Bound small hole polarons in oxides and related materials:  

strong colorations and high ionization energies

O F SchirmerO. F. Schirmer

Universität Osnabrück

central example:  acceptor Li+Zn in ZnO

O2‐O‐

O

• Small polaron: carrier localized at one of severalequivalent sites ‐ here: hole–polaron bound toacceptoracceptor

o Large polaron: usually delocalized over many equivalent sites

• Localization by short range carrier – lattice coupling

o Coupling by long range Coulomb interaction (Fröhlich‐coupling)

• Transport from a site to a neighbouring one bythermally activated hopping (incoherent)thermally activated hopping (incoherent)

o Transport by coherent bandlike tunnelingp y g

Evidence by:   

1) Paramagnetic resonance of unpaired spins:1) Paramagnetic resonance of unpaired spins:

2)  Typical optical absorption:

3) Th ti l d ti f t t d l l3) Theoretical reproduction of structure and energy levels:

Carriers are provided by:     

1) doping1) doping

2)   optical irradiation

Summarizing overviews: 

O. F. Schirmer,  J. Phys.: Condens. Matter 18, R667 (2006)

and J. Phys.:  Condens. Matter 23, 334218 (2011)

Menu:

1) Formation of small polarons, their optical absorptionsand their energies of ionization

2) Basic features of the theoretical reproduction of boundsmall polaronssmall polarons

3) Luminescence accompanying recombination with boundsmall polarons

4) E l f b d ll l i i d t ‘4) Examples of bound small polarons in ‚semiconductor‘ (high gap) materials

Bound small polarons:  

Typical features of

• their optical absorptions

• their energies of ionizationg

Optical absorption Most simple example:  2 equivalent sites(localization of a hole at one of 2 equivalent sites)  

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

1001

1001

0110 2KQVQJH ⎟⎟

⎠

⎞⎜⎜⎝

⎛LR

operating on

⎟⎟⎞

⎜⎜⎛ +

= 22 JKQVQ

H ⎟⎠

⎜⎝ +−

2KQVQJ

KVEP 4/2= KVQ 2/min ±=minimal energy at

Optical absorption:Optical absorption:

O‐ ‐ O2‐ charge transfer under Franck – Condon conditions

))4(exp()()( 222 pEwdJ −−∗∝ ωωωα hh

)8/(1 0ωhpEw =

a) high oscillator strengtha)  high oscillator strength

b) largest possible width of an absorption band for given peak energyfor given peak energy. 

c) characteristic fingerprint, indep. of EP :W2/M = ln 2 * 8 E * / 4 E ~ 0 14 eVωhW2/M = ln 2 * 8 EP *    o / 4 EP   0.14 eV

d)  width of the ‚zero‐phonon band‘ :  

ωh

2J * exp(‐ 4 EP /      o)∝ ωh

2‐well small polaron absorption

Tetrahedral symmetry:  ZnO:Li , BeO:Liy y

⎞⎛1⎟⎞⎜⎛ +2 JJJKQVQ

⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

321

⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

++

+

=−−

−2

111

2111

111

JKQVQJJJJKQVQJJJJKQVQ

Hoper‐ating    on

⎟⎟⎠

⎜⎜⎝4⎟

⎟⎠

⎜⎜⎝ +−−

2111

111

KQVQJJJQQ

))3/8(exp(/)( 22|||| JEwJd p +−−∗∝ ωωα h))23/8(exp(/)( 22 JEwJd p −−−∗∝ ⊥⊥ ωωα h

01 )3/16( ωhpEw =−

p⊥⊥

J = 0.18 eV, EP = 1.18 eV, W2/M = 0.17 eV

Energy of ionization, Eth :  

Contributions to energy of carrier: 

1) Stabilization by lattice distortion, parameter EP2) Stabilization by Coulomb potential of acceptor, parameter Edd3) Destabilization by rise of kinetic energy, typically W/2

lattice distortion, parameter EP

Coulomb potential of acceptor, parameter Ed

rise of kinetic energy typically W/2rise of kinetic energy, typically W/2

Eth =  EP +  Ed – Wb/2          Eopt = 8/3 EP   +‐ a J

Concerning the theoretical treatment of bound small polarons:structure and energy levelsgy

ZnO:Li

~ 0 7 eV

ZnO:LiZn

0.09 – 0.2 eVDFT (LDA)

~ 0.7 eV

S L (2011)S. Lany (2011)

In order to calculate the properties of a bound polaron the environment of thebinding defect, consisting of hundreds of ions, must be included. This implies a 

d h h b f lcorresponding high number of electrons.  

This Many‐Electron‐Problem can be tackled by using ‚density functional theory(DFT)‘  (W. Kohn, Nobel prize 1998).  For instance, the energy is described as a functional of the electron density:  y

E[ρ] = T[ρ] + V[ρ] + Exc[ρ] + ….. 

ρ =  …  ρi  + ρj + ….

V[ρ] = ∫∫1/(dist bet each two elec ) ( ρ + ρ + )( ρ + ρ + ) d(all elec )V[ρ] =   ∫∫1/(dist. bet. each two elec.) ( ….. ρi  + ρj + …. )(….ρi  + ρj + ….) d(all. elec.)

Coulomb‐interaction of the electrons; contains also the self‐interaction of singlelelectrons, e.g. ρi ρi .     

This is not correct physically: there is no self‐interaction. Consequence:  The electron‐electron repulsion is underestimated. Therefore it costs lessenergy to put an additional electron into the system as compared to theenergy to put an additional electron into the system, as compared to the‚correct‘ description. 

