Resistive Switching E-Book
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- Herausgeber: Wiley-VCH
- Kategorie: Fachliteratur
- Sprache: Englisch
With its comprehensive coverage, this reference introduces readers to the wide topic of resistance switching, providing the knowledge, tools, and methods needed to understand, characterize and apply resistive switching memories.
Starting with those materials that display resistive switching behavior, the book explains the basics of resistive switching as well as switching mechanisms and models. An in-depth discussion of memory reliability is followed by chapters on memory cell structures and architectures, while a section on logic gates rounds off the text.
An invaluable self-contained book for materials scientists, electrical engineers and physicists dealing with memory research and development.
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Veröffentlichungsjahr: 2015
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Table of Contents
Cover
Related Titles
Title Page
Copyright
Preface
List of Contributors
Chapter 1: Introduction to Nanoionic Elements for Information Technology
1.1 Concept of Two-Terminal Memristive Elements
1.2 Memory Applications
1.3 Logic Circuits
1.4 Prospects and Challenges
Acknowledgments
References
Chapter 2: ReRAM Cells in the Framework of Two-Terminal Devices
2.1 Introduction
2.2 Two-Terminal Device Models
2.3 Fundamental Description of Electronic Devices with Memory
2.4 Device Engineer's View on ReRAM Devices as Two-Terminal Elements
2.5 Conclusions
Acknowledgment
References
Chapter 3: Atomic and Electronic Structure of Oxides
3.1 Introduction
3.2 Crystal Structures
3.3 Electronic Structure
3.4 Material Classes and Characterization of the Electronic States
3.5 Electronic Structure of Selected Oxides
3.6 Ellingham Diagram for Binary Oxides
Acknowledgments
References
Chapter 4: Defect Structure of Metal Oxides
4.1 Definition of Defects
4.2 General Considerations on the Equilibrium Thermodynamics of Point Defects
4.3 Definition of Point Defects
4.4 Space-Charge Effects
4.5 Case Studies
References
Chapter 5: Ion Transport in Metal Oxides
5.1 Introduction
5.2 Macroscopic Definition
5.3 Microscopic Definition
5.4 Types of Diffusion Experiments
5.5 Mass Transport along and across Extended Defects
5.6 Case Studies
Acknowledgments
References
Chapter 6: Electrical Transport in Transition Metal Oxides
6.1 Overview
6.2 Structure of Transition Metal Oxides
6.3 Models of Electrical Transport
6.4 Band Insulators
6.5 Half-Filled Mott Insulators
6.6 Temperature-Induced Metal–Insulator Transitions in Oxides
References
Chapter 7: Quantum Point Contact Conduction
7.1 Introduction
7.2 Conductance Quantization in Metallic Nanowires
7.3 Conductance Quantization in Electrochemical Metallization Cells
7.4 Filamentary Conduction and Quantization Effects in Binary Oxides
7.5 Conclusion and Outlook
References
Chapter 8: Dielectric Breakdown Processes
8.1 Introduction
8.2 Basics of Dielectric Breakdown
8.3 Physics of Defect Generation
8.4 Breakdown and Oxide Failure Statistics
8.5 Implications of Breakdown Statistics for ReRAM
8.6 Chemistry of the Breakdown Path and Inference on Filament Formation
8.7 Summary and Conclusions
References
Chapter 9: Physics and Chemistry of Nanoionic Cells
9.1 Introduction
9.2 Basic Thermodynamics and Heterogeneous Equilibria
9.3 Phase Boundaries and Boundary Layers
9.4 Nucleation and Growth
9.5 Electromotive Force
9.6 General Transport Processes and Chemical Reactions
9.7 Solid-State Reactions
9.8 Electrochemical (Electrode) Reactions
9.9 Stoichiometry Polarization
Summary
Acknowledgments
References
Chapter 10: Electroforming Processes in Metal Oxide Resistive-Switching Cells
10.1 Introduction
10.2 Forming Mechanisms
10.3 Technical Issues Related to Forming
10.4 Summary and Outlook
Acknowledgments
References
Chapter 11: Universal Switching Behavior
11.1 General Properties of ReRAMs and Their Universal Behavior
11.2 Explaining the Universal Switching of ReRAM
11.3 Variable-Diameter Model
11.4 Variable-Gap Model
11.5 Coexistence of Variable-Gap/Variable-Diameter States
11.6 Summary
Acknowledgment
References
Chapter 12: Quasistatic and Pulse Measuring Techniques
12.1 Brief Introduction to Electronic Transport Testing of ReRAM
12.2 Quasistatic Measurement of Current–Voltage Characteristics
12.3 Current Compliance and Overshoot Effects
12.4 Pulsed Measurements for the Study of Switching Dynamics
12.5 Conclusions
Acknowledgment
References
Chapter 13: Unipolar Resistive-Switching Mechanisms
13.1 Introduction to Unipolar Resistive Switching
13.2 Principle of Unipolar Switching
13.3 Unipolar-Switching Mechanisms in Model System Pt/NiO/Pt
13.4 Influence of Oxide and Electrode Materials on Unipolar-Switching Mechanisms
13.5 Conclusion
References
Chapter 14: Modeling the VCM- and ECM-Type Switching Kinetics
14.1 Introduction
14.2 Microscopic Switching Mechanism of VCM Cells
14.3 Microscopic Switching Mechanism of ECM Cells
14.4 Classification of Simulation Approaches
14.5 General Considerations of the Physical Origin of the Nonlinear Switching Kinetics
14.6 Modeling of VCM Cells
14.7 Modeling of ECM Cells
14.8 Summary and Outlook
Acknowledgment
References
Chapter 15: Valence Change Observed by Nanospectroscopy and Spectromicroscopy
15.1 Introduction
15.2 Methods and Techniques
15.3 Interface Phenomena
15.4 Localized Redox Reactions in Transition Metal Oxides
15.5 Conclusions
Acknowledgment
References
Chapter 16: Interface-Type Switching
16.1 Introduction
16.2 Metal/Conducting Oxide Interfaces:
I–V
Characteristics and Fundamentals
16.3 Resistive Switching of Metal/Donor-Doped SrTiO
3
Cells
16.4 Resistive Switching of p-Type PCMO Cells
16.5 Resistive Switching in the Presence of a Tunnel Barrier
16.6 Ferroelectric Resistive Switching
16.7 Summary
Acknowledgment
References
Chapter 17: Electrochemical Metallization Memories
17.1 Introduction
17.2 Metal Ion Conductors
17.3 Electrochemistry of CBRAM (ECM) Cells
17.4 Devices
17.5 Technological Challenges and Future Directions
Acknowledgment
References
Chapter 18: Atomic Switches
18.1 Introduction
18.2 Gap-Type Atomic Switches
18.3 Gapless-Type Atomic Switches
18.4 Three-Terminal Atomic Switches
18.5 Summary
References
Chapter 19: Scaling Limits of Nanoionic Devices
19.1 Introduction
19.2 Basic Operations of ICT Devices
19.3 Minimal Nanoionic ICT
19.4 Energetics of Nanoionic Devices
19.5 Summary
Acknowledgment
Appendix A: Physical Origin of the Barrier Potential
References
Chapter 20: Integration Technology and Cell Design
20.1 Materials
20.2 Structures
20.3 Integration Architectures
20.4 Conclusions
Acknowledgments
References
Chapter 21: Reliability Aspects
21.1 Introduction
21.2 Endurance (Cyclability)
21.3 Retention
21.4 Variability
21.5 Random Telegraph Noise (RTN)
21.6 Disturb
21.7 Conclusions and Outlook
Acknowledgments
References
Chapter 22: Select Device Concepts for Crossbar Arrays
22.1 Introduction
22.2 Crossbar Array Considerations
22.3 Target Specifications for Select Devices
22.4 Types of Select Devices
22.5 Self-Selected Resistive Memory
22.6 Conclusion
References
Chapter 23: Bottom-Up Approaches for Resistive Switching Memories
23.1 Introduction
23.2 Bottom-Up ReRAM Fabrication Methods
23.3 Resistive Switching in Single (All-Oxide) NW/Nanoisland ReRAM
23.4 Resistive Switching in Axial Heterostructured NWs
23.5 Core–Shell NWs toward Crossbar Architectures
23.6 Emerging Bottom-Up Approaches and Applications
23.7 Conclusions
References
Chapter 24: Switch Application in FPGA
24.1 Introduction
24.2 Monolithically 3D FPGA with BEOL Devices
24.3 Resistive Memory Replacing Configuration Memory
24.4 Resistive Configuration Memory Cell
24.5 Resistive Configuration Memory Array
24.6 Complementary Atomic Switch Replacing Configuration Switch
24.7 Energy Efficiency of Programmable Logic Accelerator
24.8 Conclusion and Outlook
References
Chapter 25: ReRAM-Based Neuromorphic Computing
25.1 Neuromorphic Systems: Past and Present Approaches
25.2 Neuromorphic Engineering
25.3 Neuromorphic Computing (The Present)
25.4 Neuromorphic ReRAM Approaches (The Future)
25.5 Scaling in Neuromorphic ReRAM Architectures
25.6 Applications of Neuromorphic ReRAM Architectures
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction to Nanoionic Elements for Information Technology
Figure 1.1 The three most common operation modes of different types of ReRAM elements shown for the
I-V
sweep operation (left) and the pulse operation (right). Details are described in the main text. Please note that the elements are nonvolatile. At first glance, the CS (CRS) mode resembles the so-called threshold switching, which shows a hysteresis above a certain voltage bias but which disappears at voltages below this bias. The difference is the fact that the information is lost in the case of a threshold bias while it is maintained in a CRS cell and can be read out in the indicated manner.
