Kinetic Monte Carlo Simulations of Defects in Anatase Titanium Dioxide

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Kinetic Monte Carlo Simulations of Defects in Anatase Titanium Dioxide Benedikt Weiler, Alessio Gagliardi, and Paolo Lugli J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b01687 • Publication Date (Web): 30 Mar 2016 Downloaded from http://pubs.acs.org on April 20, 2016

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The Journal of Physical Chemistry

Kinetic Monte Carlo Simulations of Defects in Anatase Titanium Dioxide Benedikt Weiler,∗ Alessio Gagliardi, and Paolo Lugli Lehrstuhl f¨ ur Nanoelektronik, Technische Universit¨ at M¨ unchen, Arcisstrasse 21, 80333 M¨ unchen E-mail: [email protected] Phone: +49 9923 80033. Fax: +49 9923 80034

Abstract Anatase titanium dioxide is a highly promising material for memristors and photocatalysis. Multiple electronic transport processes are known to be influenced by defects in nanoscale anatase. Hence, in this study, we examine charge transport due to defects with respect to the fabrication of nm-thin TiO2 -films via kinetic Monte Carlo (kMC). A compact kMC-model for MOS and MOM-structures comprising TiO2 was parametrized by the electronic properties of TiO2 in agreement with literature, in particular spectroscopic studies and DFT calculations on defects in anatase. KMC-simulations of MOS-structures were refined, for the first time, by separate drift-diffusion-simulations on the band bending in p+ -Si substrates as well as by barrier heights adjusted for the Fermi-level pinning effect. Referring to the impact of specific TiO2 -film growth methods and post-growth treatments on the parameters for defect energies, in particular, electrical jV-characteristics of material stacks fabricated by PVD and CVD methods, as reported in literature, were reproduced computationally at high accuracy. ∗

To whom correspondence should be addressed

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Thus, conclusions on the dependence of electron trap levels in anatase in dependence of the sample processing could be drawn from this kMC-based computational analysis, attributing defects in TiO2 to shallow titanium interstitials Tiint or deep oxygen vacancies VO depending on the fabrication methods.

Introduction Titanium dioxide, TiO2 , as being environmentally-friendly and abundantly available, is prominent for its broad range of applications, from electronics over photovoltaics to catalysis. The high dielectric constants of ∼90(ǫ|| )-180(ǫ⊥ ) for rutile, 1–4 ∼30-40 for anatase 5–8 and ∼80 for brookite, 9 made titania natural candidates for high-k-driven device miniaturization. 2,9,10 This was promoted by research on structurally similar perovskites, like SrTiO3 or BaTiO3 having even higher k-values. 7,11–13 Additionally, the high electrochemical stability and photocatalytic activity rendered TiO2 useful for the photocatalytic reduction of CO2 14–20 and for water splitting applications, 21–23 since it charge transport and passivates the underlying Si against corrosion at the same time. 24,25 Nanocrystalline, mesoporous anatase forms the basis for dye sensitized solar cells (DSSCs) which have pushed experimental research on TiO2 extensively, starting with the seminal works by Grätzel, 26–32 evolving to hybrid solar cells 33–38 and finally to the currently increasing work on perovskite solar cells. 39,40 The properties of defects in TiO2 make it also a prototypical material for resistive switches. 41–44 However, all of the mentioned applications are strongly related to the type of defects, their energetic position in the bandgap and the spatial distribution of defects in TiO2 . This stimulated a heavy debate whether defects in TiO2 are either oxygen vacancies or Ti interstitials or both. 45–48 Repeatedly, number of transport properties in TiO2 were explained in the multiple trapping model assuming shallow traps to be distributed in exponential tails. 49–55 Nonetheless, recent experiments confirmed coexisting deep traps that contribute to electronic transport through TiO2 layers, 56 which is in accordance with DFT calculations 47 and results on other transition metal oxides, like ZrO2 , Al2 O3 or HfO2 . 57,58 Still, except for a few studies that provide 2


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systematic experiments on surfaces combined with DFT calculations, 59,60 most studies are either only experimental or ab initio calculations. In this context, we employed kinetic-Monte-Carlo (kMC) techniques to bridge the gap between the explanatory power of measurements and the relentless practical limitations of analytic ab-initio calculations. The kMC-algorithm 61 used here allows for a modular reconstruction of rate-determined systems, i.e. in particular nanoscale charge transport processes in MOM and MOS structures. By our simulations of TiO2 nanolayers we attribute experimental characteristics to certain types of defects, distinct defects and relate the type of defect to sample processing. This necessitates an accurate choice of all input parameters, especially the trap levels. Therefore, this kMC-study begins with an evaluation of the impact of trap parameters on the current density through TiO2 , the defect types and energies in TiO2 as reported in recent literature are reviewed and categorized, focusing on anatase and the dependence of oxide properties on its processing. This forms the basis of the actual kMC-model and the further parameterization in the methods section, where we validate our assumptions with respect to TiO2 . From our simulations of several jV-curves measured by other groups and by the subsequent literature survey, the nature and energies of dominant defects in TiO2 are finally linked to oxide fabrication and post-processing methods in the results section.

Defects in TiO2 Just as in other transition metal oxides, bulk defects as well as interface defects contribute notably to the current density in TiO2 . Either indirectly by influencing the charge carrier generation and recombination at the interface or in the bulk as well as by impurity scattering of electrons, or directly via the trap-specific charge transport processes TAT and PF emission, cf. methods section. This influences also the charge carrier mobility, the number of free charge carriers in the CB and thus the drift-diffusion-transport in the CB. Thus defects

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Figure 1: Overview of typical energies of the most important defect states in TiO2 anatase. Blue bars(6=lines) indicate rarer cited ranges for defect levels, values in the range of the red bars are cited more often and were considered more reliable for this reason. The blue lines indicate values cited explicitly for these defects. In terms of Tiint green lines indicate higher ionization energies of its different charge states. For the exact values, applied methods, further explanations and references see the summary in Tab. 1 and text. The grey region gives the mostly cited values for EF in intrinsic anatase assuming the existence of primarily deep traps, as focused on here, or shallow traps (accurate definition of the two terms in 62 ). in TiO2 influence the bulk fermi level EF , the interface charge neutrality level (CNL) ECN L , the interface density of states (DOS), interface band bending and finally also the conduction band offsets (CBO) between the electrode materials and TiO2 . The free carrier mobility in the oxide is ignored here due to reasons given in the methods sections. We focus on the defect density nD and the energetic positions of defect levels ED themselves - measured with respect to the CBM in the bandgap of anatase TiO2 , because these determine the electronic properties of interest here. The defect level ED is in fact the most detrimental parameter of our kMC-model, cf. methods section. To parameterize it correctly, we provide an overview of its values in anatase either taken from theoretical studies explicitly referring to anatase or experiments done on pure anatase or an anatase-rutile polymorph. On first sight, one notices in Fig. 1 that calculated and measured values are rather widely spread in a range between ∼0.4 eV and ∼1.8 eV. Referring to Tab. 1, both experiments 4


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Table 1: Overview on defect types, energies (in eV) and method (in {}-brackets) to derive them for most important electron traps in anatase. Exp. traps 0.8 {XPS,

V0O (e-trap)

XAS,

0.5

PES}

{HSE} 18,46

8,59,60,63,64

0.9 {XPS, UPS,RPS} 69,70

1.0 {XAS, RPS}

0.65-0.7 {B3LYP} 71 0.7

{GGA+U} 47

V+ O (e-trap) 0.94-0.99 {GGA+U, B3LYP} 65 1.0 {STS+ DFT+U} 72

0.72

0.51.7{PL, others} 14,16

1.17 1.08 {GGA+U} {GGA+U} 73 46,73

63,69,72,74–76

1.15, 1.48 {B3LYP} 45

1.16, 1.28 {B3LYP} 79

Ti+ int

Ti2+ int

Ti3+ int

0.0 {GGA+U}

0.5 {GGA+U}

0.6 {GGA+U}

1.2 {GGA+U}

0.39 {GGA+U} 73

-

-

47,66,67

47,66,67

47,66,67

47,66,67

-

TiT i 0.73, 0.82 {DFT+U} 68 -

0.74,0.98, 1.0,1.17, 0.82 1.71 1.09,1.73 1.0 1.74 45 45 47 1.3, 1.74 {GGA+U} {GGA+U} {GGA+U} 45 {GGA+U} {GGA+U} 45 {GGA+U} 45

