An introduction · temperature dependence of electrical resistivity • Mobility edge 1.3 eV into...
Transcript of An introduction · temperature dependence of electrical resistivity • Mobility edge 1.3 eV into...
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
• where • The free electron model: U(r)=0, and κ=κ0 • Green function becomes: • Fourier transformation gives:
)'()',,()( 22 rrrr −=+∇ δκ EG
),(0 kEG
´),(0 rr −EG
𝜅𝜅2 = 2𝑚𝑚 𝐸𝐸 − 𝑈𝑈 𝐫𝐫 /ℏ2
Uniform medium: Free electron model
• Green function formalism in free electron model • In order to avoid a divergence the small imaginary part is
added • The imaginary part of G0 is and it enumerates the states of the system • Density of states,
• which gives the usual free electron result.
122000 ))((lim),( −
→+−= ηκ
ηikEEG k
))(( 220 kEi −− κδπ
𝐷𝐷0 𝐸𝐸 = 2𝑚𝑚/ℏ2 �𝑑𝑑𝐤𝐤 8𝜋𝜋3 −1𝛿𝛿 𝜅𝜅02 𝐸𝐸 − 𝑘𝑘2
Density of states 1
• The integral can be interpreted as a Fourier transform of , when
• The density of states can then be written as
• For a crystalline material U(r) is not zero but
varies periodically • In a disordered material U(r) is varying in a more
or less random manner
),(Im 0 kEG 'rr =
𝐷𝐷0 𝐸𝐸 = −2𝑚𝑚𝜋𝜋ℏ2
Im 𝐺𝐺0 𝐸𝐸, 𝐫𝐫 = 𝐫𝐫′
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)
• Total DOS: ∫−= rr dEDVED ),()( 1
𝐷𝐷 𝐸𝐸, 𝐫𝐫 = −2𝑚𝑚𝜋𝜋ℏ2
Im 𝐺𝐺 𝐸𝐸, 𝐫𝐫 = 𝐫𝐫′
𝜌𝜌 𝐸𝐸,𝐤𝐤 = −2𝑚𝑚𝜋𝜋ℏ2
Im 𝐺𝐺 𝐸𝐸,𝐤𝐤
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
model (periodic potential) • E~k2 at low k
• Tight binding model (periodic potential), LCAO
• E=E0-B cos(ka)
First Brillouin zone
Density of states: Crystals • Free electron model: • D(E)~E-1/2 in the one-
dimensional case • D(E)~E1/2 in the three-
dimensional case • Nearly free electron
model: Peak of D(E) at Brillouin zone boundaries
• Tight binding model: • One dimension:
• Peaks at band edges
E
D(E)
20
2 )(11)(
EEBaED
−−=
π
E-E0 (a.u.)
3D
1D
Models of disorder • Anderson model • Periodic potential with
randomness of the atomic energy levels
• P(U)=1/W, -W/2<U<W/2 and 0 otherwise
• Lifshitz model • Equal atomic energy
levels, but different spacings between them (overlap)
• More suitable for disordered monatomic materials
• Effect on the DOS qualitatively similar in both cases
Crystalline vs. amorphous • Disordered 1-dim
Anderson model • Solved by CPA
• Crystalline 1-dim periodic potential
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 • Constructed by molecular
dynamics • One-band tight binding
model • Topological (site) disorder
and different degrees of bond disorder
• Broadening of features main effect of disorder
fcc
site
Increasing bond 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
Source: Myers, Introduction to solid state physics
Evac
VB
Source: Nan et al, Phys. Rev. B (1988)
UV photoelectron spectroscopy
• Photons with E<~20 eV
• Initial state: valence band
• Transport through conduction band and escape of electrons
• Measure approx.
• May be possible to deconvolute to obtain the DOS in both valence and conduction band
• Only qualitative features can be seen
• Peaks in the DOS
𝑃𝑃 𝐸𝐸,ℏ𝜔𝜔 = 𝐷𝐷𝑣𝑣 𝐸𝐸 − ℏ𝜔𝜔 𝐷𝐷𝑐𝑐 𝐸𝐸
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
photon energies • DOS comparison
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 - 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
Amorphous LixWOy.
Titanium oxide coatings
• X-ray amorphous coatings.
• Li intercalation • Proportional to
computed electron density of states (line) per spin state for anatase.
0
0.5
1
1.5
2
2.5
3
0
2
4
6
8
10
0 0.5 1 1.5 2 2.5 3
Energy (eV)
-dx/
dU (e
- /eV
/f.u.