Figure 1.2 Geometrical location of the switching event in a ReRAM cell. In the vertical direction, the switching may happen close to one or both electrodes, in the center between the electrodes, or over the entire path between the electrodes. In the lateral direction, we distinguish between a localized filamentary switching and a switching that involves the entire cross section. For example, a bipolar filamentary resistive switch constituting the majority of ReRAM devices reported to date combines a localized event in the lateral direction at/near an electrode interface in the vertical direction. Still, of course, the geometries shown here are limiting cases and intermediate situations may be encountered too.
Figure 1.3 Synopses of the ReRAM concepts, materials, generic cells, ReRAM cell types, and ReRAM technology and applications. The numbers indicate the chapters in this book.
Figure 1.4 Measured
I-V
characteristics for an Nb–Nb
2
O
5
–Bi MIM stack showing bistable resistive switching. The device is initially in a low resistance state (a) due to the previous forming operation. Reset transition to the high resistance is shown for negative applied voltage (b), while set transition to the low resistance appears at positive voltage (c). The inset shows the
I-V
curves of three stable states: a high resistance state H, a low resistance state L, and an intermediate state M (d).
Figure 1.5 The cross-point memory architecture (a) without and (b–d) with cell selection elements. (b) Represents 1T1R cells, (c) cells with a diode or varistor type selector, and (d) a CRS cell. Details are provided in Chapter 22.
Figure 1.6 Schematic of the proposed 3D cross-point architecture using the vertical ReRAM cell. The vertical ReRAM cells are formed at the intersections of each pillar electrode and each plane electrode: the resistive switching oxide layer surrounds the pillar electrode and is also in contact with the plane electrode. To enable the random access of each memory cell, three-dimensional decoding is needed through WL (decoding in
z
-direction), BL (decoding in
y
-direction), and SL of the gate of the vertical MOSFET (decoding in
x
-direction). WL, BL, and SL denote the Word Line, Bit Line, and Select Line, respectively.
Figure 1.7 Pyramidal hierarchical structure of memories and the storage devices.
Figure 1.8 Schematic illustration of the IMP gate implemented through two ReRAM switches (a) and the corresponding truth table (b). Memristive elements P and Q contain the input logic states
p
and
q
, respectively, then Q can conditionally switch to the output state
q
′.
Figure 1.9 Schematic illustration of the natural (biological) and ReRAM-based artificial synapses.
Chapter 2: ReRAM Cells in the Framework of Two-Terminal Devices
Figure 2.1 Symbols of memory circuit elements.
Figure 2.2 The combination of two or more memory-circuit elements in parallel or in series may result in the compact description of a single circuit element with nonalgebraic dependence on the control parameters [19].
Figure 2.3 Equivalent circuit model of an ionic-type ReRAM device.
Figure 2.4 Schematic of the system coupled to left (L) and right (R) electrodes. and are integration areas in Eq. (2.20).
Figure 2.5 The black-box approach (left) only considers - data, while the white-box approach considers the actual internal device physics (right). The gray-box approach is based on both the - data and some knowledge of the device physics.
Figure 2.6 ECM memristive model.
Figure 2.7 Table of typical parameters.
Figure 2.8 (a) ECM quasistatic sweep for a current compliance (CC) of 10 A. (b) Complementary resistive switch simulation for several series resistors.
Figure 2.9 (a) Single-cell - characteristic for the following sweep rates: 1, 10, and 100 V s
−1
. (b) Complementary resistive switching cell for sweep rates of 1, 10, and 100 V s
−1
.
Chapter 3: Atomic and Electronic Structure of Oxides
Figure 3.1 (a) Face-centered cubic (fcc) lattice with primitive lattice vectors and primitive cell (blue polyhedron), the conventional cubic cell being the surrounding cube. (b) Rock-salt structure: a superposition of two fcc lattices, which are shifted by . (c) Zincblende structure: a superposition of two fcc lattices, which are shifted by . If all atoms are of the same type, (c) corresponds to the diamond structure. (d) Body-centered cubic (bcc) lattice with primitive lattice vectors and primitive cell (blue polyhedron), the conventional cell being the surrounding cube.
Figure 3.2 Close-packed structures: (a) ABAB stacking (hexagonal, close-packed) and (b) ABCABC stacking (cubic, close-packed), which also corresponds to the face-centered cubic (fcc) structure. The corresponding unit cells are also shown by the black lines.
Figure 3.3 Eutactic crystal structures: (a) rock-salt structure with edge-sharing octahedra, (b) rutile structure with edge-sharing octahedra along the
c
-axis, whereas the columns are connected at the corners, (c) corundum structure with pairs of face-sharing octahedra along the
c
-axis, (d) fluorite structure with the simple cubic sublattice of the anions, (e) wurtzite structure with corner-sharing tetrahedra, and (f) spinel structure: the cations are both octahedrally and tetrahedrally coordinated by anions. The corresponding conventional unit cells are shown by the black lines. Anions are shown in red and cations in white.
Figure 3.4 Perovskite structure with large cations in between the octahedra. Without the central cation, the structure corresponds to the ReO
3
structure.
Figure 3.5 Dimer Mg
2
and clusters Mg
m
. (a) Energy level diagram for the dimer Mg
2
and linear clusters Mg
m
. The highest occupied state is the zero of the energy scale. For some states, isosurfaces of the charge density distribution are shown. Blue levels are occupied, red levels unoccupied. (b) Isosurfaces of the charge density distribution of the Mg
32
cluster.
