Vanadium Dioxide Phase Transition Modeling and Bias Control for Photodetection

VANADIUM DIOXIDE PHASE TRANSITION MODELING

AND BIAS CONTROL FOR PHOTODETECTION

Zhongnan Qu

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

The Edward S. Rogers Sr. Department of Electrical & Computer Engineering

University of Toronto

Abstract

Vanadium Dioxide Phase Transition Modeling and Bias Control for Photodetection

Zhongnan Qu

Master of Applied Science

The Edward S. Rogers Sr. Department of Electrical & Computer Engineering

University of Toronto

2020

Vanadium dioxide (VO2) is a transition material that demonstrates phase transitions between the insulator and metallic states when under thermal, electrical or optical stimuli. It is a promising material for novel devices, for instance, switches, volatile memory, oscillatory neural network units, and photodetectors. Despite being a long-standing interest in the field of condensed matter physics, the numerous observed properties of VO2 remain insufficiently explained, which impedes the progress of its utilization for electronic devices. This thesis develops theoretical explanations and corresponding analytical and numerical models that fit our fabricated VO2 devices. This thesis also investigates the feasibility of utilizing VO2 as photodetectors and develops a bias control circuit with low-cost off-the-shelf circuit components, enabling repeated operations despite the hysteresis effect, demonstrated the potential of VO2 to be integrated with existing electronics.

Acknowledgments

I thank my supervisor, Prof. Joyce Poon for granting me this opportunity as well as providing me the support and environment to conduct this research. Her guidance and encouragement throughout my master’s study have allowed it to be an enriching and fruitful experience.

I thank Junho Jeong and Dr. Youngho Jung for their device design and fabrication, on top of which this thesis is built entirely. I also thank their experimental setup design and construction, academic support, and guidance. I thank our collaborators at Max Planck Institute of

Microstructure Physics, Dr. Bin Cui and Prof. Stuart Parkin, for providing deposited material and analytical insights from the perspective of material science and physics.

I thank Chaoxin Ding for his help on device measurements and theoretical discussions, and

I thank Ankita Khanda for her inception of the circuit design.

I thank Prof. Edward Sargent, Prof. J. Stewart Aitchison and Prof. David Lie for being on my thesis defense committee. I also thank Prof. Edward Sargent and Prof. Nazir Kherani for being on my thesis proposal presentation committee.

Lastly, I thank my colleagues in Prof. Joyce Poon’s research group for research and administrative guidance, support, and companionship.

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Table of Contents

Abstract ii Acknowledgements iii Table of Contents iv List of Tables vi List of Figures vii List of Abbreviations ix 1 Introduction ...... 1 1.1 Motivation: A New Type of Transition-edge Photodetector ...... 1 1.2 Background ...... 4 1.2.1 Observed VO2 Phase Transition and Oscillation ...... 4 1.2.2 Phase Transition Mechanism ...... 5 1.2.3 VO2 Photodetector Operation and Limitation ...... 9 1.3 Thesis Objectives and Organizations ...... 11 2 VO2 Device, Experiment Setup and Properties ...... 12 2.1 VO2 Microwire Device ...... 12 2.2 Experimental Setup ...... 14 2.3 VO2 Electrical and Optical Properties ...... 15 3 VO2 Modeling ...... 24 3.1 Analytical Model ...... 24 3.1.1 Modeling Method ...... 24 3.1.2 Uncertain Parameters ...... 27 3.1.3 Parameter Optimization ...... 31 3.1.4 Results ...... 32 3.2 Circuit Equivalent Model ...... 37 3.2.1 Modeling Method ...... 38 3.2.2 Fitting Parameters...... 40 3.2.3 Results ...... 42 3.3 Grain Network Model ...... 45 3.3.1 Modeling Method ...... 45

3.3.2 Results ...... 49 3.4 Further Discussions ...... 53 4 Bias Control Circuit ...... 56 4.1 Design Metrics ...... 56 4.2 Circuit Design ...... 58 4.3 Characterization ...... 63 5 Conclusion and Future Work...... 65 5.1 Conclusion ...... 65 5.2 Future Work ...... 66 Appendix A 68 Appendix B 69 Appendix C 71 Bibliography 78

List of Tables

Table 2.1: Dimensions of the central VO2 wires vs column number. Greyed out columns do not have functioning wires after etching...... 14 Table 2.2: Ic1 mean and standard deviation, devices with different dimensions ...... 18

List of Figures

Figure 1.1: MSM photodetector, extracted from [1]...... 2 Figure 1.2: PIN photodiode ...... 2 Figure 1.3: Supercurrent-assisted hotspot formation in a superconducting strip, extracted from [3]. Arrows are the direction of the supercurrent. Sub-figures a,b,c,d illustrate the sequence of incident photon creating a hotspot of non-superconducting state and blocks supercurrent...... 3 Figure 1.4: (a) Typical VO2 thermally triggered phase transition, extracted from [13], with different phases of the material indicated. (b) Voltage triggered phase transitions. “ramp up” curve shows the insulator-to-metal transition triggered by ramping up the voltage from low to high, and “ramp down” curve shows the metal-to-insulator transition triggered by ramping down the voltage from high to low. From measurements of our devices. (c) Current triggered phase transitions (d) Optically triggered phase transition with current bias, laser on/off time = 20s ...... 5 Figure 1.5: Energy – momentum curve of materials when (a) atoms are equally spaced (b) atoms are displaced (dimerized). Extracted from [16] ...... 6 Figure 1.6: VO2 lattice structures under (a) Tetragonal rutile metal R phase (b) monoclinic insulating M1 phase. Extracted from [18] ...... 7 Figure 1.7: Repeated photodetection limited by hysteresis, current biased ...... 10 Figure 2.1: Resistance vs Temperature plot ...... 12 Figure 2.2: (a) VO2 device top view (b) VO2 Device cross-section view (c) VO2 Device under SEM (d) Part of the 12x12 devices matrix. (a) and (b) figure credit: Junho Jeong ...... 13 Figure 2.3: (a) optical system schematics. Figure credit: Dr. Youngho Jung (b) device electrical connection schematics ...... 15 Figure 2.4: (a) I bias swept up and down, 1 device (b) V bias swept up and down, 1 device. Y-axes are the current/voltage across VO2 and the series resistor. Arrows indicate the direction of the insulator-to- metal and metal-to-insulator transitions ...... 16 Figure 2.5: error bar plot of the mean and standard deviation of Ic1 ...... 18 Figure 2.6: V-I measurements of (a) one device (3μm long ×3μm wide) with consistent Ic1 across 20 sweeps (b) one device (3μm long ×2μm wide) with inconsistent Ic1 across 20 sweeps (c) 4 devices with same dimensions (5μm long ×2μm wide) demonstrate consistent Ic1 (d) 4 devices with the same dimensions (5μm long ×3μm wide) demonstrate inconsistent Ic1 ...... 19 Figure 2.7: (a) Oscillation with current biased just after Ic1 (b) Oscillation with current biased just before Ic2 (c) One cycle of the oscillation (d) Oscillation means and peak-to-peak voltages vs device current bias (e) Oscillation frequencies vs device current bias. In (d) and(e), for this specific device, Ic1 = 100μA, Ic2 = 180μA...... 21 Figure 2.8: (a) successful photodetection with current biased VO2 device (b) unsuccessful photodetection with current biased VO2 device. Voltage does not return to the initial value when the optical stimulus is taken off (c) illustration of the requirements of a successful photodetection ...... 23 Figure 3.1: Measurements of 휀푉푂2 under different frequencies and temperatures in [29]. 100Hz: •, 1kHz: ▲, 10kHz: △, 100kHz: ○ ...... 28 Figure 3.2: Measurements of 휀푉푂2 under different temperatures in [30]...... 29 Figure 3.3: carrier concentration n and carrier mobility μ with respect to temperature, measured from Hall measurement. Extracted from [32]...... 30

vii

Figure 3.4: Contour map of objective function output vs 휀푉푂2 and μ, with 5 different W values...... 33 Figure 3.5: Objective function maximum output values vs corresponding W value in eV ...... 34 Figure 3.6: Simulated vs measured curves with different εVO2, 휇, W combinations. Correlation values are indicated on top of figures. 4 devices with dimensions(a) 3μm long × 3μm wide (b) 4μm long × 3μm wide (c) 5μm long × 3μm wide (d) 6μm long × 2μm wide ...... 35 Figure 3.7: Simulated vs measured V biased curves ...... 36 Figure 3.8: Simulated required minimum optical power for transition (blue) vs optical power that enables (green) and does not enable (red) transitions ...... 37 Figure 3.9: LTSpice equivalent circuit model example schematics ...... 39 Figure 3.10: Extracting 푉퐻, 푉퐿, 퐼푐1, 퐼푐2 from device V-I curve ...... 40 Figure 3.11: (a) equivalent circuit (b) equivalent RC series circuit...... 42 Figure 3.12: Simulated vs measured V-I curves ...... 42 Figure 3.13: (a) Simulated and measured oscillations, IB = 100μA, 4 cycles (b) Simulated and measured oscillations, IB = 100μA, 1 cycle (c) Simulated and measured oscillations, IB = 180μA, 4 cycles (d) Simulated and measured oscillations, IB = 180μA, 1 cycle (e) Simulated and measured oscillation frequencies under different bias current ...... 44 Figure 3.14: 2-dimensional VO2 grain network. Contact pads – blue rectangles. Grains – white circles. Connection resistors – black rectangles...... 46 Figure 3.15: IB = 92μA (a) simulated VO2 grain network avalanche IMT in chronological order. Sub- figures are a binary representation of the network, with green grains in insulator states and blue grains in metallic states (b) VO2 voltage and resistance vs time step ...... 50 Figure 3.16: IB = 120μA (a) grain resistive states showing oscillation (b) VO2 voltage and resistance vs time step ...... 51 Figure 3.17: Hysteresis demonstrated by the grain model ...... 52 Figure 3.18: one cycle of oscillation broken down into three regions ...... 55 Figure 3.19: carrier concentration vs electric field plot with hysteresis. Oscillation cycle regions A, B, C follow the green arrows ...... 55 Figure 4.1: Block diagram of the bias control circuit ...... 57 Figure 4.2: Schematics of (a) current source configuration in [43] (b) Howland current pump (c) load-in- the-loop current source ...... 59 Figure 4.3: Current source design schematics ...... 59 Figure 4.4: (a) DC (b) transient simulations of Vin vs Iload with the load resistance swept from 5kΩ to 100kΩ...... 60 Figure 4.5: (a) Current source design with cascode current mirror (b) transient simulation with 80μA parallel bias at load ...... 61 Figure 4.6: Transition detector comparator schematics ...... 62 Figure 4.7: Experimental measurements of Vin vs IVO2 with a 1kΩ resistor, a 100kΩ resistor and a VO2 device as load ...... 64

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List of Abbreviations

MSM Metal-semiconductor-metal

TES Transition Edge Sensor

퐕퐎ퟐ Vanadium Dioxide

MIT Metal-insulator transition

IMT Insulator-metal transition

SPT Structural phase transition

ASIC Application-specific integrated circuit

SEM Scanning electron microscope

CW Continuous wave

IR Infrared

DC Direct current

AC Alternating current

TEC Thermal electric cooler

DAC Digital-to-analog converter

Chapter 1

Introduction

1.1 Motivation: A New Type of Transition-edge Photodetector

Photodetectors serve the role of interfacing the optical and electrical domains, detecting and translating optical power into electrical signals. Photodetectors have a wide range of applications in numerous fields, for instance, telecommunication and imaging, with different applications posing different requirements on its operating conditions and performance, such as operating temperature, electrical bias, responsivity, and operating speed [1]. Photoconductors, photodiodes and transition edge sensors (TES) are three of the most representative photodetector architectures. Some of the most representative designs from each type of photodetectors are presented here.

