Computational Study of Oxygen Diffusion along a[100] Dislocations in

Subscriber access provided by SUNY DOWNSTATE

Article

Computational Study of Oxygen Diffusion along a[100] Dislocations in the Perovskite Oxide SrTiO

3

Stephan P. Waldow, and Roger A. De Souza ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.5b12574 • Publication Date (Web): 28 Apr 2016 Downloaded from http://pubs.acs.org on May 4, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

ACS Applied Materials & Interfaces is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Applied Materials & Interfaces

Computational Study of Oxygen Diffusion along a[100] Dislocations in the Perovskite Oxide SrTiO3 Stephan P. Waldow and Roger A. De Souza∗ Institute of Physical Chemistry, RWTH Aachen University, 52056 Aachen, Germany E-mail: [email protected] Phone: +49 241 8094739. Fax: +49 241 8092128 Abstract We used classical molecular-dynamics simulations to study the atomistic structure of, and the diffusion of oxygen ions along, the periodic array of edge dislocations comprising a symmetrical 6.0◦ [100] tilt grain boundary in SrTiO3 . The results indicate that, at elevated temperatures, the two types of dislocation core (TiO2 -type and SrOtype) that make up the boundary are stable and that oxygen-deficient cores maintain their dissociated structures. They also confirm that oxygen vacancies prefer to reside at the cores rather than in the bulk. Tracer diffusion coefficients of oxygen were obtained for oxygen-deficient bulk and grain-boundary simulation cells at temperatures in the range of 1000 ≤ T / K ≤ 2300. Calculated values of the oxygen-vacancy diffusion coefficient for the bulk phase agree extremely well with published experimental data. Tracer diffusion coefficients obtained for the grain-boundary cell are, in comparison to those for the bulk, lower in magnitude and have a higher activation enthalpy, indicating that, relative to the bulk, the migration of oxygen ions along a[100] dislocation cores in SrTiO3 is hindered. These results provide further support for the de-coupled model of filament formation in resistively switching SrTiO3 .

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 33

Keywords SrTiO3 , dislocation, oxygen diffusion, MD simulations, resistive switching

1

Introduction

The topic of oxygen diffusion along extended defects in functional oxides is enjoying renewed interest. In particular, ion migration along dislocations and grain boundaries in oxides such as HfO2 and SrTiO3 is attracting attention, as these extended defects have been proposed to play a central role in the phenomenon of resistive switching in these oxides. 1–5 This phenomenon refers to the reversible switching of a sample’s resistance between high and low values upon application of suitable voltages, opening the door to high-density, low-power, non-volatile, rapidly switching memory devices. Extended defects are believed to serve as the active regions (the nanoscale switching filament) along which, due to ion migration, switching occurs. The idea that mass transport along extended defects is faster than in the bulk crystal comes from the extensive body of work on diffusion in metals and elemental semiconductors. 6–9 Diffusion in crystalline oxides is assumed to conform to this description, too, but in the case of fast diffusion along dislocations (pipe diffusion) in oxides, there are only a few convincing examples in the literature, namely O in Al2 O3 , 10 Ni in NiO, 11 and Mg in MgO. 12 In a previous paper 13 we found in fact a counter-example. We examined, with both experimental and computational methods, oxygen diffusion along the dislocation cores of a 6.0◦ tilt [100] grain boundary in SrTiO3 . Experimentally determined

18

O tracer diffusion

profiles provided no evidence for the fast diffusion of oxygen along the array of dislocations. This, we concluded, meant that either diffusion along the dislocations was slower than in the bulk, and/or that depletion space-charge zones prevented diffusion of oxygen out of the dislocations. Static atomistic simulations of the grain boundary provided strong evidence for space-charge formation (the preferential formation of oxygen vacancies at the dislocation 2 ACS Paragon Plus Environment

Page 3 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


cores constitutes the required thermodynamics driving energy 14,15 ); the simulations also indicated that the energetic barriers for oxygen-ion migration were higher than in the bulk. Thus, we concluded, in contrast to previous literature reports 2,16–18 (which, we argued, did not provide unambiguous results), that neither experiment nor simulation provide evidence of fast diffusion of oxygen along dislocations in SrTiO3 . One problem with static calculations, especially of ion migration along an extended defect, is that they rely on the researcher correctly identifying the most probable migration mechanism and path. This procedure is far more challenging for an extended defect than for the bulk because migration mechanisms that are improbable in the bulk may become dominant at the extended defect and because the reduced symmetry at the extended defect results in a more complex energy hypersurface. For this reason Molecular Dynamics (MD) simulations, in which the migrating ions are given the freedom to seek out the most probable migration path and mechanism, are to be preferred. A second problem with static calculations 19 is that the results obtained (atomistic structure, migration paths and barriers, etc.) refer, through the quasi-harmonic approximation, to the temperature of the lattice(s) to which the empirical pair-potentials (EPP) were fitted (generally, room temperature). Diffusion, however, is typically examined experimentally at much higher temperatures. Thus, in principle, it may take place within a different (high-temperature) atomistic structure, with concomitant different migration paths and barriers. For these reasons, we extended, in this study, the previous static calculations of the 6.0◦ [100] tilt grain boundary 13 to finite temperatures. In particular, we examined, at elevated temperatures, the dislocations’ structure, their interaction with oxygen vacancies, and their effect on the oxygen-diffusion kinetics. Finally, we discuss the implications for resistive switching.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2

Page 4 of 33

Computational Section

In this study, rigid-ion potentials, acting between ions with partial charges, were used to describe the inter-atomic interactions. The potentials take the functional form

Vij =

{[ } C ]2 zi zj e 2 ij + Dij 1 − e−aij (r−r0 ) − 1 + 12 , r r

(1)

where the first term describes the long-range Coulomb interactions; the second term is a Morse function to describe short-range interactions; and the third term is an additional repulsive term that prevents the ions approaching each other too closely in MD simulations. The values for the empirical parameters were taken directly from Pedone et al. 20 , and are summarised in Table 1. Table 1: Parameters for the empirical pair-potentials used in the simulations. Derived by Pedone et al. 20 Paira Sr+1.2 —O−1.2 Ti+2.4 —O−1.2 O−1.2 —O−1.2 a

Dij / meV 19.623 24.235 42.395 The short-range

aij / Å−1 r0 / Å Cij / eV Å12 1.8860 3.32833 3.0 2.2547 2.70894 1.0 1.3793 3.61870 22.0 potential cutoffs were set at 15 Å.

Although these potentials were originally derived for, and have been applied successfully to, modelling glassy materials, 20 they were used here for crystalline STO because they reproduce various perfect and defective lattice quantities well: 13,21 In particular, the activation enthalpies for oxygen-vacancy migration, ∆Hmig,VO ≈ 0.6 eV, and for strontium-vacancy migration, ∆Hmig,VSr ≈ 4 eV, in bulk SrTiO3 are in excellent agreement with experimental and other computational data. 22–30 In this study, simulations of bulk SrTiO3 employed a 19a×19a×19a cell containing 34295 ions, i.e. Sr6859 Ti6859 O20577 . (a is the lattice parameter of cubic SrTiO3 .) The grain-boundary simulation cell was a thicker replicate (10a instead of 2a) of the relaxed cell generated previously. 13 To be specific, the grain-boundary cell consisted of two symmetrical tilt 6.0◦


Page 5 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


[100] grain boundaries placed ≈ 77 Å apart, with the two boundaries being anti-parallel to maintain three-dimensional translational symmetry (see Fig. 1). In total the cell contained 37500 ions (i.e. Sr7500 Ti7500 O22500 ).