Figure 3.6 MgO molecule and (MgO)
m
clusters. (a) Energy level diagram for the MgO molecule and (MgO)
m
cubic clusters. Blue levels are occupied, red levels unoccupied. The highest occupied state is the zero of the energy scale, (b) total valence charge density distribution (in ) for one plane. The scale is shown in (c).
Figure 3.7 Total charge density distribution in fcc MgO. (in ) is shown for a (001) plane, and additionally, one isosurface of is plotted.
Figure 3.8 A rough sketch of the shape of the potential “seen” by an electron in a metal. To get the valence electron states of simple metals, in a first approximation, the real potential can be replaced by macroscopic potential of depth and length .
Figure 3.9 Energy (
dispersion curve
) of the electronic states in the free electron model. is quasi-continuous. The minimum distance between two allowed -points is . At 0 K, the states are occupied up to the Fermi energy . The -vector Γ = (0;0;0) is called
Γ-point
.
Figure 3.10 Band structure of MgO. The band structure is displayed for two directions in -space, from
X
= (0,5,0,0.5) to Γ = (0,0,0) to
L
= (0.5,0.5,0.5). The valence band maximum (VBM) is the zero of the energy scale.
Figure 3.11 Sketch of the sphere radii for Mg (blue) and O (red) used to calculate the partial DOS of fcc MgO. Also shown is the valence charge density distribution for one plane. The values for are given in . Note that there is almost no valence charge density in the Mg sphere.
Figure 3.12 DOS and partial DOS for fcc MgO. The highest occupied state (valence band maximum, VBM) is the zero of the energy scale. The partial DOS for Mg and O are given for (red) and (blue) partial states, where the p partial states are the sum over . The valence band maximum (VBM) is the zero of the energy scale. The partial DOS are given for two different scales in order to show more clearly the distribution of the partial DOS of the conduction states.
Figure 3.13 Crystal field splitting. The d orbitals split into e
g
and t
2g
states in an octahedral field. For an additional trigonal distortion (here compression), the t
2g
states split into e
g
and a
1g
. For a tetragonally distorted octahedral field (here, stretching), the degeneracy is lifted further.
Figure 3.14 The Hubbard model, see the main text. (a) Chain of M atoms, one orbital at each site. For half-filled states, the behavior is metal-like if the kinetic energy parameter
t
is small as compared to the onsite Coulomb repulsion parameter
U, t/U
1 (b), and for
t/U
≪ 1, there is an energy gap between the occupied and unoccupied states (c).
Figure 3.15 Electronic structure of metals. (a) Band structure of bcc Na metal for two directions in k-space, in reciprocal units from
P
= (0.25,0.25,0.25) to to
N
= (0,0,0.5). (b) Band structure of bcc Mg metal for the same two directions in k-space. (c) Density of states (DOS) of bcc Mg metal (high-pressure phase). The Fermi energy is the zero of the energy scale. The lattice constants were determined experimentally [26, 27].
Figure 3.16 Valence charge density distribution (in ) of Mg metal (high-pressure phase) in the (110) plane of the bcc conventional cell. One recognizes the constant density in between the atoms.
Figure 3.17 Valence charge density distribution (in ) of Si (diamond structure) in the (110) plane. The charge density is concentrated (yellow/green) in the nearest neighbor direction, whereas there is almost no density in the region interstitial region (deep blue).
Figure 3.18 Electronic structure of Si calculated with the HSE06 functional. (a) Total DOS on the left and Si partial DOS on the right, which is subdivided into the s (red) and p (blue) contributions. (b) Band structure for two directions in -space, in reciprocal units from
X
= (0.5,0,0.5) to Γ = (0,0,0) to
L
= (0.5,0.5,0.5). The valence band maximum (VBM) is the zero of the energy scale. The Fermi level lies within the band gap. We can see an indirect band gap of 1.1 eV and a direct gap of 3.3 eV. The lattice constants were determined experimentally [28].
Figure 3.19 One-dimensional chain. (a) Undistorted chain with primitive unit cell. (b) Chain with dimerization: the unit cell doubles.
Figure 3.20 Oxygen vacancy in MgO (63 atom cells of Mg
32
O
31
) calculated with the GGA functional. Left: DOS and partial DOS for the cell with a vacancy . The highest occupied state (valence band maximum, VBM) is the zero of the energy scale. For the partial DOS, red denoted s-like state and blue denotes p-like state. The localized states of the color center lie in the band gap of the bulk phase MgO. Right: Isosurface of the charge density distribution of the two localized electrons at the vacancy position.
Figure 3.21 (a) Structure of the (001) surface of MgO. The tetragonal simulation cell is indicated by the thick black lines. On the surfaces, one can recognize
rumpling
. (b) Structure of bulk MgO. The tetragonal simulation cell (with the same lattice vectors and as for the surface cell) is indicated by the thick black lines. (c) DOS and partial DOS of the MgO surface cell. The partial DOS for O and Mg are given for the surface layer (O
1
) and the layers below the surface layer, where bulk behavior is recovered for the second layer of oxygen and third layer of Mg. (d) Corresponding band structure for the cell with 100 surface from
M
= (0.5,0.5,0) to Γ = (0,0,0) to
Z
= (0,0,0.5). (e) Band structure of the tetragonal bulk cell of MgO with two formula units from
M
= (0.5,0.5,0) to Γ = (0,0,0) to
Z
= (0,0,0.5). The valence band maximum (VBM) is the zero of the energy scale and the calculations have been performed with the GGA functional.
Figure 3.22 Amorphous MgO. (a) DOS and partial DOS for simulated amorphous MgO. The highest occupied state is the zero of the energy scale. The partial DOS are shown for 12 different O and Mg spheres distinguished by the different colors. (b) Charge density of the first unoccupied states (CBM) in crystalline MgO. One clearly sees that the density is uniformly distributed over the whole crystal (c) charge density of the first unoccupied states in amorphous MgO. The charge is distributed in a nonuniform way.
Figure 3.23 Electronic structure of Al
2
O
3
(corundum structure) calculated with the HSE06 functional. (a) Total DOS on the left and Al and O partial DOS on the right, which are subdivided into the s (red) and p (blue) contributions. We show only one Al and one O, as the other Al and O in the cell are symmetrically equivalent to the displayed ones. The lowest displayed states are the valence bands with oxygen 2p character. (b) Corresponding band structure. The direct band gap is 8.1 eV. The valence band maximum (VBM) is the zero of the energy scale.
Figure 3.24 Electronic structure of SrO (rock-salt structure) calculated with the HSE06 functional. (a) Total DOS on the left and O and Sr partial DOS on the right, which are subdivided into the s (red), p (blue), and d (green) contributions. Notice: the scaling of the Mg partial DOS and that of O partial DOS differ by a factor of 10. The valence band has oxygen 2p character, and at the CBM, we have Sr 3d-like states. (b) Corresponding band structure. The indirect band gap is 4.8 eV. The valence band maximum (VBM) is the zero of the energy scale. The experimental lattice constant is used [47].
Figure 3.25 Electronic structure of ZnO (wurtzite structure) calculated with the HSE06 functional. (a) Total DOS on the left and Zn and O partial DOS on the right, which are subdivided into the s (red), p (blue), and d (green) contributions. The lowest displayed states are the Zn 3d states. The top of the valence band has mainly oxygen 2p character. (b) Corresponding band structure. The direct band gap is 2.5 eV. The VBM is the zero of the energy scale. The experimental lattice constant is used [50].