Photoconductor

A metal-semiconductor-metal (MSM) photodetector is a type of photoconductor device, which has two Schottky junctions at both metal-semiconductor contacts. It operates on the principle that incident photons generate charged carriers, and these carriers are swept to the terminals by an externally applied voltage bias [1] [2]. The performance of this simple photoconductor design is normally insufficient for high speed and low noise applications [1].

Figure 1.1: MSM photodetector, extracted from [1].

Photodiode

A PIN photodiode is a photodiode device with an undoped intrinsic region between a p-doped and an n-doped region, with a simple schematic in Figure 1.2. Operating under reverse biases and at room temperature, its depletion region generates electron-hole pairs under illumination, and the resulting photocurrent is measured to sense the power of the incident light. PIN photodiodes have a relatively low junction capacitance, allowing fast operation speed [2].

Figure 1.2: PIN photodiode

Transition Edge Sensor

Superconducting nanowire is a type of transition edge sensor, which is cryogenically cooled below its critical temperature to achieve a superconducting state that enables supercurrents.

Incident photons then create a local hotspot that blocks supercurrent and gives rise to a voltage

2 signal. Superconducting photodetectors can detect single photons with a response time as fast as tens of picoseconds, theoretically achieving gigahertz operation bandwidth [3] [4]. Since the first report of picosecond superconducting single-photon optical detector in 2001 [3], improvements on architecture have been made, but all of which replies on a cryogenic environment [4], since superconductors have transition temperatures near 4K.

Figure 1.3: Supercurrent-assisted hotspot formation in a superconducting strip, extracted from [3]. Arrows are the direction of the supercurrent. Sub-figures a,b,c,d illustrate the sequence of incident photon creating a hotspot of non-superconducting state and blocks supercurrent.

Vanadium Dioxide (VO2) microwire emerges as a new type of potential transition edge photodetector architecture without the need for a cryogenic environment. From theoretical and experimental results which will be elaborated in the following sections, VO2 microwire photodetectors can reach sub microwatt minimum detectable optical power with low voltage or current bias. Furthermore, reports have shown the potential of VO2 for neuromorphic computing and optical memory [5] [6] [7] [8], where optical stimuli could be an essential component. VO2- based photodetectors, in comparison to conventional photodetectors, could enable potential monolithic integration with other VO2-based components in these applications.

1.2 Background

1.2.1 Observed 퐕퐎ퟐ Phase Transition and Oscillation

VO2, along with other strongly correlated materials, has been an interest of research in the fields of condensed matter physics and electrical engineering for the past several decades due to its demonstrated property of transition between the insulator state and the metallic state. VO2 undergoes an insulator-to-metal state transition (IMT) under heating (~340K), electrical field

(~107 V/m) [9], current (~10 A/mm2) [10], optical stimulus, magnetic field [11], and strain

[12], or a combination of two or more of the above stimuli. The thermally, electrically or optically triggered phase transitions are the focus of this thesis and demonstrated in Figure 1.4.

When the stimuli are removed, VO2 reverts to the insulator state, usually with hysteresis, most noticeably shown in Figure 1.4(b). This phase transition is accompanied by a drastic change in resistivity of one to four orders of magnitude, depends on the device [10] [13]. Especially under voltage, current or optical stimuli, after the IMT has taken place, VO2 demonstrates an oscillatory property with frequencies in the kHz regime, with the oscillation frequency increases with stimuli strength, and eventually stops. These two properties make VO2 a very promising material for novel electronics such as thermally or optically controlled switches, memory, oscillators with continuous frequency tuning, oscillatory neural network units [14], and photodetectors.

Figure 1.4: (a) Typical VO2 thermally triggered phase transition, extracted from [13], with different phases of the material indicated. (b) Voltage triggered phase transitions. “ramp up” curve shows the insulator-to-metal transition triggered by ramping up the voltage from low to high, and “ramp down” curve shows the metal-to-insulator transition triggered by ramping down the voltage from high to low. From measurements of our devices. (c) Current triggered phase transitions (d) Optically triggered phase transition with current bias, laser on/off time = 20s

1.2.2 Phase Transition Mechanism

Incomprehensive understanding of the VO2 phase transition mechanism has been a major challenge for designing said novel electronics. The driving factor of the VO2 phase transition has been under debate since its first report by Morin in 1959 [15], with the two most dominant

5 theories being the Mott-Hubbard transition and the Peierls transition, arguing from the perspective of electron-electron interaction and electron-lattice interaction.

Peierls Transition

Initially proposed by Rudolf Peierls, Peierls transition is a process where lattice dimerization

(atoms forming pairs) leads to materials showing metallic characteristics under high-temperature transition to showing insulator characteristics under low temperature [16] [17]. When the temperature is lowered, lattice distortions occur due to instability caused by electron-lattice interaction. Normally atoms with equal spacings start to form pairs, leading to the opening of a bandgap, as shown in Figure 1.5. Under low temperature, electrons mostly occupy the states in the valence band, therefore the material demonstrates insulator characteristics.

Figure 1.5: Energy – momentum curve of materials when (a) atoms are equally spaced (b) atoms are displaced (dimerized). Extracted from [16]

J.B. Goodenough later attributed VO2 phase transition to Peierls transition [18]. Tetragonal rutile metallic phase (R) and monoclinic insulating phase (M1) were observed in VO2 under high and low temperatures, respectively. As depicted in Figure 1.6, Vanadium atoms are periodically spaced under the R phase, and enhanced V-V bonds were observed under the M1 phase, which was associated with the creation of the bandgap under low temperature.

Figure 1.6: VO2 lattice structures under (a) Tetragonal rutile metal R phase (b) monoclinic insulating M1 phase. Extracted from [18]

However, the Peierls transition was later found unable to explain the 0.6eV bandgap value of VO2 [17]. Moreover, other types of lattice structures other than M1 were later discovered in the insulator phase of VO2, for instance, the monoclinic M2 and M3 phases, and the triclinic T phase [17]. Rice et al. suggested that all the insulating phases should, in fact, be characterized as

Mott-Hubbard insulators instead of band insulators [19].

Mott-Hubbard Transition

N.F. Mott proposed an alternative model taking electron-electron interactions into account [20]

[21], suggesting that when the carrier concentration n reaches a critical value 푛푐 so that:

1 3 푛푐푎퐻 = 0.25 (1.1)

2 휀0휀푉푂2ħ where 푎퐻 = 4휋 2 is the Bohr radius of the material, with 휀0, 휀푉푂2 , ħ, 푚푉푂2 , 푒 being the 푚푉푂2푒 permittivity constant, VO2 relative DC permittivity, reduced Planck constant, effective mass of charged carriers (electrons) in VO2, and elementary charge. A bandgap is then created due to electron-electron correlation, and VO2 shows insulating or metallic characteristics, depending on the direction of the transition, in a discontinuous fashion. A Hubbard energy, U, needs to be introduced into the calculation of VO2 energy states to accurately describe all the insulating states [17], therefore naming the transition Mott-Hubbard transition. The wide 0.6eV bandgap in the M1 phase that cannot be explained with the Peierls transition model can be accurately calculated in the Mott-Hubbard model [17] [20].

More recently, it was experimentally shown in Raman Spectroscopy by Kim et al. that in the process of an IMT, Mott-Hubbard transition occurs first without any structural phase change, and then a transition to rutile metal phase follows [22]. An unstable monoclinic and correlated metal (MCM) phase before the structural phase transition (SPT) and after Mott-Hubbard transition was later observed through femtosecond pump-probe measurement by the same group, further confirming the picture [23], on which consensus seems to have been made in the recent years [13] [17] [24].

Instead of a purely thermally triggered IMT as suggested in the Peierls transition model, a picture of Mott-Hubbard transition followed by a subsequent SPT suggests the trigger of VO2

IMT is its carrier concentration reaching a critical value nc, which can be in turn caused by electrical, optical, or thermal stimulations. In the scope of engineering high-speed electronic devices with VO2 where accurate and predictable performance is required, this thesis mainly

8 investigates the effect of the electrical, optical and thermal stimulus, but does not lose focus on the underlying effects from carrier concentrations.

Despite the existence of theoretical models for IMT, further explanations are still needed for several other observed phenomena of VO2, such as observed two-step transition under a current bias sweep, and the oscillation under electrical or optical stimulus.

1.2.3 퐕퐎ퟐ Photodetector Operation and Limitation

Abrupt phase transitions have always been favoured by sensor applications [25]. Since IMT/MIT can be triggered by a combination of electrical and optical stimuli, VO2 photodetector operates based on the principle that if the device is electrically biased right before the point of transition

(the critical current Ic or voltage Vc), incident light with low power can trigger an abrupt drop in voltage (or a raise in current) which can be detected.

In practice, the bias point proves very challenging to locate. Same VO2 microwire devices could demonstrate different critical voltages or currents across tests, and different VO2 microwire devices with the same geometries could demonstrate different critical currents or voltages as well. These differences are usually on the order of 1-10μA and made it very difficult to achieve microwatt power optical detection in our lab.

Hysteresis between IMT and MIT is also a major challenge for high sensitivity photodetection. Illustrated in Figure 1.7, when conducting a photodetection test without any auxiliary circuitry, if the bias point is chosen in close proximity to the critical current (point A) within the range of hysteresis for low optical power detection, simply removing the optical stimulus cannot trigger the device back into the insulator state (point A) for the next detection.

Instead, the device stays in the metallic state (point C). The voltage bias hysteresis can reach as

9 high as 2V. Although hysteresis under current bias is significantly smaller than that of voltage bias, it still poses a barrier towards achieving low detectable power and repeated operations.

Figure 1.7: Repeated photodetection limited by hysteresis, current biased

In addition, the photodetector devices cannot be designed with confidence if the transition mechanism is not well understood, for instance, the oscillation phenomenon suggests the existence of an unstable phase, which has the potential to disrupt the readout and repeated detection of incident optical stimulus. Scientific justification for utilizing IMT and MIT for photodetection and other device functions is needed.

Due to the unstable nature and hysteresis of the critical voltages and currents, a control circuit is needed to find, control and reset the bias of VO2. For the photodetector, this bias control circuit will be able to locate the bias before every detection and reset the bias after every detection to negate the effect of hysteresis.

1.3 Thesis Objectives and Organizations

This thesis investigated VO2 phase transition mechanism based on the existing theory of electron correlation effect triggered Mott-Hubbard IMT followed by subsequent Joule heating triggered

SPT. Three simulation models are developed with the experimental observations of VO2 microwires as the baseline. An analytical model is developed to reaffirm the transition mechanism and provide a method to analytically estimate the critical stimuli. A circuit equivalent model and a numerical model are developed to support an explanation of the oscillation mechanism and the two-stage transitions under current bias. Supported by the theory and the circuit equivalent model, a bias control circuit is developed to conveniently locate and control the bias for the VO2 microwire devices for repeated photodetection as well as other potential applications.

