Bound small hole polarons in oxides and related materials: strong colorations and high ionization energies
O F. Schirmer
O.
F Schirmer
Universität Osnabrück
central example: acceptor Li+Zn in ZnO
O‐
O2‐
• Small polaron: carrier localized at one of several
equivalent sites ‐ here: hole–polaron bound to
acceptor
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 by
thermally activated hopping (incoherent)
o Transport by
p
y coherent bandlike tunneling
g
Evidence by: 1) Paramagnetic resonance of unpaired spins:
2) Typical optical absorption:
3) Th
3) Theoretical
ti l reproduction
d ti off structure
t t
and
d energy levels:
l l
Carriers are provided by: 1) 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 absorptions
and their energies of ionization
2) Basic features of the theoretical reproduction of bound
small polarons
3) Luminescence accompanying recombination with bound
small polarons
4) EExamples
l off bound
b
d smallll polarons
l
i
in ‚semiconductor‘ i d t ‘
(high gap) materials
Bound small polarons: Typical features of
• their optical absorptions
• their energies
g of ionization
Optical absorption Most simple example: 2 equivalent sites
(localization of a hole at one of 2 equivalent sites) 0⎞
⎛0 1⎞
⎛1 0 ⎞
2⎛1
⎟⎟ + VQ⎜⎜
⎟⎟ + KQ ⎜⎜
⎟⎟ operating on
H = J ⎜⎜
1
0
0
−
1
0
1
⎝
⎠
⎝
⎠
⎝
⎠
⎛VQ + KQ 2
H = ⎜⎜
J
⎝
⎞
⎟
2⎟
− VQ + KQ ⎠
EP = V 2 / 4 K
minimal energy at
J
Qmin = ±V / 2 K
⎛ R⎞
⎜⎜ ⎟⎟
⎝ L⎠
Optical absorption:
Optical absorption:
O‐ ‐ O2‐ charge transfer under Franck – Condon conditions
α (ω ) ∝ ( J 2 d 2 hω ) ∗ exp(− w(hω − 4 E p ) 2 )
w = 1 /(8E p hω0 )
a) high oscillator strength
a) high
b) largest possible width of an absorption band for given peak energy. energy
c) characteristic fingerprint, indep. of EP :
W2/M = ln
= ln 2 * 8 E
2 * 8 EP * * hωo / 4 E
/ 4 EP ~ 0.14 eV
0 14 eV
d) width of the ‚zero‐phonon band‘ : hωo)
∝ 2J * exp(‐ 4 EP / 2‐well small polaron absorption
Tetrahedral symmetry: ZnO:Li
y
y
, BeO:Li
⎛VQ111 + KQ 2
⎞
J
J
J
⎜
⎟
2
J
VQ−111 + KQ
J
J
⎜
⎟
H =⎜
⎟
2
J
J
VQ
Q
+
KQ
Q
J
−1−11
⎜
⎟
2⎟
⎜
J
J
J
VQ1−1−1 + KQ ⎠
⎝
oper‐
ating on
⎛1⎞
⎜ ⎟
⎜ 2⎟
⎜ 3⎟
⎜ ⎟
⎜ 4⎟
⎝ ⎠
α || ∝ ( Jd || ) 2 / ω ∗ exp(− w(hω − 8 / 3E p + J ) 2 )
α ⊥ ∝ ( Jd ⊥ ) 2 / ω ∗ exp(− w(hω − 8 / 3E p − 2 J ) 2 )
w −1 = (16 / 3) E p hω 0
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 EP
2) Stabilization by Coulomb potential of acceptor, parameter Ed
3) Destabilization by rise of kinetic energy, typically W/2
lattice distortion, parameter EP
Coulomb potential of acceptor, parameter Ed
rise of kinetic energy, typically
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
gy levels
ZnO:LiZn
~ 0 7 eV
~ 0.7 eV
0.09 – 0.2 eV
DFT (LDA)
S L
S. Lany
(2011)
In order to calculate the properties of a bound polaron the environment of the
binding defect, consisting of hundreds of ions, must be included. This implies a corresponding
d high
h h number
b off electrons. l
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
V[ρ] = ∫∫1/(dist. bet. each
elec ) (
ρi + ρ
+ ρj + …. )(….ρ
+ )( ρi + ρ
+ ρj + ….) d(all. elec.)