Figure 3.26 Electronic structure of TiO (rock-salt structure) calculated with the HSE06 functional. (a) Total DOS on the left and partial DOS of Ti and O on the right, which are subdivided into the s (red), p (blue), and d (green) contributions. The lowest displayed states are the O 2p-like states. Above these states lie the Ti 3d-like states. These states are occupied up to the VBM, which is the zero of the energy scale. The actual sp-conduction states are located above the Ti 3d-like states. (b) Corresponding band structure. The experimental lattice constant is used [54].
Figure 3.27 Electronic structure of Ti
2
O
3
(corundum structure) calculated with the HSE06 functional. (a) Total on the left and partial DOS of Ti and O on the right, which are subdivided into the s (red), p (blue), and d (green) contributions. One clearly sees the occupied states from −1 eV to the VBM. (b) Corresponding band structure. The VBM is the zero of the energy scale. The experimental lattice constant is used (low-temperature phase) [57].
Figure 3.28 Metal–metal bonding in Ti
2
O
3
. (a) Due to the trigonally distorted octahedra in Ti
2
O
3
, the crystal field splitting yields low lying a
1g
and e
g
levels. (b) The metal–metal bonding of the Ti along the
c
-axis is reflected in bonding and antibonding a
1g
levels. (c) With increasing temperature
T
, the lattice parameter increases and the metal–metal overlap decreases. Therefore, the bands become narrower, and the bonding level is shifted upward and finally overlaps with the e
g
level. The material becomes metallic.
Figure 3.29 Electronic structure of TiO
2
(rutile structure) calculated with the HSE06 functional. (a) Total DOS and partial DOS of Ti and O, which are subdivided into the s (red), p (blue), and d (green) contributions. The partial DOS for the different O are very similar. Therefore, we have plotted only one O. The lowest displayed states are the O 2p-like states between −6 and 0 eV. The conduction band is mainly of Ti 3d character and the calculated band gap is 3.2 eV. (b) Corresponding band structure. The VBM is the zero of the energy scale.
Figure 3.30 Electronic structure of SrTiO
3
(perovskite structure) calculated with the HSE06 functional. (a) Total DOS and partial DOS of Sr, Ti, and the three oxygen, which are subdivided into the s (red), p (blue), and d (green) contributions. The lowest displayed states are the O 2p-like states, followed by the unoccupied Ti 3d-like states in CBM. The calculated band gap is 3 eV. (b) Corresponding band structure. The VBM is the zero of the energy scale.
Figure 3.31 Schematic plot of superexchange. The up and down arrows represent the up and down spin electrons, respectively, and the upper line of arrows corresponds to the
antiferromagnetic order
. Based on the Pauli exclusion principle, only in this spin configuration, the electrons can be transferred between the orbitals (without spin flip) to yield the spin configurations in the lines below. These additional configurations lower the energy of the system and stabilize the antiferromagnetic order.
Figure 3.32 Electronic structure of NiO (rock-salt structure with antiferromagnetic ordering of type II) calculated with the HSE06 functional. Here, we perform a
spin polarized
calculation, that is, each band can be occupied by just one electron. (a) Total DOS and partial DOS of Ni (spin up and down) and oxygen, which are subdivided into the s (red), p (blue), and d (green) contributions. The lowest displayed states are the valence bands formed by a considerable hybridization of oxygen 2p and Ni 3d states. The calculated band gap is 4.6 eV and the net magnetic moment on the Ni is 1.6 µ
B
. (b) Corresponding band structure. The VBM is the zero of the energy scale.
Figure 3.33 Electronic structure of MnO (rock-salt structure with antiferromagnetic ordering of type II) calculated with the HSE06 functional. Here we perform a
spin polarized
calculation, that is, each band can be occupied by just one electron. (a) Total DOS and partial DOS of Mn (spin up and down) and oxygen, which are subdivided into the s (red), p (blue), and d (green) contributions. The calculated band gap is 3 eV and the magnetic moment on the Ni is 4.5 µ
B
. (b) Corresponding band structure. The VBM is the zero of the energy scale.
Figure 3.34 Electronic structure of NbO
2
calculated with the GGA +
U
functional [22] with
U
−
J
= 3. (a) Monoclinic NbO
2
(low-temperature phase). Total DOS and partial DOS of Nb and O, which are subdivided into the s (red), p (blue), and d (green) contributions. We have an insulator with a band gap of 0.8 eV. The VBM is the zero of the energy scale. The lattice constants were determined experimentally [101]. (b) Rutile NbO
2
(high-temperature phase). Total DOS and partial DOS of Nb and O, which are subdivided into the s (red), p (blue), and d (green) contributions. We observe metallic behavior, as the Fermi energy lies within the Nb 4d states. The Fermi energy is the zero of the energy scale.
Figure 3.35 Electronic structure of Ta
2
O
5
( phase, orthorhombic structure, space group
Pbam
[109]) calculated with the HSE06 functional. (a) Total DOS on the left and Ta and O partial DOS on the right, which are subdivided into the s (red), p (blue), and d (green) contributions. The lowest displayed states are the O 2p-like states and the unoccupied Ta 5d-like states. The top of the valence band has mainly oxygen 2p character. (b) Corresponding band structure. The direct band gap is 4.0 eV. The VBM is the zero of the energy scale.
Figure 3.36 Electronic structure of Ta
2
O
4
(rutile structure) calculated with the HSE06 functional. (a) Total DOS on the left and Ta and O partial DOS on the right, which are subdivided into the s (red), p (blue), and d (green) contributions. The lowest displayed states are the O 2p-like states and the partly occupied Ta 5d-like states. (b) Corresponding band structure. The direct band gap is 4.0 eV. The Fermi energy is the zero of the energy scale.
Figure 3.37 Temperature dependence of the standard free reaction enthalpy Δ
G
0
(
T
) for the reaction for various transition metal oxides M
y
O. The energy axis is also shown in equivalent (formal) equilibrium oxygen partial pressures. The color code shows phases of identical cations, from Ref. [117].
Chapter 4: Defect Structure of Metal Oxides
Figure 4.1 Schematic two-dimensional representation of some of the defects present in a solid of the type
MX.
Figure 4.2 Simplified representation of (a) an edge dislocation with and (b) a screw dislocation with .
Figure 4.3 The change of free energy expressed as a function of the number of point defects exhibits a minimum, which explains why defects are always present in a solid at nonzero temperatures.
Figure 4.4 Brouwer diagram determined at an arbitrary temperature for an oxide of the type in which .
Figure 4.5 Charge concentration profiles calculated for SrTiO
3
as a function of the distance from a grain boundary (
x
= 0) in the Mott–Schottky situation (acceptor concentration equal to 0.1 mol%): (a) log-scale and (b) linear-scale representation (electrons and holes are here out of scale). Note that due to their higher charge, oxygen vacancies are stronger depleted than holes. The parameter used are
T
= 600 °C, , , .
Figure 4.6 Brouwer diagram of nominally pure TiO
2
.
Figure 4.7 Electrical conductivity as a function of oxygen partial pressure of a pristine TiO
2
single crystal (squares) compared with a plastically deformed crystal (triangles and circles) upon compression along the [001] direction. Positions 1 and 2 refer to different locations in the deformed crystal, which contained a different density of dislocations.
Figure 4.8 Electrical conductivity as a function of the oxygen partial pressure of (i) the bulk contribution of a microcrystalline sample (open squares), (ii) a nanocrystalline sample with an average grain size of 50 nm (gray diamonds), and (iii) a nanocrystalline sample with an average grain size of 30 nm (solid squares) measured at 544 °C. All the samples were prepared from the same batch of SrTiO
3
powder containing about 100 ppm Fe impurities. The solid gray line represents a plateau at ranging between 10
−20
and 10
−10
atm corresponding to the conductivity calculated from the defect chemistry model of Ref. [11] for a concentration of Fe impurities of 100 ppm.