63,74–76

{GGA+U} 77

Tiint

1.84 1.46 1.16 0.98 {GGA+U} 78 {GGA+U} 78 {GGA+U} 78 {GGA+U} 78

-

1.35 {GGA+U} 45

1.5 {GGA+U} 78

(PL, PES, XPS, UPS, RPS, XAS) and calculations (DFT, e.g. LDA, GGA+U, B3LYP, sX, HSE06) agree at least that the defects in TiO2 , which are located around 1 eV below the CB are Ti3+ -type, 8,14,63,70,80 show d-d transitions/3d-character and an EPR signal of paramagnetic Ti3+ . 81,82 Measured peaks suffer from ∼ 0.5 eV FWHM, with the maximum mostly between 0.5 to 1.7 eV, 14,16,18,63,69,72,74–76 so that the defects in TiO2 can be caused either by oxygen vacancies, VO , or Ti interstitials, Tiint , and titania layers could correspondingly be a reduced form of TiO2 , i.e. an O-poor TiO2−x , or a Ti-poor Ti1−x O2 . Since, consecutively, distinguishing between those two defect types experimentally by localizing them in the bandgap, the debate on the nature of defects in TiO2 , found in the references of this paragraph, still goes on. This issue has also high significance for the structural properties, as the oxygen vacancies would be accompanied by a Schottky disorder resulting only into deep

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defects, while Tiint are associated with Frenkel disorder and would induce deep or shallow traps with the latter ones most probably dominating the electrical conductivity of TiO2 . 83,84 The deep VO are mostly reported to be dominant in density, but also Tiint are considered to contribute to the conductivity of TiO2 . 85,86 In UPS, XPS, XAS or PL measurements mostly peaks with the maximum around 0.8-0.9 eV are reported, as seen in Tab. 1. However, the major drawback of these techniques, especially of EELS, for example, is their limitation to surface states. 18 Exemplarily, Tang, et al. attributed the 0.8 eV state measured by XPS to Ti3+ defects from VO , 5,8 Wendt et al. exposed rutile surfaces to oxygen flows of 0 L, 4 L, 40 L and 420 L, monitored the defects by PES and the surface by STM in parallel and still observed a Ti3+ peak at 0.85 eV despite of optimal oxygen bridging rows. Finally, they concluded that defects should be due to near-surface Tiint rather than bridging oxygen vacancies on the surface. 60 In a similar study Wang et al. determined chemically reactive Ti3+ -type defects on anatase surfaces by XPS/UPS and, assisted by STM images, they suggested a model of sub-surface Ti interstitials (intercalated Ti-pairs). 59 Like most of the surface-restricted experiments, also Henderson claimed Tiint to be dominant on ionsputtered TiO2 surfaces, 18 while other studies, such as the ones by Kr¨ uger et al. or Hengerer et al. with Grätzel favored oxygen vacancies in anatase at ∼0.9 eV. 63,70,74 Former theoretical models also favored oxygen vacancies as dominant defect. 60,87 To resolve contradictions to some of the experiments, new DFT calculations on TiO2 were performed, but one has to note that in DFT the exact location of the calculated defect levels depends strongly on the method used and on the choice of parameters for the model, e.g. the U value in GGA+U, used to account for an underestimation of the exchange-correlation in DFT resulting into a too small bandgap. 78 Thus defect values are widely spread in TiO2 , as indicated in Fig. 1. For example, Robertson et al. computed the defect levels in rutile by sceened exchange (sX) hybrid density functional methods resulting into a localized state at 0.7 eV for VO and Tiint distributed between 0.7 eV and 1.3 eV in the bandgap for its four different positive charge states. 48 Di Valentin et al. - as one of the few - determined values for self-trapped electrons

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TiT i on hydroxylated rutile surfaces with energy levels of ∼0.73 eV and 0.82 eV, similar to VO values, as they stated that additional hydrogen atoms could have a similar effect like the removal of oxygen atoms. 87 On amorphous TiO2 , Pham and Wang reported states at 1.6 eV and 1.9 eV for oxygen vacancies much deeper than for rutile or anatase. 88 Concerning 0/+

our material of focus, anatase, di Valentin, Pacchioni and Selloni derived VO

2+/+

and VO

charge states at ∼1.2 eV and 1.3 eV as well as ∼0.7 eV by the B3LYP-method, 71 whereas by GGA+U, with U=3 eV, they reported both a shallow energy level, 0.15 eV below CBM, delocalized, and a deep one, 0.94 eV below CBM, localized, for the same defects. 65 Janotti et al. also reported a state at 0.9 eV below CBM attributed to oxygen vacancies (charge state unspecified) with the electrons localized at the nearest neighbours of the vacancy, derived using HSE06 functionals. By the same method Deák and Frauenheim found one electron +/0

trap due to VO

2+/+

at 0.5 eV and one due to VO

at 1.3 eV. 46 Morgan and Watson published

two studies on anatase providing vertical transition values (which differ by about ∼0.15-0.25 +/0

eV for the two studies, see Tab. 1) of 1.15 eV and ∼1.35 eV for VO

2+/+

and VO

which

should be dominant for O-rich conditions meaning low T or high oxygen partial pressure during growth or annealing. 45,78 In an extended study on both rutile and anatase Mattioli et al. attributed deep defects in anatase at 0.7 eV and 1.0 eV to VO , stating like most of the authors mentioned above that electrons hop through those localized states as small polarons. 47 +/0

The defect levels of VO

2+/+

and VO

in anatase from a GGA+U study by Yamamoto et al.

are comparably, located at 1.08 eV and 1.17 eV. 73 In most of the mentioned studies also Ti interstitials were calculated and are mostly categorized to be shallow, just as in the latter one, with only one value at 0.39 eV. 73 Especially for Tiint we will refer to this comprehensive and extensive study in the subsequent sections. Bonapasta et al. and Mattioli et al. reported from LSD+U calculations also one delocalized, shallow state at 0.1 eV and one localized, deep state at 0.9 eV in bulk anatase in good agreement with experimental observations and close to former values for Tiint derived by them and placed as single states at 0 eV, 0.5 eV and 0.6 eV. 66,67 Only their Ti3+ defect value was energetically deeper with 1.2 eV below

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CBM. 47 Morgan and Watson reported four occupied states associated to a mixing of four different occupied Kohn-Sham orbitals contributing to the neutral Tiint state resulting into the values for the peaks of 0.74 eV, 1.00 eV, 1.09 eV and 1.71 eV, indicated by blue bars in Fig. 1 stretching over the range of vertical transition energies needed to bring the specific charge state into a state of higher valence. 45,78 They expected the Tiint states to become comparable in concentration to VO under O-poor conditions and state that they are particularly favored only in rutile. Furthermore they consented with Deskins and Dupuis that electrons in TiO2 form localized small polarons and are transported via hopping 89 which we hence implemented into our model, see methods section. Thus, the reviewed results on defects in TiO2 can be summarized as follows: From spectroscopy (PES, UPS, XPS) one can, also in anatase, find basically only a broad peak with a maximum at 0.8 0.9 eV, partly also 0.5 eV, below CBM in the bandgap, attributable to Ti3+ nature of the defects, but not to the specific chemical origin of the defect. Ab initio methods can differentiate between defect types, esp. VO and Tiint , by choosing suitable models, methods and parameterizations. We refer to basically three groups of defect energies as resulting from DFT which will be the point of reference for our trap levels in the simulations: (i) Deep, positively charged oxygen vacancies V+ O that can act as donors or can also accept an electron themselves, with 0.7 eV < ED < 1.2 eV. (ii) Shallow1 0/1+ Tiint , with ED < 0.6 eV, which stem either from excess Ti or are induced by oxygen vacancies. (iii) Deep, 0/1+, 1+/2+, 2+/3+ Tiint with 1.0 eV < ED < 1.75 eV. The latter ones are harder to distinguish from the oxygen vacancies, but in the end they should also depend strongly on the chemical processing (the history) of the sample and be much lower in concentration the deeper they are located. Since the sample processing is highly important for the dominant type of defects, also in other oxide metals, Robertson et al. 90 summarized the qualitative effects of the most common metal oxide growth techniques which we reproduce here for the readers convenience in Tab. 2. The quality measures of the four given physical properties must be understood

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Table 2: Overview of the impact of common fabrication methods for nm-thin TiO2 films on fundamental physical properties. Fabrication technique MOCVD PVD+therm.Oxid. Sputtering ALD