Figure 1: Projection of the simulation cell containing two complementary 6.0◦ [100] tilt grain boundaries (one in the middle of the cell, the other at the edge) after full ion relaxation. The viewing direction is the [100] tilt axis. Sr ions are shown in blue, Ti ions in grey and O ions in red. The Lammps (Large-scale Atomic/Molecular Massively Parallel Simulator) 31,32 code was used to realize the MD simulations. All simulations were performed within the NpT ensemble: the system’s temperature T , pressure p and number of constituent particles N were fixed, while its energy U and volume V were allowed to vary. The system’s temperature (pressure) was controlled by using a Nose–Hoover thermostat (barostat), as implemented in Lammps. 32 For each calculation, the simulation cell was first allowed to equilibrate for 50 ps (after which U and V were observed to only fluctuate around constant values). Thereafter, the simulation was run for 100 ps to produce data.

3 3.1

Results And Discussion Results and discussion of bulk properties: Validation

In order to examine the fidelity of these EPP 20 for MD calculations, we consider, first, selected thermal properties of the perfect lattice, namely the thermal expansion coefficient α and the heat capacity at constant pressure, Cp ; and second, a kinetic property of the 5 ACS Paragon Plus Environment


defective lattice (oxygen-deficient, bulk SrTiO3 ), namely, the diffusivity of oxygen vacancies, DV . 3.1.1

Thermal properties

In Fig. 2 we compare the normalised change in lattice parameter for stoichiometric, bulk SrTiO3 obtained from our MD simulations with the EPP of Pedone et al. 20 with data from one experimental study 33 and from two other classical MD simulations employing more complex EPP. 34,35 The linear increase in ∆a/a with temperature is reproduced qualitatively, with a single thermal expansion coefficient, α = a−1 (∂a/∂T )p , describing the entire temperature range. In comparison with literature data (see Table 2), our value of α = (12.2 ± 0.2) × 10−6 K−1 for the temperature range 300 ≥ T /K≥ 1000 is a slight overestimate of experiment. Nevertheless, given that these EPP were not specifically derived to describe a(T ), the level of agreement is surprisingly good. 0.010 A B C D

a a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 33

0.005

0.000 400

600

800

1000

T/K Figure 2: Normalised change of the lattice parameter ∆a/a versus temperature T . A, this study; B, simulation from Tosawat et al. 34 ; C, simulation from Goh et al. 35 and D, experimental data from Bunting et al. 33 Combining the definition of the heat capacity at constant pressure with the definition of


Page 7 of 33

160

-1

/ J mol K

-1

140 120 100

Cp

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


80

A B C

60

D

1000

1500

2000

T / K

Figure 3: Heat capacity at constant pressure, Cp , versus temperature T . The black line is a guide to the eye. A, this work; B, Tosawat et al. 34 ; C, Goh et al. 35 ; and D, de Ligny and Richet 36 The inset is a plot of internal energy U against temperature T for the temperature interval 780 K to 820 K, from which Cp is ultimately obtained. enthalpy H, one can write ( Cp = ( =

∂H ∂T ∂U ∂T

) )

p

( +p

p

∂V ∂T

) (2) p

The first term in eq (2) was calculated for a given T by conducting MD simulations at five closely spaced temperatures (T , T ± 10 K and T ± 20 K), 35 and then extracting (∂U/∂T )p from a linear fit of U against T (see inset of Fig. 3). The second term in eq (2) was calculated from the data shown in Fig. 2: at ambient pressure this second term was found to be more than six orders of magnitude smaller than the first term, and it was therefore neglected. In this manner we obtained the data shown in Fig. 3, Cp as a function of T . Also shown are literature data from other MD simulations 34,35 and from experiment. 36 Our results display the same trend of a weak increase in Cp with T , but again, the absolute value is overestimated with regard to the other simulations and experiment.



Table 2: Thermal expansion coefficient α for the temperature range 300 ≥ T /K ≥ 1000 for bulk SrTiO3 . α / 10−6 K−1 12.2 ± 0.2 9.3 ± 0.4 11.1 ± 0.4 11.4 ± 0.5

Method MD MD MD X-ray diffraction

Ref. This work 34 35 33

4.5 4.0 3.5

2

2.5

2

3.0

2.0

rO / Å

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 33

1000 K 1100 K 1200 K 1300 K 1400 K

1.5

1500 K 1600 K

1.0

1700 K 1800 K

0.5

1900 K 2000K

0.0 0

50

100

t / ps

2 Figure 4: Mean-squared displacement of the oxygen ions, ⟨rO ⟩, as a function of time t at various temperatures for oxygen-deficient bulk SrTiO3 . Site fraction of oxygen vacancies, nV = 0.333%.

3.1.2

Diffusion in bulk SrTiO3

We examined oxygen diffusion in six different oxygen-deficient bulk cells, namely, cells with three different vacancy site fractions of nV = 0.333%, nV = 0.167% and nV = 0.049%, together with the charge of the vacancies being compensated either by a uniform background charge or by lowering the charge of all Ti cations. In all cases, oxygen ions were removed at random from the bulk simulation cells, and then the cells were allowed to equilibrate. From each MD production run we computed the mean squared displacement of the oxygen 2 ions, ⟨rO ⟩, according to 2 ⟩ ⟨rO

N 1 ∑ = [rO,i (t) − rO,i (0)]2 . N i

(3)

In Fig. 4 we plot, exemplarily for the case of the highest vacancy site fraction and a uni-


Page 9 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 form background charge, ⟨rO ⟩ as a function of time t for various temperatures. At each 2 ⟩ is seen to increase linearly with time. Fitting the standard relation, temperature ⟨rO

2 ∗ ⟨rO ⟩ = 6 · DO · t + B,

(4)

∗ to the data yields the oxygen tracer diffusion coefficient DO and the parameter B, which

describes thermal vibrations. The further analysis only considers those data for which a fitting correlation coefficient larger than 0.95 was obtained for linear regression to eq (4). 2 2 ⟩, were confirmed ⟩ and ⟨rTi Note: the mean squared displacements of the Sr and Ti ions, ⟨rSr

to exhibit constant values as a function of time in all simulations (not shown). Such behaviour is expected, since no cation defects were present in the N pT simulation cells, and hence no cation diffusion can occur. ∗ Fig. 5 shows values of DO obtained for the six different cases. For these data, only ∗ the site fraction of vacancies plays a role: isothermal values of DO increase linearly with

increasing nV (see below). Clearly, the manner in which the charge of the vacancies is compensated (uniform background charge or lowering the charge of all Ti ions) does not affect the oxygen diffusivity for the three site fractions of vacancies that were examined. An alternative method for charge-compensating the oxygen vacancies would have been to substitute at random specific Ti cations with lower valent cations (acceptor dopants). We did not pursue this alternative here, because acceptor-dopant cations, such as Fe3+ or Ni2+ , are known to interact strongly with oxygen vacancies, resulting in the vacancies’ dynamics being strongly modified. 37 In order to extract reliable conclusions on vacancy behaviour around dislocations, it is preferable to have relatively uncomplicated vacancy dynamics in the bulk cell. In Fig. 6 we plot the individual mean squared displacements of oxygen ions along the x, y and z directions obtained from one specific MD simulation. The respective tracer diffusion ∗,x ∗,y ∗,z coefficients are DO = 3.2 · 10−7 cm2 s−1 , DO = 3.3 · 10−7 cm2 s−1 and DO = 3.6 · 10−7