Chapter 2 describes the fabricated VO2 microwire devices, the experimental setup, and the measured thermal, electrical and optical properties. Chapter 3 presents the analytical model, circuit equivalent model and numerical model of the VO2 microwires in terms of their theoretical basis, modeling methods and connection to observed experimental properties, providing an explanation to the oscillation. Chapter 4 presents the design and characterization of the bias control circuit and the associated photodetection performance. Chapter 5 provides a summary of this thesis and future directions of research.

Chapter 2

퐕퐎ퟐ Device, Experiment Setup and Properties

2.1 퐕퐎ퟐ Microwire Device

The devices were designed and fabricated by Junho Jeong and Youngho Jung. 360nm thick VO2 thin films were deposited onto TiO2 substrates using pulsed laser deposition by Bin Cui, in Prof.

Stuart Parkin’s department at the Max Planck Institute of Microstructure Physics. High- resolution x-ray diffraction and Rutherford backscattering spectroscopy show the VO2 films are single crystalline. Atomic force microscopy shows cracks on the thin films, possibly due to large tensile misfit strain. This added tensile strain potentially caused the lower insulator-to-metal transition temperature between 310K and 320K of the prepared devices.

Figure 2.1: Resistance vs Temperature plot

Figure 2.2(a) and (b) show the top view and cross-section view of a designed device. Half of the devices have the H-shaped VO2 channel as indicated with the dashed lines and the central

12 yellow segment, and the other half only has the line-shaped channel with only the central segment. Palladium (Pd) is chosen as the contact material, covering the dashed region.

The devices were fabricated by electron beam lithography and CF4 based dry etching. The

VO2 channel area was first covered by electron beam resist (ZEP-520A), and the surrounding film was etched in an inductively coupled plasma reaction ion etcher (ICP-RIE). The Pd contacts were formed using electron beam lithography, thermal evaporation deposition, and lift-off. The scanning electron microscope (SEM) image of a fabricated device is shown in Figure 2.2(c).

Shown in Figure 2.2(d), a 12 by 12 matrix of devices were fabricated in total, with row 1-6 having the H-shaped VO2 channels and row 7-12 line-shaped channels. Dimensions of the central

VO2 wires vs column numbers are shown in table 2.1. The 1μm wide wires were completely etched away during fabrication. The H-shaped wires proved to have formed more stable contact with the palladium pads and were chosen to be the focus of this thesis.

More details on VO2 thin film deposition can be found in [26].

Figure 2.2: (a) VO2 device top view (b) VO2 Device cross-section view (c) VO2 Device under SEM (d) Part of the 12x12 devices matrix. (a) and (b) figure credit: Junho Jeong

Column# 1 2 3 4 5 6 7 8 9 10 11 12 Length(um) 3 3 3 4 4 4 5 5 5 6 6 6 Width(um) 1 2 3 1 2 3 1 2 3 1 2 3 Table 2.1: Dimensions of the central VO2 wires vs column number. Greyed out columns do not have functioning wires after etching.

2.2 Experimental Setup

Figure 2.3(a) shows the optical system that focuses the continuous wave (CW) laser onto

VO2 devices, as well as provide real-time visibility in visible and infrared (IR) wavelengths for device and probe manipulation. A CW 1550nm laser is used to enable a continuous and controlled optical stimulus to VO2 films. The laser output is first collimated to reduce the angle of divergence, then redirected by a beam splitter into the 100X objective and focused onto the

VO2 thin film held by a sample stage. The transmitted 1550nm light is then picked up by a 50X objective and redirected into an IR camera with a mirror for real-time recording on a computer.

Visible wavelength illumination is directly applied from the environment onto the sample using a pair of gooseneck illuminators. Reflected visible light is collected by the 100X objective and directed to the CMOS camera above.

The VO2 devices are electrically addressed by two tungsten probes placing on each palladium pad, as shown in Figure 2.3(b), with the probes being manually controlled by mechanical manipulators with three degrees of freedom. The entire sample is placed on a sample stage, which in turn placed on a motorized stage controlled by a computer so that different devices can be placed under the objective lens.

Figure 2.3: (a) optical system schematics. Figure credit: Dr. Youngho Jung (b) device electrical connection schematics

2.3 퐕퐎ퟐ Electrical and Optical Properties

Electrical measurements were conducted on fabricated devices. As shown by the arrows in

Figure 2.4(a), when the VO2 microwire device is under a current bias that first ramps up from 0A to a maximum value, then ramps back to 0A, a two-step “transition” can be observed. Note that the two “transitions” here under current bias are not both associated with the Mott-Hubbard transition followed by the SPT discussed previously, since the Mott-Hubbard transition should only happen once when current is ramped up unidirectionally. For the convenience of the discussion, this thesis refers to these two observed discontinuities “1st transition” and “2nd transition”. For this specific measured 5μm long and 3μm wide device, the 2nd transition demonstrates a greater hysteresis effect about 100μA, and the 1st transition demonstrates a significantly lower hysteresis effect about 10μA. In Figure 2.4(b), the same VO2 device

15 demonstrates only one transition when its bias voltage was swept up and down in the same pattern, with a hysteresis about 2V. Other devices with different dimensions have a hysteresis effect on the same order of magnitude.

Figure 2.4: (a) I bias swept up and down, 1 device (b) V bias swept up and down, 1 device. Y-axes are the current/voltage across VO2 and the series resistor. Arrows indicate the direction of the insulator-to- metal and metal-to-insulator transitions

We chose the current bias approach instead of the voltage bias approach for photodetection purpose mainly due to the reason that the 1st current biased transition shows the smallest hysteresis. Without an external bias control circuit, this property allows the potential for multiple photodetection operations with relatively low optical power, without the need to reset bias before every detection.

Moreover, the current biased transition has a negative feedback effect, while the voltage biased transition has a positive feedback effect, which despite produces a higher current increase, it has the potential to damage the VO2 microwire. According to Ohm’s law

푉 = 퐼 ∙ 푅 (2.1) and the law of Joule heating

푉2 푃 ∝ (2.2) 푅

푃 ∝ 퐼2 ∙ 푅 (2.3)

When under voltage bias, the Joule heating power before and after the transition is described in equation 2.2. With applied voltage being constant, an abrupt resistance decrease leads to an abrupt Joule heating power increase, which could potentially anneal the microwire and make it lose the transition property. When under current bias, with the Joule heating equation in 2.3, transition leads to an abrupt decrease in Joule heating power. This negative feedback effect protects the limited number of devices during experiments.

All fabricated 2μm and 3μm wide H-shaped devices were screened for their first transition currents Ic1 with 3 current sweeps, then 20 more sweeps were conducted on devices with consistent transition currents. Mean and standard deviation values were calculated across all current sweeps for devices with the same dimensions and summarized in table 2.2, with the spread visualized in Figure 2.5. Figure 2.6(a)(b) shows one device can demonstrate different Ic1 across different sweeps, with 2.6(a) showing a device with consistent Ic1 while 2.6(b) showing a device with inconsistent Ic1. Figure 2.6(c)(d) shows different devices with the same dimensions can demonstrate consistent or inconsistent Ic1.

Length (μm) Width (μm) Mean Ic1_avg (μA) Standard Deviation Ic1_σ (μA) 3 2 56.4 12.1

4 2 59.2 1.7

5 2 56.1 6.7

6 2 58.0 10.0

3 3 116.8 2.6

4 3 98.5 5.8

5 3 90.2 5.9

6 3 90.4 3.2

Table 2.2: Ic1 mean and standard deviation, devices with different dimensions

Figure 2.5: error bar plot of the mean and standard deviation of Ic1

Figure 2.6: V-I measurements of (a) one device (3μm long ×3μm wide) with consistent Ic1 across 20 sweeps (b) one device (3μm long ×2μm wide) with inconsistent Ic1 across 20 sweeps (c) 4 devices with same dimensions (5μm long ×2μm wide) demonstrate consistent Ic1 (d) 4 devices with the same dimensions (5μm long ×3μm wide) demonstrate inconsistent Ic1

As shown in Figure 2.7(a)-(c), oscillation with a frequency in the kHz regime can be observed between the first and second transitions under current bias, and after the transition under voltage bias, provided the series resistance is of an appropriate value. In the case of a 5μm long by 3μm wide device and under current bias, oscillation first occurs right after the first transition with a mean amplitude of 6.124V, a peak-to-peak amplitude of 7.6V and initial frequency of 4.8kHz. Figure 2.7(d)(e) shows that as the bias current increases, the mean

19 amplitude decreases, oscillation frequency increases, and the peak-to-peak amplitude remains approximately the same. The oscillation frequency increases by a factor of 3 until right before the second transition and disappears when the second transition takes place. The mechanism and the conditions for the oscillation to happen will be further explained in detail in Chapter 3.

All voltage and current sweep measurements shown in this thesis were obtained from DC measurements with a Keithley 2632A sourcemeter. In Figure 2.4(a), when investigated with AC measurements using a Keysight DS0X3104A oscilloscope, the measured current using the sourcemeter in the the region between 1st and 2nd transition is the DC average of the oscillation.

In this region, the VO2 device rapidly switches between the insulator state and metallic state, with each cycle following the previously discussed Mott-Hubbard transition - SPT model, and stays in the metallic state after the 2nd transition [27]. In the current sweep measurements, a slight voltage increase after the first transition is observed from DC measurements but not from the mean value of AC measurements, suggesting it is likely a measurement artifact.

Figure 2.7: (a) Oscillation with current biased just after Ic1 (b) Oscillation with current biased just before Ic2 (c) One cycle of the oscillation (d) Oscillation means and peak-to-peak voltages vs device current bias (e) Oscillation frequencies vs device current bias. In (d) and(e), for this specific device, Ic1 = 100μA, Ic2 = 180μA.

Figure 2.8(a) shows a repeatable photodetection. The device used in this experiment has an averaged Ic1 of 84μA. The device was biased at 80μA, and the laser power was 20mW. In comparison, Figure 2.8(b) shows an unrepeatable attempt on the same device with the same optical power, but with current biased at 83μA, closer to Ic1. The transition was triggered when optical power was incident, but the device failed to recover to the initial state when the optical power was taken off. As illustrated in Figure 2.8(c)(d), the unrepeatable case follows the path

A->B->C in Figure 2.8(c), and the repeatable case follows A->B->A in Figure 2.8(d). To ensure a repeatable photodetection, the bias point should be placed outside of the hysteresis zone, and the optical power should be large enough such that the gained equivalent current due to the photo injection effect is larger than the hysteresis. The hysteresis inherently places a lower limit on the detectable optical power of a VO2 photodetector that allows repeatable operations, if a bias control scheme is not introduced.

Figure 2.8: (a) successful photodetection with current biased VO2 device (b) unsuccessful photodetection with current biased VO2 device. Voltage does not return to the initial value when the optical stimulus is taken off (c) illustration of the requirements of a successful photodetection

Chapter 3

퐕퐎ퟐ Modeling

3.1 Analytical Model

In Chapter 1, a carrier concentration triggered Mott-Hubbard transition followed by a Joule heating triggered SPT model has been summarized, providing some insight into the underlying physics of VO2 phase transitions. However, there is still a gap between the theoretical understanding and experimental properties of VO2. Moreover, VO2 devices cannot be designed before fabrication if its properties cannot be predicted with known design parameters, for instance, microwire dimension, VO2 crystallinity, substrate material, etc. This thesis developed an analytical model based on the work from A.L. Pergament et al. [28] that predicts the critical current and voltage of a VO2 microwire with specific dimensions and fabrication processes.

