1. Electronic structure of transparent conducting oxides (TCO’ s)
Christoph Janowitz
Humboldt University Berlin

2. TCO-overview Composition space for conventional TCO materials
(ITO, IZO, GIZO)
Ginley, Hosono, Paine,
Handbook of Transparent Conductors
Compound Annual Growth Rate
SID.org

3. Outline Electronic structure of Ga2O3 and In2O3 single crystals Donors
Character of states at EF
Polaronic states
Conclusions
From Ginley, Hosono,Paine,
Handbook of Transparent
conductors

4. the Czochralski method Institute for crystal growth ( IKZ) Berlin
β- Ga2O3
Crystals grown by
the Czochralski method
Institute for crystal growth ( IKZ) Berlin
Transporting agents: I2+S2
T1 = ºC
T2 = 1000 ºC
CVT
4 mm
2 mm
Sample cut
along (100)
Z. Galazka et al., Cryst. Res. Technol. 45 pp (2010)

5. Crystal structure of β-Ga2O3
From J. Ahman, Acta Cryst. C52, 1336(1996).
From E. G. Vı´llora et al., Phy. Rev. B 68, (2003).
4 Ga2O3 per unit cell = 20 atoms

6. cleaved (100) surface of β-Ga2O3 crystalsM A
The (100) surface of
the Brillouin zone G Z b* c*
Typical values
n = 5.8 x 1017 cm-3
(300K)
ρ = (Ω.cm)
μ = 239 cm2/Vs

8. BEST at BESSYII 2 gratings 7eV < hν < 40eV
C. Janowitz et al., Nucl. Instr. and Methods in Phys. Res A 693, 160 (2012)

9. β-Ga2O3 experimental k┴ band structure
𝑘 ⊥
Binding Energy [eV]
(Å-1)
hν (eV)
Symmetric dispersion
of emission maxima
Γ-point around 30 eV
Intensity [a.u.]
Binding Energy [eV]
Mohamed , Janowitz et al., Appl. Phys. Lett. 97, (2010)

10. β-Ga2O3 comparison with DFT calculationZ Γ A Γ A M
Binding Energy [eV] Z M Z
Brillouin zone periodicity
in the cleavage plane
DFT Theory: John Varly, C. Van de Walle, University of California, Santa Barbara

11. β-Ga2O3 experimental k|| band structure, indirect band gapEg Z G M A
No states in the energy gap
Eg (direct) is eV
The VB width is about 7.37 eV
Eg (indirect) is eV
VBM near the symmetry point M
Valence band width
Binding Energy [eV] G M Z b* c* A

12. β-Ga2O3 : effect of Sn doping (800ppm Sn), n = 1 x 10 19 cm-3
An emission from the conduction band minimum (CBM)
Deep states around eV binding energy
CBM
-2.0
-1.0
0.0
-4.0
-3.0
Binding Energy [eV] EF
Intensity [a.u.]
Sn-doped β-G2O3
Undoped β-Ga2O3
Deep states
At photon energy of 30 eV
(G-point)

13. In2O3 single crystals CVT grown In2O3 (HUB) Melt grown In2O3 (IKZ)
Typical properties at 300K
In2O3
resistivity
8.5 x 10-3 Ωcm
conductivity
120 S/cm
mobility
66 cm2/Vs
carrier density
1.2 x 1019 cm-3
Z. Galazka et al.
Journal of Crystal Growth 363, 349 (2013)
IKZ: Institut für
Kristallzüchtung
Berlin

14. In2O3 crystal structure Bixbyite (bcc) unit cell
(80 atoms, a= Å)
Bixbyite local Structure
New Journal of Physics 13 (2011)
melt
CVT
Laue-picture [111]-
und [100]- direction

15. Donors, n-type conductivity
Tempering in Oxygen at 1000°C
O-vacancies
get filled
Phys. Status Solidi A, 1–7 (2013)
Saturation
O on intersites

16. In gap states by low energy ARPES

17. Deep in gap states by low energy ARPES & STS

18. Donors, Theory Oxygen vacancies not primary shallow donor
Defect formation energies for oxygen vacancies too low
DFT calculations with hybrid functionals (HF exact exchange) for oxygen rich/poor TCO‘s candidates: Ge, Sn, H
J. Varley et al. Appl. Phys. Lett. 97, (2010)
Alternative : Intrinsic doping
Shallow donors by charge transfer states O2p->Metal-d
Resonant photoemission study at O1s and Cu2p thresholds
BL: U49/2-PGM2 at BESSY II in Berlin
J. Haeberle, M. Richter, Z. Galazka, C. Janowitz, D. Schmeißer; Thin Solid Films 555 (2014) 53.