-6

10

A B C D E F

10

* O

2

/ cm s

-1

-7

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 33

-8

10

-9

10

0.6

0.8 3

T

10 ·

-1

1.0 -1

/ K

∗ Figure 5: Oxygen tracer diffusion coefficients DO as a function of inverse temperature obtained from MD simulations of bulk SrTiO3 with three different site fractions of oxygen vacancies, and two different methods for compensating the vacancies’ charge. A and B: nV = 0.333%; C and D: nV = 0.167% and E and F: nV = 0.049%. For A, C and E the charge is compensated by an uniform background charge; for B, D and F the charge is compensated by lowering the charge of all Ti cations.

cm2 s−1 . Hence, as expected for a cubic system, in which all possible migration jumps are identical, the three individual mean squared displacements, as well as their evolution with time, are essentially the same. There are of course some small variations that arise from the statistical nature of the simulations. Nevertheless, the data show that there is no preferred direction for oxygen migration. Since oxygen vacancies are present as a dilute solution (Fig. 5) and execute a threedimensional random walk in a cubic crystal (Fig. 6), one can relate the tracer diffusion coefficient of oxygen to the diffusion coefficient of the vacancies, DV , through

DV =

∗ DO , f ∗ nV

(5)

where f ∗ is the tracer correlation coefficient and equal to f ∗ = 0.69 for a dilute solution of oxygen vacancies in a cubic AB O3 perovskite. 38 Values of DV calculated for the three different values of nV are compared in Fig. 7 with each other and with literature data. 25,39,40 The two important results emerge from this comparison. First, the vacancy diffusiv10 ACS Paragon Plus Environment

Page 11 of 33

1.6 x direction y direction z direction

1.2

2

/ Å

2

1.4

rO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.0

0.8

0

50

100

t / ps

Figure 6: Individual mean squared displacements of oxygen ions along the x, y and z directions of the oxygen-deficient bulk cell. Site fraction of oxygen vacancies, nV = 0.333%, at a temperature of T = 2000 K. ity obtained from our MD simulations for three different vacancy site fractions are essenbulk tially the same, and exhibit the same activation enthalpy, ∆Hmig,V = (0.60 ± 0.05) eV bulk (not shown). That is, DV and ∆Hmig,V are, as expected for a dilute solution of non-

interacting vacancies, independent of nV . Considered together, one obtains from the datasets bulk ∆Hmig,V = (0.62±0.02) eV. Second, there is excellent agreement between our simulated data

and the vacancy diffusivity obtained from the analysis of experimental data, 25 both in terms bulk of the absolute values and in terms of the activation enthalpy of vacancy migration, ∆Hmig,V .

There is also good agreement with activation enthalpies of vacancy migration predicted from static simulations. 13,22–24,41 The MD data of Schie et al. 39 and of Marrocchelli et al. 40 show significantly worse agreement with experimental data, and we attribute this to the use in both cases of a different set of EPP. 42 Finally, we inspected the individual oxygen-ion trajectories: only jumps along the ⟨110⟩ directions were observed; no jumps along the longer ⟨100⟩ path took place. These two results are consistent with the results of static simulations 13 using the same EPP 20 in which the ⟨100⟩

⟨100⟩ jump was found to have a significant higher migration barrier, ∆Hmig,V = 4 eV, than ⟨110⟩

the ⟨110⟩ jump, ∆Hmig,V = 0.6 eV. On the timescale of our MD simulations, oxygen-ion



A H

mig

= (0.62 ± 0.02) eV

B

-4

C D

H

-5

mig

10

= 0.62 eV

V

2

/ cm s

-1

10

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 33

-6

10

H

mig

= 0.9 eV

-7

10

0.4

0.5

0.6

0.7 3

10 ·T

0.8 -1

0.9

1.0

1.1

-1

/ K

Figure 7: Diffusion coefficients of oxygen vacancies, DV , obtained for bulk SrTiO3 as a function of reciprocal temperature. A, this work with vacancy site fractions of (nV = 0.333%), (nV = 0.167%) and (nV = 0.049%), in each simulation the charge was compensated by an uniform background charge; B, from tracer diffusion experiments by De Souza et al. 25 ; C, from MD simulations by Schie et al. 39 ; and D, from MD simulations by Marrocchelli et al. 40 . jumps along the ⟨100⟩ directions are thus extremely unlikely to take place. 3.1.3

How good are these EPP? A summary

In general, one set of simple EPP will be not able to reproduce all physical and chemical properties of an oxide lattice equally well. Certain sets of EPP will perform better than others in specific cases. The EPP set derived by Goh et al. 35 , for instance, reproduce the heat capacity at constant pressure, Cp , and the thermal expansion coefficient, α, of SrTiO3 very well. Evidently, the EPP set of Pedone et al. 20 reproduces defect kinetics of SrTiO3 , such as the oxygen vacancy diffusivity DV , extremely well, at the expense of minor deviations for other properties, such as α and Cp . Given that the focus of this study is the kinetic behaviour of oxygen vacancies, the EPP set of Pedone et al. 20 provides a solid foundation.


Page 13 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.2 3.2.1

Results and discussion of the 6.0◦ grain boundary Grain-boundary structure and oxygen-vacancy distribution

MD simulations up to T = 2300 K of the stoichiometric GB cell indicated no change in the atomistic structure of the interfaces. The dislocations remained equidistantly spaced from one another and aligned along the interface. Their structure alternated between two different types, termed the TiO2 -type core and the SrO-type core. These results are consistent with those from the previous static simulation study 13 and the room-temperature High-Resolution Transmission Electron Microscopy (HR-TEM) by Zhang et al. 43 We did observe oxygen ions jumping into the cores from the surrounding oxygen ion positions. Once in the cores, however, these ions did not migrate along the cores but jumped back to their original sites (see Fig. 8), thus not contributing to long-range diffusion.

Figure 8: Ion trajectories obtained from an MD simulation of the stoichiometric 6.0◦ [100] tilt grain boundary. The SrO-type core is perpendicular to the viewing direction. The orange lines indicate successful ions jumps. Note: There are only jumps in and out of the core (along the viewing direction); there are no successful ion jumps along the core. For the oxygen-deficient GB cells, two sets of MD simulations were conducted. In the first approach, ten oxygen vacancies were placed randomly throughout the GB simulation cell. During the simulations they tended to segregate to the dislocation cores, residing at the positions marked in Fig. 9 or directly next to these positions. (The marked positions were discovered in previous static calculations 13 to be the energetic most favorable locations for oxygen vacancies.) The presence of an oxygen vacancy at a core induced the dissociation of the dislocation core, as described in detail elsewhere. 13,43 In the second approach, ten 13 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

vacancies were placed at one dislocation core (this corresponds to the removal of an entire column of oxygen ions from the simulation cell). Roughly half (5 ± 1) of these vacancies were still there at the end of the equilibration period; this number remained constant during the production run. It is stressed that value does not correspond to the equilibrium site fraction of vacancies within the core because the simulations were performed in the N pT (and not µpT ) ensemble. The other oxygen vacancies were found to have escaped into the bulk phase, that is, they were more than 5 jumps away from the dislocations core. Hence, we confirm the previous description obtained from static calculations 13 —that a[100] dislocation cores in SrTiO3 trap oxygen vacancies and are dissociated if an oxygen vacancy is trapped—and we now predict that this remains the case, even at elevated temperatures.

