Modeling and Finite Element Simulations of Phase Change Memory Materials and Devices Jake Scoggin University of Connecticut - Storrs, Jacob.Scoggin@Uconn.Edu

University of Connecticut OpenCommons@UConn

Doctoral Dissertations University of Connecticut Graduate School

8-8-2019 Modeling and Finite Element Simulations of Phase Change Memory Materials and Devices Jake Scoggin University of Connecticut - Storrs, [email protected]

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Recommended Citation Scoggin, Jake, "Modeling and Finite Element Simulations of Phase Change Memory Materials and Devices" (2019). Doctoral Dissertations. 2256. https://opencommons.uconn.edu/dissertations/2256 Modeling and Finite Element Simulations of Phase Change Memory Materials

and Devices

Jake Scoggin, PhD

University of Connecticut, 2019

I simulate phase change including stochastic nucleation and growth of discrete grains with a finite element framework which captures the latent heat of phase change and heterogeneous melting. I model the electrical conductivity, thermal conductivity, Seebeck coefficient, and specific heat in TiN, SiO2, and the face centered cubic and amorphous phases of Ge2Sb2Te5 as well as the thermal boundary conductance at interfaces of these materials over the temperatures and fields encountered during phase change memory device operation. Simulations show reset, set, and read of phase change memory devices including mushroom cells, encapsulated pillar cells, double mushroom cells, and threshold switch controlled mushroom cells. My results show that the latent heat of phase change increases the temperature at crystalline-amorphous interfaces during crystallization, heterogeneous melting can help account for the experimentally observed improvement in reset and set speeds as grain size decreases, and that threshold switching can be captured using a closed form model of independent thermal and electric field contributions to electrical conductivity.

Modeling and Finite Element Simulations of Phase Change Memory Materials

and Devices

Jake Scoggin

B.S., Louisiana Tech University, 2012

M.S., Louisiana Tech University, 2014

A Dissertation

Submitted in Partial Fulfillment of the

Requirements for the Degree of

Doctor of Philosophy

at the

University of Connecticut

2019

APPROVAL PAGE

Doctor of Philosophy Dissertation

Modeling and Finite Element Simulations of Phase Change Memory Materials

and Devices

Presented by

Jake Scoggin, B.S., M.S.

Major Advisor ______

Ali Gokirmak

Associate Advisor ______

Helena Silva

Associate Advisor ______

Don Sheehy

University of Connecticut

2019

Acknowledgements

My first thanks go to my family: Mom, for teaching me to read; Dad, for demonstrating diligence, kindness, and wisdom; Dan, for getting me through Mt. Moon; and Kate, for the lectures on responsibility. For all the time and distance, I am still most at home when we are together. Jake

Terracina and Avery Green, our book club is the best part of my week. You keep me sane. Ali

Gokirmak and Helena Silva, you have treated me with kindness and respect throughout the highs and lows that make up a PhD. I could not ask for better advisors. My labmates have improved this work through technical discussions and their friendships. I list them here alphabetically for want of a way to put them all first: Elizabeth Baranovic, David Blaine, Noah Del Coro, Adam Cywar,

Austin Deschenes, Md. Tashfiq Bin Kashem, Raihan Khan, Sadid Muneer, Nafisa Noor, Brittany

Smith, A.B.M. Hasan Talukder, Nicholas Williams, and Zach Woods.

Thank you all.

‘Men work together,’ I told him from the heart,

‘Whether they work together or apart.’

From The Tuft of Flowers by Robert Frost

iii

Table of Contents

Acknowledgements ...... iii

Table of Contents ...... iv

List of Figures ...... vi

List of Tables ...... xiii

1. Introduction ...... 1

2. Finite Element Framework ...... 3

2.1 Adaptive Meshing...... 7

2.2 Dynamic Grains ...... 9

2.3 Cycle-to-Cycle Variations ...... 10

3. Thermodynamics of Phase Change ...... 11

3.1 Thermodynamic Description of Phases ...... 11

3.2 Coupling the Latent Heat of Phase Change and Specific Heat ...... 14

3.3 Heterogeneous Melting ...... 23

4. Material Parameters ...... 31

4.1 Measuring Material Parameters ...... 31

4.2 Electrical Conductivity ...... 33

4.3 Thermal Conductivity ...... 37

4.4 Seebeck Coefficient ...... 38

4.5 Specific Heat ...... 43

4.6 Thermal Boundary Conductance ...... 45

5. Ovonic Threshold Switching ...... 47

5.1 Computational Model ...... 52

5.2 Simulations ...... 54

6. Conclusion ...... 60

7. References ...... 62

List of Figures

Figure 1: Control functions for (a) stbl and (b) diff. The stability well near 0 in (a) prevents small perturbations from triggering crystallization, requiring nucleation or templated growth to reach the stability point at CD = 1...... 4

Figure 2: GST anneal with close-ups showing mesh elements. Adaptive (a,c) and static (b,d) meshing techniques produce similar results, but adaptive meshing reduces computation time