19. In2O3 electronic structure
a) Brillouin zone
b) VB ARPES in normal emission
c) Near EF-ARPES
d) Intensity map of the gap region,
VB and CB
In2O2 DFT
bandstructure
J.E. Medvedeva and
C.L. Hettiarachchi
PRB 81, (2010)
C. Janowitz et al. New Journal of Physics 13 (2011)

20. In2O3 electronic structure
Eg1 = eV
Dipole forbidden
Eg2 =
dipole allowed
ARPES at 18 eV
Allowed and forbidden transitions
(Walsh et al., PRL 100, (2008)

21. In2O3 effective masses by ARPES
kp direction
(normal emission)
kN direction
(off normal emission)
exp. effective masses
along the ΓP direction
mVB*=(0.27±0.02)m0 and
mCB*=(0.18±0.02)m0
Theory: low anisotropy
mVB 0,28 mo
0,18 mo < mCB < 0,22 mo
J.E. Medvedeva and C.L. Hettiarachchi
PRB 81, (2010)
exp. effective masses
along the ΓN direction
mVB*=(0.28±0.05)m0 and mCB*=(0.24±0.02)m0

22. Fermi surface Fermi surface cut in ΓM plane
-Near EF state is three dimensional
-No surface state
-No surface charge accumulation layer state
as in thin films
(PDC King et al. PRL 101, (2008)
V. Scherer, C. Janowitz, A. Krapf, H. Dwelk, D. Braun, and R. Manzke, Appl.Phys.Lett. 100, (2012).

23. Methods for the determination of a Fermi level crossing
max [I (EF, k)] method: The criterion for kF can be formulated so: If a band crosses the Fermi-edge then the intensity at EF is exactly at maximum when k = kF [1,2].
[1] Santoni, A.; Terminello, L.; Himpsel, F. & Takahashi, T., Applied Physics A, Springer-Verlag, 52, (1991)
[2] S. Legner, S. V. Borisenko, C. Dürr, T. Pichler, M. Knupfer, M. S. Golden, J. Fink, G. Yang, S. Abell, H. Berger, R. Müller, C. Janowitz, and G. Reichardt, Phys. Rev. B 62, 154 (2000)
ii) max [∇Iint(k)] method: The basic idea is that by the integrated intensity Iint(k) of an energy distribution curve (EDC) (typically within an energy range between 300 and 800 meV) the occupied, k-resolved density of states of a band n(k) can be equated. Under this condition by applying a sum rule [2,3] it can be derived that k = kF is valid, if Iint(k) has dropped to the half of the maximum value [2,3].
[3] M. Randeria et al., Phys. Rev. Lett. 74, 4951(1995)

24. Methods for the determination of a Fermi level crossing
iii) symmetrization method: Here kF is determined by first mirroring every EDC at EF = 0, so Isymm(k, E) = I(k, E) + I(k, -E). For momenta where the occupied states have not yet reached kF the intensity Isymm(k, E) forms a local minimum around EF. The momentum, where this minimum disappears, is kF [4].
[4] J. Mesot et al., Phys. Rev. B 63, (2001)

25. Methods for the determination of a Fermi level crossing
iv) the leading edge method: The energetic position of the center of the front flank (tangent) of the single EDC spectrum i.e. the change of the slope is used. The energy position of this inflection point must coincide with the Fermi energy.

26. So simply a degenerated semiconductor?
The elementary model of a degenerate semiconductor describes the bandwidth below the Fermi energy. The electron concentration n in the conduction band is given by
NC effective density of states in the conduction band, EC - conduction band minimum energy and EF the Fermi energy.
at 300K an energy minimum below the Fermi energy (ΔE = EF – EC) of
ΔE = eV for n=1.8·1018 cm-3 and ΔE = eV for n=1.3·1019 cm-3.
More elaborated model
including CB nonparabolicity
(M. Feneberg, priv. com.)
M. Feneberg et al.
PHYSICAL REVIEW B 93, (2016)

27. Comparison to experimental values
State is blurred
Doping independend
ΔΕ = 280meV – (500meV)

28. Polaronic gas in Anastase TiO2
body centered tetragonal
d not s
„large polaron quasiparticle“
intermediate between localized
small polaron and free electron

29. ARPES on Anatase TiO2, ne ~ 3.5 x 10 19 cm-1
shallow QP band at 40meV
renormalisation m*/m ~ 1,7
Replicas at
~ 100 meV
3-dim
Franck Condon
Electron coupled to vibrational mode: zero phonon peak + vibronic satellites
Raman spectroscopy (PRB 55, 7014 (1997)): LO Eu phonon mode at 108 meV
Satellites disappear at high doping (3,5 x 1020cm-3)