Figure 9: Atomistic structures of the two types of dislocation core found for the 6.0◦ [100] tilt grain boundary. (a) SrO-type core, (b) TiO2 -type core. The marked oxygen ion positions are the energetically most preferable positions for oxygen vacancies.

3.2.2

Oxygen tracer diffusion in the grain-boundary cell

2 For the stoichiometric GB cell, that is, without any oxygen vacancies, ⟨rO ⟩ maintained a

constant value during every MD simulation. That is, no long-range diffusion of oxygen took place. As noted above and indicated in Fig. 8, some oxygen ions did execute jumps into and out the free space offered by the dislocation cores, but they did not migrate along the cores. Evidently, despite there being space available for ions to migrate along the core, this alone is not sufficient to bring about fast ion migration along a dislocation in an oxide. We argued previously 13 that this is due to Coulomb interactions between the migrating ion and 14 ACS Paragon Plus Environment

Page 14 of 33

Page 15 of 33

1000 K

0.6

1100 K 1200 K 1300 K

0.5

1400 K 1500 K

0.4

1600 K

1800 K

0.3

1900 K 2000 K

2

/ Å

2

1700 K

rO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.2

0.1

0.0 0

20

40

60

80

100

120

140

t / ps

2 Figure 10: Mean squared displacement of the oxygen ions, ⟨rO ⟩, as a function of time t at various temperatures for the oxygen-deficient grain-boundary cell containing two anti-parallel dislocations arrays. Site fraction of oxygen vacancies, nV = 0.044%.

the surrounding ions. In this specific case, oxygen-ion migration along ⟨100⟩ in the AB O3 perovskite structure is characterised by a prohibitively high migration energy. 2 Fig. 10 shows values of ⟨rO ⟩ determined from MD simulations of the oxygen-deficient

grain-boundary cell, in which roughly half the vacancies reside at a dislocation core and the 2 remaining vacancies are distributed in the bulk phase. ⟨rO ⟩ is seen to increase linearly with ∗,cell increasing time. As before, we applied eq (4) to the data to obtain DO . Again, only those

data were considered for which a fitting correlation coefficient larger than 0.95 was obtained for linear regression to eq (4). These data are shown in Fig. 11 for the two GB simulation cells (roughly five vacancies either at a TiO2 -type core or at an SrO-type core during the simulations). Because oxygen vacancies not only reside at one dislocation but also in the bulk phase in ∗,cell these MD simulations, the tracer diffusion coefficients DO refer to all oxygen ions in the

simulation cell: those around the dislocation and those in the bulk. In order to extract from ∗,cell ∗,dv DO the tracer diffusion coefficient of oxygen in the d islocation’s v icinity, DO , we use a



A -7

B

s

-1

10

D*O / cm2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 33

-8

10

H

mig, A

H

mig, B

= (0.97 ± 0.05)

= (0.97 ± 0.05)

-9

10

0.5

0.6 3

10 ·

T

-1

0.7 -1

/ K

∗,cell Figure 11: Oxygen tracer diffusion coefficients, DO , obtained as a function of reciprocal temperature for the oxygen-deficient grain-boundary cell. A, SrO-type core; B TiO2 -type core. 10 vacancies placed initially at each core; roughly half of them present at the core during the production run.

Maxwell-Garnett approach. 44,45 ∗,cell DO =

∗,dv ∗,bulk ∗,dv DO [(3 − 2g)DO + 2gDO ] ∗,dv ∗,bulk DO (3 − g) + gDO

,

(6)

∗,dv where g is the volume fraction of material within which diffusion takes place with DO . As

there is no simple way to place an exact value on g, we consider a range of values. As a lower limit we take a radius of 0.6 nm, corresponding to the immediate vicinity of the core: this gives g = 0.039. As an upper limit we take a radius of 1.4 nm, corresponding to five jump distances away from the core: this gives g = 0.212. In this way we obtain two values of ∗,dv ∗,dv DO for each core type. We emphasise that, as DO refers to tracer diffusion in the vicinity

of the core and not along the core itself, it should be regarded as a bulk tracer diffusion coefficient that is modified by the presence of a dislocation. Furthermore, it is the tracer diffusion coefficient obtained from an N pT ensemble simulation; what we really require is the tracer diffusion coefficient from a µpT ensemble simulation, which, though difficult to implement, would include an equilibrium space-charge tube depleted of oxygen vacancies surrounding the dislocation.


Page 17 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


We now make the assumption that f ∗ does not change from the bulk value of 0.69 in order dv via eq (5). We are aware that f ∗ may differ from the bulk value owing to the to obtain DV

presence of a dislocation, but taking this effect into account is outside the scope of this study. dv bulk In Fig. 12 we compare results with values of Dbulk V . We find that D V is lower than D V

for both limiting values of g. In addition, the activation enthalpies of vacancy migration dv are also significantly higher, ∆Hmig,V = (1.5 ± 0.1) eV, than the value obtained for the bulk, bulk = (0.62 ± 0.03) eV. By examining the individual mean squared displacements of ∆Hmig,V

oxygen ions along the x, y and z directions of the cell, we can make sense of this behaviour. These individual values are plotted in Fig. 13; the corresponding tracer diffusion coefficients ∗,x ∗,y ∗,z are DO = 5.5 · 10−8 cm2 s−1 ; DO = 5.2 · 10−8 cm2 s−1 and DO = 5.6 · 10−8 cm2 s−1 . These

are seen to be essentially the same, indicating that there is no preferred diffusion direction, despite the presence of the four dislocation cores (two in each boundary). (Again, no ion jumps along the cores were observed.) This leads us to conclude that, as vacancies reside prefentially at the dislocation cores, the dislocations act as one-dimensional traps for oxygen vacancies. This is analagous to the case of acceptor dopants binding oxygen vacancies, 37 but here the trapping center is not another point defect, but a line defect. Sometimes the vacancies escape from the trap, migrate for a while in the bulk phase before being trapped again. The higher activation enthalpy is thus an effective value due to the (strong) trapping ,dv of the vacancies at the core. That values of DV for the two cores are essentially the same

supports our assertion that it is the bulk vacancy dynamics that are being modified by the dislocations.