(Table I)...... 8

Figure 3: Grain maps after an initial anneal (a), during boundary amorphization (b), and after regrowth (c)...... 9

Figure 4: Grain maps for set (a, c) and reset (b, d) states in a 10 nm mushroom cell. Cycle-to- cycle resistance for set (blue) and reset (red) states over 10 cycles is shown in (e). Voltage and max temperature within the GST for the first cycle are shown in (f)...... 10

Figure 5: Schematic illustration showing metastable (liquid and fcc) and stable (hcp) phases in

GST at some temperature below the melting point. The amorphous glass is not at a minima and so is not a true thermodynamic phase, but atoms tend not to re-configure because of the high viscosity.

...... 11

Figure 6: (a) cp,a(T) and cp,c(T) models and measured values [16]. The shaded area is the enthalpy difference between phases minus Δhcrys(Tglass). (b) Enthalpy difference between phases...... 16

Figure 7: Simulated (a) crystal growth and (b) amorphization with initial and boundary temperatures (Tinit, Tbnd) fixed at (a) 600 K and (b) 900 K. Out-of-plane depth is 20 nm...... 17

Figure 8: Average grain radius (blue markers, left axis) depends on Iss(T) and vg(T) [4] (solid and dashed lines, scaled by maximum values, right axis). Radii are from 300 nm x 300 nm simulations using the model in this work, see Figure 9 for simulation details...... 18

Figure 9: Snapshots of simulated grain (left column) and temperature (right column) maps when annealing with Tinit = Tbnd = 650 K comparing (a,b) this work, where QH gives rise to ~180 K increase in local temperature, to (c,d) Woods et al.[3] and (e,f) neglecting the latent heat of phase change. (g) shows grains from 5 simulations at each temperature and QH model ranked by size.

Out-of-plane depth is 20 nm. Grain analysis is performed with ImageJ[9]...... 19

Figure 10: (a) Reset of a confined cell (45 nm out-of-plane depth) with a square pulse Vapp(t), amplitude Vreset = Vdd = 6 V. (b) Current and (c) temperature in the GST lag slightly when modeling the latent heat of fusion as in this section (yellow spheres) or as in [3] (blue spheres) compared to when the latent heat is neglected (solid line). Woods et al. [3] releases the latent heat of fusion when cooling, giving a relatively flat maximum temperature from 8 - 10 ns (c). The model in this section uses a higher cp,a(T) for Tmelt > T > Tglass, resulting in slower cooling and more time spent near the peak nucleation temperature (highlighted region in (c))...... 21

Figure 11: Reading and writing a PCM mushroom cell (10 nm out-of-plane depth) using (a-d) the models in this section and (e-h) the models in [3]. The models in this section use a higher cp,a(T) for Tmelt > T > Tglass, resulting in slower cooling and more probable nucleation after reset (c,g).

This “pre-nucleation” reduces set time, as evidenced by the third read operation: a 40 ns pulse sets only the pre-nucleated device (d,h,i). Pre-nucleation decreases set time; however, it is a stochastic process. Read, reset, and set are performed with square Vapp(T) pulses with magnitudes Vread = 0.1

V, Vreset = Vdd = 2 V, and Vset = 1.3 V...... 22

Figure 12: (a) General behavior of volume (dashes), surface area (dash-dots), and total (solid) contributions to the Gibbs free energy of a crystal. (b) rc increases with T, diverging to infinity at

Tmelt...... 24

vii

Figure 13: (a) The integral of Δcp,lc accounts for the difference in the latent heats of fusion and crystallization (Δhlc(916 K) = 128.9 J/g and Δhlc(431 K) = 34.2 J/g, respectively[10]). (b)

Calculated thermodynamic parameters. (c) Phase change velocities for bulk, 7.5 nm (dash-dots), and 2.5 nm (dashes) radius grains and that calculated by Burr et. al[4]. Negative values denote melting. vkinetic stabilizes with the liquid viscosity at higher temperatures; vkinetic(T > 2000 K) ≈ 60 m/s...... 26

Figure 14: Crystalline vectors defined by (a) a 3-vector with CD = Σ(CDi) as in Woods et. al [3] and (b) a 2-vector with CD = ||CD||2 as used here. The cut plane and dotted line show allowed crystalline values...... 27

Figure 15: Control functions for (a) stbli when T < Tmelt, (b) diffi, and (c) stbli when T > Tmelt. A stability well in (a)/(c) prevents small perturbations from triggering crystallization/melt, respectively...... 28

Figure 16: (a) A polycrystalline GST square is brought to 900 K at t0. (b) Liquid forms at grain boundaries in 1 ns and (c) melts at 1.3 nm/ns...... 29

Figure 17: (a) Schematic illustration of geometry and initial and boundary conditions used in simulations. Iapp is a square pulse with 0.1 ns rise and fall times and a magnitude of 500 μA for reset or 50 μA for set. (b) Crystal maps of initial conditions and after reset with 3 different pulse durations (τReset) for 7.5 nm, 15 nm, and 25 nm grains. The amorphous area (and hence reset resistance) increases with τReset. The Initial column is used for RInitial in Figure 18 and Figure 19.