Coverage very good average average very good

Crystal Phase Purity average good good average

Defects

Thickness

good (few) average bad (many) good

good good n/a very good

Used in Ref. 91 92,93

n/a 94,95

relative to each other. The scale is ”graded” like very good, good, average, bad, n/a (no information). ”Coverage” means homogeneity and conformality of a film, e.g. no pin-holes as for molecular organic chemical vapor deposition (MOCVD) or atomic layer deposition (ALD) is ”good”, while homogeneity and number of pinholes of films from physical vapor deposition (PVD) or sputtering is just ”average”. ”Crystal Phase purity” means purity of the crystalline oxide, definition of the phase and crystalline order. So ”good” as for PVD or sputtering means well-defined phase and crystallinity and the better it is the more certain is the phase and at the same time the fewer the number interstitials due to Frenkel disorder. This implies less Ti interstitials, for example. ALD and MOCVD can show a mixture of phases making these methods ”average” and proposing more disorder and more Ti interstitials. For the ”defect” density ”good” means few defects ∼ 1018 cm−3 ) like for MOCVD or ALD, ”average” is ∼ 1019 cm−3 like for PVD, sputtering is ”bad” causing up to ∼ 1020 cm−3 and requiring careful annealing to reduce them. Fourth, ”thickness” means control of thickness, e.g. atomic monolayer-∼nm range, like in ALD, is ”very good”, circa ±2 nm as possible with PVD or MOCVD is still ”good”. Thus, Tab. 2 serves as a basic reference for the discussion in the results section, to analyze the consistency of the results of our simulations with respect to the chemical or physical treatment used for the computationally fitted experiments.

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Figure 2: Exemplary MOS-(accumulation) or MOM-bandstructure: The five transport channels through the oxide are given, i.e. SE, DT, FN, PF, TAT plus hopping through oxide traps, focus on injection into traps and variable (related) parameters of the model, for details see text.

Methods The kMC-model for TiO2 To begin with the core of this work, the basis of our kMC-model, as firstly introduced in, 96,97 shall be briefly reviewed: The simulation of the current density consists of five, partly correlated, charge carrier transport processes, namely Schottky Emission (SE), Direct Tunneling (DT), Fowler-Nordheim-Tunneling (FN), Poole-Frenkel-Emission (PF) and TrapAssisted-Tunneling (TAT). These processes compete and are selected statistically in each step of the kMC-simulation. 97 Three of the five processes, i.e. SE, DT and FN, are computed by the Tsu-Esaki-Formula 98 corrected by us for asymmetric electrode contacts with different

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effective masses

jDT e 4π 3 ~3

e = 3 3 4π ~ Z

∞ 0

Z

∞

dEx T (Ex ) 0

Z

"

∞ 0

dE ′ [m∗ca fca − m∗an fan ] =

e(EF,ca −Ex )/kB T

1+ dEx T (Ex ) ln 1 + e(EF,ca −eVox −Ex )/kB T

# m∗∗ca

man

(1)

where e is the electric charge, T temperature, m∗ca and m∗an are the effective electron masses in the cathode and the anode respectively, EF,ca is the Fermi-level in the cathode, Ex is the kinetic energy of an electron and Vox is the voltage drop over the oxide. Generally, the cathode is metallic with EF,ca (dashed line left in Fig. 2) being taken as 0-eV-reference in kMC. The CBM in the metallic cathode is 3-9 eV below the cathode Fermi level EF,ca , thus defining the cathode material. We assumed that the cathode Fermi level EF,ca is pinned due to interface traps indicated by the two black interface trap distributions (donor-like and acceptor-like) in Fig. 2 so that charge neutrality is given at the interface, cf. also the section on the metal-induced-gap-states-model (’MIGS’-model’) later. The CBO to TiO2 , EB , together with a linear voltage drop V /d · x of EF,T iO2 in the oxide starting at EF,ca , determines EF,an . For two metallic contacts with effective mass 1 m∗0 there is no difference between Eq. 1 and the original Tsu-Esaki formula, while for a semiconductor, like Si with m∗an = 0.32m∗0 , the DT current is corrected by the corresponding factor m∗ca /m∗an . The temperature T was fixed at 298 K and the electric fields were higher than 0.1 MV/cm in all simulations, so thermionic emission is improbable and we expect to see only DT and FN. Additionally, to account for the surface potential in the substrate at a given gate voltage Vg , which is present also under accumulation conditions, i.e. negative substrate bias, the voltage drop Vox over the MOS-contacts in the results section was recalibrated in dependence of the gate voltage Vg by drift-diffusion simulations with TiberCAD. 99 This was done for negatively biased p-type Si substrates using the same material parameters as for the kMCsimulations and all relevant combinations of offsets of p+ Si to TiO2 , cf. parameterization

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Figure 3: Supplementary DD simulations of a Pt-TiO2 -p+ -Si contact under accumulation, CB, VB and EF for Vg = −2.0 V (red), Vg = −1.27 V (green) and Vg = −0.55 V (blue), e.g. Pt with EB,ca ≈ 1.1 eV and EB,an ≈ 0.4 eV according to the MIGS-model with pinning factor ST iO2 = 0.3 and ECN L,T iO2 = 1.0 eV, ECN L,Si = 0.8 eV. The band bending in TiO2 is negligible. The inset shows the derived calibration of the oxide voltage Vox in dependence of the gate voltage for such a MOS-contact. section. This recalibration resulted into flat band voltages of only about VF B ≈-0.1 V and a surface potential under accumulation of φs ≈-0.3 eV for Vg = −0.55 V, φs ≈-0.6 eV for Vg = −1.27 V and φs ≈-0.8 eV for Vg = −2.0 V, for example. Thus, for 0.0 V> Vg > −2.0 V the curve of the oxide voltage Vox in dependence of the gate voltage Vg given in the inset of Fig. 3 was used to recalibrate the experimental data in the results section before fitting. For Vg < −2.0 V, the surplus in gate voltage exceeding φs ≈-0.8 eV drops over the oxide and we find Vox = Vg − φs ≈ Vg + 0.8 eV. Although a possible Fermi-level bending inside the TiO2 is sketched in Fig. 2, it is assumed to be negligible according to the DD-simulations shown exemplarily in Fig. 3. The DD-simulations exhibited a band bending inside T iO2 only in a neglibly small region next to the anode. Thus Vox is assumed to drop approximately linearly across the oxide.

Next, the energy-dependent transmission coefficient T (Ex ) is calculated in WKB-approximation:

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"

T (E) ≈ exp −2

Z

x1

dx x0

r

2m∗ox (V (x) − E) ~2

#

(2)

where V (x) is the local potential as a function of the coordinates x of an electron, tunneling between two x-positions x0 and x1 with an energy E. For the effective mass in TiO2 a parabolic dispersion was assumed. The image potential of the electrodes is said to be omitted for tunneling electrons which ist justified e.g. by the discussion by Weinberg, Hartstein and Schenk. 100–103 So assuming a linear potential drop V (x) = φB − eVox /d · x, with φB being the barrier height (parametrization determined by the MIGS-model provided two sections from here), the tunneling integral was evaluated analytically resulting into the simple general expression:

T (E) ≈ e

4d − 3eV

ox

q

2m∗ ox ~2

x1 −E)3/2 −(φB −e Vox x0 −E)3/2 ] [(φB −e Vox d d

(3)

Complementary numerical evaluations of the integral by our previously demonstrated kMCsimulator 97 showed that the classical image terms caused a barrier reduction of ∼0.1 eV for lowest voltages, i.e. Vox . 0.5 V, increasing linearly with voltage to ∼0.2 eV for Vox & 3.0 V, and remaining constantly at this value for higher voltages. This true at a high accuracy for all cases of simulated parameters, while computations get much more intense if the image potential is included by calculating T (E) numerically. Hence we perform simulations without the image potential and are aware tha barrier heights in our fits in the results section are on average ∼0.15 eV higher, if the image potential was included. 100–102 Each kMC-sampling step starts with an electron of energy Ex is injected from the cathode into the anode, i.e. by SE, DT, FN, or into trap levels at ED (w.r.t. the CBM of TiO2 ). This occurs either elastically or inelastically, i.e. phonon-assisted (energy ~ω, indicated by green arrows, in Fig. 2). These injection rates depend on EB , DOS(E) (i.e. m∗ca ) and the oxide trap parameters ED , xD , m∗ox and nD . Injection is favored for ”resonance” of EF,ca and ED , i.e. there is a trade-off between pre-defined trap levels ED and (random) trap locations

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xD . Similar rates as the ones here for the elastic process have already been derived 104–106 and implemented 13,57,97,107–113 for metals, but we tested an alternative formula for elastic injection confectioned to semiconductor electrodes derived by Svensson and Lundstroem for the first time 114

REl,D m∗El 5/2 × m∗ox

3/2

m∗ 5/2 = El m∗ox !

