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lectronic structure of surfaces: Simple models Bedřich Velicý III. NVF 54 Surface Physics Winter Term 3-4 Troja 8 th October 3 This class explores the general properties and important

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lectronic structure of surfaces: Simple models Bedřich Velicý III. NVF 54 Surface Physics Winter Term 3-4 Troja 8 th October 3 This class explores the general properties and important features of the electronic structure of solid surfaces using the simplest possible semi-realistic models, mostly -dimensional and tractable analytically lectronic structure within the orbital theories ITHR An explicit solution of the one-electron Schrödinger equation ( V( r)) ( r) ψ ( r) m e + ψ = eigen-states ( orbitals) eigen-values ( energy levels) OR ψ ( r) α α α α Other characteristics, lie the density of states (DOS), local DOS, projected DOS, spectral densities etc. α 3 lectronic structure within the orbital theories ITHR ( + ( r) ) ψ ( r) = ψ ( r) m e An explicit solution of the one-electron Schrödinger equation eigen-states ( orbitals) eigen-values ( energy levels) ψ ( r) OR α V α α α Other characteristics, lie the density of states (DOS), local DOS, projected DOS, spectral densities etc. α basic quantity specifies the system either self-consistent (ab initio) or modelled - present case 4 T:Why everybody lies the orbital theories. a universal simple method: instead of SR for N (e) =, 9,, 8 electrons ( I. ) ( II.) (III.) (IV.) 5 one-electron Schrödinger equation ( ( )) e m + V r ψα( r) = αψα( r) Many-electron state sequence of occup. numbers { nασ } add spin σ =, Pauli principle nασ Aufbau principle fill from the bottom up Charge balance defines Fermi energy n = f ( ) = + e ασ FD α β ( F ) ( e) ( e) ( e) n = N + N = N = Z ασ Observables local particle density n σ local spin density α... HOMO highest occupied... LUMO lowest unoccupied n( r) = n ( r) + n ( r) m( r) = n ( r) n ( r) ( r) = nˆ σ ( r) = nασ ψασ ( r) orbital interpretation double average J What we are going to do ( I. ) ( II.) 6 solve the one-electron Schrödinger ( ( )) ( ) ( m + V ψ e α = αψα r r r) equation Many-electron state sequence of occup. numbers { nασ } add spin σ =, Pauli principle nασ Aufbau principle fill from the bottom up ( e) ( e) ( e) (III.) Charge balance nασ = N + N = N = Z defines Fermi energy... HOMO highest occupied n = ϑ( F ) ασ α... LUMO lowest unoccupied (IV.) Observables local particle density n ( r) = n ( r) + n ( r) local spin densit y m( r) = n ( r) n ( r) n σ ( r) = nˆ σ ( r) = nασ ψασ ( r) orbital interpretation double average J One-electron potential at the surface atomic around each site periodic in the bul the zero fixed by the barrier with respect to infinity in the vacuum PSUDOPOTNTIAL VACUUM BARRIR One-electron potential at the surface atomic around each site periodic in the bul the zero fixed by the barrier with respect to infinity in the vacuum QUSTION: WHAT IS TH PSUDOPOTNTIAL VACUUM BARRIR Digression: PSUDOPOTNTIALS Na atom 3s core levels s s Atoms: valence states core states Valence states: extended shallow participate in bonding, transport sensitive to environment oscillate in order to be orthogonal to core states gain enough inetic energy Core states: compact deep typically inert to valence processes PSUDOPOTNTIAL IDA Get rid of strong potential