)
Total DO
S (states/eV
/f.u.)
niklasson_fig1
Titanium oxide
exp
Theory
Strømme, Ahuja and Niklasson, Phys. Rev. Lett. 2004
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 • Fraction of atoms participating in a state 0
(participation ratio)
)2exp(~2 rαψ −
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
Source: Klingshirn, Semiconductor Optics
• (a) Bloch function – crystalline material
• (b) Extended state – disordered material
• (c) Localized state
)(exp()( r)uir kk rk •=ψ
)()( ∑ −=i
iicr Rrϕψ
)/exp()()( 1−−−−= ∑ αϕψ 0rrRri
iibr
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 – states at band edges localized
• Mobility edges separate localized and extended states
• W/B>1 – all states localized
W
Source: Klingshirn, Semiconductor Optics W/B<1 W/B>1
Localization transition • Usual situation in practise: Localized states near band
edges and extended states in the mid of the band • At a critical value of the localization criteron all states
become localized, (W/zV)crit~2-2.5 • As w=W/zV approaces the critical value from above the
localization length (α−1) diverges
• Similarly as E approaches the mobility edge Ec from the localized side
• Exponents ~1-1.5 • If the Fermi level can be made to cross a mobility edge
(by doping, applying pressure…), there is a non-metal – metal transition
'1 )(~ να −− − critww
να −− − )(~1 EEc
Why are states in the middle of a band more likely to be extended?
• Illustration: Multicomponent percolation
• An electron can move between sites if ∆E <~B/z
• Divide DOS into slices of width ~B/z
• If the sites in a range are connected through the sample the states are extended
• D(E) large – percolation more probable – extended states
• D(E) small – finite clusters of states – localized states
Source: Zallen
Ex: Li-WOy thin films • Most experiments on the
localization transition have been done on heavily doped semiconductors
• Only a few experiments on localized band states of metal oxides
• WO3 intercalated by Li-ions • Intercalation spectroscopy
together with analysis of temperature dependence of electrical resistivity
• Mobility edge 1.3 eV into the conduction band Ec = 1.27+/-0.05 eV ν=1.25+/0.3
Defects in a CRN: Gap states
• Defects in a Continous Random Network leads to localized states in the band gap
• Dopants, impurities • Dangling bond (D0) – atom absent from a bond – one
electron • Another electron can pair up with it (D-); this costs an
interaction energy U • Empty dangling bond (no electron) – D+
• Disorder: Broadened levels, often close to mid-gap • D0 D+ + D- may be energetically favorable
Amorphous semiconductors
• Band tails from doping and spatial potential fluctuations
• Dangling bond states close to mid-gap
• Other defect types: • Vacancies V2+,V+,V0 • Interstitials
Source: Klingshirn, Semiconductor Optics
Example: a-Si:H
• P-doped (n-type) • Compensated
Source: Elliott, Physics of amorphous materials
Example: a-Al2O3
• Thin film produced by sputtering
• Amorphous, low density • Near edge X-ray absorption
fine structure (NEXAFS) • Conduction band of Al2O3 • Shifts to lower energies in
amorphous state • Localized states are visible in
the band gap • Interpreted as due to O-O
bonded defects – peroxide ions
Source: Cecilia Århammar et al (2011)
Gap state spectroscopies
• Optical absorption – ex. photothermal spectroscopy: band tails, polarons, defect states
• Photoluminescence • Transient photoconductivity – electrons excited into
conduction band by laser pulse and are then trapped by empty gap states – measure current due to an applied field
• Electrical techniques – • Space charge limited currents, thermally stimulated
current/capacitance, C-V field effects • Intercalation spectroscopy
Electrical methods
• Different methods give very different results
• At best qualitative observation of peak positions
• Ex: a-Si:H
Source: Elliott, Physics of amorphous materials
Transient photoconductivity • Laser pulse: Injection of
charge carriers at one end of the sample
• Measure photocurrent I(t) • tT – transit time of the
current pulse through the sample
• Drift mobility µd=L/EtT • Shape of I(t) depends on
trapping of electrons in empty gap states
• Schematic picture
t
I
tT
log t
log I Slope: -1+α
Slope: -1-α
Multiple trapping 1 • Very broadened I(t) curves
are interpreted in terms of multiple trapping in a broad distribution of localized states
• t=0: electrons start to be trapped
• Any t: electron distribution is peaked at energy Ep, where the release time from the traps is equal to t
• As time increases states lower and lower in the band gap are probed
-
Trap states
EF
photon
Multiple trapping 2 • Consider the case of an exponential density of states • Release rate of electrons from traps (∆E=Ec-E)
• Distribution of transition rates with α=T/T0 • Energy – time relationship
• Finally one obtains I(t)~t-1+α
)/)(exp()( 00 kTEENED c −−=
)/exp()( kTEEr ∆−=∆ ν1~)( −αrrP
)ln()( tTktEp ν=∆
Example: a-As2Se3 • Exponent s = -1+T/T0 • Power law with temperature
dependent exponent
Source: Monroe
Exponential band tail!
• Power law over >8 decades in time – probing energies from 0.3 to 0.86 eV from the conduction band edge.
• The band gap is about 2 eV
Source: Monroe and Kastner, PRL (1986)