The energy distance to the excited states is too low:The energy distance to the excited states is too low:If the next higher state is not populated, this cor‐responds to a hole with a low ionization energy. 

Since recently there are tricks by which thecompensation of the erroneous self‐interaction can be described exactly. y

One adds a repulsive additional potential, which strongly raises energy of next electronwhich strongly raises energy of next electron. A hole, captured in the corresponding level, then has a high ionization energy with respectto the occupied statesto the occupied states.  

Since the hole is expected to be localized at an O2‐ ion, the electronpotential at this ion is raised correspondingly:p p g y

This raises the ionization energy of the holeThis raises the ionization energy of the hole.

The strength of the potential is not arbitrary, however. The parameter λhs ischosen in such a way that all effects of self‐interaction are removed Thischosen in such a way, that all effects of self interaction are removed. Thisleads to exact solution of the problem.  

In addition there are a series of approximation procedures which have aIn addition there are a series of approximation procedures, which have a good predictive power.  

R l h l l li d O Mi i i iResult:  hole localized as O‐.  Minimizationof energy also with respect to latticedistortion yields polaronic state andionization energy.   

Luminescence:  recombination of electron with bound polaron

f h h l l h f d hSo far: Transitions within hole polaron system with fixed chargestate (LiZn0 (h)).  

When recombining with an electron the charge state changes : (e + h) What is the corresponding complete level scheme?(e + h).  What is the corresponding complete level scheme?

ZnO:LiZnO:Cu2+

Emax = 1.95 eV

ZnO:LiZnO:Cu

D. Zwingel, J. Luminescence (1972)  Emax = Egap‐ 2 Eion→→ Eion ~  0.7 eV 

Examples of bound small polarons in semiconductor (orinsulating)  materials:

Identification by: 

•EPR

•Optical absorption•Optical absorption

•Theoretical simulation

Oxides  ‐ acceptor defects binding small hole polarons:  

Tetrahedral coordination (4 O neighbors)  

ZnO LiZn NaZn V Zn NOBeO Li VBeO LiBe VBe

SiO2 AlSi GeSi2 Si Si

Bi12MO20 (M = Ti or Si or Ge)

BiM (antisite defect)

ZnO:LiZn Eionization = 0.64 eV      (Lany 2009)

ZnO:NaZn Eion = 0.79 eV (Lany 2009) Zn ion ( y )

ZnO:VZn Eion = 0.95 eV (Chan, Lany, Zunger 2009) 

ZnO:NO Eion =  1.62  eV     (Lany 2011)

SiO : Al (smoky quartz) ‐ Gillen Robertson 2012SiO2 : Al    (smoky quartz)   ‐ Gillen, Robertson 2012

Ei = 2 5 eVEion   2.5 eV

For comparison:   oxygen vacancy, VO , in ZnO Clark et al.  20112011

Electron(s): widely extended (no small polaron)Electron(s):  widely extended (no small polaron)

Further example for tetrahedral coordination

Bi12TiO20 BiTii12TiO20        iTi

Oxides 

Octahedral coordination (6 O neighbors): 

MgO LiMg NaMg VMgCaO LiCaO LiCaSrO LiSr

LiNbO3 VLiAl2O3 MgAl

CeO2 LaCeCu O VCu2O VCu

oxides

perovskitesperovskites

BaTiO3 :  NaBa CaTi3 Ba,  Ti

SrTiO3 :  FeTi AlTi

CaTiO3 :  Vti

LaMnO : SrLaMnO3 : SrLa

Tellurides, selenides, sulfides: 

Tetrahedral coordination

ZnTe,  ZnSe:          VZnZn

NitridesNitrides

Tetrahedral coordinationTetrahedral coordination

GaN:   MgGa ,   BeGa ,  ZnGaGa Ga Ga

GaN:Mg deep: Eion =   0.18 eV (Lany, Zunger 2010)

•polaronic•responsible for blue luminescenceof GaN (LED !!)

GaN:Mg shallow:   Eion = 0.15 eV (Lany, Zunger 2010)

d h lldeep shallow

GaN:Be 0 45 eV 0 15 eV (Lany Zunger 2010)GaN:Be 0.45 eV 0.15 eV (Lany, Zunger 2010)

GaN:Zn 0.32 eV     0.25 eV   (Lany, Zunger 2010)

Bound small O‐ polaron also in topaz:Bound small O polaron also in topaz:  

Summary:

•Small (hole) polarons bound to acceptor defects are often found in•Small (hole) polarons bound to acceptor defects are often found in semiconductor (or high gap) materials.   

•Such polarons are characterized by intense wide absorption andluminescence bands with maxima in the visible spectral range. 

•The interaction with the lattice is rather strong; EP  (typ.) ~ 1 eV

•This implies high ionization energies and low p‐conductivity

•Reliable ab‐initio predictions of the features of bound smallpolarons have become possible recently

Holes ‚self‐trapped‘, without attraction by acceptor:

Theory: van de Walle et al (Phys Rev B 85 081109 (2012))Theory: van de Walle et al. (Phys. Rev. B 85, 081109 (2012))

TiO2TiO2Al2O3Ga2O3In2O3

a‐SiO2 O‐ hole small polaron, self‐trappeda SiO2 O hole small polaron, self trapped

No ‚self‐trapped‘ holes in ZnO, cryst‐SiO2