3.3

Discussion of literature data

It was discussed previously 13,46,47 in detail the reasons why experimental reports 2,16–18 of fast diffusion of oxygen along dislocations in SrTiO3 are not unambiguous. Recently Marrocchelli et al. 40 examined a[110] dislocations in SrTiO3 using classical simulation techniques. They found, confirming the results of earlier studies, 13–15 that oxygen vacancies formed prefer17 ACS Paragon Plus Environment


-3

10

-4

10

-5

V

2

/ cm s

-1

10

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 33

-6

10

-7

10

Bulk SrO core,

-8

10

SrO core, TiO core, 2

TiO core, -9

2

g g g g

= 0.039, = 0.212, = 0.039, = 0.212,

H H H H

mig

= (1.5 ± 0.2) eV

mig

mig

mig

= (1.6 ± 0.2) eV = (1.4 ± 0.1) eV = (1.5 ± 0.1) eV

10

0.4

0.5

0.6 3

10 ·

T

-1

0.7 -1

/ K

Figure 12: Vacancy diffusivities obtained for diffusion in the vicinity of dislocation cores, bulk Ddv V , and in the bulk, D V , as a function of inverse temperature. Data points obtained at the lowest temperatures are not considered in the analysis due to the large scatter. entially at the dislocations, making the dislocations easier to reduce and leading to the formation of space-charge zones.1 Their simulations indicated D∗,dis > D∗,bulk (though only O O dis bulk by a factor of four at T = 1200 K) but ∆HD ∗ > ∆HD ∗ . Using a definition of pipe diffuO O

sion employed sometimes in the literature—one that is based on activation barriers—, they concluded that there is no pipe diffusion. This definition of pipe diffusion should be avoided because it leads to inconsistencies (no pipe diffusion despite Ddis > Dbulk ) and because it conflates the definition of fast diffusion with the reason why fast diffusion may occur. Although the results of Marrocchelli et al. 40 apparently do not accord with the data we obtained in this study, it is evident from the values of D∗,dis and ndis V reported by Marrocchelli O bulk et al. 40 that, also in that case, Ddis V < D V . That is, even though the type of dislocation ex-

amined by Marrocchelli et al. 40 , an a[110] dislocation, has a different atomistic structure 48,49 to the type of dislocations examined here (a[100] 13,43 ); and even though Marrocchelli et al. 40 used a different set of EPP 42 to the EPP used here; 20 both studies agree that there is no accelerated diffusion of oxygen vacancies along dislocations in SrTiO3 . 1 The predictions of Marrocchelli et al. 40 concerning vacancy concentrations at the core and in the attendant space-charge zone, as well as the extent of the space-charge zone, should be regarded with caution. The values were obtained directly from N pT (rather than µpT ) simulations and thus do not refer to equilibrium. In addition, data were interpreted using expressions derived for a 1D Cartesian system, whereas a single isolated dislocation conforms in the ideal case to 1D radial symmetry.


Page 19 of 33

0.20 x direction y direction z direction

2

0.15

rO / Å

0.10

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.05

0.00 0

20

40

60

80

100

t / ps

Figure 13: Individual mean squared displacements of oxygen ions along the x, y and z directions of the oxygen-deficient grain-boundary cell. Site fraction of oxygen vacancies, nV = 0.044%, at a temperature of T = 2000 K. There is a small question that remains concerning the absolute values predicted by Marrocchelli et al. 40 because the simulation cell they employed contained two complementary dislocations, rather than two arrays as low-angle tilt grain boundaries. Consequently, there is a substantial strain field along and across the cell that results from the requirement of periodic boundary conditions (that is, a strain field additional to the individual strain field of each dislocation). If one compares diffusion coefficients for this cell (bulk + dislocations + additional strain field) with diffusion coefficients for a bulk cell (only bulk), one has two possible origins for any differences observed: the dislocation itself, and the effect of the additional strain field on the dislocation. Although this effect may be small, it is nevertheless unclear at present how small it is; and a decrease in activation energy of only 0.15 eV through strain 50,51 would be sufficient to increase the diffusion coefficient in the core at T = 1200 K by a factor of four. Alternative cell geometries should be considered, therefore, when investigating pipe diffusion. It is worth emphasising one point regarding the experimental detection of fast diffusion. A variety of experimental and theoretical studies indicate directly or indirectly that both edge and screw dislocations in SrTiO3 are enveloped in space-charge tubes, in which oxygen



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

vacancies are strongly depleted. 14,15,43,52–59 Detecting experimentally the fast diffusion of oxygen along such dislocations, if such diffusion does indeed occur, will be very difficult. Since the flux of oxygen isotope is prevented by the depletion space-charge tube from leaking out of the core, one needs to be able to detect the miniscule amount of isotope sitting at the dislocation core. The classical dislocation diffusion experiment (fast diffusion, no depletion space-charge zone) is not bound by this requirement, for it is the diffusion of isotope out of the dislocation (with the bulk diffusion coefficient) that provides at large penetration depths sufficient isotope for the fast path to be detected. 60–64

3.4

Filament formation in resistively switching oxides: decoupled ionic and electronic paths

Although resistive switching in SrTiO3 was not studied in this paper, our results have implications for the formation of switching filaments. We have shown here and elsewhere 13,47 that there is no evidence of fast diffusion of oxygen along dislocations in SrTiO3 . Evidently, the fast diffusion of oxygen along dislocations is not a prerequisite for resistive switching to occur. Switching filaments can form and switching can take place in regions in SrTiO3 that are free of dislocations. 65 And only bulk diffusion coefficients of oxygen are necessary to decribe quantitatively the kinetics of resistive switching. 66 To form a filament, all that is required initially is a path along which a substantial electronic current can flow. 67 This path can indeed be provided by a dislocation; or to be precise, by the accumulated electrons in the surrounding space-charge tube (see Fig. 14). It is unclear, but relatively unimportant, if electron transport occurs along the actual dislocation core (as opposed to within the space-charge zone); the electron mobility within the core may be significantly diminished due to the strong perturbation of the bulk structure. The important point is that the substantial electronic current that flows through the space-charge tube around the extended defect raises the temperature of the surrounding material through Joule heating. Somewhere in these surroundings, outside the space-charge tube that is depleted of oxygen vacancies, 20 ACS Paragon Plus Environment

Page 20 of 33

Page 21 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


there are sufficient vacancies and their mobility has been increased sufficiently by the rise in temperature, so that a significant oxygen-ion conductivity results. The transport of ions along this path leads to the local reduction of the oxide and thus to the formation of the conductive filament. The paths for electrons and ions during filament formation are, therefore, decoupled. In this regard it is worth noting that a dislocation in a real crystal is unlikely to be an entirely straight line. It may contain jogs and kinks, and its edge/screw character may vary along its length. Such structural complexities will conceivably affect oxygen diffusion close to the dislocation core much more strongly than the electron transport within the attendant space-charge tube. Oxygen-ion migration is sensitive to the local crystal structure around the core, and it may even be impeded by the structural complexities. In contrast, although the space-charge tube may be modified by the structural complexities of the dislocation core, the electronic path is unlikely to be disrupted completely, as long as the space-charge screening length is long compared with the extent of the structural complexity and as long as there are sufficient traps for oxygen vacancies along the entire length of the dislocation. This is essentially an argument in favour of de-coupled paths for electrons and ions in filament formation. Without a dislocation or other extended defect present, there seems to be no mechanism to stabilise the high concentration of oxygen vacancies. An extended defect provides a structural perturbation to which oxygen vacancies will segregate, forming attendent depletion space-charge zones in the surrounding bulk phase, if the vacancy formation energy is lower at the defect than in the bulk. 14,15 In the absence of an extended defect a different mechanism may take over, stabilising the high concentration of oxygen vacancies within the filament: electron trapping. Whereas two doubly charged vacancies are predicted to repel each other, 39 two neutral vacancies are predicted to associate into a cluster. 41 It is an open question, however, whether one may extrapolate the behaviour of binary clusters at zero Kelvin to the defect concentrations at finite temperatures relevant to a switching filament. In this



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 14: (a) Equilibrium concentrations of point defects at a dislocation in SrTiO3 plotted as a function of distance from the center of the dislocation, r, for T = 800 K, aO2 = 10−18 and cdop = 5 · 1017 cm−3 with a thermodynamic driving force for space-charge formation of ∆g{VO·· } = −1.7 eV. Data calculated according to a thermodynamically self-consistent procedure; a brief summary in given in the Appendix. (b) Radial distribution of local equilibrium conductivities calculated using the concentration profiles shown in (a). regard, detailed experimental studies and also further computational studies that examine × ·· · the interactions between oxygen vacancies of various charge states (VO , VO , VO ) and at

various concentrations are required.