Figure 4.9 Concentration of the minority charge carriers as a function of the oxygen partial pressure. Note that the oxygen-vacancy concentration in 8YSZ is on the order of 2·10
21
cm
−3
.
Chapter 5: Ion Transport in Metal Oxides
Figure 5.1 Solutions of the diffusion equation for the cases of an instantaneous source (a) and a constant source (b) plotted as normalized concentration against depth for various values of . The units of and are arbitrary but identical (if is expressed in cm, has the unit s
−1
and has the unit s).
Figure 5.2 A single random walk of a single particle on an empty two-dimensional lattice, with intersite jump distance of . After jumps, the particle has achieved a displacement . Repeating the random walk of jumps many times yields but . Note: for each random walk, the particle covers a distance of ; after many random walks, however, its root-mean-square displacement is . For this two-dimensional case, the diffusion coefficient follows as , with .
Figure 5.3 Common mechanisms of atomic migration in a binary compound with mobile species (red) and immobile species (blue): (a) interstitial, (b) interstitialcy, and (c) vacancy.
Figure 5.4 Simple schematic illustration of the ion movements involved in the jump of an ion to a neighboring vacant site (migration mediated by a vacancy mechanism). (a) Initial configuration; (b) Saddle-point configuration; (c) Final configuration. (d) The change in the system's Gibbs energy associated with the ion movements in (a)–(c). is the Gibbs activation energy of migration.
Figure 5.5 The chemical diffusion experiment consists of, for example, instantaneously increasing the activity of oxygen in the surrounding gas phase (a) and then monitoring the change in a characteristic sample property (b) as a function of time, that is, one that depends on the oxygen content of the sample (c), such as the electrical conductivity, as the sample attains the new equilibrium with the gas phase.
Figure 5.6 The tracer diffusion experiment [12]: (a) An oxide sample is subjected to pre-annealing in oxygen of normal isotopic abundance at given temperature and oxygen activity in order to equilibrate the sample with the surrounding atmosphere. (b) Subsequently, it is annealed, at the same temperature and oxygen activity, in an O-enriched gas for a given time. At the sample surface, the dynamic equilibrium between gaseous oxygen and oxygen in the sample leads to the incorporation of O and the removal of O (no net incorporation or removal of oxygen, only the exchange of one isotope into another). Subsequent diffusion of O away from the interface and into the solid produces an oxygen isotope profile. (c) The isotope profile in the oxide is commonly determined by an ion beam analysis method, such as secondary ion mass spectrometry (SIMS).
Figure 5.7 Schematic illustration of a positive interstitial ion overcoming an activation barrier of migration : (a) Gibbs energy profile at zero applied field. (b) Gibbs energy profile at nonzero applied field.
Figure 5.8 Mass-transport processes in a polycrystal. (a) Cross section of a polycrystalline solid. (b) The brick-layer model, an idealized representation of the microstructure shown in (a). The arrows indicate possible transport processes, with the length of the arrows being inversely proportional to the resistance of the associated process. A – hindered transport across a surface; B – transport in the grain bulk; C – hindered transport across a grain boundary; D – enhanced transport along a grain boundary.
Figure 5.9 Diffusion in a polycrystalline sample with grain size wherein transport across grain boundaries is slow () and diffusion along grain boundaries is negligible. (a) Cross section of a polycrystal: diffusant enters the sample from the left. (b) Concentration profiles across the polycrystal, showing the decrease in concentration () at the grain boundaries (whose positions are indicated by the dashed, vertical lines).
Figure 5.10 Illustration of three regimes of diffusion kinetics in a polycrystal identified by Harrison [30]. The polycrystal consists of grains of width and diffusion coefficient and grain boundaries of width and diffusion coefficient . The three regimes are defined by the inequalities shown in the bottom line of the Figure ( is the diffusion time).
Figure 5.11 Simulated concentration profiles obtained at various times for a single crystal with dislocation density cm, dislocation radius nm, bulk diffusion coefficient cm
2
s
−1
, and dislocation diffusion coefficient cm
2
s
−1
.
Figure 5.12 Isotope transport through an equilibrium surface space-charge layer depleted of oxygen vacancies [34, 35]. (a) Local variation of the oxygen tracer diffusion coefficient, , which arises from oxygen vacancy depletion near the surface (see Section 4.4). Solving Eq. (5.3) with this spatially variant yields the isotope profile shown in (b) [the first 40 nm] and in (c) [the entire profile].
Figure 5.13 Comparison of vacancy diffusion coefficients obtained experimentally for and . Oxygen-vacancy diffusion in : (a) [35], (b) [51], (c) [52]. Oxygen-vacancy diffusion in , (d) [53]. Strontium-vacancy diffusion in : (e) [54], (f) [55], (g) [55]. Titanium-vacancy diffusion in , (h) [56].
Figure 5.14 SIMS analysis of a cross section of a polycrystalline sample exposed to an O / O anneal at K for 1 h. The grain boundaries are seen to hinder oxygen transport. Fast transport of oxygen along grain boundaries is not observed [76]. Reproduced with kind permission of Trans Tech Publications.
Figure 5.15 Comparison of oxygen and cation diffusion coefficients obtained experimentally for yttria-stabilized zirconia with about 9.5 mol% O. (a) Oxygen tracer diffusion [91] based on data from Refs [89–91]; (b) zirconium tracer diffusion [97]; (c) - zirconium tracer diffusion [98]; (d) yttrium tracer diffusion [99].
Figure 5.16 Tracer diffusion data for Al and O in -AlO. Al diffusion: (a) [112], (b) [114], (c) [115]. d is a fit to the three experimental datasets (a–c). O diffusion: (e) [118], (f) [119], (g) [120], (h) [121], (i) [122], (j) [123], (k) [124].
Figure 5.17 Diffusion data for oxygen in L-TaO obtained from electrical conductivity measurements at high oxygen partial pressures ((a) [136], (b) [137], (c) [138], (d) [139], (e) [140], (f) [141], and (g) [142]) and from tracer diffusion measurements ((h) [143]). Note: The tracer data [143] are not corrected for tracer correlation, as for L-TaO is not known.
Chapter 6: Electrical Transport in Transition Metal Oxides
Figure 6.1 The B-site positions of spinel AB
2
O
4
. The unit cell is shown by dotted lines.
Figure 6.2 (a) Metal–oxygen octahedron with the local
x, y, z
coordinates defined. Schematics of (b) σ and (c) π bonding between the metal 3d orbitals and oxygen 2p orbitals, creating e
g
and t
2g
states, respectively, as well as (d) direct metal–metal bonding in a 90° metal–oxygen–metal bond.
Figure 6.3 Schematic of small polaron hopping. The carrier hops from (a) one site to (c) another via (b) an intermediate state with a thermally activated excited lattice configuration.
Figure 6.4 (a) Schematic of a spatially random trapping potential landscape created by defects. (b) Hypothetical defect density of states with respect to energy.
Figure 6.5 Formation energies of native donor defects in (a) SnO
2
and (b) TiO
2
. Fermi level of 0 represents the valence band edge, and the highest value represents the conduction band edge [24].
Figure 6.6 (a) Calculated band structure of SnO
2
. Charge density contours (in steps of 0.4 electrons per cubic angstrom) near the conduction band minimum of (b) pure, (c) Sb-doped, and (d) F-doped SnO
2
[26].
Figure 6.7 Hall and drift electron mobility of electron-doped TiO
2
[9].