8Ex √ 3~ ED

f (Ex )e

3/2

8Ex √ 3~ ED n

− 43

2m∗ ox ~2

!

f (Ex )T (Ex ) =

wt −E)3/2 −(φB −E)3/2 ] [(φB −e Vox d

(4) o

with index El = {ca, an}, wt being the distance of the trap from the electrode, f (Ex ) being the Fermi-function, i.e. the probability that an electronic state in the electrode is occupied by an electron with total energy Ee . Correspondingly, by the complementary probability 1 − f (Ex ) replacing f (Ex ) the extraction from a trap at ED to an unoccupied state at Ex in the electrode was modelled. Now, to simulate semiconductor contacts, Ex was set to the Fermi-energy in the free electron approximation of a degenerate semiconductor

EF − EC = kB T ln {n/NC } + 2−3/2 · (n/NC )

(5)

with n being the (approximate) density of conduction electrons in the semiconductor substrate, EC the CBM in Si, NC the effective electron density in the CB of Si. For a highlydoped p-type Silicon substrate in strong inversion this value takes on magnitudes of several tens to hundreds of meV, e.g. ∼200 meV, for an effective mass in the Si electrode of ∼0.3 m0 (NC ∼ 1022 cm−3 ) and an electron density of 5 × 1019 cm−3 at the inverted interface, which we use as fixed value for Ex . Inelastic, i.e. multiple-phonon-assisted, injection and emission rates were employed in the same way as outlined in 104,106,115 and tested e.g. in. 13,57,97,107 Defining

14


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F=

kB T √ β F

2 "

1+

! √ β F − 1 exp kB T

√ !# β F 1 + kB T 2

(6)

electrons injected into an initially positively charged trap can be emitted by PF according to

RP F

ED = ν · exp − kB T

×F

(7)

where ν is the typical phonon interaction frequency (set to a standard value of 1013 Hz), q e3 β = , with ǫopt being the optical dielectric constant (reference value: ∼5.8 1,116 πǫ0 ǫopt andF = V ox/d is the electric field, i.e. oxide voltage drop Vox over oxide thickness d.

The investigated deep defects of TiO2 are localized, cf. the overview on defects in the introduction, so transport occurs via small polarons. Accordingly, we refer to the MillerAbrahams-hopping rates for TAT processes 117    exp − ∆E , −2rij kB T · Rij = ν exp  rD  1,

if ∆E > 0

(8)

otherwise

where ∆E = Ef − Ei the energy difference between initial and final state, rij the traptrap-distance and rD = √

~ 2m∗ox ED

the localization radius. We assume a delta-like defect-

potential 118 for the traps, so the Coulomb-potential of initially positively charged traps is neglected. This is reasonable to save computational costs, but stay accurate, because hopping is only dependent on the mere difference between the initial and the final state and for PF emission the Coulomb contributions are inherently included. This approximate trap radius, the localization length, depends on the specific effective mass and the defect depth. Thus it reflects the specific material and its defects. Finally, transport of the electrons to the electrode or even retrapping, which would lead to multiple phonon-assisted ionization processes, as discussed in, 106 could generally be neglected, too, in our kMC-simulations, since all processes leading to electrons in the CB, such 15



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as thermal SE or PF-emission from defects, occur on much longer time-scales than the CBrelated processes. The transit time through TiO2 is considered to be ∼0.1 ps for a ∼10 nm TiO2 slab, while trapping (or recombination) needs at least tens of ps to several ns, 119,120 even µs are reported, 121 times which are all on the upper edge of all five processes in our simulations, cf. also Ref. 96 So once an electron is emitted to the conduction band by SE or PF, it can be assumed to have readily reached the anode. Therefore the computation of the drift-diffusion-movement in the CB would not alter our results and, combined with the DD-results above, band bending effects could be neglected in the simulations. Eventually, also hole tunneling transport between the electrodes is neglected due to the generally high valence band offsets of TiO2 of at least 1.6 eV in the most conservative estimate on p+ -Si.

Parameterization - Constant Parameters of the Model Since the model is highly sensitive to its input parameters, a thorough parameterization is essential. The two for TiO2 most controversial parameters of focus, the trap energies and densities, have already been discussed in the section on defects. Defect levels in amorphous TiO2 or rutile are structurally and energetically comparable to the ones in anatase which is an advantageous property for the parameterization. 88 Hence, in agreement with our review in , we tested parameter values for ED of 0.2 to 1.5 eV, i.e. from shallow to deep states, with step sizes as fine as 0.02 eV. The defect densities (on which the corresponding current components depended linearly) were varied between 1 × 1018 cm−3 and 1 × 1020 cm−3 , i.e. for example minimum 1 up to ∼100 defects in a simulation volume with dox = 10 nm, times squared lateral dimensions of 10 nm x 10 nm. Regarding the further parameters, as introduced in the methods sections, most of them can be treated quickly, since appropriate literature values are available, such as for the electron masses in the electrodes, which are 1 m0 for metals and ∼0.28 m0 for Si. We varied the effective mass in TiO2 from 0.5 m0 or 1 m0 up to 5 m0 (step size ∼0.2-0.5 m0 ), which is the most frequently reported range, 5,122 although literature values range from 0.35 m0 123 up to 15 m0 . 5,18 The optical dielectric constant ǫopt of 16


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TiO2 is generally well-reported in a tight range of ∼6.8-8.4 for rutile and ∼5.6-5.8 116,124,125 for anatase. Staying consistent this way, these independent literature values were used for all samples. The bandgap Eg of anatase is commonly known to be 3.2 eV 116 (rutile: 3.0 eV) and was a fixed constant in all simulations. Same holds for the static dielectric constant which was fixed to a value of 30. 5–8 The CBO to the anode φB,an is mainly determined by the charge neutrality level (CNL) of TiO2 in the MIGS-Model, outlined in the methods sections, and could therefore usually be kept at ∼0.4-0.6 eV. One has to note that our kMC-model is less sensitive to the last four, better reported quantities (ǫopt , Eg , ǫ, φB,r ) in contrast to the more controversial ones mentioned before. This high sensitivity to the less certain physical quantities is a welcome property of the simulator, because as soon as a specific fit is achieved it supports the validity of the parameterization.

Barrier Heights by MIGS-Model and Fermi-Level Pinning The CBO to the cathode φB,ca , is a very sensitive and crucial parameter of the simulation and must be accurately parameterized. With respect to experimental references, metal-oxide and semiconductor-oxide interfaces are very sensitive to the specific chemical and physical treatments which induce different interface defect densities, defect levels, electron affinities, Fermi-levels and work functions of the contacts at their interfaces and thus also band bending. Apparently, such processes can hardly be controlled exactly, so should always be determined experimentally. Reported CBOs of MOM or MOS-structures with TiO2 as oxide layer and electrical contacts, such as Al, Ti, Au, Pt or Si vary broadly and are much lower than expected from Mott-Schottky-theory which gives the barrier:

EB = φB = φm − χT iO2

(9)

By kMC-simulations this problem of the Mott-Schottky-approximation can be circumvented by performing an extensive parameterization-analysis resulting into a best fit that gives the

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Page 18 of 62

CBO. This approach was pursued here. To obtain the best approximation of the CBOs in advance, they were calculated by the well-known MIGS-model for the Fermi-level-pinning caused by interface defects on metal oxide surfaces, studied into detail by J. Robertson et al., 58,116,126,127 Mnch et al. 128 and Cowley and Sze. 129 It modifies the CBOs via a pinning factor S, e.g. assuming a Silicon electrode: φB = (χSi − φCN L,Si ) − (χT iO2 − φCN L,T iO2 )+

(10)

S · (φCN L,Si − φCN L,T iO2 ) The charge neutrality level (CNL) is the level below which all interface defect states must be filled to obtain charge neutrality at the interface. 130,131 If the Fermi-level of a material lies below CNL the interface is positively charged, whereas it is negatively charged, if EF is above CNL. The CNL of TiO2 has been reported by Robertson to lie at 1.0 eV below CBM. 127 To determine the pinning factor S we employ the empirical model by Mnch using the optical dielectric constant of 5.8, cf. above: 116,128

