Electronic structure An introduction Outline • Density of states – formalism • One-dimensional models • Computations • Experimental techniques • Some examples • Localization • States in the band gap • Determination of localized density of states Amorphous materials • No periodicity, like in a crystal • No reciprocal space, no Brillouin zone • Electron wavefunctions are not of the Bloch form • No bandstructure, E(k); it is replaced by a spectral function ρ(E,k) • The density of states, D(E) is an important quantity • Bands and band gaps are present • Localization of states occurs, specially near band edges • Gap states due to defects, impurities etc. Schrödinger equation • Schrödinger equation • Green function – solution of the Schrödinger equation with δ excitation (∇2 +κ 2 )G(E,r,r') = δ (r − r') • where = / 2 2 • The free electron 2 model − : U( r)=0,ℏ and κ=κ0 • Green function becomes: G 0 (E,r − r´) • Fourier transformation gives: G0 (E,k) Uniform medium: Free electron model • Green function formalism in free electron model 2 2 −1 G0 (E,k) = lim(κ0 (E) − k + iη) η→0 • In order to avoid a divergence the small imaginary part is added 2 2 • The imaginary part of G0 is − iπ δ (κ0 (E) − k ) and it enumerates the states of the system • Density of states, = / 8 2 3 −1 2 2 • which gives0 the 2usualℏ free� electron result 0. − Density of states 1 • The integral can be interpreted as a Fourier transform of Im G 0 ( E , k ) , when r = r' • The density of states can then be written as = Im , = 2 ′ • For a crystalline0 − material2 U(0 ) is not zero but ℏ r varies periodically • In a disordered material U(r) is varying in a more or less random manner Density of states 2 • The relation between the density of states and G(E,r=r’) holds also for disordered materials • But now the DOS depends on position r! • We define a local density of states D(E,r) and a spectral function ρ(E,k) , = Im , = 2 ′ − 2 , = ℏ Im , 2 • Total DOS: − 2 D(E) =ℏV −1 D(E,r) dr ∫ Density of states 3 • The spectral function is a continuous function of k for a given E – no E(k) dispersion relation • G cannot be obtained exactly for an amorphous material • Numerical methods necessary • Configuration averaging necessary • Coherent Potential and Effective Medium approximations (mostly used for alloys – tight binding alloy model) • Ab initio claculations for small clusters of atoms – density functional theory with various approximations…… One-dimensional case • Nearly free electron • Tight binding model model (periodic potential) (periodic potential), LCAO 2 • E~k at low k • E=E0-B cos(ka) First Brillouin zone Density of states: Crystals • Free electron model: • Tight binding model: • D(E)~E-1/2 in the one- • One dimension: dimensional case 1 1 1/2 D(E) = • D(E)~E in the three- πa 2 2 B − (E − E0 ) dimensional case • Nearly free electron • Peaks at band edges model: Peak of D(E) at Brillouin zone boundaries D(E) 3D 1D E E-E0 (a.u.) Models of disorder • Anderson model • Lifshitz model • Periodic potential with • Equal atomic energy randomness of the atomic levels, but different energy levels spacings between them • P(U)=1/W, -W/2<U<W/2 (overlap) and 0 otherwise • More suitable for disordered monatomic materials • Effect on the DOS qualitatively similar in both cases Crystalline vs. amorphous • Crystalline 1-dim • Disordered 1-dim periodic potential Anderson model • Solved by CPA E-E0 (a.u.) E-E0 (a.u.) Source: Ping Sheng, Introduction to wave scattering, localization and mesoscopic phenomena Three dimensional case • Simple cubic lattice • Simple cubic lattice with disorder E-E0 (a.u.) E-E0 (a.u.) Source: Ping Sheng, Introduction to wave scattering, localization and mesoscopic phenomena Example: Amorphous metal • Model of amorphous metal: 500 atom cluster fcc • Constructed by molecular dynamics • One-band tight binding site model • Topological (site) disorder bond disorder Increasing and different degrees of bond disorder • Broadening of features main effect of disorder Source: Rehr et al, Phys. Rev. B 16(1977) Example: a-Si • 4096 atom model of amorphous Si • Tight binding model • One s and three p basis functions per site • Valence and conduction band • Band tails into the band gap Source: Drabold et al, J. Mol. Graph. Model. (1999) Example: a-SiO2 • Tight binding model • Molecular dynamics with simulated quenching from melt • Crystalline DOS –a • Amorphous models – b,c • Faster quenching – dotted line in c Source: Koslowski et al. Phys. Rev. B 56 (1997) Spectral function • Calculation for a-Si • Broadened ”bandstructure” Source: Elliott, Physics of amorphous materials Experimental techniques • X-ray photoelectron spectroscopy (XPS) (~keV) - filled states, valence band • Ultraviolet photoelectron spectroscopy (UPS), (5-50 eV), convolution of valence