3.1.1 Modeling Method

Carrier concentration in VO2 can be generated by both Joule heating and field-induced generation. Field-induced generation is a phenomenon due to autoionization caused by Coulomb barrier lowering, analogous to the Poole-Frenkel effect, which conveniently ties electric field and temperature to the carrier concentration of VO2:

푊−훽√퐸 푛 = 푁0푒푥푝 (− ) (3.1) 푘푇푉푂2

푒3 where 훽 = √ (3.2) 휋휀푉푂2휀0

Here 푁0 is a constant approximately independent of field and temperature, W is the conductivity activation energy, β is the Poole-Frenkel constant, E is the electric field, k is the Boltzmann

constant, 푇푉푂2 is the microwire temperature.

When the carrier concentration reaches a critical value nc purely due to heating to the transition temperature 푇푐 = 340K, where there is no electric field present, equation (3.1) can be rewritten as

푊 푛푐 = 푁0푒푥푝 (− ) (3.3) 푘푇푐

푛푐 can be calculated from the Mott criterion in equation (1.1). By dividing (3.1) and (3.3) we will eliminate unknown constant 푁0 and arrive at

푊 푇푐 훽√퐸 푛 = 푛푐푒푥푝 {− [ (1 − ) − 1]} (3.4) 푘푇푐 푇푉푂2 푊

With electrical bias present, 푇푉푂2 will always be higher than the environment or substrate

temperature due to Joule heating. Since 푇푉푂2 is unable to be measured directly (only substrate temperature is accessible via a Thermal Electric Cooler (TEC)), therefore it must be calculated based on known or measurable parameters. To provide a reasonable estimation, consider the situation where 푛 = 푛 at the electrical bias point and 퐸 = 퐸 = 푉푐 (l is the length of the wire) so 푐 푐 푙

that 푇푉푂2 = 푇푉푂2_푐 < 푇푐. From (3.4):

훽 퐸 푒푥푝 {− 푊 [ 푇푐 (1 − √ 푐) − 1]} = 1, (3.5) 푘푇푐 푇푉푂2_푐 푊 which can be simplified to

푉푐 훽 퐸 훽√ 푇 = 푇 (1 − √ 푐) = 푇 (1 − 푙 ) (3.6) 푉푂2_푐 푐 푊 푐 푊

Additionally, according to Fourier’s law of heat conduction,

푑푄 = 푃 − 푘 (푇 − 푇 ) (3.7) 푑푡 푗표푢푙푒 푒푓푓 푉푂2 퐴

where Q is the heat of the system, 푃푗표푢푙푒 the power of Joule heating, 푘푒푓푓 the effective thermal conductance, and 푇 the environment or substrate temperature. At a state of equilibrium, 푑푄 = 0, 퐴 푑푡

(3.7) becomes

푃푗표푢푙푒 푇푉푂2 = + 푇퐴 (3.8) 푘푒푓푓

At the same electrical bias point as in (3.5) where 푉 = 푉푐 such that 푇푉푂2 = 푇푉푂2_푐 , (3.8) can be transformed into

2 푉푐 푃푗표푢푙푒 푅푐 푘푒푓푓 = = , (3.9) 푇푉푂2_푐−푇퐴 푇푉푂2_푐−푇퐴

where 푅푐 is the resistance of the device at transition. Substitute in (3.6), it can be derived that

2 푉푐 푅푐 (3.10) 푘푒푓푓 = 푉 훽√ 푐 푇 (1− 푙 )−푇 푐 푊 퐴

푘푒푓푓 here is a function of only measurable variables 퐼푐, 푉푐, 푅푐 and known constants. Real-time

device temperature 푇푉푂2 can then be estimated with (3.8).

Combining (3.4) with the relationship between current density J and electric field E in an insulator

퐽 = 휎퐸 = 푒 ∙ 휇 ∙ 푛 ∙ 퐸 , (3.11)

where e is the elementary charge, μ is the carrier mobility in VO2, a system with two equations

(3.4) and (3.11) and two variables n and E can be formed. With accurate estimations of the constants, n and E can be calculated based on applied current density J.

VO2 photo-absorption process can also be modeled. Approximately 10% of the laser power

푃푙푎푠푒푟 is incident on the surface of VO2 due to loss from the optical setup, and among which 10% is absorbed by VO2 according to FDTD simulations. If the quantum efficiency of photo-injection effect is η, the photo-induced carrier concentration 훥푛 can be expressed as

훥푛 = 퐺 ∙ 휏 = 휂∙1%푃푙푎푠푒푟 ∙ 휏 ∙ 1 , (3.12) 푒 ℎ푓 푤푙ℎ

where 퐺푒 is the electron-hole pair generation rate, 휏 is the carrier lifetime, h is the Planck’s constant, f is the laser frequency, w, l, h are the width, length, and height of the VO2 device, respectively. The remaining power 휂 ∙ 1%푃푙푎푠푒푟 can be attributed to photo-thermal effect heating power 푃푝ℎ표푡표푡ℎ푒푟푚푎푙, which can be added to equation (3.8) to become

푃푗표푢푙푒+푃푝ℎ표푡표푡ℎ푒푟푚푎푙 푇푉푂2 = + 푇퐴 (3.13) 푘푒푓푓

3.1.2 Uncertain Parameters

All but three parameters, 휀푉푂2 , u, and W, can be derived or measured as described above. These parameters vary depends on factors including VO2 quality, fabrication method, and substrate materials. As a result, ranges instead of single values have been reported.

Methods to calculate or measure 휀푉푂2 has been reported since VO2 has gained research interest.

A. Mansingh et al. conducted low-frequency dielectric constant measurements on bulk VO2 crystals with a Schering Bridge [29]. The measurements were only conducted under low temperatures because their sample had the tendency to shatter near the insulator-to-metal

transition. The results are described in Figure 3.1. 휀푉푂2 was estimated to be 100, based on the saturation region at high temperatures. Z. Yang et al. utilized gated capacitor devices consisting

of HfO2 and VO2 and measured that 휀푉푂2 increases from 36 at room temperature to more than

6 × 104 at 373K, with results in Figure 3.2 [30]. These previous reports give an approximate

range of 휀푉푂2 to be (36, 100) from room temperature to critical transition temperature 340K.

Figure 3.1: Measurements of 휀푉푂2 under different frequencies and temperatures in [29]. 100Hz: •, 1kHz: ▲, 10kHz: △, 100kHz: ○

Figure 3.2: Measurements of 휀푉푂2 under different temperatures in [30].

[31] used a 휀푉푂2 value of 100 and equation (1.1) and arrived at a critical carrier

18 −3 concentration 푛푐 = 2.8 × 10 [cm ]. This report suggested 푛푐 should be on the order of 푛푠, the equilibrium electron density in the conduction band of VO2 in the insulator phase, which was

18 −3 experimentally measured to be 10 [cm ] and agree with the calculated value of 푛푐. [32]

19 −3 obtained 푛푐 on the order of 10 [cm ] from Hall measurement with the results in Figure 3.3.

By feeding this value back into equation (1.1), 휀푉푂2 as low as 36 is possible at near transition temperature.

Figure 3.3: carrier concentration n and carrier mobility μ with respect to temperature, measured from Hall measurement. Extracted from [32].

[32] measured μ to be 0.15~0.2cm2/Vs, and from [21] and [31] μ is suggested to be

퐸 0.3cm2/Vs. Typically 푊 = 푔 at the transition temperature, with 퐸 the bandgap of VO 2 푔 2 calculated to be from 0.6eV to 1eV [17] [28], W has the range of 0.3eV to 0.5eV. [28] suggested

W can be as low as 0.15eV when the temperature is closer to room temperature.

Due to the uncertainty of these parameters, this model aims to search the space of the three

2 uncertain parameters: 휀푉푂2 ∈(36,100), 휇∈(0.15,0.3) [cm /Vs], W∈(0.15,0.5) [eV], and find the best combination that simulates the V-I curves of devices with different dimensions well. The goodness of the model can be evaluated by the correlation between the simulated and measured

V-I curves of each device.

3.1.3 Parameter Optimization

4 devices from columns 3, 6, 9, 11 (Table 2.1) with the most consistent V-I curves across 20 runs were chosen as the comparison baseline. These 4 devices have widths varying from 2μm to 3μm and length varying from 3μm to 6μm, covering all the possible dimensions of our fabricated devices (except 1μm wide wires that are completely etched away). One most representative V-I curve has been chosen from each of the 4 devices. Data points with bias currents greater than the first transition current Ic1 have been omitted for comparison with the simulated results.

The objective function of this optimization is defined as the averaged correlations between

corresponding simulated and measured V-I curves. With one combination of the 휀푉푂2 , 휇 , and W values, the simulated V-I curves are one-dimensional arrays S1, S2, S3, S4, and the measured curves are M1, M2, M3, M4. The output value O of the objective function is then:

푂 = 휮푛(푐표푟푟(푆푛, 푀푛))/4 , (3.14) with corr being the correlation function. An O value of 1 indicates a 100% fit between simulated and measured results.

A linear sweep of all three variables was used in this problem, with each variable space divided into 10 steps. One iteration of the objective function takes approximately 10 minutes, therefore common global optimization algorithms that involve many starting points and iterations, for instance, genetic algorithm and particle swarm optimization, are not practical for this thesis. Furthermore, since the comparison baseline itself is not perfectly consistent across multiple measurements, this model can at best provide qualitative but not quantitative simulations, therefore very accurate and time-consuming parameter search is not necessary.

3.1.4 Results

The parameter sweep outputs can be rendered into a series of contour maps to intuitively show the trend of the goodness of fit with respect to each parameter, as in Figure 3.4. Only the five most representative figures out of ten totals are shown. Yellow “peaks” can be visible from each contour plot, suggesting the location of the parameter combinations that fit the comparison baseline best. The data points without values (indicated by the white regions on the plots) are combinations that could not generate definite numerical solutions.

Figure 3.4: Contour map of objective function output vs 휀푉푂2 and μ, with 5 different W values

The maximum output values corresponding to each swept W value are demonstrated in

Figure 3.5. It can be concluded from Figures 3.4 and 3.5 that on W axis there is only one local

and global maximum, whereas on 휀푉푂2 and μ axes there are multiple (3 observable ones from the parameter sweep). From the linear sweep, three local minima are visible in the three-dimensional parameter space.

Figure 3.5: Objective function maximum output values vs corresponding W value in eV

After further manual optimization on the best parameter combination from the center

2 “peak” of the W=0.189 eV contour plot, 휀푉푂2 = 56, 휇 = 0.22 [cm /Vs], 푊 = 0.18[eV] can be obtained that achieves an averaged correlation value of 99%. Combinations from 2 other local minima are also collected, and the three combinations are plotted against measured V-I curves, shown in Figure 3.6. From visual comparison, this method fails to completely capture the shape of the I-V curves before transition but predicts the critical currents with decent accuracy. The overall lower predicted transition voltage is possibly due to the contact resistance between the

34 palladium contacts and the VO2 microwire which is not taken into account by this analytical model, as it depends greatly on the deposition method and contact forming.