S=

1 [1 + 0.1(ǫopt,T iO2 − 1)2 ]

≈ 0.3

(11)

The relevant contact materials here were Al, Ti, Au, Pt and Si. The work functions of Al and Ti are similar and range between 4.1 and 4.3 eV as found in literature, 132,133 the one of Au between 4.8 and 5.4 eV 133,134 and the one of Pt between 4.9 and 5.7 eV. 41,133 The electron affinity of Si is well-known to be ∼4.05 eV, while the work function depends on doping via EF,Si , which we assumed to be ∼1.0 eV below CBM for highly p-type Si, resulting into φSi = 5.05eV and the CNL of Si lies at 0.3 eV. 127 A more controversial parameter, however, remains χT iO2 , since values found in literature are broadly distributed from 3.2 to 5.0 eV and accordingly spread are the work functions of TiO2 between 4.4 to 5.8 eV, while, additionally, DFT calculations do not agree at all with experimental values. 135–137 But mostly values around 3.6 and 3.9 eV (or 4.0) are cited for χT iO2 58,90,126,138,139 making these values a reliable benchmark for the electron affinity. Using typical S-factors for TiO2 of at least 0.1 (strong 18


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pinning) and 0.3 (strong to moderate pinning), as presented above, one can thus narrow the variation ranges for the CBO to TiO2 depending on the contact material to ∼0.4-0.7 eV for Ti or Al, ∼0.6-0.9 eV for Au, ∼0.7-1.0 eV for Pt and ∼0.35 to 0.7 eV for p+ -Si. So, to summarize, the issue of the quite uncertain parameter space for the CBO between the cathode and TiO2 is tackled in kMC by simply varying the CBO φB,l in a range from 0.3 to 1.2 eV with fine step sizes down to 0.01-0.05 eV for all investigated samples, as documented in Tab. 2. This variation of φB,l , indicated in the sketch of the bandstructure, Fig. 2, is also in qualitative agreement with experimental values for barrier heights on TiO2 , which are maximally 1.2 eV for all electrode materials considered in this study. By the pinning of the cathode Fermi level in the bandgap at the CNL of TiO2 according to Eq. (10) also the Fermi level in TiO2 is fundamentally defined over the whole oxide structure. This is indicated by the slight non-linear drop of the Fermi levels in the very vicinity of the TiO2 metal-interface in Fig. 2 and 3. The first image is a sketch overemphasizing this drop (under accumulation conditions in p+ -Si) to illustrate it better, but the latter one is the result of the DD-simulations of the band bending in TiO2 performed with TiberCAD. 99 As described in the methods sections,these DD-simulations show that charge compensation effects can be neglected and the whole oxide can be assumed to be neutral in the kMC-model neglecting drift-diffusion of charge carriers in its CB at the same time. Thus the Fermi-level in TiO2 can be assumed to drop constantly with voltage throughout the oxide. To determine this voltage drop, the Vox in our model accurately in case of MOS samples, like sample #3 and sample #4 in the results section,2 we accounted for the band bending in Si by the results of the DD-simulations for the Pt/TiO2 /p+ -Si structure shown in Fig. 3. Therefore, the bias in the experimental curves samples #3 and #4 were rescaled to the simulation parameter according to Vox = 0.107Vg2 + 0, 4315Vg − 0, 0192 to account for the voltage drop in Si. Thus the linear voltage drop in TiO2 is considered an accurate assumption and the Fermi-levels in the materials are determined basically by the CBOs according to the MIGS-model. Besides, defects are assumed to be positively charged traps at the beginning of each simulations. This

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is also in accordance with defect formation energies in dependence of the Fermi level in TiO2 given in most important atomistic calculations cited here, i.e. by Yamamoto et al., Morgan et al. and Deák et al. which usually favor Tiint and VO , partly also TiT i as dominant defects for a Fermi level between 0.3-1.2 eV (=CBO) below CBM. 46,3

Results and Discussion Table 3: Overview on fixed or varied parameter values of the kMC-model for the 6 fitted curves of the 5 samples. Reference Variation Parameter (cf. range text) φB,l [eV]

0.3-1.2 18

0.3-1.5 18

kMC-Fit #1 Al/TiO2 /Al ALD (a) 0.48 // (b) 0.52

kMC-Fit #2 Pt/TiO2 /Pt ALD

kMC-Fit #3 pSi/TiO2 /Pt e-beam+Ox

kMC-Fit #4 pSi/TiO2 /Pt MOCVD

kMC-Fit #5 pSi/TiO2 /Al e-beam

0.6

1.0

1.01

0.88

5 × 1018

2 × 1019

1 × 1019

5 × 1018

5 × 1018

0.4

>1.1

>1.06

>0.65

3

0.48

0.5

1.0

nD [cm−3 ]

10 1020

10 1020

ED [eV]

0.3-1.3

0.2-1.5

m∗ox [m0 ] mcat [m0 ] ǫopt,T iO2 φB,r [eV] Eg [eV]

0.5-15

0.5-5.0

(a) 0.24 // (b) 0.38 1.5

0.28

0.3

1

1

0.3

0.3

0.3

5.8 0.5-0.7 3.2 From spec. study

5.8±1 0.3-1.5 3.2

5.8 0.4 3.2

5.8 0.5 3.2

5.8 0.4 3.2

5.8 0.4 3.2

5.8 0.4 3.2

Ref.±20%

15

60

18

19

15

d [nm]

KMC-Fits and Fabrication of Five TiO2 Samples By the described compact model and accurate parametrization we were able to accomplish fits to six curves taken from literature, shown in Fig. 4a-5c. We took samples of nm-thin TiO2 films deposited by different methods to check, if the parameters for the best fits are consistent with the defect levels in TiO2 , compiled in our introduction on defects, Tab. 2, and with the qualitative expectations for the specific way of fabrication, particularly in terms 20


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of purity of the crystalline phase and the defects, cf. Tab. 2. The five samples were deposited and oxidized at temperatures below 700 ◦ C. This favors the formation of anatase crystals, which the parameterization was focused on. Sample #1, Fig. 4a and 4b, is a polymorph evaporated by plasma-enhanced atomic layer deposition (PEALD) at 180 ◦ C between two Al contacts. 94 Sample #2, Fig. 4c, is reported to be polycrystalline anatase evaporated by PEALD at 250 ◦ C with two Pt contacts. 95 Sample #3, Fig. 5a, was e-beam evaporated from Ti followed by thermal oxidation at 500 ◦ C favoring the formation of anatase. 93 Sample #4, Fig. 5b was deposited by MOCVD, subsequently annealed in O2 atmosphere and showed also predominantly anatase. 91 Sample #5, Fig. 5c, was also e-beam evaporated, just as sample #3, however, followed by annealing in O2 environment at a temperature of 700 ◦ C, which induced typical characteristics of pure anatase, as reported by the authors, and a low density of defects, as expected from Tab. 2. 92 Note that also sample #5 has Al as injecting electrode, just as sample #1, while samples #2, #3 and #4 have Pt as injecting cathode for the given biasing conditions. The resulting fitting parameters for the jV-profiles of all these samples, given in Fig. 4 and 5, are summarized in Tab. 3. Firstly, one can see that most fitting parameters show an excellent agreement with literature values provided in the sections on defects and on the parametrization of the model. Particularly, the thickness was rigorously fixed to the experimental values in the specific studies. For the CBOs fitting values were obtained that are considerably lower than the simple Mott-Schottky-barriers for all stacks. But they agree well with the ones expected from the MIGS-model, as documented in the methods sections and Tab. 3. The effective oxide mass was set to 3 m0 for sample #2, which deviates slightly from the other effective masses but is still well inside the range of literature values. All further, material specific parameters, i.e. the bandgap, the CBO to the anode, the electrode mass and the optical dielectric constant could be fixed to the reference values. This proves the validity and consistency of our kMC-model and parametrization. Secondly, it is consistent that the defect densities are comparably small in all samples,

21



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because all the deposition methods (PEALD, PVD, i.e. e-beam or thermal oxidation, and MOCVD) are considered to keep defect densities rather low in general. From the two curves of the ALD-grown sample #1 we extracted a defect density of 5 × 1018 cm−3 , in agreement with the statement that ALD keeps particularly defects low, cf. Tab. 2. Sample #2 has a higher defect density of 2 × 1019 cm−3 in our simulations which might be because it was not annealed. Besides, for such a high thickness higher defect densities still do not influence current transport via TAT. Sample #3 is e-beam-deposited, a PVD-technique, which is supposed to induce more defects and correspondingly ends up into 1 × 1019 cm−3 . For sample #4 we report only 5 × 1018 cm−3 again, which agrees well with MOCVD to give less defects. Also for sample #5 best fits needed a defect density as low as 5 × 1018 cm−3 , although it was e-beam-deposited. But this sample was annealed in O2 atmosphere which decreases defects, or oxygen vacancies, in particular. In total, we could justify in the first place the relation between defect densities and fabrication methods for the specific samples.