Figure 6.8 Schematics of (a) the ground state and (b) an excited state of a system with one electron and one orbital per lattice site described by Hubbard Hamiltonian for
U
t
> 0.
Figure 6.9 The combination of (a) thermopower
S
and (b) resistivity
ρ
measurements is used to calculate (c) the temperature dependence of
μ
in intrinsically p-type MnO. The different letters corresponding to each curve denote different samples. The mobility curve in (c) was extracted from the data points shown as symbols in (a) and (b). [39].
Figure 6.10 Hall (A) and drift (1 and 2) mobilities of 0.09% Li-doped NiO. The drift mobilities were calculated from the same data with two different assumptions: (1)
N
= constant and (2) (see explanation in the main text) [45].
Figure 6.11 A t
2g
3
e
g
2
system doped with electrons, and the extra carrier is shown as a dashed arrow. (a) The carrier can transfer from one site to an adjacent site with parallel spin alignment (b) but not to one with antiparallel spin alignment.
Figure 6.12 (a) Hall mobility values of epitaxial n-type (Fe
1−
x
Ti
x
)
2
O
3
films at different dopant percentages (
x
*100%). (b) Resistivity curve of a
x
= 0.03 doped film fitted to the functional forms for two-dimensional (2-D) Mott variable-range hopping (vrh) and small polaron hopping (sph) [54].
Figure 6.13 Temperature-dependent resistivity of several titanium and vanadium oxides [58].
Figure 6.14 Energy splitting of the t
2g
levels in a rutile structure. Thick (thin) lines represent levels that can hold two (one) electrons per cation.
Figure 6.15 (a) Setup of planar electrical switching experiment of a VO
2
film deposited on
c
-plane sapphire. (b) IV trace showing threshold switching. (c) Time-resolved voltage ramp-up and response characteristics of the VO
2
film, magnified in (d) [74].
Figure 6.16 IV sweeps of a vertical
n
-Si/VO
2
/Pd structure at different temperatures, showing voltage-induced threshold switching at temperatures below the transition temperature of VO
2
[76].
Figure 6.17 Antiferromagnetic structure of V
2
O
3
consistent with neutron scattering, wherein the dark sites and light sites represent V
3+
cations of opposite spin [84].
Figure 6.18 Temperature-dependent resistivity of nearly stoichiometric (I) and slightly overoxidized (II) Fe
3
O
4
[91].
Figure 6.19 Current–voltage sweeps with (a) a staircase and (b) a pulsed voltage source on planar Fe
3
O
4
-based devices grown on (001) MgO [100].
Chapter 7: Quantum Point Contact Conduction
Figure 7.1 Discovery of conductance quantization: conductance as a function of the width of a point contact in a two-dimensional electron gas, formed by means of control of the gate potential. The inset shows the arrangement of the contacts and the gate electrodes.
Figure 7.2 Schematic representation of an adiabatic constriction in 2D (top) and the resulting profile of the bottom of the subbands (below).
Figure 7.3 (a) Examples of conductance curves for Au, Al, and Pt taken while breaking contacts at 4.2 K. (b) Conductance histograms formed from hundreds of breaking curves.
Figure 7.4 Superconducting subgap structure for Al atomic-size contacts. Al has a critical temperature of about 1 K, and measurements were performed at 0.1 K. Inset: Theoretical
I-V
curves for a junction formed between two superconductors by a single conductance channel, for transmission probabilities ranging from 1 (top) to 0.1 (bottom curve).
Figure 7.5 Scanning electron microscopy (SEM) images for a Pt/H
2
O/Ag cell showing (a) the
off
state of the device with shorter and smaller Ag dendrites and (b) the same cell in the
on
state with longer and larger Ag dendrites obtained by applying a positive voltage on the Ag electrode.
Figure 7.6 Preswitching (a) and full bipolar switching (b) current–voltage characteristics of a Ag/Ag
2
S/Pt (nanocontact) system. In the full bipolar switching case, the transition from off to on state and back is explained by the formation and dissolution of a metallic nanowire.
Figure 7.7 (a) Steps in conductance measured in an Ag/AgI/Pt cell during a current sweep. (b) Conductance histogram constructed from cumulative data of 65 devices.
Figure 7.8 Steps in conductance traces of Ag
2
S devices become visible when breaking the conductive path at low bias voltages (<100 mV). (a) Breaking trace with three clear conductance steps of approximately 1
G
0
observed at the last stages of breaking. The inset shows a zoom of the steps. (b) Three breaking traces with atomic conductance steps having different lengths in time. The upper trace shows two-level fluctuations that are typical for atomic-size contacts and are attributed to single atoms oscillating near the contact. (c) Continuous-conductance traces and mixed-conductance traces observed in the same devices.
Figure 7.9
I-V
characteristics and differential conductance as a function of applied voltage corresponding to two consecutive HBD events detected on the same MOS device.
Figure 7.10 Fresh (thin solid lines) and post-SBD
I
(
V
) characteristics (symbols) of three MOS devices with different thicknesses and areas.
Figure 7.11 Schematic energy diagram of the constriction and its associated potential barrier profile.
Figure 7.12 SBD
I-V
characteristics corresponding to first events measured on different MOS devices. Φ is the barrier height associated with the first subband level and
α
is a parameter related to the shape of the barrier.
Figure 7.13 Schematic energy diagram for a filamentary structure with multiple subbands.
E
i
represents the top of each subband.
Figure 7.14 (a)
I-V
curves measured during the application of reset voltage ramps in Pt/HfO
2
/Pt structures. The dashed line corresponds to a linear
I
(
V
) with conductance equal to
G
0
= 2
e
2
/
h
. (b) Detail of the current–voltage evolution during the last phase of the reset transients. (c) Histogram of conductance at the final reset point; the inset shows the histogram of conductance readings during 100 successive conductance-time traces.
Figure 7.15 Histogram of the conductance values for ITO/ZnO/ITO devices obtained by sweeping the bias to successive levels of maximum voltage. Similar results were obtained for Nb/ZnO/Pt.
Figure 7.16 Histogram of conductance changes collected for 1000 conductance steps observed in silicon-oxide resistive memory devices. Peaks are obtained at half-integer multiples of
G
0
.
Chapter 8: Dielectric Breakdown Processes
Figure 8.1 (a) Experimental BD distributions for SiO
2
-based MOS structures with
T
ox
= 2.7 nm and five different areas (see legend). (b) Normalization of the same
t
BD
statistical data to a reference area of 8.4 × 10
−7
cm
2
, according to the scaling law derived from the weakest-link property. Straight lines in the Weibull plot demonstrate that the Weibull model is adequate for the time-to-BD distribution. Overlapping of the data for structures with different areas demonstrates the area scaling property.
Figure 8.2 A schematic representation of the projection procedures used for gate oxide breakdown reliability assessment. First, data from strongly accelerated BD experiments are extrapolated to operating conditions using voltage and temperature acceleration models. Second, using the weakest-link property, the results are extrapolated to chip total oxide area. Finally, the results are extrapolated to low failure percentiles using the Weibull distribution model.
Figure 8.3 Schematic representation of the different processes involved in defect generation according to current energy-based models such as the AHR model.
ζ
1
represents the efficiency of H release by incoming electrons,
ζ
2
is the defect creation efficiency by the released species, and ζ
3
is the probability of the released species escaping to the electrodes without causing damage in the oxide.
Figure 8.4 Comparison of charge-to-BD data as a function of voltage and temperature with a complete BD model combining percolation statistics and anode hydrogen release model for defect generation. (a) 2.67 nm SiO
2
in p
+
-poly/SiO
2
/n-Si structures under substrate injection (positive gate voltage). (b) 2.23 nm SiO
2
in n
+
-poly/SiO
2
/p-Si structures under gate injection (negative gate voltage). Oxide area is 10
−4
cm
2
in both cases.