in the core region core states unwieldy oscillations of valence states 9 Digression: PSUDOPOTNTIALS Na pseudoatom PSUDOPOTNTIAL IDA Get rid of strong potential in the core region core states unwieldy oscillations of valence states 3s r c Today, the (norm-conserving) pseudopotential is defined by a selected radius conditions. the valence energy preserved. the valence pseudo-state coincides with the true state outside the radius 3. is smooth and node-less inside 4. The norm of the pseudo-state is equal the norm of the true state One-electron potential at the surface atomic around each site periodic in the bul the zero fixed by the barrier with respect to infinity in the vacuum PSUDOPOTNTIAL VACUUM BARRIR MODLS I. AND II. One-electron potential at the surface atomic around each site periodic in the bul the zero fixed by the barrier with respect to infinity in the vacuum not everybody uses pseudopotentials, but for simple metals, it is a good idea mae a Fourier expansion of the bul pseoudopotential. th term -- MODL I. th and st term -- MODL II. PSUDOPOTNTIAL BARRIR VACUUM MODLS I. AND II. One-electron potential at the surface atomic around each site periodic in the bul the zero fixed by the barrier with respect to infinity in the vacuum not everybody uses pseudopotentials, but for simple metals, it is a good idea mae a Fourier expansion of the bul pseoudopotential. th term -- MODL I. th and st term -- MODL II. BARRIR VACUUM MODLS I. AND II. MODL I. D Sommerfeld with abrupt barrier F I. Fermi sea Φ relevant data... see next slide: Tae them over from realistic values for simple metals V II. F x + Φ = V RATIONAL Sommerfeld in the bul successful: transport, pseudopotential argument experiment: soft X ray spectr. all for simple metals reasonable physical justification abrupt barrier - simplest confinement, single parameter exact form of the barrier not critical just an expedient, maes calculations doable by hand MODL I. D Sommerfeld with abrupt barrier F I. Fermi sea Φ relevant data... see next slide: Tae them over from realistic values for simple metals volume in -space cell volume per -vector V II. F 4π 3 x + Φ = ( π ) Ω 3 F 3 V N ( e) RATIONAL Sommerfeld in the bul successful: transport, pseudopotential argument experiment: soft X ray spectr. all for simple metals reasonable physical justification abrupt barrier - simplest confinement, single parameter exact form of the barrier not critical just an expedient, maes calculations doable by hand = n Ω = n 3π 4π 3 3 F ( π ) 3 Ω 3 F F π m m = = (3 n) F = 3 bcc fcc 3 é 3 3 π N ů é e 3N 3π nů ę ú é = e ů 3 m m m F 3 F ( π ) = ę = ë úű a ę a ęëπ ë úű úű 6 7 MODL I. D Sommerfeld with abrupt barrier I. Φ II. x F Fermi sea V F + Φ = V Solution of the Schrödinger equation by matching A. SMI-INFINIT SAMPL ix ix I. ψ = α + β = + bound states e e V V κ x I. I ψ = γ e κ = V matching ψ x= = ψ x= + α + β = γ ψ = ψ α β = γ i κ x= x= + m + κ = m m MODL I. D Sommerfeld with abrupt barrier 3 I. Φ II. x F Fermi sea V F + Φ = V Solution of the Schrödinger equation by matching A. SMI-INFINIT SAMPL ix ix I. ψ = α + β = + bound states e e m V + κ = m V V κ x I. I ψ = γ e m κ = matching i α = α e tan = κ ψ x= = ψ x= + α + β = γ i ψ x ψ x α β γ i κ β = α e sin = κ = = = + = γ = α cos cos = π MODL I. D Sommerfeld with abrupt barrier 4 F I. Fermi sea V II. F x + Φ = V RSULT standing wave in region