An introduction - teknik.uu.se dependence of electrical resistivity • Mobility edge 1.3 eV into...

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Electronic structure An introduction

Transcript of An introduction - teknik.uu.se 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)