Figure 3.6: Simulated vs measured curves with different εVO2, 휇, W combinations. Correlation values are indicated on top of figures. 4 devices with dimensions(a) 3μm long × 3μm wide (b) 4μm long × 3μm wide (c) 5μm long × 3μm wide (d) 6μm long × 2μm wide

By slightly increasing 휀푉푂2 to 66 while keeping μ and W the same, a good fit for voltage biased I-V curve can be obtained, shown in Figure 3.7. Previous reports have shown that single- crystalline VO2 nanobeam deposited on SiO2 substrate by physical vapor deposition with VO2 bulk powder as target demonstrates a similar x-ray diffraction pattern to our sample. From this

VO2 nanobeam parameter values 휂 = 0.06% and 휏 = 4μs were estimated from scanning

35 photocurrent microscopy [33] [34]. This unusually long carrier lifetime has been confirmed by extrapolating frequency-dependent capacitance measurements on GaN − VO2 p-n junctions [35].

The minimum optical power that can trigger an IMT under different bias voltages is simulated and plotted in Figure 3.8 along with the measured lower and upper bounds of these minimum transition optical power under different voltage biases. The agreement demonstrated between simulations and measurements shows that this analytical model can also simulate VO2 properties under voltage bias, and can quantitatively predict optically triggered transition effects. However, this thesis acknowledges that 0.06% quantum efficiency is possibly inaccurate since the photothermal effect was not taken into consideration in [35], and the experiment was conducted with a 532nm laser instead of 1550nm. Carrier lifetime also varies depends on the fabrication method and quality. Nevertheless, it can be confirmed that a product of η and τ on the order of

−9 10 can accurately model the optical response of our VO2 devices.

Figure 3.7: Simulated vs measured V biased curves

Figure 3.8: Simulated required minimum optical power for transition (blue) vs optical power that enables (green) and does not enable (red) transitions

This photo-absorption model can be utilized to determine the VO2 photodetector minimum detectable optical power. With a temperature of 300K, measurement bandwidth of 100kHz

(greater than the VO2 oscillation frequency), a resistance of ~100kΩ, and a current of ~100μA, the thermal noise, and shot noise can be calculated to be 16μV/105pA and 268μV/1.8nA, showing a shot noise dominated system. Due to the nature of VO2 that any voltage/current spike can trigger an IMT, 3 standard deviations (99.7%) should be taken to determine the noise floor, which leads to a shot noise of 5.4nA. The model simulates a 400nW incident optical power can create an equivalent amount of photocurrent, which should be on the same order of magnitude as the minimum detectable optical power.

3.2 Circuit Equivalent Model

Analogous to MOSFETs, circuit equivalent models are essential to the application of novel devices, allowing circuit-level simulations of VO2 in a network of conventional electronics

(resistors, capacitors, etc.) or other VO2s. For instance, the coupling between VO2 devices can be easily investigated with circuit equivalent models, paving the way for VO2 based oscillatory neural networks [36].

3.2.1 Modeling Method

Inspired by the driving point equivalent model developed by Maffezzoni et al. [37], this thesis developed a simplified version specifically for the LTSpice environment, minimizing simulation time and the potential for convergence issue. Essential parameters, for instance, critical currents and voltages, insulator and metallic state resistances, and switching times, can be easily tuned to fit different measured electrical properties of each device.

The schematics of the equivalent circuit are shown in Figure 3.9. The VO2 microwire is modeled as a two-terminal device with a switchable resistance in parallel with a capacitance

퐶푉푂2 . In insulator state, the switch S is open circuit and the device resistance 푅1 + 푅2 holds the value of the parameter 푅푖푛푠 which is extracted from measurements. In the metallic state, switch S closes and shorts out 푅1, leaving device resistance 푅2 = 푅푚푒푡, the measured metallic state resistance.

Figure 3.9: LTSpice equivalent circuit model example schematics

To model the hysteresis, a bi-threshold control scheme for switch S is introduced. The switch closes when the voltage across the capacitor 퐶0 is 1V, and opens when the voltage is 0V.

Variable comp1 holds the voltage across the VO2, and comp2 holds the device state and phase transition trigger value.

At the beginning of the simulation with no electrical bias, comp2 holds the value of the insulator-to-metal critical voltage VH, suggesting the device is in the insulator state. Voltage source 퐵1 is 0V. When a sufficiently large bias (either voltage or current) is applied such that the voltage across VO2 is greater than 푉퐻, i.e. 푉푐표푚푝1 > 푉푐표푚푝2, 푉푐표푚푝2 switches to the metal-to- insulator transition critical voltage 푉퐿, suggesting the device is entering the metallic state. 퐵1 raises from 0V to 1V, and charges 퐶0 up with a time constant 휏 = 푅0퐶0. When 퐶0 is charged up

so that 푉퐶0 = 1V, switch S closes and shorts out 푅1, reducing VO2 resistance to 푅푚푒푡. When the device is in the metallic state and 푉푐표푚푝1 < 푉푐표푚푝2 = 푉퐿, comp2 switches to 푉퐿, the device

39 reverts to the insulator state. 퐵1 drops to 0V and discharges 퐶0, then S closes, increasing device resistance back to 푅푖푛푠.

3.2.2 Fitting Parameters

By choosing appropriate values for parameters 푅푖푛푠, 푅푚푒푡,푉퐻, 푉퐿, the simulated V-I characteristics can resemble that of a measured device. As shown in Figure 3.10, 푉퐻 and 푉퐿 can be extracted from the measured device, and 푅푖푛푠 and 푅푚푒푡 can be determined by the relationships

푉퐻 푅푖푛푠 = (3.15) 퐼푐1

푉퐿 푅푚푒푡 = (3.16) 퐼푐2

Figure 3.10: Extracting 푉퐻, 푉퐿, 퐼푐1, 퐼푐2 from device V-I curve

As the only capacitive component in the circuit, device capacitance 퐶푉푂2 dictates the

oscillation frequency. To estimate 퐶푉푂2 , the equivalent circuit in Figure 3.11(a) can be simplified

to an RC series circuit in Figure 3.11(b) using Thevenin Theorem, with 푉푇퐻 = 퐼퐵푅푇퐻 = 퐼퐵푅푉푂2 ,

and time constant 휏 = 푅푇퐻퐶푉푂2 = 푅푉푂2퐶푉푂2 . At bias voltage IB, immediately after VO2 switches into the metallic state, 퐶푉푂2 discharges from 푉퐻 to 푉퐿 with 푅푉푂2 = 푅푚푒푡, 휏푑푖푠푐ℎ푎푟푔푒 = 푅푚푒푡퐶푉푂2 .

Immediately after VO2 switches into insulator state, 퐶푉푂2 charges from 0 to V_H with 푅푉푂2 =

푅푖푛푠, 휏푐ℎ푎푟푔푒 = 푅푖푛푠퐶푉푂2 . The respective equations are

푡푑푖푠푐ℎ푎푟푔푒 푉퐿 = 푉푇퐻 ∙ 푒푥푝(− ) (3.17) 휏푑푖푠푐ℎ푎푟푔푒

푡푐ℎ푎푟푔푒 푉퐻 = 푉푇퐻 ∙ [1 − 푒푥푝 (− )] (3.18) 휏푐ℎ푎푟푔푒 which lead to

푉퐿 푡푑푖푠푐ℎ푎푟푔푒 = −휏 ∙ 푙표푔 ( ) (3.19) 푉푇퐻

푉퐻 푡푐ℎ푎푟푔푒 = −휏 ∙ 푙표푔 (1 − ) (3.20) 푉푇퐻

푡푐ℎ푎푟푔푒 + 푡푑푖푠푐ℎ푎푟푔푒 = 푇, (3.21)

where T is the period of the oscillation. A proper 퐶푉푂2 value needs to satisfy all the above conditions.

Figure 3.11: (a) equivalent circuit (b) equivalent RC series circuit

3.2.3 Results

To fit the simulated results to a 5μm long by 2μm wide device, parameters 푅푖푛푠, 푅푚푒푡, 퐶푉푂2 ,푉퐻,

푉퐿 are chosen to be 93kΩ, 18kΩ, 1050pF, 8.5V and 3.6V, respectively. The simulated and measured V-I curves are shown in Figure 3.12. Note that in the plot of the simulated results, the oscillation region between the first and second transitions simply takes the mean value, while the measured results in this region are from the measurement sourcemeter averaging algorithm, so they do not necessarily match up.

Figure 3.12: Simulated vs measured V-I curves

The simulated and measured oscillations are analyzed in the time domain. Figure 3.13(a)(b) show the simulated oscillation fitted to measurement, when IB = 100μA. The goodness of fit is not maintained when the bias current is increased to 180 μA, as in Figure 3.13(c)(d). Simulated and measured oscillation frequencies are plotted in Figure 3.13(e), and it is visible that as the bias current increases, this fitted model gradually loses the goodness of fit. By observing the shape of the measured oscillations at 100 μA, it is obvious that the rising section of each cycle resembles the charging section of its simulated counterpart, however, the falling section does not resemble the discharging section in the model. When IB is low (100 μA), 푡푐ℎ푎푟푔푒 dominates

푡푑푖푠푐ℎ푎푟푔푒 in the simulation, and dominates the falling time of each cycle in the measurement, therefore 푇푠푖푚푢푙푎푡푒푑 ≈ 푇푚푒푎푠푢푟푒푑. When IB is high (180 μA), it takes significantly less time for the capacitor to charge to 푉퐻, the fall time of each cycle needs to be considered. 푡푐ℎ푎푟푔푒 is still consistent between simulated and measured results, however 푡푑푖푠푐ℎ푎푟푔푒 is an overestimation of the fall time. This suggests the falling section of the VO2 is possibly due to other mechanisms.

Figure 3.13: (a) Simulated and measured oscillations, IB = 100μA, 4 cycles (b) Simulated and measured oscillations, IB = 100μA, 1 cycle (c) Simulated and measured oscillations, IB = 180μA, 4 cycles (d) Simulated and measured oscillations, IB = 180μA, 1 cycle (e) Simulated and measured oscillation frequencies under different bias current

3.3 Grain Network Model

Due to the inability of the circuit equivalent model to accurately simulated the oscillation waveform, the need for a more in-depth analysis of the IMT process arises. This thesis developed a numerical model for the insulator-to-metal phase transition on a grain level that in conjunction with the other models, aims to provide a better understanding of the transition mechanism.

3.3.1 Modeling Method

Adapted from the model proposed in [27], although our fabricated VO2 thin films are single crystalline, the entire thin film can be divided into a 2-dimensional n rows by m columns network of “grains” in Figure 3.14, analogous to the crystal structure of a polycrystalline film, since both VO2 structures have qualitatively similar properties under electrical, thermal or electrical biases [10]. The connections between grains are modeled as resistors that can switch between the insulator and metallic states, depends on the voltage across. The entire network is connected to two contact pads, modeled as nodes in the circuit, with one pad connected to the bias current, and the other pad connected to the ground.

Figure 3.14: 2-dimensional VO2 grain network. Contact pads – blue rectangles. Grains – white circles. Connection resistors – black rectangles.