Figure 8.5 Schematic representation of the formation of the BD path in the cell-based percolation model of Ref. [52].
Figure 8.6 Application of the percolation model to the SET transition in ReRAM. A cell-based geometrical model is proposed to describe the insulator gap in the broken CF. SET transition is modeled as the BD of the insulator gap. With the percolation approach, the
R
OFF
dependence on the scale and shape factors of the SET voltage distribution are well captured.
Figure 8.7 Application of the BD statistical methodology to deal with the SET speed/READ disturb trade-off in ReRAM. From the measured time-to-SET distribution, and having characterized the voltage acceleration power law, the distribution of time-to-SET (
t
set
) and time-to-disturb (
t
dis
) can be obtained for any SET and READ voltages, respectively. The high percentile region of the
t
set
distribution is important to ensure a highly efficient SET process, while the low-percentile tail of the
t
dis
distribution is crucial to ensure low disturb probability, as discussed in detail in Ref. [10].
Figure 8.8 Application of the BD statistical methodology allows for establishing the design requirements to ReRAM trade-offs [10]. In this figure, the minimum Weibull slope of the SET voltage distribution required for an efficient SET with low READ disturb is plotted versus the scale factor of the
V
set
distribution under VRS conditions. Only above both the solid and the dashed lines is it possible to simultaneously meet the specifications for reliable SET and low disturb, thus defining a useful design area.
Figure 8.9 Application of the successive BD theory to consider ReRAM failure due to the formation of a second CF during SET/RESET cycling. The relevant time to failure is the time between the first BD (under forming at
V
forming
) and the second one (during cycling at
V
set
).
Figure 8.10 (a,b) Oxygen deficiency count in the percolated region of the dielectric in poly-Si–SiON based gate stack. With higher compliance, the percolation region laterally dilates in size and the central core of the percolation path becomes increasingly Si-rich with the oxygen stoichiometry shifting from
x
∼ 2 to
x
∼ 0 [67, 68]. (c,d) Physical TEM micrograph showing the formation of the Si nanowire [23] and DBIE epitaxial defect due to silicon protrusion from the substrate (maintaining the same substrate orientation) [70]. (e) TEM micrograph of HBD (
I
comp
∼ 500 μA) for a NiSi–HfSiON–SiO
x
(bilayer)–Si stack showing Ni spiking into the dielectric and all the way into the substrate in the (111) direction [71]. Note that the size of the filament is as small as 2–3 nm. (f) TEM micrograph showing another failure mechanism in the NiSi gate stack involving the lateral diffusion of Ni from the source/drain contacts into the channel region for
I
comp
∼ 500 μA [71]. This lateral diffusion is more clearly detected in the high angle annular dark field (HAADF) image in the inset of (f). (g) TEM micrograph of filamentation in the TaN (TiN)–HfO
2
–Si stack showing Ta having migrated into the substrate forming a bowl-shaped defect. Note that the size of the Ta filament is much larger with a diameter of ∼20–30 nm [73]. In this case, it is to be noted that although the chosen
I
comp
is as low as 8 μA, there is a compliance overshoot effect as the TDDB transient was too fast to be adequately controlled by the external compliance setting.
Chapter 9: Physics and Chemistry of Nanoionic Cells
Figure 9.1 Equilibrium phase diagram of the Zr–O system as a function of the temperature. Subphases can be observed at higher metal content, corresponding to lower oxidation state of the Zr ion. rt and ht in the diagram denote room temperature and high temperature, respectively. L denotes liquid phase. The diagram is modified from Ref. [7]. The phase diagram of Hf–O is also quite similar.
Figure 9.2 (a)
T-x
phase diagram for a simple binary system assuming ideal solubility of the phases. L and S denote the liquid and solid phase, respectively. (b) The corresponding change in the Gibbs energy
G
. The Figure is adapted from Ref. [2].
Figure 9.3 Ordered arrays of oxygen vacancies (left) are eliminated by the formation of shear planes (right), such as Wadsley defects, in which the cations move into interstitial positions [8].
Figure 9.4 Equilibrium phase diagram for TiO
x
system. The formation of a number of Magnéli phases separated by small Gibbs formation energies can be observed at lower oxygen potentials. a) in the entire range of Ti/O ratio and b) in a selected region zoomed around . The Figure is adapted from Ref. [2].
Figure 9.5 Sketch of different material classes used for ReRAM and the transition of their transport properties, depending on the thermodynamic conditions and operation (pre)history.
Figure 9.6 Different models for the formation of the space-charge layer (electrical double layer) described in the main text. The corresponding charge and electrical potential distributions are also presented. A positively charged metal is assumed on the left side.
Figure 9.7 Electrostatic potential
ϕ
across an electric double layer (space charge layer) (a) at the electrode–electrolyte interface according to the Stern model and (b) for the situation in the nanoworld where the electrolyte thickness is comparable to twice the Debye length (
d
< 2
λ
D
).
Figure 9.8 Schematic presentation of the Gibbs free energy for nucleation of a hemispherical critical nucleus
N
c
at an electrode surface. The influence of the surface free energy Φ and the applied cathodic potential Δϕ is also illustrated. The Figure is adapted from Ref. [43].
Figure 9.9 Atomistic theory of nucleation. Binding energy of atomic assemblies based on atomic bond energies in a vacuum. The bond energies between neighboring Me atoms and the substrate are denoted as
Ψ
1
and
Ψ
sub
, respectively.
Figure 9.10 Comparison between the classical and atomistic treatment of the nucleation process. The free reaction enthalpy change (a) and the change in the electrochemical potential (b) as a function of the number of constituent atoms
N
are schematically represented.
Figure 9.11 Schematic representation of the distribution of the potentials in a symmetric solid-state cell with predominant oxygen ion conductivity. The electrochemical potentials of the individual components are split into chemical and electrical parts in accordance to Eq. (9.16).
Figure 9.12 Formation of the diffusion potential. The Figure is adapted from Ref. [48].
Figure 9.13 (a–d) Situation in insulating materials where the opposite charges are attracted. Situation (d) represents a model ECM cell based on SiO
2
solid electrolyte and Cu active electrode.
Figure 9.14 Different mechanisms of formation of oxide thin films on metal surfaces. (a) Formation of porous films, typically with poor adhesion to the metal substrate. (b) Formation of different types of dense films. (c) Growth mechanism accounting for growth; (i) general processes; (ii) growth toward the metal substrate (anion dominated); and (iii) outward growth (cation dominated). The markers are used to determine the direction of film growth.
Figure 9.15 Different growth modes for oxide layers on metal substrates. The Figure is adapted from Ref. [49].
Figure 9.16 Current–voltage dependence during oxidation and passivation of metal electrodes. The metal is in contact with an aqueous electrolyte, and it is positively polarized.
Figure 9.17 (a) Potential energy profile of an electron-transfer reaction at the electrode–electrolyte interface. The gray line represents equilibrium conditions. The red line shows the situation with applied potential. (b) Electrical potential distribution within the inner dense part of the Helmholtz double layer. From Ref. [13].
Figure 9.18 Current density versus overpotential in accordance to Butler–Volmer equation. The individual partial currents are represented by dashed lines. The total current is the sum of the partial currents.
j
total
=
j
a
+
j
c
.