30. Polaron band in In2O3 ? (different material (electronic structure, phonon spectrum))
no one to one correspondence with In2O3
-> no renormalisation observed
-> no substructure observed
-> In2O3 with several phonons between 37 meV and 62 meV by Raman
(White&Keramidas, Spectr. Chim. Acta A, 28, 501 (1972)
and by absorption 31,4meV-38,6 meV , (Irmscher et al., Phys.Stat.Sol A 211,54 (2014)
„Multiphonon Generation during Photodissociation of Slow Landau-Pekar Polarons“
E.N. Myasnikov et al., J. Exp. Theo. Phys. 102, 480 (2006)
strong electron phonon interaction
- Fröhlich Hamiltonian (large polaron)
- longitudinal polarization interactions
- polarisation field quantized
- Single polaron can photodissociate to one or many phonons
- Polaron band forms after decay of a single polaron

31. Halfwidth of polaron band
For the effective carrier mass of m* = (1-3) me the halfwidth of the polaronband can be estimated to
3.4 Ep < ħΩ1/2 < 5.6 Ep  
with Ep the polaron energy.
Assuming that the polaron energies are proportional to the phonon energies and by using the phonon energies between 37meV and 62 meV
111meV < ħΩ1/2 < 347 meV
Compares to experimental value of 280+ meV from photoemission spectroscopy.
V. Scherer, C. Janowitz, Z. Galazka, M. Nazarzadehmoafi, R. Manzke, EPL, 113 (2016) 26003

32. Summary Valence bands well described by DFT
(band dispersions, effective masses…)
Classical degenerate semiconductor model possibly not sufficient
No two dim. states like charge accumulation layers on UHV cleaved single crystals
Incorporation of electron phonon interaction (polarons)

33. Thank you Collaborators Valentina Scherer Zbigniew Galazka John Varley
Mansour Mohamed
Leibniz-Institut
für Kristallzüchtung
Department of Materials, University of California, Santa Barbara
Assiut university, Egypt
BESSY II
EES group
Thank you

36. Landau
Pekar
Feynman
Cardona (bandgap)

37. Some estimates for Fröhlich polarons
Fröhlich coupling constant α
∝= 𝑒 2 ℎ𝑐 √ 𝑚𝑏 𝑐 2 2ℎ𝜔𝐿𝑂 ∙( 1 𝜀∞ - 1 ε𝑜 )
ε0 Static dielectric constant
ε∞ High freq. Diel. Constant
mb bandstructure effective mass
ωLO phonon frequency
(α ~ x total number of phonons in cloud around electron, => Polaron effects α >1)
Polaron effective mass m* =~ mb (1+ α/6) for α<<1; m* ≈ mb 0.02 α4 for α >>1
(„Feynman approx“. J.T. Devreese „Polarons“ Digital Encycl. Of Appl. Phys. Ed. GL Trigg, Wiley)
μ ~ exp Θ/T; Θ Debeye Temperature. better: μ ~ F(α) exp Θ/T
( J.T. Devreese „Polarons“ Digital Encycl. Of Appl. Phys. Ed. G.L. Trigg, Wiley,
C. Kittel, Quantum Theory of solids Wiley 1987))

38. Kristallzüchtung, XRD
CVT
from melt
In2O3 Kristalle gezüchtet aus Schmelze
Z. Galazka et al., Journal of Crystal Growth ,362, 349 – 352 (2013)
XRD-Ergebnisse von den Kristallen aus
beiden Züchtungsarten
zeigen eine Einkristallphase

39. Transporteigenschaften
Ladungsträgerdichte von Kristallen aus CVT überschreitet den kritischen Wert des Mott-Kriteriums (nc=2,7·1018cm-3)
Eine Entartung liegt vor
Kristalle aus der Schmelze unterscheiden sich stark untereinander, so dass Mott-Kriterium nicht immer erfüllt ist
n-Leitung
219K
Transporteigenschaften bei Raumtemperatur
Proben aus CVT
Proben aus Schmelze
Spezifischer Widerstand
8,5 mΩ·cm
182 mΩ·cm
Leitfähigkeit
117 S/m
5,5 S/m
Mobilität
66 cm2/V·s
164 cm2/V·s
Ladungsträgerdichte
1,3·1019 cm-3
2,1·1017cm-3
150 K
10% Änderung
200 K
70 cm2/V·s
V. Scherer et al., Appl.Phys.Lett. 100, (2012).

40. Charge accumulation in thin films
P.D.C. King et al. PRL 101, (2008)
(Here undoped is 1019cm-3)
charge accumulation layer
 Strong band bending
charge accumulation only near surface
Band bending changes with doping