Based on the method reported in [38], the first step of the model is to calculate the voltage distribution of the grain network with given bias current and insulator/metallic state profile of all the resistors. The voltage of each node 푉푖,푗 can be obtained by solving Kirchhoff current law:

푑 푟 푑 푟 (푉푖−1,푗 − 푉푖,푗)퐺푖−1,푗 + (푉푖,푗−1 − 푉푖,푗)퐺푖,푗−1 = (푉푖,푗 − 푉푖+1,푗)퐺푖,푗 + (푉푖,푗 − 푉푖,푗+1)퐺푖,푗 (3.22)

푑 푟 Where 퐺푖,푗 is the conductance of the resistor to the bottom of node i,j, and 퐺푖,푗 the conductance of the resistor to the right of node i,j. Currents going from lower row number to higher row number, and currents from lower column number to higher column number are denoted positive.

The positive contact voltage 푉푉푂2 can be solved by:

푟 휮푖(푉푉푂2 − 푉푖,1)퐺푖,0 = 퐼퐵 (3.23)

The system with 푚 × 푛 + 1 variables and 푚 × 푛 + 1 equations from (3.22) and (3.23) can be organized into the form of matrix multiplication

퐴 × 푉 = 퐵, (3.24) where

푀1 퐿1 0 0 ⋯ 0 0 푅1 퐿 푀 퐿 0 ⋯ 0 0 푅 1 2 2 2 0 퐿 푀 퐿 ⋯ 0 0 푅 2 3 3 3 0 0 퐿 푀 ⋯ 0 0 푅 퐴 = 3 4 4 , (3.25) ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 0 0 ⋯ 푀푚−1 퐿푚−1 푅푚−1 0 0 0 0 ⋯ 퐿푚−1 푀푚 푅푚 [ 퐶1 퐶2 퐶3 퐶4 ⋯ 퐶푚−1 퐶푚 퐶푅 ] with

푟 퐶푖 = [−퐺푖+1,1 0 ⋯ 0] , (3.26)

푟 퐺푖+1,1 푅 = [ 0 ] , (3.27) 푖 ⋮ 0

푟 퐶푅 = 훴푖퐺푖,1 , (3.28)

퐺푑 0 0 ⋯ 0 푖,1 푑 0 퐺푖,2 0 ⋯ 0 푑 퐿푖 = 0 0 퐺 ⋯ 0 , (3.29) 푖,3 ⋮ ⋮ ⋮ ⋱ 0 푑 [ 0 0 0 0 퐺푖,푛]

푟 −푊푖,1 퐺푖,1 0 0 ⋯ 0 푟 푟 퐺푖,1 −푊푖,2 퐺푖,2 0 ⋯ 0 푟 푟 0 퐺푖,2 −푊푖,3 퐺푖,3 ⋯ 0 푀푖 = 푟 , (3.30) 0 0 퐺푖,3 −푊푖,4 ⋯ 0 푟 ⋮ ⋮ ⋮ ⋮ ⋱ 퐺푖,푛−1 푟 [ 0 0 0 0 퐺푖,푛−1 −푊푖,푛]

푑 푟 푑 푟 푊푖,푗 = 퐺푖−1,푗 + 퐺푖,푗−1 + 퐺푖,푗 + 퐺푖,푗, (3.31)

47 and

푉 1,1 푉 1,2 ⋮ 푉 푉 = 2,1 , (3.32) 푉2,2 ⋮

푉푚,푛

[푉푉푂2 ]

0 0 퐵 = [ ] , (3.33) ⋮ 퐼퐵 all nodal voltages in the 푚푛 × 1 matrix can then be solved.

To simulate the progression of grain phase transitions in time, a time-stepping method is adopted, with voltage distribution calculated using the above method each time step. At the end of each time step, the phase of each resistor needs to be updated (insulator or metallic state) according to the voltage across it and its temperature. The length of each time step is left undefined in this simulation, but the overall transition time should be on the order of picosecond or faster [13]. According to [27], the randomness of grain phase transition can be simulated by

푖,푗 assigning a Gaussian distribution to the critical grain transition voltages and temperatures 푉퐼푀푇,

푖,푗 푖,푗 푖,푗 푉푀퐼푇, 푇퐼푀푇, 푇푀퐼푇 for each grain in each time step:

0 2 푃(푉푖,푗 = 푥) = 1 푒푥푝 [− (푥−푉퐼푀푇) ] (3.34) 퐼푀푇 √2휋휎2 2휎2 푉퐼푀푇

0 2 푃(푉푖,푗 = 푥) = 1 푒푥푝 [− (푥−푉푀퐼푇) ] (3.35) 푀퐼푇 √2휋휎2 2휎2 푉푀퐼푇

0 2 푃(푇푖,푗 = 푥) = 1 푒푥푝 [− (푥−푇퐼푀푇) ] (3.36) 퐼푀푇 √2휋휎2 2휎2 푇퐼푀푇

0 2 푃(푇푖,푗 = 푥) = 1 푒푥푝 [− (푥−푇푀퐼푇) ], (3.37) 푀퐼푇 √2휋휎2 2휎2 푇푀퐼푇

0 0 0 0 where 푉퐼푀푇 , 푉푀퐼푇 , 푇퐼푀푇, 푇푀퐼푇 is the mean IMT and MIT critical voltages, and 휎 the standard deviations of the distributions, all left as fitting parameters. The temperature of each grain can be estimated with equation (3.8), with adjusted 푘푒푓푓 for VO2 grains.

3.3.2 Results

When under sufficient bias current, due to the normal distribution of critical voltage, at least one grain will go through IMT. Increased grain conductance will cause the voltage to be re- distributed to its surrounding grains, leading to more grains going through IMT, eventually avalanche into a random metallic path of least resistance and reach a steady state. This process is demonstrated in Figure 3.15(a) in a 50x50 grain network with minimum bias current 92μA that can trigger IMT. Figure 3.15(b) shows the full insulator state, transition state and steady-state with insulator grains co-existing with the metallic path.

Figure 3.15: IB = 92μA (a) simulated VO2 grain network avalanche IMT in chronological order. Sub- figures are a binary representation of the network, with green grains in insulator states and blue grains in metallic states (b) VO2 voltage and resistance vs time step

A phenomenon similar to this simulated randomly formed metallic path has been observed from other researches in our lab – when polycrystalline VO2 microwires were undergoing annealing process with high bias current, randomized paths of crystalline VO2 formed, demonstrated in the SEM image in Figure 3.15 (c). Under this experiment, bias current also induced a randomized polycrystalline metallic path which then became crystalline due to re- crystallization [10], agreeing with the simulation.

With a stronger bias current, the entire network oscillates between insulator state and metallic state. The grain resistive states and network voltages are shown in Figure 3.16. This

50 oscillation eventually stops at IB ≈ 300μA, when the bias current is enough to support the VO2 grains to stay in the metallic state.

Figure 3.16: IB = 120μA (a) grain resistive states showing oscillation (b) VO2 voltage and resistance vs time step

This model agrees with the measured VO2 transition and oscillation mechanism reasonably well, and provides an explanation to the IMT process: instead of discharging the device capacitance, the capacitance breaks down at the moment the metallic path is formed, and the voltage across the VO2 microwire drops. As a result, the time required is significantly shorter than discharging an equivalent capacitor – it is measured to be ~125ns in our lab. This explanation can be supported by observing the experimentally measured oscillation waveform, as

51 the fall time is almost instantaneous compared to the rise time which is associated with capacitive charging.

Hysteresis can also be demonstrated by this two-dimensional grain model. This specific parameter configuration shows ~67μA hysteresis, as in Figure 3.17 when the bias current is ramped up and down from 10μA to 92μA. Note that since the time step is undefined and every iteration solves for the steady-state value, this hysteresis can be held permanently as long as the bias is kept. This feature has the potential for applications such as hysteresis based electro-optic memory and grants it the advantage of not needing to refresh the state of VO2 every few microseconds as proposed in [39].

Figure 3.17: Hysteresis demonstrated by the grain model

3.4 Further Discussions

Numerous different explanations to VO2 oscillation have been reported previously [27] [39] [40].

This thesis provides an explanation that is best supported by our own evidence from the above three models.

As in Figure 3.18, one cycle of the oscillation can be divided into three regions. These three regions can be mapped onto the carrier concentration vs electric field plot with hysteresis in

Figure 3.19 [41]. Region A is the capacitive breakdown region, illustrated by the conductive path forming in section 3.3. When the bias current IB (f(I1) in Figure 3.19 is large enough such that the induced carrier concentration of a grain satisfies the Mott criterion in equation (1.1), IMT takes place, the carrier concentration jumps due to VO2 bandgap collapse. Grains will go through

IMT in an avalanche manner and form a conductive path. At the moment of conductive path formation, device junction capacitance breaks down to almost non-existent, VO2 voltage drops within 125ns.

However, I1 is not necessarily large enough to sustain the grains’ metallic states. Electric field drops as a result of the voltage drop, therefore the carrier concentration of these metallic grains is below 푛푐2, the critical concentration for MIT, and MIT takes place. Throughout region

B, capacitance forms, possibly in a way similar to the localized capacitance formation pattern described in [40]. Eventually, the device capacitance returns to its full insulator state value. It can be observed from the AC oscillation measurements and the DC measurements of the current biased sweep (Figure 2.4 and 2.6) that the metallic state voltage is lower during oscillation between the 1st and 2nd transitions than in the steady metallic state region after the 2nd transition.

[40] and [42] attribute this phenomenon as the “elastic restoring force”, however, this difference

53 is not observed from the oscillation measured from the external resistor with the device connected in series, suggesting it is possibly due to the temporary redistribution of voltage from the device to the resistor due to a sudden decrease in VO2 resistance.

Region C is the exponential capacitive charging period, in which VO2 stays in the insulating phase. After enough time, VO2 returns to the exact same condition, the oscillation cycle repeats.

If the bias is large enough so that the metallic state carrier concentration is above nc2, MIT would not happen, and the oscillation cycle breaks.

After establishing the insulator and metallic regions of the oscillation cycles, it is more obvious that the hysteresis observed at the 1st and 2nd current bias transition (Figure 1.4(c)) is, in fact, a hysteresis between the insulator state and the oscillation state, instead of between the insulator state and the metallic state (or IMT and MIT). If this is a hysteresis of IMT and MIT, then whether a hysteresis effect is shown should be dependent on the moment stimulus is removed: if the stimulus is removed in region C, VO2 is in the insulator phase and will stay in the insulator phase. Since region C contributes to the majority of the oscillation cycles, a 1st current biased transition with no hysteresis should be measured most of the time. However, in reality, the opposite is measured: hysteresis is always measured at the 1st transition. The same reasoning applies to the 2nd current biased transition as well, but a repeatable hysteresis effect with significantly larger hysteresis value has been experimentally measured at the 2nd transition. At the moment this thesis cannot provide a theory for the oscillation hysteresis. Nevertheless, either type of hysteresis has the same effect on VO2 operating as a photodetector They both pose a limit on the minimal detectable optical power, which requires a bias control scheme to bypass.

Figure 3.18: one cycle of oscillation broken down into three regions

Figure 3.19: carrier concentration vs electric field plot with hysteresis. Oscillation cycle regions A, B, C follow the green arrows

Chapter 4

Bias Control Circuit

As described in chapter 2, all the experimental control and measurements are conducted with a computer and a sourcemeter through serial ports, which has a latency on the order of milliseconds. When iterative operations are needed, for example, repeated photodetection and bias point locating and resetting, this becomes a potential time-consuming problem. This chapter describes a bias control circuit that locates the bias current of VO2 devices, detects IMTs and

MITs, and reset the bias for repeated photodetections even if the device is biased within the hysteresis region. The circuit consists of only off-the-shelf electronics and requires no custom- designed integrated circuits, demonstrating the potential of VO2 to be integrated into existing electronic systems with low cost.