4

Conclusions

The present study examined, by means of molecular dynamic simulations with empirical pair potentials, the structural properties of, and point-defect behaviour at, a[100] edge dislocations in SrTiO3 at finite temperatures. To this end, we conducted simulations of cells containing two anti-parallel 6.0◦ [100] tilt grain boundaries, to take advantage of the fact that a low-angle tilt grain boundary consists of a periodic array of edge dislocations. The main results are summarized as follows: • There is no transport of oxygen ions along a[100] dislocations in either stoichiometric or oxygen-deficient SrTiO3 , according to (a) the jump trajectories of oxygen ions; to (b) the tracer diffusion coefficients obtained; and to (c) the individual mean squared displacements along x, y and z directions. 22 ACS Paragon Plus Environment

Page 22 of 33

Page 23 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


• Oxygen diffusion in the dislocations’ vicinity (dv) was observed, and this was characterised by lower tracer diffusion coefficients and a higher activation enthalpy relative to the corresponding values for the bulk phase (e.g. ∆HOdv = 1.5 eV versus ∆HObulk = 0.6 eV). This behaviour was explained in terms of a[100] edge dislocations acting as one-dimensional traps for oxygen vacancies. • A de-coupled model of filament formation in resistively switching SrTiO3 was proposed, in which ions and electrons move along different paths. The proposal is based on (a) the presence of space-charge zones surrounding the dislocations, in which oxygen vacancies are depleted but electrons are accumulated, and (b) dislocations exhibiting no accelerated diffusion of oxygen.

5

Appendix

In order to predict the spatial dependence of point-defect concentrations around a dislocation in SrTiO3 , we assume that the system is in electrochemical equilibrium. Consequently, having specified (i) equilibrium point-defect concentrations in the bulk phase, cbdef , for the given temperature T , oxygen activity aO2 and dopant concentration cdop ; 68 and (ii) the thermodynamic driving forces for space-charge formation, ∆g{def} , that connect the defect chemistry of the extended defect to that of the bulk phase; one can obtain, with the relevant electrostatic relationship, the space-charge potential Φ0 and the charge at the extended defect, Qdis , in a self-consistent manner. 14,15,68,69 We derive the variation in the electrical potential around a charged, isolated dislocation within the depletion approximation. To this end we assume (a) that there is no variation in potential along the dislocation tube, such that a 1D radial treatment is sufficient; (b) that the positive charge of the dislocation core is confined to a tube of radius rdis ; (c) that the dopant is immobile; and (d) that the space-charge zone extends out from rdis to rscz , and in this region the concentration of charge-compensating oxygen vacancies can 23 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 33

be neglected (depletion approximation). The relevant Poisson equation is therefore (with dielectric permittivity ϵr ϵ0 ) 1 d r dr

( ) dϕ ecdop r =+ , dr ϵr ϵ0

(7)

which can be solved analytically to give     Φ0 , 0 ≤ r ≤ rdis     2 )−1−ln(r 2 /r 2 ) scz ϕ(r) = Φ0 (r2 2 /rscz rdis ≤ r ≤ rscz 2 )−1−ln(r 2 /r 2 ) , (rdis /rscz  dis scz      0, rscz < r

(8)

Applying Gauss’ law to eq (8) provides the electrostatic coupling condition,

Qdis = 4ϵr ϵ0 πΦ0

2 2 1 − (rdis /rscz ) ; 2 2 2 2 ) (rdis /rscz ) − 1 − ln(rdis /rscz

global charge neutrality yields

(9)

√ rscz =

Qdis 2 + rdis . ecdop π

(10)

For T = 800 K, aO2 = 10−18 and cdop = 5 · 1017 cm−3 with a thermodynamic driving force for space-charge formation of ∆g{VO·· } = −1.7 eV, we obtain Φ0 = 0.38 V and Qdis = 6.7 × 10−10 C m−1 . The point-defect concentrations shown in Fig. 14 follow from [

cdef (r) =

cbdef

] zdef eϕ(r) exp − , kB T

(11)

and the conductivities, from σdef (r) = cdef (r)zdef eudef , with mobilities udef taken from Refs. 25,70


Page 25 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Acknowledgement The authors thank A. H. H. Ramadan and B. Yildiz for useful discussions. The authors acknowledge funding from the German Science Foundation (DFG) within the collaborative research centre, SFB 917 ‘Nanoswitches’ and the provision of computational time from the IT-Center of RWTH Aachen University.

References (1) Waser, R.; Dittmann, R.; Staikov, G.; Szot, K. Redox-Based Resistive Switching Memories − Nanoionic Mechanisms, Prospects, and Challenges. Adv. Mater. 2009, 21, 2632– 2663. (2) Szot, K.; Speier, W.; Carius, R.; Zastrow, U.; Beyer, W. Localized Metallic Conductivity and Self-Healing during Thermal Reduction of SrTiO3 . Phys. Rev. Lett. 2002, 88, 075508. (3) Szot, K.; Speier, W.; Bihlmayer, G.; Waser, R. Switching the Electrical Resistance of Individual Dislocations in Single-Crystalline SrTiO3 . Nat. Mater. 2006, 5, 312–320. (4) McKenna, K.; Shluger, A. The Interaction of Oxygen Vacancies with Grain Boundaries in Monoclinic HfO2 . Appl. Phys. Lett. 2009, 95, 222111. (5) Lanza, M.; Zhang, K.; Porti, M.; Nafría, M.; Shen, Z. Y.; Liu, L. F.; Kang, J. F.; Gilmer, D.; Bersuker, G. Grain Boundaries as Preferential Sites for Resistive Switching in the HfO2 Resistive Random Access Memory Structures. Appl. Phys. Lett. 2012, 100, 123508. (6) Le Claire, A. D. In Diffusion in Solid Metals and Alloys; Mehrer, H., Ed.; LandoltBörnstein New Series III; Springer-Verlag: Berlin Heidelberg, 1990; Vol. 26; Chapter 11, p 626. 25 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(7) Kaur, I.; Mishin, Y.; Gust, W. Fundamentals of Grain and Interphase Boundary Diffusion; J. Wiley & Sons: Chichester, 1995. (8) Peterson, N. Grain-Boundary Diffusion in Metals. Int. Mater. Rev. 1983, 28, 65–91. (9) Mehrer, H. Diffusion in Solids; Springer: Berlin Heidelberg New York, 2007. (10) Nakagawa, T.; Nakamura, A.; Sakaguchi, I.; Shibata, N.; Lagerlöf, K. P. D.; Yamamoto, T.; Haneda, H.; Ikuhara, Y. Oxygen Pipe Diffusion in Sapphire Basal Dislocation. J. Ceram. Soc. Japan 2006, 114, 1013–1017. (11) Atkinson, A.; Taylor, R. I. The Diffusion of Ni in the Bulk and along Dislocations in NiO Single Crystals. Phil. Mag. A 1979, 39, 581–595. (12) Sakaguchi, I.; Yurimoto, H.; Sueno, S. Self-Diffusion along Dislocations in SingleCrystals MgO. Solid State Commun. 1992, 84, 889 – 893. (13) Metlenko, V.; Ramadan, A. H. H.; Gunkel, F.; Du, H.; Schraknepper, H.; HoffmannEifert, S.; Dittmann, R.; Waser, R.; De Souza, R. A. Do Dislocations act as Atomic Autobahns for Oxygen in the Perovskite Oxide SrTiO3 ? Nanoscale 2014, 6, 12864– 12876. (14) McIntyre, P. C. Equilibrium Point Defect and Electronic Carrier Distributions near Interfaces in Acceptor-Doped Strontium Titanate. J. Am. Ceram. Soc. 2000, 83, 1129– 1136. (15) De Souza, R. A. The Formation of Equilibrium Space-Charge Zones at Grain Boundaries in the Perovskite Oxide SrTiO3 . Phys. Chem. Chem. Phys. 2009, 11, 9939–9969. (16) Waugh, J. S.; Paladino, A. E.; di Benedetto, B.; Wantman, R. Effect of Dislocations on Oxidation and Reduction of Single-Crystal SrTiO3 . J. Am. Ceram. Soc. 1963, 46, 60.