and conduction band • Inverse photoemission – conduction band • X-ray absorption techniques – unfilled states • X-ray emission techniques – filled states • Optical absorption – UV and visible-NIR wavelength ranges (0.5 – 6 eV) X-ray photoelectron spectroscopy • Experiments usually in UHV • Very surface sensitive • Escape depth of the photoelectrons ~20-50 Å for XPS • Excitation of an electron by an X-ray photon from an initial state to continuum above the vacuum level • Transport of the electron through the solid - scattering • Escape through the sample surface • The distribution function of photoelectrons excited by an incoming photon, P(E,hν) is modified by a transmission and escape function T(E) • Studies of core levels as well as the valence band DOS Example • Schematic XPS process • XPS and valence band UPS (40 eV), ZrNi alloy VB Evac Source: Myers, Introduction to solid state physics Source: Nan et al, Phys. Rev. B (1988) UV photoelectron spectroscopy • Photons with E<~20 • May be possible to eV deconvolute to obtain • Initial state: valence the DOS in both band valence and • Transport through conduction band conduction band and • Only qualitative escape of electrons features can be seen • Measure approx. • Peaks in the DOS , = ℏ − ℏ Schematics of UPS • Photoelec- trons from surface region ~5Å • Complex analysis of data Source: Elliott, Physics of amorphous materials Ex: Amorphous Ge • UPS spectra, different • DOS comparison photon energies between a-Ge and C-Ge Source: Elliott, Physics of amorphous materials Optical absorption • Described by the imaginary part of the complex dielectric function ε2(ω). • Proportional to joint density of states • E ~ 0.5-6 eV • Vacuum UV above 6 eV • Example: Ge Source: Elliott, physics of amorphous materials Intercalation spectroscopy • Gives a qualitative picture of DOS of certain materials, • Intercalation of small ions (protons, Li+,…) together with electrons that are inserted into empty electron states • Limited to states within 1-2 eV from the band edge (or Fermi level) • Conduction band states for materials where ions can be inserted • Valence band states for materials where ions can be extracted The intercalation process • Ion/electron extraction – Fermi level moves into valence band • Ion/electron insertion – Fermi level moves into conduction band • How far into the conduction (valence) band is the mobility edge situated? • Slow kinetics prevent intercalation into all electronic states • Only a fraction of them are available Tungsten oxide coatings • Electrochemical methods - Amorphous LixWOy. measure the potential, U, as a function of Li/metal ratio, x. • Chronopotentiometry - a very low constant current • GITT – current pulses • ”Electrochemical density of states” • Compared to DOS of crystalline monoclinic WO3 Niklasson, Physica Scripta 2015 Titanium oxide coatings • X-ray amorphous 3 10 Titanium oxide 2.5 (states/eV/f.u.) DOS Total coatings. 8 • Li intercalation 2 6 /eV/f.u.) - • Proportional to 1.5 Theory 4 computed electron -dx/dU (e 1 2 density of states (line) 0.5 exp per spin state for 0 0 anatase. 0 0.5 1 1.5 2 2.5 3 Energy (eV) Strømme, Ahuja and Niklasson, Phys. Rev. Lett. 2004 niklasson_fig1 Localization • Electronic states can be localized at certain sites or in certain regions of a disordered material • More band states localized the higher the disorder – most probable close to band edges • Semiconductors and insulators: Fluctuations in short- range order (bond lengths, bond angles) lead to band tails extending into the energy gap • Band tails from valence and conduction bands may overlap • Dangling bonds, vacancies, interstitials – localized states in the band gap Criteria for localization • Ioffe-Regel criterion: Electron mean free path L<1/kmax~a (distance between atoms) • Absence of diffusion at T=0. Localized if there is a finite probability of finding the particle at the origin as time goes to infinity • Wave function ψ(r)~exp(-αr) • Probability ψ 2 ~ exp(−2αr) • Fraction of atoms participating in a state 0 (participation ratio) Mott transition • Bandwidth B (due to overlap energy) • Electron-electron interaction energy (on a site) U • B>U – extended states • Localization of states occurs when B<U Source: Zallen Anderson transition • Distribution of energy levels with width W • Bandwidth due to overlap B • B>W – extended states • B<W – localized states Source: Zallen Wave functions • (a) Bloch function – crystalline material ψ k (r) = exp(ik •r)uk (r) • (b) Extended state – disordered material ψ (r) = ∑ciϕ(r − Ri ) i • (c) Localized state −1 ψ (r) = ∑biϕ(r − Ri )exp(− r − r0 /α ) i Source: Klingshirn, Semiconductor Optics Anderson model • P(U)=1/W, -W/2<U<W/2 and 0 otherwise • Bandwidth B=2zV, where V is the overlap energy • W/B<1