4.1 Design Metrics

The block diagram of the designed circuit is presented in Figure 4.1. This design aims to drive the VO2 with a voltage-controlled current source, with the control voltage supplied by a microcontroller. The transition detector detects the moment when VO2 goes through IMT or MIT and sends a signal to the microcontroller to determine subsequent actions.

Figure 4.1: Block diagram of the bias control circuit

The system should not become the limit of the operation speed of VO2 – 휏푐ℎ푎푟푔푒 associated with VO2 intrinsic capacitance should still be the dominant time constant. The system supply voltage should be no higher than the mean voltage required to drive VO2 through IMT.

Due to the potential for VO2 to be annealed and lose its transition property under high current and temperature, the current source should provide variable bias current that is only dependent on the microcontroller voltage output and independent of VO2 resistance changes through phase transitions. No current surges above VO2 safety threshold should be produced.

450μA has been experimentally proven to be a safe high bias current for VO2 (Figure 2.5(b)).

The current source should also provide a bias current with resolution smaller than the minimum bias current standard deviation 1.68μA demonstrated by our devices (Table 2.2). The current source needs to have to ability to drive the VO2 in parallel with another DC current source, which

57 requires the current source output impedance to be much greater than VO2 resistance to avoid current backflow from the DC source into the current source.

The microcontroller should be able to provide a true analog voltage output with sufficient resolution to control the current source.

The transition detector should be able to detect both the IMT and MIT and output a digital signal for the microcontroller. The parallel connection of the transition detector and the VO2 should not interfere with VO2 electrical characteristics, which requires the input impedance of the transition detector to be much greater than VO2 resistance.

4.2 Circuit Design

Current Source

Several common amplifier-based voltage-controlled current source designs are explored, for example, the op-amp and instrumentation amplifier current source configuration in [43],

Howland current pump [44], and load-in-the-loop current source [45], with their schematics shown in Figure 4.2. However, none of these designs can satisfy high output impedance and low current resolution at the same time. This thesis proposes a design with an op-amp driven nmos current source with a pmos Wilson current mirror as shown in Figure 4.3. The current source output 퐼 = 푉푖푛 is determined by microcontroller DAC voltage 푉 and resistor R, independent 표푢푡 푅 푖푛

of load resistance, and the Wilson current mirror copies 퐼표푢푡 to 퐼푙표푎푑. With 푅1 = 20kΩ and 푅푉푂2 modeled from 5k (metallic state) to 100k (insulator state), the current source characteristics from

DC and transient simulations are shown in Figure 4.4, demonstrating excellent linearity and load resistance independence. The largest 0-3.3V full swing settling time is approximately 100μs, still

58 less than 휏푉푂2 . The current mirror can be cascoded to achieve a high output impedance, as shown in Figure 4.5.

Figure 4.2: Schematics of (a) current source configuration in [43] (b) Howland current pump (c) load-in- the-loop current source

Figure 4.3: Current source design schematics

Figure 4.4: (a) DC (b) transient simulations of Vin vs Iload with the load resistance swept from 5kΩ to 100kΩ.

Figure 4.5: (a) Current source design with cascode current mirror (b) transient simulation with 80μA parallel bias at load

Transition Detector

The translation from analog VO2 voltage to digital signal representing the state of the VO2 can be simply performed by a comparator. Figure 4.4 shows the connection of a 10ns comparator IC

[46]. With the threshold voltage value at the negative input port of the comparator chosen between the insulator state voltage and metallic state voltage, VO2 phase can be detected with negligible delay. Its 250MΩ input resistance will ensure a good parallel connection with the load.

Figure 4.6: Transition detector comparator schematics

Microcontroller

The microcontroller is chosen to be Kinetics K64F with 120MHz clock speed guaranteeing sub- microsecond level operations. The analog voltage output to drive the current source will be provided by its 12-bit digital-to-analog converter (DAC). With 4096 steps and typical VO2 critical current less than 150μA, current resolution of 36nA and lower can be theoretically achieved [47]. The maximum 30μs full swing settling time of the DAC combined with the 100μs max 0-3.3V full swing settling time of the current source is smaller than τcharge, ensuring timely

VO2 bias current change. Teensy 3.5 development board is chosen for easy interfacing between

Kinetics K64F microcontroller and PC.

To locate the bias point, the microcontroller sweeps the DAC voltage linearly until transition detector output is sensed by the microcontroller. Then the microcontroller sweeps the

DAC voltage from 0 to one code before the last iteration. If a transition can still be detected, this process is repeated until no transitions are present. For repeated photodetection operations with bias within the hysteresis region, the microcontroller reset the DAC voltage from 0 at the detection of a transition.

4.3 Characterization

The voltage-current performance of the current source design is tested with a 1kΩ resistor, a

100kΩ resistor, and a VO2 device as load, with the results shown in Figure 4.7. Due to width and length mismatch of the transistor used, 1:1 current mirroring was not achieved. However, the linearity of the current source is maintained, demonstrated by the 1kΩ and 100kΩ curves in

Figure 4.7. With the varying resistance of VO2, same current source performance is maintained until the first transition point where VO2 resistance starts to oscillate, demonstrated by the complete overlapping of the three curves.

Figure 4.7: Experimental measurements of Vin vs IVO2 with a 1kΩ resistor, a 100kΩ resistor and a VO2 device as load

However, despite the output voltage and current of the bias control circuit are tested to be well within the limit, the devices still sometimes demonstrate properties the same as if they are short-circuited by high current or temperature. This is possibly due to unwanted instantaneous current spikes introduced by VO2 as an unstable load, and this is not the case when biasing with a source meter because 1) the source meter has over-current protection, but the bias control circuit does not and 2) the bias control circuit current ramp-up speed is 1000 times faster than the source meter. Unfortunately, due to a shortage of fabricated devices, bias finding and photodetection tests were not able to be conducted.

Chapter 5

Conclusion and Future Work

5.1 Conclusion

The work in this thesis developed models to help understand the transition mechanism of VO2 on a macroscopic and behavioral level, suitable for system-level simulations, as well as on a microscopic and physical level, suitable for device-level simulations. We have presented to our knowledge the first model that simulates VO2 electrical and optical characteristics based on carrier concentration and Mott criterion. With a more methodological method of obtaining

material-specific parameters (휀푣표2, μ, W), this model will be able to accurately simulate 푉푂2 electrical and optical properties with arbitrary microwire designs, allowing designs before fabrications. Combined with a behavioral circuit model and a microscopic grain network model, this thesis is able to map the device electrical and optical properties to its instantaneous states, providing the limitations and non-limitations for proposed photodetection operations.

The work in this thesis also proposed the first control circuit for VO2 electrical biasing for rapid bias point locating and repeatable photodetections despite hysteresis. This bias control circuit can be operated with a supply voltage no higher than nominal VO2 bias, does not limit

VO2 operation speed, and allows parallel current source connection in case of potentially different bias current due to different VO2 fabrication process.

5.2 Future Work

Photodetection is only one of the many potential applications of this transition material. A thorough understanding of its transition mechanism can provide the groundwork for other abrupt transition or oscillation-based applications as previously introduced, for instance, neuromorphic computing and optical memory. An accurate carrier concentration and Mott criterion model can create the possibility of device electrical performance design before fabrication, and an accurate circuit model can enable the design of integration between VO2 and conventional control electronics, or between coupled VO2 devices.

This thesis acknowledges that improvements can be made with further studies, including:

i) In the analytical model, either theoretical or experimental studies can be done to

draw the relationships between unknown parameters 휀푉푂2, 휇, 푊 and device

factors, for instance, VO2 crystallinity, substrate, dimensions, and deposition

method.

ii) In the case where the relationships are not possible to deduce, a thorough

parameter search with a better optimization method can be done to find

potentially better fitting parameter combinations, with enough computational

power and time provided.

iii) In the circuit equivalent model, the capacitive breakdown region can be simulated

in the SPICE circuit as a time-varying capacitance to simulate the oscillation with

correct frequency from the beginning to the end of the oscillation region.

iv) If the capacitance value is not dependent on fitting to experimental oscillation

frequency but instead can be deduced from device parameters, the circuit model

can be universally applied to any VO2 with different dimensions and fabrication

processes. v) Over-current protection can be introduced to the bias control circuit to eliminate

potential current spikes to protect the VO2 devices.

Appendix A

Circuit Equivalent Model LTSpice Netlist

R7 N002 V_VO2_1 5k

R1 N001 V_VO2_1 {Rins-Rmet}

R2 N001 0 {Rmet} R4 N004 0 {R0} B1 N003 0 V=if(V(comp1)>V(comp2),1,0) C1 N004 N003 {C0} S1 N001 V_VO2_1 N003 N004 MYSW1 R5 comp2 0 1 B2 comp2 0 V=if(V(comp1)>V(comp2),V_L,V_H) R6 comp1 0 1 B3 comp1 0 V={V(V_VO2_1)-V(0)} C2 0 V_VO2_1 1.05n I1 0 N002 PWL(0 0 10m 0 10.001m 100u 20m 100u 20.001m 110u 30m 110u 30.001m 120u 40m 120u 40.001m 130u 50m 130u 50.001m 140u 60m 140u 60.001m 150u 70m 150u 70.001m 160u 80m 160u 80.001m 170u 90m 170u 90.001m 180u 100m 180u) .tran 0 100m 0 1u .param Rmet=18k .param Rins=93k .model MYSW1 SW(Ron=1 Roff=10000Meg Vt=0.5 Vh=0.45) .param V_L=3.6 .param V_H=8.5 .param R0=1 .param C0=10n .backanno .end

Appendix B Analytical Model Objective function

function [correlations_4_devices, V_VO2_voltages_4_devices] = I_sweep_opt_func(epsilon_VO2, u_cm, W_eV) %% Physical Constants h_bar = 1.054e-34; e = 1.6e-19; k = 1.38e-23; epsilon_0 = 8.854e-12; m_e = 9.11e-31; T_A = 295; T_c = 340;

%% Deduced Constants m_VO2 = 3 * m_e; beta = (e^3/(pi*epsilon_0*epsilon_VO2))^0.5; a_H = 4*pi*(epsilon_VO2*epsilon_0*h_bar^2)/(m_VO2*(e^2)); n_c = (0.25/a_H)^3;

h = 360e-9; u = u_cm*1e-4; gamma = 0.03;

%% Fitting parameters W = W_eV*e;

I_c = 115e-6; V_c = 6.017; R_c_i = 52310;

steps = 70;