+ ) d(all elec )
Coulomb‐interaction of the electrons; contains also the self‐interaction of single
electrons, e.g. ρ
l
i ρi . This is not correct physically: there is no self‐interaction. Consequence: The electron‐electron repulsion is underestimated. Therefore it costs less
energy to put an additional electron
an additional electron into the system, as
system as compared to the
‚correct‘ description. The energy
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 the
compensation of the erroneous self‐
interaction can be described exactly. y
One adds a repulsive additional potential, which strongly raises energy of next electron. electron
A hole, captured in the corresponding level, then has a high ionization energy with respect
to the occupied states. states
Since the hole is expected to be localized at an O2‐ ion, the electron
potential at this ion is raised correspondingly:
p
p
gy
This raises the ionization energy of the hole.
hole
The strength of the potential is not arbitrary, however. The parameter λhs is
chosen in such a way, that
in such a way that all effects
all effects of self
self‐interaction
interaction are removed. This
removed This
leads to exact solution of the problem. In addition
In
addition there are a series
a series of approximation procedures, which
procedures which have a a
good predictive power. Result: hole localized
R
l h l l li d as O‐. Minimization
Mi i i i
of energy also with respect to lattice
distortion yields polaronic state and
ionization energy. Luminescence: recombination of electron with bound polaron
So far: Transitions within
f
h hole polaron
h l
l
system with
h fixed
f d charge
h
state (LiZn0 (h)). When recombining with an electron the charge state changes : (e + h) What is the corresponding complete level scheme?
(e + h). What
ZnO:Cu2+
ZnO:Li
Emax = 1.95 eV
D. Zwingel, J. Luminescence (1972) Emax = Egap‐ 2 Eion
→ Eion ~ 0.7 eV Examples of bound small polarons in semiconductor (or
insulating) materials:
Identification by: •EPR
•Optical absorption
•Optical absorption
•Theoretical simulation
Oxides ‐ acceptor defects binding small hole polarons: Tetrahedral coordination (4 O neighbors) ZnO
BeO
LiZn
LiBe
SiO2
AlSi
Bi12MO20
NaZn
VBe
GeSi
(M = Ti or Si or Ge)
BiM (antisite defect)
V Zn
NO
ZnO:LiZn Eionization = 0.64 eV (Lany 2009)
ZnO:NaZn
Eion = 0.79 eV
((Lanyy 2009) )
ZnO:VZn
Eion = 0.95 eV
ZnO:NO
Eion = 1.62 eV (Lany 2011)
(Chan, Lany, Zunger 2009) SiO2 : Al (smoky
: Al (smoky quartz) ‐
quartz) ‐ Gillen, Robertson 2012
Gillen Robertson 2012
Eion
= 2.5 eV
2 5 eV
i
For comparison: oxygen vacancy, VO , in ZnO Clark et al. 2011
Electron(s): widely extended (no small polaron)
Electron(s): widely
Further example for tetrahedral coordination
Bii12TiO20 BiiTi
20
Oxides Octahedral coordination (6 O neighbors): MgO
CaO
SrO
LiMg
LiCa
LiSr
LiNbO3
Al2O3
VLi
MgAl
CeO2
Cu2O
LaCe
VCu
NaMg
VMg
oxides
perovskites
BaTiO3 : NaBa, Ba CaTi
SrTiO3 : FeTi
CaTiO3 : Vti
LaMnO3 : Sr
: SrLa
AlTi
Tellurides, selenides, sulfides: Tetrahedral coordination
ZnTe, ZnSe: VZn
Nitrides
Tetrahedral coordination
GaN: MgGa , BeGa , ZnGa
GaN:Mg deep: Eion = 0.18 eV (Lany, Zunger 2010)
•polaronic
•responsible for blue luminescence
of GaN (LED !!)
GaN:Mg shallow: Eion = 0.15 eV (Lany, Zunger 2010)
d
deep
GaN:Be 0.45 eV
0 45 eV
shallow
h ll
0 15 eV (Lany, Zunger
0.15 eV
(Lany Zunger 2010)
GaN:Zn 0.32 eV 0.25 eV (Lany, Zunger 2010)
Bound small O‐ polaron also in topaz: also in topaz:
Summary:
•Small (hole) polarons
•Small
(hole) polarons bound to acceptor defects are often found in in
semiconductor (or high gap) materials. •Such polarons are characterized by intense wide absorption and
luminescence 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 small
polarons 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))
TiO2
Al2O3
Ga2O3
In2O3
aa‐SiO
SiO2
O‐ hole
O
hole small
small polaron,
polaron, self‐trapped
self trapped
No ‚self‐trapped‘ holes in ZnO, cryst‐SiO2