Figure 9.19 Calculated time evolution of the oxygen vacancy profiles after applying a DC voltage
V
to a 0.1 at % acceptor-doped SrTiO
3
crystal of thickness
d
= 0.1 cm at
T
= 500 K for two different voltages. The electrodes are assumed to block the ion transfer. (a)
V
= 3 mV, within the validity range of the linear transport theory and (b)
V
= 1 V, leading to highly nonlinear concentration profiles. Please note the very different scales of the ordinate: (a) a small interval on the linear scale and (b) several orders of magnitude on the logarithmic scale. For the concentration symbol, the bracket [ ] notation is used. (From Ref. [55]; details are provided in Ref. [56].)
Figure 9.20 Calculated steady-state demixing profiles,
x
B
(
z
), of the dopant B in an oxygen ion conductor AO
2
(+B
2
O
3
) for a ratio of the diffusion coefficients
γ
=
D
A
/
D
B
= 0.5 and different values of ratio of oxygen partial pressures on both sides of the cell
p
O
2
(2)/
p
O
2
(1). The initial dopant fraction is
x
B
= 0.1. Details are provided in Ref. [58].
Chapter 10: Electroforming Processes in Metal Oxide Resistive-Switching Cells
Figure 10.1 Schematics of the current–voltage behaviors undergoing electroforming with applied (a) voltage sweeps and (d) current sweeps. The applied input sweeps in a time domain for each case are illustrated in (b) and (e), respectively. The corresponding output current and voltage in the time domains are also illustrated in (c) and (f), respectively.
Figure 10.2 Bipolar switching behaviors depending on the forming bias polarity. (a) The positive constant current (500 μA) application to a Pt/TiO
2
/Ti/Pt cell in vacuum showed the forming
V-t
profile and (b) consequently led to the bipolar switching
I-V
behavior: reset and set switching under a positive and a negative current, respectively. (c) The negative constant current (−500 μA) application in vacuum, however, showed the forming
V-t
profile and (d) consequently led to the opposite bipolar switching behavior: reset and set switching under a negative and a positive current, respectively.
Figure 10.3 Forming processes required for Pt/TiO
2
/Pt switching cells. Potentiodynamic methods (i.e., (a) a negative voltage sweep and (b) a positive voltage sweep) and a potentiostatic method (i.e., application of constant currents alternating the polarity at 2000 s) were employed to form the cells. (a)
I-V
curve representing a single forming process indicated by “Forming ON.” (b)
I-V
curves representing a multiple forming process (i.e., first, a positive, and second, a negative voltage sweep, denoted by “Forming OFF” and “1st ON,” respectively). (c)
V-t
profile exhibiting a multiple forming process. (d) Bipolar switching
V-I
curves of the cell that underwent the forming process shown in (c).
Figure 10.4 (a) Illustration of the setup employed in the investigation of the oxygen partial pressure dependence of the resistance of the formed Fe-doped SrTiO
3
-based switching cell. (b) The measured resistance change with respect to time (
R-t
) upon the change of atmosphere (first Ar/H
2
and then O
2
at 50 min). The initial and transitional behaviors upon the atmosphere change are enlarged in (c) and (d), respectively.
Figure 10.5 Voltage-pulse-induced forming in Pt/TiO
2
/Pt switching cells at three different temperatures (25, 62.5, and 100 °C). For each temperature, the relationship between forming time and voltage pulse height is plotted. The power was evaluated from the measured voltage responses to the applied voltage pulses. The short-dashed vertical line at approximately 5.5 V demarcates the dielectric breakdown (right side) and bipolar-switching-stable (left side) regions.
Figure 10.6 Forming voltage scaling with the thickness of the HfO
x
films in the 10 × 10 and 40 × 40 nm
2
sized TiN/HfO
x
/Hf/TiN switching cells. The inset shows a comparison between the forming and the subsequent set switching voltages for the 10 × 10 nm
2
sized cell employing a 2 nm thin HfO
x
film.
Figure 10.7 Current-sweep-driven forming for vertical asymmetric Pt/TiO
2
/Ti/Pt cells (from the bottom). The square-shaped cells have five different pad sizes: 100 × 100, 200 × 200, 300 × 300, 400 × 400, and 500 × 500 nm
2
.
Figure 10.8 Optical microscope image of an as-formed Si/SiO
x
/Au switching cell. A number of bubbles are shown and placed at the anode.
Figure 10.9 Optical microscope images of (a) an as-fabricated Pt/TiO
2
/Pt switching cell (its schematic is shown in the inset), the same cell with (b) a negative voltage, (c) a positive voltage, and (d) a higher and longer voltage applied to the top electrode. The schematic mechanisms for the bubble formation under the negative and positive voltages are illustrated in the insets of (b) and (c), respectively.
Figure 10.10 Atomic-force-microscope topographic images of (a) a pristine Pt/TiO
2
/Pt switching cell and (b) the same cell after a forming process.
Figure 10.11 Schematics of the point-dynamics-triggering CF formation sequence for (a) hyperstoichiometric NiO and (b) hypostoichiometric TiO
2
.
Figure 10.12 High-resolution TEM images of the CFs of (a) a single Magnéli phase (
n
= 4; Ti
4
O
7
) and (b) two different Magnéli phases (
n
= 4, 5; Ti
4
O
7
and Ti
5
O
9
, respectively).
Chapter 11: Universal Switching Behavior
Figure 11.1 Measured
I-V
characteristics for a HfO
x
-based ReRAM with bipolar switching. In positive-voltage set transition, the current is limited by a compliance current
I
C
, which controls the resistance in the set state
R
ON
. In negative-voltage reset transition, the reset current
I
reset
is equal to
I
C
.
Figure 11.2 Measured
R
(a) and
I
reset
(b) as a function of
I
C
for different materials and switching concepts, including TCM cells based on NiO [13], VCM cells based on HfO
x
[8, 19], HfO
x
/ZrO
x
[17], and TiO
x
[16], and Cu ECM with various electrolytes, for example, Cu:SiO
2
[18] and Ag:GeSe [18].
Figure 11.3 Nonlinear switching kinetics based on voltage pulse experiments for various VCM cells (a), namely HfO
x
[15, 27–29], HfO
x
-AlO
x
[23], TiO
x
[24], TaO
x
[22], and SrTiO
x
[30], and for various ECM cells (b), namely RbAg
4
I
5
[31], Ag
2
S [32], Cu
2
S [33], AgI [25], and Ag/GeS
2
/W [34–36].
Figure 11.4 (a) Calculated resistance for variable-diameter type using Fuchs–Sondheimer formula [38, 39] and a resistivity of 400 μΩ cm (red) and 200 μΩ cm (blue), corresponding to metallic Hf and Cu filaments, respectively. (b) Calculated resistance for the variable-gap type using Simmons equation [40] for electron tunneling. A barrier height of 3.6 eV, a filament diameter of 4 nm, and a relative electron effective mass of 1 are assumed.
Figure 11.5 Schematic illustration of the set transition in a bipolar RRAM. In the reset state, defects are initially accumulated at the top electrode (a). Application of a positive voltage to the top electrode induces defect migration toward the bottom electrode, resulting in CF formation (b) and growth (c,d).
Figure 11.6 Measured and calculated
R
(a) and voltage across the ReRAM (b) as a function of time, during the set transition with a load resistance
R
L
= 1kΩ for increasing voltage
V
A
. Data and calculations are also shown for
R
L
= 2.2 kΩ (c,d) and
R
L
= 5 kΩ (e,f). The voltage across the ReRAM follows a universal behavior with time, supporting the voltage-driven model of Eqs. (11.1)(11.2)(11.3).
Figure 11.7 Experimental setup for data shown in Figure 11.6, consisting of a ReRAM in series with a load resistor
R
L