I = zeros(1,steps); N = zeros(1,steps); Sigma = zeros(1,steps); R_VO2 = zeros(1,steps); V_VO2 = zeros(1,steps);

w_l = [3 3;3 4;3 5;2 6]; w_l = w_l*1e-6;

voltages = cell2mat(struct2cell(load('voltages.mat')));

correlations_4_devices = zeros(1,5); V_VO2_voltages_4_devices = zeros(steps,4);

for ii = 1:4 w = w_l(ii,1); l = w_l(ii,2); Area = w*h; k_eff = I_c^2*R_c_i/(T_c*(1-beta*sqrt(V_c/l)/W)-T_A);

transition_step = 0; for i = 1:steps I(i) = i*2.5e-6; J = I(i)/Area;

if i == 1 T_vo2 = T_A; else T_vo2 = T_A + I(i)^2*R_VO2(i-1)/k_eff; end

syms E N_this; eqn1 = N_this - n_c*exp(-(W/k/T_c)*((T_c/T_vo2)*(1-… beta*sqrt(E)/W)-1)); eqn2 = J - e * u * N_this * E; S = solve(eqn1,eqn2,E,N_this); N_this = double(vpa(S.N_this,4)); E = double(vpa(S.E,4));

if N_this >= n_c transition_step = i; N_this = N_this*exp(gamma*e/(k*T_vo2)); end

N(i) = N_this; Sigma(i) = e * u * N_this; R_VO2(i) = l/(Area*Sigma(i)); V_VO2(i) = I(i)*(R_VO2(i));

if transition_step ~= 0 break end end

V_VO2((transition_step+1):steps) = 0;

correlations_4_devices(ii) = corr(V_VO2.',voltages(:,ii)); V_VO2_voltages_4_devices(:,ii) = V_VO2.';

figure plot(linspace(0,125e-4,50),V_VO2(1:50),'LineWidth',3) hold on VV = voltages(:,ii); plot(linspace(0,125e-4,50),VV(1:50),'LineWidth',3) title(correlations_4_devices(ii)) xlabel('Current (A)') ylabel('Voltage (V)') legend('Simulated','Measured','Location','northwest')

end

correlations_4_devices(5) = correlations_4_devices(1) + correlations_4_devices(2) + correlations_4_devices(3) + correlations_4_devices(4); end

Appendix C Grain Network Model

%% Inputs num_rounds = 20; m = 50; n = 50;

G_i = 1/100000; G_m = 1/100;

I_B = 92e-6;

%% Fitting parameters V_IMT = 0.2; V_MIT = 0; var_IMT = 0.00002; var_MIT = 0;

%% Variables % -1-unused 0-open circuit 1-insulator 2-metallic. Binary matrices Gr = ones(m+2,n+2); Gd = ones(m+2,n+2); Gd(1,:) = 0; Gd(m+1,:) = 0;

% unused grains Gr(1,:) = -1; Gr(m+2,:) = -1; Gd(:,1) = -1; Gd(:,n+2) = -1;

% Array to store V_VO2 calculated for each round V_VO2s = zeros(1,num_rounds); % Array to store R_VO2 calculated for each round R_VO2s = zeros(1,num_rounds); % Array of matrices to store voltage distributions for each round Vs = zeros(m,n,num_rounds); % Array of matrices to store Gr,Gd matrices and 2 combined for each round Grs = zeros(m+2,n+2,num_rounds); Gds = zeros(m+2,n+2,num_rounds); % Gs = zeros(m+n+3,m+n+3,num_rounds); Gs = zeros(2*(m+2),n+2,num_rounds); % Array of matrices to store currents going through Gr,Gd matrices and 2 combined for each round IGrs = zeros(m+2,n+2,num_rounds); IGds = zeros(m+2,n+2,num_rounds); % Is = zeros(m+n+3,m+n+3,num_rounds); Is = zeros(2*(m+2),n+2,num_rounds);

% Array to store dV_maxs and dV_mins calculated for each round dV_maxs = zeros(1,num_rounds); dV_mins = zeros(1,num_rounds);

%% Function handles Calc_network = @f_two_port_network_resistance; % calculate the two port resistance and voltage distribution of the network Update_G = @f_update_grains; % check thresholds of grains and change between insulator and metallic states Combine_r_d = @f_combine_r_d; % combine Gr and Gd or IGr and IGd Calculate_current = @f_calculate_current; % calculate the currents going through all the grains

%% Loop for round = 1:num_rounds fprintf('%d ',round);

% Calculate V_VO2, network voltage distribution, and total resistance [V_VO2, V, R_VO2] = Calc_network(Gr,Gd,G_i,G_m,m,n,I_B);

% Combine Gr and Gd G = Combine_r_d(Gr,Gd,m,n);

% Calculate currents going through all grains [IGr, IGd] = Calculate_current(V,V_VO2,Gr,Gd,G_i,G_m,m,n); I = Combine_r_d(IGr,IGd,m,n);

% Record V_VO2s(1,round) = V_VO2; R_VO2s(1,round) = R_VO2; Vs(:,:,round) = V; Grs(:,:,round) = Gr; Gds(:,:,round) = Gd; Gs(:,:,round) = G; IGrs(:,:,round) = IGr; IGds(:,:,round) = IGd; Is(:,:,round) = I;

% Compare voltage distribution to IMT amd MIT thresholds and change G % dV_max and dV_min are the maximum and minimum dVs of each grain from this round % this helps determine V_IMT and V_MIT [Gr,Gd,dV_max,dV_min] = Update_G(Gr,Gd,m,n,V_IMT,V_MIT,var_IMT,var_MIT,V_VO2,V); dV_maxs(1,round) = dV_max; dV_mins(1,round) = dV_min; end function [V_VO2, V, R_VO2] = f_two_port_network_resistance(Gr_b,Gd_b,G_i,G_m,m,n,I_B) %% Translate binary G matrices to real G matrices Gr = zeros(m+2,n+2); Gd = zeros(m+2,n+2); for i =1:m+2 for j = 1:n+2 if Gr_b(i,j) == 1 Gr(i,j) = G_i; elseif Gr_b(i,j) == 2 Gr(i,j) = G_m; end

if Gd_b(i,j) == 1 Gd(i,j) = G_i; elseif Gd_b(i,j) == 2 Gd(i,j) = G_m; end end end

%% Build Matrices L % third dimension L1,L2,...,Lm-1 L = zeros(n,n,m-1); for k = 1:m-1 for i = 1:n for j = 1:n if i == j L(i,j,k) = Gd(k+1,j+1); end end end end

%% Build Matrices W W = zeros(m,n); for i = 1:m for j = 1:n W(i,j) = Gd(i,j+1) + Gr(i+1,j) + Gd(i+1,j+1) + Gr(i+1,j+1); end end

%% Build Matrices M % third dimension M1,M2,M3... M = zeros(n,n,m); for k = 1:m for i = 1:n for j = 1:n if i == j M(i,j,k) = -W(k,j); end if (j-i) == 1 && i ~= n M(i,j,k) = Gr(k+1,i+1); end if (i-j) == 1 && j ~= n M(i,j,k) = Gr(k+1,j+1); end end end end

%% Build Matrix A zero_block = zeros(n,n); for i = 1:m % Each row of matrix A % First block if i == 1 A_row_i = M(:,:,1); elseif i == 2

A_row_i = L(:,:,1); else A_row_i = zero_block; end % Concatenate for j = 2:m if i == j + 1 A_row_i = cat(2,A_row_i,L(:,:,i-1)); elseif i == j A_row_i = cat(2,A_row_i,M(:,:,i)); elseif i == j - 1 A_row_i = cat(2,A_row_i,L(:,:,i)); else A_row_i = cat(2,A_row_i,zero_block); end end if i == 1 A = A_row_i; else A = cat(1,A,A_row_i); end end

%% Add terms for V_VO2 add_row = zeros(1,m*n); for i = 1:m*n if mod(i,n) == 1 add_row(1,i) = -Gr(ceil(i/n)+1,1); end end A = cat(1,A,add_row);

add_col = zeros(m*n+1,1); for i = 1:m*n if mod(i,n) == 1 add_col(i,1) = Gr(ceil(i/n)+1,1); end end add_col(m*n+1,1) = sum(Gr(2:m+1,1)); A = cat(2,A,add_col);

%% Solve B = zeros(m*n+1,1); B(m*n+1,1) = I_B; V_vec = linsolve(A,B);

%% Results V = vec2mat(V_vec(1:m*n),n); V_VO2 = V_vec(m*n+1); R_VO2 = V_VO2/I_B; end function [Gr,Gd,dV_max,dV_min] = f_update_grains(Gr,Gd,m,n,V0_IMT,V0_MIT,var_IMT,var_MIT,V_VO2,V) % dV_max and dV_min are the maximum and minimum dVs of each grain from this round % this helps determine V_IMT and V_MIT

74 dV_max = 0; dV_min = inf;

%% Process Gr % very first and last rows not in use for i = 2:m+1 for j = 1:n+1 % Threshold voltage different for each grain V_IMT = normrnd(V0_IMT,sqrt(var_IMT)); V_MIT = normrnd(V0_MIT,sqrt(var_MIT));

if j == 1 % left contact column V_high = V_VO2; V_low = V(i-1,j); elseif j == n+1 % ground column V_high = V(i-1,j-1); V_low = 0; else % ones in the middle V1 = V(i-1,j-1); V2 = V(i-1,j); V_high = max(V1,V2); V_low = min(V1,V2); end

dV = V_high - V_low; if dV > dV_max dV_max = dV; elseif dV < dV_min dV_min = dV; end

% If dV greater than IMT threshold and Gij is in insulator phase if dV > V_IMT && Gr(i,j) == 1 Gr(i,j) = 2; end

% If dV less than MIT threshold and G is in metal phase if dV < V_MIT && Gr(i,j) == 2 Gr(i,j) = 1; end

end end

%% Process Gd % very first and last columns not in use % very first and last rows won't change for i = 2:m for j = 2:n+1 % Threshold voltage different for each grain V_IMT = normrnd(V0_IMT,sqrt(var_IMT)); V_MIT = normrnd(V0_MIT,sqrt(var_MIT));

V1 = V(i-1,j-1); V2 = V(i,j-1); dV = abs(V1 - V2);

if dV > dV_max dV_max = dV; elseif dV < dV_min dV_min = dV; end

% If dV greater than IMT threshold and Gij is in insulator phase if dV > V_IMT && Gd(i,j) == 1 Gd(i,j) = 2; end

% If dV less than MIT threshold and G is in metal phase if dV < V_MIT && Gd(i,j) == 2 Gd(i,j) = 1; end

end end end function A = f_combine_r_d(Ar,Ad,m,n) A = zeros(2*(m+2),n+2); for i = 1:2*(m+2) if mod(i,2) == 1 A(i,:) = Ar((i+1)/2,:); else A(i,:) = Ar(i/2,:); end end end function [IGr,IGd] = f_calculate_current(V,V_VO2,Gr,Gd,G_i,G_m,m,n) IGr = zeros(m+2,n+2); IGd = zeros(m+2,n+2);

% Gr for i = 2:m+1 for j = 1:n+1 if Gr(i,j) == 1 G_grain = G_i; elseif Gr(i,j) == 2 G_grain = G_m; end

if j == 1 % 1st col IGr(i,j) = (V_VO2 - V(i-1,j))*G_grain; elseif j == n+1 % last col IGr(i,j) = V(i-1,j-1)*G_grain; else IGr(i,j) = abs(V(i-1,j-1) - V(i-1,j))*G_grain; end end end

% Gd for i = 2:m

for j = 2:n+1 if Gd(i,j) == 1 G_grain = G_i; elseif Gd(i,j) == 2 G_grain = G_m; end IGd(i,j) = abs(V(i-1,j-1) - V(i,j-1))*G_grain; end end end

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