I. exponential leaing into region II. energy dependent phase shift needed for smooth matching Solution of the Schrödinger equation by matching A. SMI-INFINIT SAMPL ix ix I. ψ = α + β = + bound states e e V V V κ x I. I ψ = γ e κ = i α = α e tan = κ ψ ( x) = Acos( x+ ) x i β = α e sin = κ κ x = Acos e x γ = α cos cos = m + κ = m m MODL I. D Sommerfeld with abrupt barrier 5 Solution of the Schrödinger equation by matching B. FINIT SAMPL SLAB III. I. MATCH HR and HR II. ψ ( x) ψ ( x) = Acos( x+ ) x = Acos e x at the same κ x time = Bcos( x + L ) x L κx+ κl = Bcos e x L Discrete and (quantization) = arcsin ither B = A cos( L ) = cos = π L p + L or (shown above) B = A cos( L ) = cos( + π) = π ( ) L p + + L For L = Na about N values [, π / a]... grin of the Cheshire cat Normalization of ψ for L a + dx ψ ( x) = 4A dx cos ( x + ) + O() = A L + O() A = L L ( ) MODL I. D Sommerfeld with abrupt barrier 6 Band filling N ( e) = N( ) = ϑ( ) F F F general definition { p () ( π ) m = m = π L s+ L s= s L s p + π s( ) m L for L a suitable for smoothing N ( e) in particular π N π ( ) = m L, = m L F DOS density of states hustota stavů IDOS Integrated DOS dn N ( ) dηg( η) g ( ) = general definition DOS d ϑ( ) = δ( ) g ( ) = δ( ) basic form IDOS d d m h g( ) = L our model Intermezzo: comments on the density of states DOS density of states hustota stavů IDOS Integrated DOS IDOS d d dn N ( ) dηg( η) g ( ) = general definition DOS d ϑ( ) = δ( ) g ( ) = δ( ) basic form m h g( ) = L our model ψ For large samples surface plays a minor role, in dimension D, D D L : L = / L CHCK: Born-Karman periodic boundary conditions in the bul: ± ix ± il = e, e =, = πs/ L π s( ) = m L π π N ( ) = m L = m L as before IDOS and DOS for various D D ( ), ( ) D Γ = C g = C / D/ D D D D D 3 C D g 3 D γ πγ πγ γ = m h γ ( ) / h = π πγ 4π 3 γ 3 MODL I. D Sommerfeld with abrupt barrier 7 Local particle density without spin x F nx ( ) = ψ ( x) n = ψ ( ) ϑ( ) LDOS local density of states LDOS nx ( ) = ψ ( x) dηδ( η ) dη gx (, η) gx (, ) = ψ ( x) δ ( ) dx g( x, ) F = dx ψ ( x) δ ( ) = g ( ) NOT: LDOS in 3D defined in the same way F SUM RUL I. SUM RUL II. g( r, ) = ψ ( r) δ( ), ζ complete quantum numbers ζ ζ ζ MODL I. D Sommerfeld with abrupt barrier 8 Local particle density x F LDOS local density of states without nx ( ) = ψ ( x) n = ψ ( ) ϑ( ) LDOS nx ( ) = ψ ( x) dηδ( η ) dη gx (, η) gx (, ) = ψ ( x) δ ( ) dx g( x, ) LDOS f or our model F spin = dx ψ ( x) δ ( ) = g ( ) F ( ) ( x) δ ψ( ) ( x) gx (, ) = ψ ( ) = g ( ) gx (, ) = L x m h SUM RUL I. SUM RUL II. cos ( + ) L = m cos ( x + arcsin ) π see next slide LDOS FOR D SOMMRFLD MODL WITH INFINIT BARRIR LDOS computed as a function of energy for x = n a, n =,,...,8 for some arbitrary lattice constant from m g( x, ) = cos ( x arcsin ) π + = m with Note the increasingly rapid oscillations. The LDOS tends to the bul density B in a singular manner. In higher dimensions contributions from various transverse wave vectors suppress these oscillations. 