Current Transport Channels and Defects in ALD-grown TiO2 In the following, the five implemented current channels for all five kMC-simulated TiO2 samples are analyzed in terms of defect energy levels and the correlation of the defects to the processing of the TiO2 films. Starting with the ALD-grown samples #1 and #2, which are memristor samples, 94,95 the formation of energetically shallow Tiint is approved corresponding to the expectations from the less pure fabrication method ALD. Generally, ALD allows for a good control of the thickness with few extrinsic defects due to single-layer-growth, but cannot guarantee the best crystallinity as compared to PVD-methods, cf. Tab. 2. To validate the simulated curves physically, we start from Fig. 4a, curve (a) of sample #1. DT prevails below 2 V without TAT and with only slight PF contributions for the lowest bias. Then, above 2 V, it is surpassed by TAT. The kMC-simulations show that only TAT via shallow traps at ED = 0.24−0.26 eV and a moderate defect density of only 5×1018 cm−3 22


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(a) Sample #1, forward bias, symmetric Al- (b) Sample #1, reverse bias, symmetric Alcontacts to ALD-grown TiO2 . contacts to ALD-grown TiO2 .

(c) Sample #2, symmetric Pt-contacts to ALDgrown TiO2 .

Figure 4: jV-characteristics of ALD-grown MOM memristor structures. For details on the fabrication process and discussion refer to text. can explain the upswing in current density by more than two orders of magnitude above 2 V. The Al/TiO2 /Al-junction in the cited study is a memristor sample and the upswing resembles the switching from ”OFF” to ”ON” state when applying forward bias starting from high resistance (OFF) state in curve (a), Fig. 4a. For the ON state in resistive switching studies current transport is assumed to occur through conductive filaments. These are essentially pictured as understoichiometric spatial regions of TiO2−x stretching from anode to cathode inside TiO2 of only nm-size lateral extensions. In principle, the ON state is obtained, if a

23



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strong accumulation of active, unfilled VO+ oxygen vacancies in such a regions extends completely to the cathode for the respective basing. The OFF-state is obtained, if there is still at least a nm-thin layer of insulating TiO2 in between the filament and the counter-electrode. Thus, simulating defects distributed equally over the whole oxide range at high density in our model applies to the ON state. Thus for the ON-state region even for high defect densities up to 1020 cm−3 which marks the transition into the Ti4 O7 Magnli phase, the correct current densities could not be achieved in our simulations, if typical defect levels of oxygen vacancies are used, i.e. such ones deeper than ∼0.5 eV in the bandgap. The latter value is also below the border of deep defects in our categorization according to the literature values in the introductory section on defects. Thus, we argue TAT through shallow traps with 0.2 eV. ED . 0.6 eV is accountable for the high currents observed e.g. in the ON state of TiO2 memristor structures, as long as only ”physically viable” defect densities - resembling still TiO2−x and not Ti4 O7 already - between 1018 cm−3 and 1020 cm−3 are allowed. The given energy range corresponds exactly to Tiint according to our literature review. Curve (b) of sample #1, Fig. 4b, is the high resistive state of the same sample for reversed polarity. For this OFF-state, we also assumed the defective region to stretch over the whole oxide as a first approximation for the OFF-state. Furthermore, we assume that the defect density must not change by more than 2 orders of magnitude compared to curve (a), i.e. range between 1 × 1018 cm−3 to 1 × 1020 cm−3 , since for stronger changes are physically not viable, if the phase should not turn from TiO2−x into Ti4 O7 . Thus, however, to fit curve (b) of sample #1 by kMC, DT must be the dominant current channel over the whole measurement range up to 4 V. Under the given restrictions, we found DT to dominate the fit to curve (b) of sample #1 using trap levels no shallower than ED = 0.38 eV, keeping the same defect density as for curve (a), i.e. 5 × 1018 cm−3 , and reducing the injection barrier only slightly to φB,l = 0.48 eV. Otherwise, for only slightly lower defect energies or higher defect densities, the blue TAT branch in Fig. 4b would dominate. Thus, we argue that TAT through shallow defects is the transport channel responsible for the high-conductive

24


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ON-state in memristors. This channel through shallow defects must be deactivated when switched to OFF state assuming physically viable defect densities between 1018 cm−3 and 1020 cm−3 . Moreover, the fits to the higher voltage range of curve (a), sample #1 are attributed to Miller-Abrahams-hopping, meaning that small polaron hopping prevails in TiO2 and that hopping can explain the trap-related transport through the TiO2 films. This is in accordance with studies by Setvin et al. and Henderson. 18,72 Next, the jV-curve-fitting for sample #2 presented in Fig. 4c refers also to an OFF-state of a TiO2 memristor stack. To obtain the fit, traps shallower than ED = 0.38 eV cannot be present or are not active at defect densities of 1 × 1018 cm−3 . nD . 1 × 1020 cm−3 in the insulating layer of the ALD-grown sample #2. Only this way the blue TAT branch in Fig. 4b does not alter the fit. Deeper traps would give too low currents at such physically viable densities, while such ones as shallow as only ED ≈ 0.2 eV would require unphysically low densities of defects already, i.e. virtually 0, and for intermediate energy ranges no fit could be found with a different set of the other parameters. Thus, sample #2, Fig. 4c, showed also shallow traps at ED ≈ 0.4 eV, which we attribute to Tiint . With 60 nm thickness sample #2 was the thickest of all simulated TiO2 structures. The defect density of 2 × 1019 cm−3 corresponded to 120 defects in the total simulation volume which is close to the computational limit of this kMC-method.4 Comparing the total jV-characteristics and single current components of this MOM-structure, sample #2, to the two MOM-curves of sample #1, one clearly observes a reduced SE/DT/FNchannel. Taking into account the CBO of 0.6 eV of #2, which is comparable to the one of ∼0.5 eV of sample #1, this observation agrees with the four times larger thickness, since for a 60 nm thick oxide DT is expected to be exponentially less relevant. Most remarkably, consistent with expectations the FN channel starts at voltages above the voltage equivalent of φB,l ≈ 0.6 eV, rises quickly for higher voltages and becomes higher than the SE/DT-part at about 2 V here, marked by the step-up in slope for the red curve in Fig. 4c. This step-up is noteably smaller in sample #2 than for sample #1, whereas the current density of sample

25



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#2 is dominated and clearly explained by PF emission from shallow traps located ∼0.4 eV below CBM. This is consistent, too, because sample #2 has a 0.1 eV higher CBO due to the high work function electrode material Pt in contrast to the Al as cathode of sample #1 and it is four times thicker. Thus DT must decrease, TAT through multiple traps is also improbable and only PF from shallow traps can be responsible. From these validations one can see that the kMC-results on sample #1 and #2 are plausible with respect to expectations. The derived shallow traps levels are located at ED = 0.24 eV and ED & 0.38 by the TAT branch in curve (a) and (b), respectively, of sample #1 and at ED = 0.4 eV for sample #2 by the dominating PF branch. These values agree both with the energetic region A of 0.2-0.6 eV under which the Tiint values were subsumed from our review in the section on defects. Besides, from the less pure chemical-growth-method, ALD, worse oxide crystallinity and intrinsic, i.e. not impurity-induced, Ti interstitials were expected 5

. Due to this reason, due to the good fits to the jV-curves by our kMC-parametrization

and due the agreement of the parameters with literature values, we attribute the defects present in the ALD-grown memristor samples to Tiint and not VO . This will be discussed two sections from here.