Page 26 of 33

Page 27 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(17) Paladino, A. E.; Rubin, L. G.; Waugh, J. S. Oxygen Ion Diffusion in Single Crystal SrTiO3 . J. Phys. Chem. Solids 1965, 26, 391–397. (18) Amante, J. C.; Cawley, J. D.; Kim, J.; Lemberger, T. R. In Point Defects and Related Properties of Ceramics, Ceram. Trans.; Mason, T. O., Routbort, J. L., Eds.; American Ceramic Society: Westerville, Ohio, 1991; Vol. 24; pp 303–312. (19) Minervini, L.; Zacate, M. O.; Grimes, R. W. Defect Cluster Formation in M2 O3 -Doped CeO2 . Solid State Ionics 1999, 116, 339 – 349. (20) Pedone, A.; Malavasi, G.; Menziani, M. C.; Cormack, A. N.; Segre, U. A New SelfConsistent Empirical Interatomic Potential Model for Oxides, Silicates, and SilicaBased Glasses. J. Phys. Chem. B 2006, 110, 11780–11795. (21) Yeandel, S. R.; Molinari, M.; Parker, S. C.; Freer, R. unpublished work. (22) Walsh, A.; Catlow, C. R. A.; Smith, A. G. H.; Sokol, A. A.; Woodley, S. M. Strontium Migration Assisted by Oxygen Vacancies in SrTiO3 from Classical and Quantum Mechanical Simulations. Phys. Rev. B 2011, 83, 220301. (23) Mizoguchi, T.; Takahashi, N.; Lee, H.-S. First-Principles Study on Migration Mechanism in SrTiO3 . Appl. Phys. Lett. 2011, 98, 091909. (24) Lontsi-Fomena, M.; Villesuzanne, A.; Doumerc, J.-P.; Frayret, C.; Pouchard, M. A Density Functional Theory Study of Oxygen Diffusion in LaAlO3 and SrTiO3 . Comput. Mater. Sci. 2008, 44, 53–60. (25) De Souza, R. A.; Metlenko, V.; Park, D.; Weirich, T. E. Behavior of Oxygen Vacancies in Single-Crystal SrTiO3 : Equilibrium Distribution and Diffusion Kinetics. Phys. Rev. B 2012, 85, 174109. (26) Hackmann, A.; Kanert, O. NMR Investigation of Defect Properties in Single Crystal SrTiO3 . Radiat. Eff. Defects Solids 1991, 119-121, 651–656. 27 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(27) Cordero, F. Hopping and Clustering of Oxygen Vacancies in SrTiO3 by Anelastic Relaxation. Phys. Rev. B 2007, 76, 172106. (28) Schwarz, D. B.; Anderson, H. U. Determination of Oxygen Chemical Diffusion Coefficients in Single Crystal SrTiO3 by Capacitance Manometry. J. Electrochem. Soc. 1975, 122, 707–710. (29) Gömann, K.; Borchardt, G.; Schulz, M.; Gömann, A.; Maus-Friedrichs, W.; Lesage, B.; Kaitasov, O.; Hoffmann-Eifert, S.; Schneller, T. Sr Diffusion in Undoped and La-Doped SrTiO3 Single Crystals under Oxidizing Conditions. Phys. Chem. Chem. Phys. 2005, 7, 2053–2060. (30) Meyer, R.; Waser, R.; Helmbold, J.; Borchardt, G. Observation of Vacancy Defect Migration in the Cation Sublattice of Complex Oxides by O18 Tracer Experiments. Phys. Rev. Lett. 2003, 90, 105901. (31) Plimpton, S. Fast Parallel Algorithms for Short Range Molecular Dynamics. J. Comput. Phys. 1995, 117, 1–19. (32) http://lammps.sandia.gov. (33) Bunting, E. N.; Sheldon, G. R.; Creamer, A. S. Properties of Barium-Strontium Titanate Dielectrics. J. Am. Ceram. Soc. 1947, 4, 114–125. (34) Tosawat, S.; Gjindara, W.; Chanchana, T.; Vittaya, A. Molecular Dynamics Simulation of Strontium Titanate. Chin. Phys. Lett. 2010, 27, 026501. (35) Goh, W. F.; Yoon, T. L.; Khan, S. A. Molecular Dynamics Simulation of Thermodynamic and Thermal Transport Properties of Strontium Titanate with Improved Potential Parameters. Comput. Mater. Sci. 2012, 60, 123 – 129. (36) de Ligny, D.; Richet, P. High-Temperature Heat Capacity and Thermal Expansion of SrTiO3 and SrZrO3 Perovskites. Phys. Rev. B 1996, 53, 3013–3022. 28 ACS Paragon Plus Environment

