Polarons An introduction to Basic concepts and Recent findings Ulrike Diebold Michael Schmid Martin Setvin Igor Sokolović Cesare Franchini Georg Kresse Thomas Hahn Michele Reticcioli
PolaronsAn introduction toBasic concepts
andRecent findings
Ulrike DieboldMichael SchmidMartin SetvinIgor Sokolović
Cesare FranchiniGeorg KresseThomas HahnMichele Reticcioli
Quasi-particle:Electronic carrier + altered phonons
[Emin (2013)]
Polaron
Delocalized electron Polaron
Charge carriers –-> polaron formation
[Setvin, PRL (2014)]
Path-integral formalism [Feynman (1955)]
Self trapping [Landau (1933)]
Continuum approximation [Pekar (1946)]:
- effective mass
- coupling constant
The weak coupling limit (α<1)[Frölich (1954)]
A polaron story
The strong coupling limit[Holstein (1959)]
The weak coupling limit (α<1)[Frölich (1954)]:
- Second quantization, perturbation theory
- Cloud thickness:
- Effective mass:
A polaron story
The strong coupling limit[Holstein (1959)]:
- Electron hopping:
[Devreese (2010)]
Solvable in the adiabatic and non-adiabatic limits:renormalized hopping integral t
The weak coupling limit (α<1)[Frölich (1954)]:
Polaron features
The strong coupling limit[Holstein (1959)]:
Large polaron(or continuum polaron)
Small polaron(or lattice polaron)
Long-range electron-phonon interaction
Radius >> lattice parameter
Shallow state (~10 meV)
Coherent motion(scattered occasionally by phonons)
Mobility >> 1 cm2/V/secHeavy massWeakly scattered by phonons
Falling mobility with increasing Temperature
Short-range electron-phonon interaction
Radius ~ lattice parameter
Mid-gap state (~1 ev)
Incoherent motion(assisted by phonons)
Mobility << 1 cm2/V/sec
Arising mobility with increasing Temperature
Observation techniques
Charge carriers introduced by:- irradiation- lattice defects (dopants, vacancies, ...)
Polarons probed by- Scanning Tunneling Microscopy and Spectroscopy (STM and STS)- Angle-resolved photoemission spectroscopy (ARPES)- Electron Paramagnetic Resonance (EPR)- Infra-red (IR) spectroscopy- Time-resolved Kerr spectroscopy- Current measurements
Computationally:- DFT (DFT+U, HSE, Molecular Dynamics)- Diagrammatic Quantum Monte Carlo
Where?Oxides, Perovskites, Organic materials, DNA, ...
Polarons in bulk TiO2
[Setvin, PRL (2014)]
Devreese, Bull. Soc. Belge Phys. (1963)Nagels, Denayer, Devreese, Sol. St. Commun. (1963)
First observation of polarons[according to Stoneham (2007)]
Hole small polaronsin UO
2-x
Measured quantity:current
Charge carrier origin:O vacancies
[Crevecoeur, Wit, J. Phys. Chem. Solids (1970)]
Hole small polaronsin Li-doped MnO
Measured quantity:current
Charge carrier origin:Li dopants
Small polaron hopping via Molecular Dynamics
[Kowalski, PRL (2010)][Reticcioli, hopefully 2018]
[Setvin, PRL (2014)]
[Reticcioli, PRX (2017)]
Measurement technique:STM/STS
Charge carrier origin:O vacancies
Electron Small polarons in r-TiO2
~70% Experimental STM
[Setvin, PRL (2014)]
Measurement technique:STM/STS
Charge carrier origin:Nb dopants
Electron Large polaronlike states in a-TiO2
[Miyata, Science Advances (2017)]
Measurement technique:Time-resolved opticalKerr effect (TR-OKE)
Charge carrier origin:irradiation
Hole polaron: HSE results for CsPbBr3
CH3NH
3PbBr
3 and CsPbBr
3
TR-OKE on CH3NH
3PbBr
3
Large polarons on PbBr sublattice
>Egap
The cell is too small for electron polarons that are more delocalized because of the different states in the CB and VB
τ1,2
= 0.3, 3.4 ps (phonon+photoinjection)
[Verdi, Nature Com. (2017)]
[Moser, PRL (2013)]
Measurement techniqueARPES
Charge carrier origin:O vacancies
a-TiO2
increasing doping
CB
satellites
CB
satellites
CB only
Electron Large polarons
Measurament technique:EPR
Charge carrier origin:defects
[Yang, PRB (2013)]
[Lenjer, PRB (2002)]
[Possenriede, Ferroelectricts (1994)]
BaTiO3
r-TiO2
Measurament technique:EPR
Charge carrier origin:Irradiation
[Yusupov, PRB (2011)]Nb doped (1.2%) KTaO
3
Attempt of interpretation:Signal I: also in pure KTO => trapped chargeSignal II: only in doped KTO => associated to Nb-O electron and hole states
[Sezen, Sc. Reports (2015)]
Measurement technique:IR-adsorbtion spectroscopy
Charge carrier origin:UV-irradiation
orH adatoms
Small polarons in r-TiO2
[Freytag, Sc. Reports (2016)]
Measurement technique:Mid-IR spectroscopy
Charge carrier origin:irradiation
Δ νOH=−3 cm−1O Hole small polarons => Δ El. Pot. =>
near stoichiometric lithium niobate LiNbO3
[Cao, Sc. Reports (2017)]
Measurement technique:IR spectroscopy
Charge carrier origin:O vacancies
NO adsorbed on r-TiO2
No el. polaron: 1870
With el. polaron: 1751
νNO
Small polarons
“Brave” assumptions:- no CO at V
O’s
- ΔνNO
not affected by VO’s
(cm-1)
AFM STM (filled states)
CO at Ti5c rows, with CO at VO in between
CO at VO
Bright: CO at Ti5c with S0 polaron.Darker: CO at Ti5c with no polaron.
(a)
(b)
(c)
CO adsorption on r-TiO2
[Reticcioli, hopefully soon]
Measurement technique:STM
Charge carrier origin:O vacancies
Small polarons
S0-NNN to VO
S0-NNN to VO
Filled state STM
CO on VO
S1 polaron
r-TiO2
Polarons affected by defects but not confined.
SrTiO3 [Hao, PRB (2015)]
Charge carrier origin:O vacancies
r-TiO2
Polarons affected by defects but not confined.
r-TiO2
Repulsion between polarons.
TiO2stoichiometric
(110) Rutile surface
sputteringannealing
TiO
S1
S0
O
S0
Reduced Surface
VO
Oxygen Vacancy
AFM
O2c
rows
O2c
(1x2) Structural Reconstruction
High T
Critical VO= 16.7% ML
AFM
STM, empty states
Exp. Critical VO
DFT Phase Diagram
No agreementwith experiment
- High Vo concentration
- Ti2O
3 not stable
Exp. Critical VO
ST
M, m
id-g
ap s
tate
s
[001]
[001]
3x1 periodicityfor V
O=16.7%
Related topics:
- Magnetic polarons & Colossal Magnetoresistance
- Bipolarons and High Temperature Superconductivity
- Polarons and quantum dots
- Polaronic transport in DNA
[Emin (2013)]
PolaronsAn introduction toBasic concepts
andRecent findings
Ulrike DieboldMichael SchmidMartin SetvinIgor Sokolović
Cesare FranchiniGeorg KresseThomas HahnMichele Reticcioli