Current Transport Channels and Defects in PVD and MOCVDgrown TiO2 For the jV-profiles of the anatase films on p+ -Si in sample #3 and sample #4 the experimentally applied gate voltages had to be recalibrated to the oxide voltage drop by the formula from the drift-diffusion-simulations given in the methods sections to account for the band bending in the accumulated Si substrate. They were fabricated by the purest methods and show jV-characteristics similar to each other corresponding to their nearly identical thicknesses, 18 nm and 19 nm. Above ∼0.3 V for sample #3 and ∼0.5 V for sample #4 respectively, DT and FN dominate the current densities over the remaining bias range. Although their thickness was comparable to or even lower than for the other three samples, 26


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(a) Sample #3, with its DD-recalibration ac- (b) Sample #4, with its DD-recalibration according to Fig. 3, asymmetric p+ -Si and Al- cording to Fig. 3, asymmetric p+ -Si and Alcontacts to PVD-grown TiO2 . contacts to MOCVD-grown and annealed TiO2 .

(c) Sample #5, asymmetric p+ -Si and Alcontacts to PVD-grown and low-T-annealed TiO2 .

Figure 5: jV-characteristics of three MOS structures, two PVD-, one CVD-grown. For details on the fabrication process and discussion refer to text. only DT/FN over higher CBOs of 1.0 eV for sample #3 and 1.01 eV for sample #4 and, in particular, PF from traps deeper than 1.05 eV could explain the low current densities for samples #3 and #4. This stands in contrast to the trap parameters of the other samples, in particular the ones from ALD in the previous section. The jV-fits give also almost identical further fitting parameters: The effective mass was varied only by 0.02 m0 , and especially trap levels were located deeper than 1.1 eV for sample #3 and 1.06 eV for sample #4. Such 27



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traps levels deeper than 1.1 eV and 1.06 eV, respectively, are necessary to reduce PF and TAT sufficiently, so that they do not superimpose on the fitting DT-branch at lowest or highest bias, respectively. Defect densities above ∼ 1019 cm−3 can lead to a fit, too, but they increase PF and TAT only linearly according to the model. Contrarily, the sensitive parameter, defect energy ED , must not be lowered by more than ∼ 0.05 eV, as otherwise the fit is not possible anymore with physically viable defect densities below ∼ 1020 cm−3 . According to the literature review, cf. Tab. 1 these reported defect energies required for the fits correspond to deep oxygen vacancies VO while shallow Tiint , like in the ALD-grown samples, must be excluded from the fits. The low defect densities of 1 × 1019 cm−3 for sample #3 and 5 × 1018 cm−3 for sample #4, which will be discussed in the next paragraph, are also in qualitative agreement with the fabrication methods of these samples. E-beam evaporation combined with thermal oxidation for sample #3 and MOCVD for sample #4 are purer and less-defective than ALD and restrict the defect densities to more moderate values. Correspondingly, the e-beam-PVD method of sample #3 results into a moderate defect density and indicates high crystal phase purity. Direct e-beam-deposition of the oxide from the precursor combined with an annealing step, like for sample #5, must be considered the purest of all methods and showed the least defects according to our parameterization. Thus, these qualitative expectations from the fabrication methods agree with the findings from the kMC-determined parameters and the comparison to literature values in so far that the existence of deep oxygen vacancies at low density for the e-beam-grown anatase film of sample #3 is favored rather than the existence of (shallow or deep) Tiint . Sample #4 was produced by MOCVD which is in fact considered moderately pure, cf. Tab. 2. 140 However, it was combined with a high-T annealing step, reducing extrinsic and intrinsic defects and favoring the crystal formation. Thus the crystallinity is improved, Tiint interstitials are reduced and deeper VO vacancies are pronounced. Hence, the sample treatment agrees with potential energetic location of traps below 1.06 eV in this sample. Defects shallower than ED . 1.05 eV, which would be necessary to be Tiint , for example, according to

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literature, would influence transport in both sample #3 and sample #4 and thus cannot be present. Moreover, values for ED around 0.9-1.1 eV lie perfectly in the range of DFT calculations for oxygen vacancies in the V+ O state, see introductory section on defects and Tab. 1, and the peak commonly observed by spectroscopic measurements (mostly PES, XPS, UPS) which report states at ∼0.7-1.2 eV, or more precisely, at ∼0.9-1.1 eV, see Fig. 1. Thus, from this agreement of kMC-model and literature we argue that, given defects are present at physically viable densities of 1 × 1018 cm−3 . nD . 1 × 1020 cm−3 in sample #3 and sample #4, these defects can only be deep oxygen vacancies at ∼0.9-1.1 eV. Similarly, by the kMC-results in Fig. 5c representing the MOS-contact of sample #5, shallow traps can be determined as insignificant, again in agreement with the higher purity expected from this direct PVD growth method combined with an annealing step. First, one has to note that this curve, which exhibits a remarkably low current density, is well explained by a dominating DT transport mechanism despite of the relatively high oxide thickness of 15 nm and the high barrier of 0.88 eV for this fit which actually reduces DT. DT can only be visible, if rather shallow traps have either vanished or are sufficiently low in density for this sample, so that PF or TAT are insignificant. Few shallow traps can be understood also in terms of sample processing which was e-beam-evaporation, i.e. a PVD-method, of TiO2 and subsequent O2 -annealing. 92 The first method implies higher crystal purity, cf. Tab. 2. This means less structural defects and less extrinsic dopants, so less intrinsic or extrinsic Tiint in the end, while annealing must reduce the VO s. Thus in agreement with expectations from Tab. 2, the kMC-fits were only possible for a defect density of only 5×1018 cm−3 with defects located deeper than 0.65 eV, ED > 0.65 eV. To potentially obtain a fit to the jV-curve by only 0.1 eV shallower defects the number of defects would have to be decreased to virtually 0 (lower limit of physical defect density reached). Otherwise the slightly contributing PF branch in Fig. 5c would get too strong, i.e. comparable in strength to DT, and thus alter the simulated trend from the experimental one. If one assumes a higher defect density, e.g. 1 × 1019 cm−3 the traps must in turn be even deeper than 0.65 eV to avoid an experimentally 29



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not visible PF- or TAT-like-trend. Hence, defects cannot be closer to the CBM than 0.65 eV in this sample. Referring to the introduction on defects, this excludes the measureable presence of Tiint energy levels, which were localized by the cited references in the energetic region A, i.e. around 0.2-0.6 eV. This agreement provides evidence that the higher purity expected from PVD growth implies less shallow Tiint caused by intrinsic defects or extrinsic impurities and supports the similar arguments provided above for sample #3 and #4. We note that still deeper defect energies would not alter the fits and are possible, too. Hence, we report the value of ED > 0.65 eV of sample #5, which clearly is not opposed to our findings above, that for PVD-grown samples oxygen vacancies are dominant and they are supposed to be located around 0.9-1.1 eV. But shallow Tiint are definitely excluded for the PVD and MOCVD-grown samples, in contrast to the results from the fits for ALD-grown samples #1 and #2 which both required shallow traps and whose fabrication method is supposed to favor the formation of interstitials. In total, these simulations approved that PVD-grown TiO2 is supposed to have less shallow Tiint induced by intrinsic or extrinsic defects. Futhermore, if e-beam evaporation is combined with an annealing step, as for sample #5, the best sample quality of all methods enlisted in Tab. 2 in terms of crystallinity and defects is produced.