Page 28 of 33

Page 29 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(37) Schie, M.; Waser, R.; De Souza, R. A. A Simulation Study of Oxygen-Vacancy Behavior in Strontium Titanate: Beyond Nearest-Neighbor Interactions. J. Phys. Chem. C 2014, 118, 15185–15192. (38) Ishigaki, T.; Yamauchi, S.; Kishio, K.; Mizusaki, J.; Fueki, K. Diffusion of Oxide Ion Vacancies in Perovskite-Type Oxides. J. Solid State Chem. 1988, 73, 179 – 187. (39) Schie, M.; Marchewka, A.; Müller, T.; De Souza, R. A.; Waser, R. Molecular Dynamics Simulations of Oxygen Vacancy Diffusion in SrTiO3 . J. Phys.: Condens. Matter 2012, 24, 485002. (40) Marrocchelli, D.; Sun, L.; Yildiz, B. Dislocations in SrTiO3 : Easy To Reduce but Not so Fast for Oxygen Transport. J. Am. Chem. Soc. 2015, 137, 4735–4748. (41) Cuong, D. D.; Lee, B.; Choi, K. M.; Ahn, H.-S.; Han, S.; Lee, J. Oxygen Vacancy Clustering and Electron Localization in Oxygen-Deficient SrTiO3 : LDA+U Study. Phys. Rev. Lett. 2007, 98, 115503. (42) Thomas, B.; Marks, N.; Begg, B. Developing Pair Potentials for Simulating Radiation Damage in Complex Oxides. Nucl. Instrum. Methods Phys. Res., Sect. B 2005, 228, 288 – 292. (43) Zhang, Z.; Sigle, W.; Kurtz, W.; Rühle, M. Electronic and Atomic Structure of a Dissociated Dislocation in SrTiO3 . Phys. Rev. B 2002, 66, 214112. (44) Kalnin, J.; Kotomin, E.; Maier, J. Calculations of the Effective Diffusion Coefficient for Inhomogeneous Media. J. Phys. Chem. Solids 2002, 63, 449 – 456. (45) Belova, I.; Murch, G. Diffusion in Nanocrystalline Materials. J. Phys. Chem. Solids 2003, 64, 873 – 878. (46) De Souza, R. A. Oxygen Diffusion in SrTiO3 and Related Perovskite Oxides. Adv. Funct. Mat. 2015, 25, 6326–6342. 29 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(47) Schraknepper, H.; De Souza, R. A. Competing Descriptions of Diffusion Profiles with Two Features: Surface Space-Charge Layer versus Fast Grain-Boundary Diffusion. J. Appl. Phys. 2016, 119, 064903. (48) Jia, C. L.; Thust, A.; Urban, K. Atomic-Scale Analysis of the Oxygen Configuration at a SrTiO3 Dislocation Core. Phys. Rev. Lett. 2005, 95, 225506. (49) Hirel, P.; Mrovec, M.; Elsässer, C. Atomistic Simulation Study of Dislocations in Strontium Titanate. Acta Mater. 2012, 60, 329 – 338. (50) Yang, Q.; Cao, J. X.; Ma, Y.; Zhou, Y. C.; Jiang, L. M.; Zhong, X. L. Strain Effects on Formation and Migration Energies of Oxygen Vacancy in Perovskite Ferroelectrics: A First-Principles Study. J. Appl. Phys. 2013, 113, 184110. (51) Al-Hamadany, R.; Goss, J. P.; Briddon, P. R.; Mojarad, S. A.; Al-Hadidi, M.; Oapos; Neill, A. G.; Rayson, M. J. Oxygen Vacancy Migration in Compressively Strained SrTiO3 . J. Appl. Phys. 2013, 113, 024108. (52) De Souza, R. A.; Fleig, J.; Maier, J.; Kienzle, O.; Zhang, Z.; Sigle, W.; Rühle, M. Electrical and Structural Characterization of a Low-Angle Tilt Grain Boundary in Iron-Doped Strontium Titanate. J. Am. Ceram. Soc. 2003, 86, 922–928. (53) De Souza, R. A.; Fleig, J.; Maier, J.; Zhang, Z.; Sigle, W.; Rühle, M. Electrical Resistance of Low-Angle Tilt Grain Boundaries in Acceptor-Doped SrTiO3 as a Function of Misorientation Angle. J. Appl. Phys. 2005, 97, 053502. (54) Zhang, Z.; Sigle, W.; De Souza, R. A.; Kurtz, W.; Maier, J.; Rühle, M. Comparative Studies of Microstructure and Impedance of Small-Angle Symmetrical and Asymmetrical Grain Boundaries in SrTiO3 . Acta Mater. 2005, 53, 5007–5015. (55) De Souza, R. A.; Maier, J. Capacitance of Single Crystal and Low-Angle Tilt Bicrystals of Fe-Doped SrTiO3 . Faraday Discuss. 2007, 134, 235–245. 30 ACS Paragon Plus Environment

Page 30 of 33

Page 31 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(56) Guo, X.; Zhang, Z.; Sigle, W.; Wachsman, E.; Waser, R. Schottky Barrier Formed by Network of Screw Dislocations in SrTiO3 . Appl. Phys. Lett. 2005, 87, 162105. (57) Zhang, Z.; Sigle, W.; Kurtz, W. HRTEM and EELS Study of Screw Dislocation Cores in SrTiO3 . Phys. Rev. B 2004, 69, 144103. (58) Jia, C. L.; Thust, A.; Urban, K. Atomic-Scale Analysis of the Oxygen Configuration at a SrTiO3 Dislocation Core. Phys. Rev. Lett. 2005, 95, 225506. (59) Waser, R.; Hagenbeck, R. Grain Boundaries in Dielectric and Mixed-Conducting Ceramics. Acta Mater. 2000, 48, 797–825. (60) Le Claire, A. D. The Analysis of Grain Boundary Diffusion Measurements. Br. J. Appl. Phys. 1963, 14, 351–356. (61) Le Claire, A. D.; Rabinovitch, A. A Mathematical Analysis of Diffusion in Dislocations. I. Application to Concentration ’Tails’. J. Phys. C: Solid State Phys. 1981, 14, 3863– 3879. (62) Atkinson, A. Diffusion Along Grain Boundaries and Dislocations in Oxides, Alkali Halides and Carbides. Solid State Ionics 1984, 12, 309–320. (63) De Souza, R. A.; Martin, M. Probing Diffusion Kinetics with Secondary Ion Mass Spectrometry. MRS Bull. 2009, 34, 907–914. (64) Fielitz, P.; Borchardt, G.; Schmücker, M.; Schneider, H. How to Measure Volume Diffusivities and Grain Boundary Diffusivities of Oxygen in Polycrystalline Oxides. Solid State Ionics 2003, 160, 75–83. (65) Kamaladasa, R. J.; Noman, M.; Chen, W.; Salvador, P. A.; Bain, J. A.; Skowronski, M.; Picard, Y. N. Dislocation Impact on Resistive Switching in Single-Crystal SrTiO3 . J. Appl. Phys. 2013, 113, 234510.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(66) Menzel, S.; Waters, M.; Marchewka, A.; Böttger, U.; Dittmann, R.; Waser, R. Origin of the Ultra-Nonlinear Switching Kinetics in Oxide-Based Resistive Switches. Adv. Funct. Mater. 2011, 21, 4487–4492. (67) Lenser, C.; Connell, Z.; Kovács, A.; Dunin-Borkowski, R.; Köhl, A.; Waser, R.; Dittmann, R. Identification of Screw Dislocations as Fast-Forming Sites in Fe-Doped SrTiO3 . Appl. Phys. Lett. 2013, 102, 183504. (68) De Souza, R. A.; Gunkel, F.; Hoffmann-Eifert, S.; Dittmann, R. Finite-Size versus Interface-Proximity Effects in Thin-Film Epitaxial SrTiO3 . Phys. Rev. B 2014, 89, 241401. (69) De Souza, R. A.; Munir, Z. A.; Kim, S.; Martin, M. Defect Chemistry of Grain Boundaries in Proton-Conducting Solid Oxides. Solid State Ionics 2011, 196, 1 – 8. (70) Moos, R.; Härdtl, K. H. Defect Chemistry of Donor-Doped and Undoped Strontium Titanate Ceramics between 1000◦ and 1400◦ C. J. Am. Ceram. Soc. 1997, 80, 2549– 2562.


Page 32 of 33

Page 33 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Graphical TOC Entry


Computational Study of Oxygen Diffusion along a[100] Dislocations in

Recommend Documents