) ( ( )) ( ( ) ( Number of states i eergy iterval dε per uit volume, d d 1 1

22 Atom-Projected Desity Of States (APDOS) Decompositio of DOS ito cotributios from differet atomic fuctios i : gi ( ) i ( r) ( r) d r ( ( )) d BZ O(3d) Mg(3d) Mg(3p) Mg(3s) O(2p) O(2s) Recovery of the chemical iterpretatio i terms of orbitals Qualitative aalysis tool; ambiguities must be resolved by trucatig the r-itegral or by Löwdi orthogoalizatio of i

23 Potetial of a array of poit charges + Total potetial at r=(0.05,0.05,0.05) Å, ev V ( r) R i1 Number of poit charges N N q i, q r r R Covergece of the potetial with umber of charges is extremely slow i i1 i 0

24 V ( r) 1 i, R V ( r) q i R Ewald summatio N i1 q i r r R screeig gaussia charge distributio erfc r ri R r ri R i + 2 V ( r) 4 ( r) (Poisso's equatio) / V exp ( r r ) 2 i / ( r) q i exp ig ( r r ) 2 i i, G0 G 4 G Decays fast with R Decays fast with G Diverges at G 0 is cacelled for, but divergece There is o uiversal potetial eergy referece (lie vacuum level) for periodic systems importat whe comparig differet systems i q i 0

25 Modelig surfaces, iterfaces, ad poit defects the supercell approach

26 The supercell approach Ca we beefit from periodic modelig of o-periodic systems? Yes, for iterfaces (surfaces) ad wires (also with adsorbates), ad defects (especially for cocetratio or coverage depedeces) Supercell approach to surfaces (slab model) Approach accouts for the lateral periodicity supercell Sufficietly broad vacuum regio to decouple the slabs Sufficiet slab thicess to mimic semi-ifiite crystal Semicoductors: saturate daglig bods o the bac surface No-equivalet surfaces: use dipole correctio Alterative: cluster models (for defects ad adsorbates)

27 Surface bad structure Example: fcc crystal, (111) surface z z z M Γ M Γ K surface Brilloui zoe

28 Surface bad structure of Cu(111) 2 ε F 1-1 Shocley surface state Tamm surface state Eergy, ev M Γ Cu(111) M

29 For ear-free electros: Shocley surface states ( r) ~ exp( i r) matchig coditio ~ exp[ z] ~ exp[ i( i ) z] 0 potetial Decayig states ca be treated as Bloch states with complex (W. Koh, Phys. Rev., 115, 809 (1959)) Complex bad structures ca give useful iformatio about coductace through iterfaces ad molecular juctios z

30 Tamm surface states I the tight-bidig (localized orbital) picture, surface states may appear due to daglig orbitals split off from the bad edge

31 Surface recostructio ad bad structure Dimerizatio at (001)-surface of group IV-elemets [001] side view top view [110] [110] bul-termiated atomic structure side view

32 Surface recostructio ad bad structure Buclig of dimers at Si (100) surface π-bod see, e.g., J. Dabrowsi ad M. Scheffler, Appl. Surf. Sci , 15 (1992) re-hybridizatio ad charge trasfer (from dow to up)

cotour plot of electro desity differece with respect to free Si atoms (dashed =

33 Surface recostructio ad bad structure symmetric dimer model (SDM) asymmetric dimer model (ADM) Experimetal results from agular-resolved photoemissio spectroscopy cotour plot of electro desity differece with respect to free Si atoms (dashed = decrease) total desity cotour plot P. Krüger & J. Pollma, Phys. Rev. Lett. 74, 1155 (1995)

34 Cocludig remars 1) Periodic models ca be efficietly used to study cocetratio/coverage depedece, icludig ifiitely dilute limit (low-dimesioal systems, defects, etc.) 2) A lot of useful ad experimetally testable iformatio o material s properties ca be obtaied from the aalysis of its electroic structure (bad structure, DOS, APDOS, etc.) 3) A lot of developmet (i both computatioal methods ad code efficiecy) is still ecessary to go beyod stadard DFT for periodic systems, ad to approach accuracy that ca be achieved owadays for molecules

35 Recommeded literature Neil W. Ashcroft ad N. David Mermi, Solid state physics Axel Groß, Theoretical surface sciece: A microscopic perspective Roald Hoffma (1981 Nobel Prize i Chemistry (shared with Keichi Fuui)): 1) How Chemistry ad Physics Meet i the Solid State, Agew. Chem. It. Ed. Egl. 26, (1987) 2) A chemical ad theoretical way to loo at bodig o surfaces, Reviews of moder physics, 60, (1988)

Surface modelig: importat issues 1) Fiite slab thicess (surface-surface iteractio) 2) Fiite vacuum layer thicess (image-image iteractios) 3) Log-rage iteractios (charge, dipole momet) φ - + -

36 Surface modelig: importat issues 1) Fiite slab thicess (surface-surface iteractio) 2) Fiite vacuum layer thicess (image-image iteractios) 3) Log-rage iteractios (charge, dipole momet) φ periodic boudary coditios artificial electric field φ ) Surface polarity Nb (+5) Li (+1) Δφ O (-2) +q -q +q -q+q -q +q -q +q -q +q -q

37 From molecules to solids Electroic bads as limit of bodig ad ati-bodig combiatios of atomic orbitals: bads bad gap R. Hoffma, Solids ad Surfaces - A chemist s view of bodig i exteded structures, VCH Publishers, 1998

38 Shocley surface states For early-free electros: ( ) ~ exp[ z] ~ exp[ i( i) z] gap 2 V G i 0 z 0 a

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka) 7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.