Implications for Defect Levels in ALD- and PVD-grown TiO2 films Finally, one can derive from the presented kMC-simulations that TiO2 films formed by a well-controlled e-beam evaporation-process followed by an optional low temperature annealing, but also MOCVD-grown-TiO2 accompanied by an obligatory annealing step, favor the formation of anatase films with (i) deep defects at (ii) sufficiently low density. PVD methods in general, as confirmed by the model, keep defects deep and low in density, if combined with annealing, and thus prevent TAT or PF current channels from increasing the current density. Combined with a high CBO, achieved by providing clean surfaces and low pinning of the Fermi-levels of the (quasi-)metallic substrates in the CNL of TiO2 , located 1.0 eV below CBM, we suppose PVD-methods to be preferential to guarantee low current densities and 30


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(electronic) dominance of deep oxygen vacancies, esp. when combined with annealing. By relating our kMC-results to defect formation energies from DFT studies and processing conditions of the investigated samples, one can draw further conclusions on type and density of defects. Firstly, considering samples #3-#5, oxygen atmosphere (i.e. O-rich conditions reported for #3 and #4) at moderate temperatures (350 ◦ C, sample #5, to 750 ◦ C, sample #4, 7,92,93 avoids the formation of shallow traps, no matter if VO or Tiint , for all three different TiO2 growth methods. Thus we saw, VO s shallower than 1.1 eV cannot be more than 1 × 1020 cm−3 in density or, if they were at 0.9 eV, they cannot be even more than 1 × 1018 cm−3 , since otherwise they would contribute to transport. Analogously, there were no defects shallower than 0.65 eV at densities higher than 1 × 1018 cm−3 in sample #5.6 In agreement with these conclusions from the kMC-results, the key theory DFT papers cited above predict high defect formation energies of VO and even higher ones for Tiint under O-rich and high temperature conditions, and the formation energies generally increase with increasing temperature for anatase, i.e. one expects less defects in the temperature range around 350 ◦ C to ∼ 750 ◦ C. 45,73,78,141 Assuming that the Fermi energy is consistent with the kMC-parameters we determined for samples #3-#5, i.e. about 0.8 eV to 1.0 eV below CBM for Eg = 3.2eV , in the specific temperature range, O-rich conditions, the formation energy 73 ∼4 eV 45 or ∼4.5 eV, 141 as well.7 The high formation enerfor V2+ O is, for example, ∼3 eV,

gies from DFT confirm our kMC-results that the density of VO must be low, e.g. less than 1 × 1018 cm−3 for trap levels at 0.9 eV, for sample #3-#5 which had O-rich and moderate to high temperature processing conditions, cf. above. In agreement with their even higher formation energy, especially, shallow Tiint that would dominate current transport are even lower in density. Therefore, in practical terms growth or annealing of nm-thin TiO2 -films under oxygen atmosphere at ∼350 ◦ C to 750 ◦ C minimizes leakage current whose lower limit is then given by DT instead of TAT or PF. This is beneficial e.g. for insulating high-k dielectric layers or for certain photocatalytic applications. In contrast to this, for TiO2 grown by ALD, which is less pure in terms of crystallinity, the electronically dominant traps are

31



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clearly categorized as shallow and they do not vanish by annealing, as seen for sample #1. Our kMC-approach provides evidence that the shallow traps in anatase films are Tiint and, most particularly, ALD-growth at low temperature favors their formation in agreement with the defect overview in the section on defects. Most prominently, in a recent extensive PESstudy by Reckers et al. shallow gap states were once more located at ∼0.2 eV below CBM in annealed, low-temperature-ALD-grown anatase films. 56 Presumably, similar experimental conditions apply for the fabrication of samples #1 and #2 which were both ALD-grown at low temperature. Since shallow traps dominate the transport even for ∼60 nm thick films, as in sample #2, or thicker ones, ALD-growth is not the method of choice, if leakage currents should be kept low for the specific application, such as in high-k gate dielectrics. Deep traps might be present there, too, but their contributions to the defect related current channels are orders of magnitude lower. Finally, we relate the kMC-results to processing conditions and predictions by DFT for samples #1 and #2, as done already for #3-#5. The ALD-growth at low T and the electroforming step of #1 and #2 is supposed to create understoichiometric regions of (high) oxygen deficiency, i.e. with high VO concentration, which is qualitatively comparable to O-poor growth conditions with low temperature. 94 The kMC-analysis showed that traps at 0.2-0.4 eV dominate current which we attributed to shallow Tiint in accordance with literature, while deep VO are irrelevant for current transport even at extremely high densities above 1 × 1020 cm−3 . Now, assuming temperatures below ∼300 ◦ C, a shallower Fermi energy around 0.5-0.6 eV at Eg = 3.2 eV from the kMC-fits of #1 and #2 and O-poor/Ti-rich conditions the defect formation energy of both VO and Tiint are comparable in magnitude with values around 0.5 eV 73,85 to 1.0 eV for VO 45,47,78 and also 0.5 eV 47 to 1.0 eV for Tiint . 45,78 According to these way lower DFT formation energies than for #3-#5 45,73,78 one expects both more VO and Tiint at lower T and O-poor atmosphere during growth. However, VO s are too deep to contribute notably to TAT and PF. Since Tiint have a similar formation energy, so have comparable density, they dominate TAT and PF conduction through these

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shallow defects, as analyzed by kMC. Thus, these expectations from DFT formation energies due to O-poor processing agrees with our kMC-results which needed shallow defects of 0.2-0.4 eV at high densities to explain the jV-profiles, i.e. 5 × 1018 cm−3 for sample #1 and 2 × 1019 cm−3 for #2. Therfore, O-poor and low-temperature conditions should be avoided during processing, if TAT and PF through shallow Tiint are not desired, as for example for insulating dielectric layers in contrast to e.g. resistive switches. In total, by our kMC-model the shallower traps in TiO2 , i.e. those in the range of 0.2-0.4 eV, are attributed to Ti interstitials, while deep defects, i.e. those around ∼0.9-1.1 eV, are attributed to oxygen vacancies. Further defect types which were not dominant in these samples can, of course, still coexist in the titanium oxide layers and are not accessible for investigations by our method as only the hypothetically dominant defect type is reflected by the model. Thus, by the simulations of jV-profiles reported in external studies we found a consistency of (i) the literature on DFT calculations and spectroscopic experiments on defects in TiO2 , (ii) expectations from fabrication methods and (iii) the kMC-model.

Conclusions To conclude, for the first time, a general and compact kMC-model of MOM and MOS contacts with high-k anatase TiO2 has been employed successfully to compute jV-characteristics such structures. The validation of the model from by the fits to the fabricated samples show that electronic transport through anatase nanolayers can be modelled properly by kMCsimulations. Most importantly, however, we have seen by the kMC-analysis that the energy of the dominant defects in TiO2 depends on the specific fabrication method: Such producing more crystalline materials (PVD-methods, e.g. e-beam-evaporation, at best combined with high-T annealing or MOCVD with a mandatory annealing step) result into deep traps, while less pure ones (ALD without or just with low-T annealing) produce shallow ones dominating the jV-characteristics. Considering the fabrication techniques the shallower levels at ∼0.2433



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0.4 eV were attributed to Ti interstitials, while the deep ones ∼0.9-1.1 eV should be oxygen vacancies. This means as well that it is the balanced interplay of CBOs, defect levels and the oxide thickness that determines the current characteristics. Thus kMC is able to discriminate between electrically dominant defect types of differently processed samples making it a useful tool, also for experimentalists, to check the specific fabrication and sample processing methods for their impact on defect configurations and evaluate their usefulness with respect to the particular applications.

Acknowledgements This work was partially supported by the ”Nanosystems Initiative Munich” (NIM) and the ”Munich School of Engineering” (MSE). The authors thank Dipl.-Phys. Tim Albes for fruitful discussions.

Notes 1

The authors are aware that the term ”shallow” is used differently for example in EPR-studies where it

refers to defects that are located at an energy below CBM in the thermal range, i.e. 0.025 eV. However, also this definition is not unambiguous. For an accurate definition of ”shallow” and ”deep” defects found in literature, refer to. 62 2

Sample #5 had already been corrected for the band bending in Si by the authors in the respective study.

3

While some groups reported the Fermi level in TiO2 to be rather shallow ∼0.25-0.4 eV below CBO

(which might be the case for illuminated TiO2 in DSSCs, since EF − EF o =qVoc ), there is in agreement with our assumptions a consent by a number of different active groups who calculated or measured the bulk Fermi-level of TiO2 (static, i.e. without photogenerated or other injected charge carriers) by different methods at about 0.8-1.2 eV below CBM. 139,142,143 Such a deep value is also consistent with the average difference between the reported widely spread electron affinities (3.2-5.0 eV) and work functions (4.4-5.8 eV) in TiO2 , the typically reported dominant defect levels in the bulk influencing EF which are located also around 0.8-1.2 eV below CBM, cf. introduction on defects, as well as the interface CNL of 1.0 eV, a kind of interface Fermi-level.

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4

But still the highest number related defect sites found in state-of-the-art literature on kMC-simulations

of jV-curves. 5

These stand also few 0.1 eV deeper in energy than the extrinsic Ti interstitials, cf. section on defects.

6

For numerically lower densities there is less than one defect on average in a typical simulation volume

of 10x10x10 nm. 7

Since partial pressures were never documented in the studies we took the sample data from, a similar

kMC-study of defect types and concentrations in anatase samples of systematically documented growth temperature and oxygen partial pressure would be considered beneficial by the authors.

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Kinetic Monte Carlo Simulations of Defects in Anatase Titanium Dioxide

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