9 MODL I. D Sommerfeld with abrupt barrier 9 Local particle density inside the sample x nx ( ) = dη gx (, η) F V+ d V m F F g( x, ) = cos ( x + arcsin ) π n( x) = d cos ( x arcsin ) π + = π ( + cos(x + arcsin )) For high barriers, F, approx. arcsin sin( F [ x+ ]) nx ( ) = nb Friedel oscillations F ( x+ ) MODL I. D Sommerfeld with abrupt barrier 9 Friedel oscillations For high barriers, sin( F [ x+ ]) nx ( ) = nb F ( x+ ) F, PROPRTIS. characteristic appearance of F x. sharp Fermi surface necessary 3. the charge leas exponentially out into the barrier region 4. net depletion of the surface... NOT neutral 5. not an D artifact, similar oscillations and depletion in 3D 6. depletion can be mended by an additional parameter of the barrier but the only systematic way to neutral surfaces is self-consistency MODL II. D NF model, abrupt barrier F I. Φ V x II. Bul periodic potential V( x) = V( x+ a) periods: lattice a, reciprocal lat. B = NF model (of the potential) single Fourier component V = V e iϕ ibx iϕ ibx V( x) = V + V {e e + e e } iϕ π a a d V( x) V ( x) = V + V cos( Bx + ϕ) = V + V cos( Bx [ d]) altogether 5 parameters: V, B( a), dv, ; F BULK BAND STRUCTUR IN NF ix ix i( + B) x * i( B) x = V + V e + V e SMALL COMPONNTS V( x)e e TRIAL BASIS start from plane waves MODL II. D NF model, abrupt barrier BULK BAND STRUCTUR IN NF start from plane waves ix ix i( + B) x * i( B) x = + + SMALL COMPONNTS V( x)e V e Ve V e TRIAL HAMILTONIAN TRIAL BASIS C ( + B) V = V C ( B) H * V C V * C = m NF condition V C B INTRSTING POINTS: CROSSINGS POINT BZ / C note I. II. III. IV. = π / 4 π / 4 B a B B= a B B minimum at BZ center critical point at BZ edge equivalent with II. next cr. point at BZ center st band nd band 3 rd band MODL II. D NF model, abrupt barrier 3 BULK BAND STRUCTUR IN NF start from plane waves ix ix i( + B) x * i( B) x = e + e + e SMALL COMPONNTS V( x)e V V V INTRSTING CAS TRIAL BASIS I. I (quasi-degenerate) = B+ q, q B H C ( + B) V NF condition C m V C B V C ( B) = * V C V = * H C( B q) V = + V ene ry g + * V C( B q) MODL II. D NF model, abrupt barrier 4 C( B q) V H energy = + V secular equation + * V C( B q) ( )( C( B+ q) C( B q) ) 4 = CB ± V + C B q V = MODL II. D NF model, abrupt barrier 5 = 4 CB ± V + C B q RUNNING WAV FUNCTIONS iq ( + Bx ) iq ( Bx ) ψ = αe + βe ix = e { α + βe } periodic ibx... Bloch function MODL II. D NF model, abrupt barrier 6 = 4 CB ± V + C B q RUNNING WAV FUNCTIONS iq ( + Bx ) iq ( Bx ) ψ = αe + βe ix = e { α + βe } periodic ibx... Bloch function SURFACS by matching In the bands: similar to Sommerfeld, but slightly more complicated In the gap: new feature -- possibility of a surface state MODL II. D NF model, abrupt barrier 6 IN TH BULK, BUT TOWARDS TH SURFAC STATS Complex band structure: find complex q for which the energies are real in the bands, in the gap, for real q for imaginary q =± iw, w w q on real axis on imaginary axis q on real axis = 4 CB ± V + C B q RUNNING WAV FUNCTIONS = 4 CB ± V C B w ATTNUATD WAV FUNCTIONS iq ( + Bx ) iq ( Bx ) ψ = αe + βe wx iq ( + Bx ) iq ( Bx ) ψ = αe + βe ix = e { α + βe } periodic i Bx ibx i Bx = e { αe + βe }... Bloch function MODL II. D NF model, abrupt barrier 6 TOWARDS TH SURFAC STATS Matching: ψ( ) = ψ( + ) ψ ( ) = ψ ( + ) Many factors intervene: the cutoff position decisive the barrier height also important even if the potential is real, it may be positive or negative in the bands two Bloch waves matched for any energy in the gap only one wave surface state exceptional BAND STATS XTND THRU TH WHOL CRYSTAL In general, the surface states, their occurrence and properties are very sensitive to details of the surface condition. SURFAC STATS AR LOCALIZD NAR TH SURFAC MODL II. A sample matching procedure 7 SYMMTRIC TRMINATION iq ( + Bx ) iq ( Bx ) ( Bx ξ ) attenuated wave functions ψ = αe + βe = e { αe + βe = e cos [ ] should be matched to e wx wx κ x d = or d = a i Bx i Bx } w = ---- w d = d = a V cos Bx V cos Bx For d = surface a... attractive surface well sta te exists For d =... repulsive surface hump surface state not possible 4 CB a d V MODL II. Surface state -- results 8 + V λ d / a V 4 V CB λ Position of the surface state in the gap as a function of the termination variable d for several heights of the barrier 4 CB a d V Complex band structure of silicon A realistic complex band structure shows: similarity with the D model - imaginary loops at high symmetry points effects of degeneracy additional complex branches at other extrema A detailed description is given below in Czech Complex band structure of silicon A realistic complex band structure shows: similarity with the D model - imaginary loops at high symmetry points effects of degeneracy additional complex branches at other extrema nglish translation Fig.. Complex band structure of silicon at the center of the surface Brillouin zone of the () surface [3]. Full lines represent the classical real band structure along the Γ X axis ( zr (, π / a), zi = ). Dotted lines indicate the complex band structure. The lines to the left give real energies at purely imaginary z = izi. Two such loops bridge over the forbidden band between the top of the valence band Γ 5 and the local extrema of the conduction band Γ5, Γ. From the conduction band minimum a complex line comes out; is seen directly, is given by numbers. zr z i Last time Plane wave vs. atomic orbital expansion Last time Plane wave vs. atomic orbital expansion quivalent description LCAO realistic picture NF LCAO provides a simple lin to chemical bond & atomic properties Shocley theory of surface states in semi-conductors (939) 3 band structure of a chain acc. to Shocley p s lattice spacing a critical value γ hopping integral between the neighbors hopping integral dimension-less parameter γ = = U sp splitting c a c separates the regions lattice spacing HG & surface states NHG no surface states a a a a a c c c γ γ γ γ γ c c c U 4 band structure of a chain acc. to Shocley hybridization gap p surface states s non-hybridization gap lattice spacing a critical value γ hopping integral between the neighbors hopping integral dimension-less parameter γ = = U sp splitting c a c separates the regions lattice spacing HG & surface states NHG no surface states a a a a a c c c γ γ γ γ γ c c c U 5 p s explanation on a biatomic chain A A - + A U U U U - A odd solution + p s Hˆ = U U s p U U p s U U s U U U ψ = as s + ap p + as s + ap p mirror symmetry ψ = a s + b p + a s + b p ψ = a s + b p + a s b p u g ± = s + ± 4 + ± = s + + ± 4 + p even solution U U U U explanation on a biatomic chain p s - + U U U U - + p s Hˆ = mirror symmetry U U s p U U p s U U s U U U p even solution odd solution g u ± = s + + ± 4 + ± = s + ± 4 + U U U U NHG HG p band s band p s U U U + U U + U p s U s + p s + p ( ) ( ) 8U + U conduction band valence band 7 explanation on a biatomic chain p s - + U U U U - + p s Hˆ = mirror symmetry U U s p U U p s U U s U U U p even solution odd solution g u ± = s + + ± 4 + ± = s + ± 4 + U U U U NHG HG p band s band p s U U U + U U + U p s U s + p s + p ( ) ( ) 8U + U conduction band valence band 8 The end

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