A ~20x increase in cell resistance requires a 4.1 ns reset pulse for 7.5 nm grains and a 4.2 ns reset pulse for 15 nm or 25 nm grains (dashed borders). (c) Quenched-in crystals embedded in the amorphous GST in the r = 7.5 nm case grow simultaneously, allowing for shorter set times. .... 29

viii

Figure 18: ‘Reset’ resistance after a square current pulse (500 μA magnitude and τReset duration).

Smaller grains allow for faster reset, while larger grains have a wider pulse duration window for intermediate values. RInitial corresponds to the cells in the Initial column in Figure 17b...... 30

Figure 19: ‘Set’ resistance after a square current pulse (50 μA magnitude and τSet duration) beginning with ~20x (dashed lines) and ~100x (solid lines) reset resistances for each grain size

(Figure 18). Smaller grains allow for faster set when the initial reset resistance is low. Set time becomes less dependent on grain size as fewer quenched-in crystals remain in the amorphous region and reset resistance increases; crystallization is achieved through templated growth from the same number of similarly spaced crystal-amorphous interfaces. RInitial corresponds to the cells in the Initial column in Figure 17b...... 30

Figure 20: Thermal and electric field contributions to electrical conductivity in a-GST. Measured data on thin films (spheres above Tmelt) and nanowires (spheres below Tmelt and circles above Tmelt) included for reference...... 34

Figure 21: (a) Resistivity measurements on metastable a-GST nanowires [4] show an exponential dependence with T that intercepts thin film measurements at Tmelt = 858 K [3]. (b) [5] used (a) to calculate a temperature dependent band diagram for metastable aGST. I use that diagram to calculate (c) Peltier and (d) Seebeck coefficients for electrons and holes, finding close agreement with S measurements on as-deposited a-GST thin films (spheres [3] and crosses [6])...... 40

Figure 22 Snapshots of a transient reset simulation (a-d) with and (f-i) without modeling thermoelectric effects. Thermoelectrics result in more heating in the top and less heating in the bottom phase change layer, as evidenced by the size of the amorphous areas after quenching. Vdd

= 1.9 V for both devices, resulting in a reset-to-initial resistance ratio of 703x with and 1040x

ix without thermoelectrics. The device resistance is lower during melt without thermoelectrics, resulting in a slightly higher current (e,j)...... 41

Figure 23 Heat at the peak of reset (Fig. 2c) due to (a) Joule, (b) thermoelectric, and (c) interfacial thermoelectric contributions. Interfacial heating is calculated assuming the heat at material interfaces (W/μm2) is released throughout a 1 nm thick volume. There is significant Joule heating at the lower GST/heater interface (~10s of W/μm3), but this is countered by thermoelectric cooling, so the GST near that interface remains crystalline. Heating and cooling at liquid-crystal interfaces due to the difference in Sh and Sc at TMelt can be seen in (b): the magnitude of S is small, but QTE is large because the narrow liquid-crystal interfaces (~1 nm) produce a high S gradient...... 42

Figure 24: (a) Specific heat in a-GST (blue) and c-GST (red) and (b) the corresponding latent heat released during phase change...... 44

Figure 25: Thermal boundary conductance at (a) GST-SiO2 and (b) GST-TiN interfaces due to phonons (dashed), electrons (dotted), and the sum of both (solid) for c-GST (red) and a-GST (blue).

Measured data from [11] (triangles), [9](spheres), and [13] (stars) are included for reference. .. 46

Figure 26: A schematic illustration of a cross-point cell with an Ovonic threshold switch in series with a phase change memory element. The shown cell structure is used for 2D analysis to compare modeling results to the experimental results in [58]...... 47

Figure 27: (a) 3D and (b) 2D-rotational OTS geometries used in this work. T = 300 K is used as the initial condition and as the boundary condition at the top and bottom of the TiN contacts. (c)

When a 3 V / 60 ns triangular pulse is applied at Vapp, the device (i) is highly resistive until Vswitch

~ 1.75 V, (ii) switches on in ~10 ps, (iii - iv) remains on until the voltage and current drop below

Vhold ~ 0.4 V and Ihold ~ 0.25 mA, and (v) returns to the high resistance off-state. Symmetric switching behavior is observed when a negative ramped pulse is applied. The 2D-rotational and

3D simulations give similar results, as expected for rotationally symmetric filamentary switching.

Inset in (c) is the first quadrant in logarithmic scale...... 49

Figure 28: (a) T and 퐸 dependent contributions to electrical conductivity in (1); temperature is uncertain for T > Tmelt. (b) Conductivity-Field behavior using the model in this work (metastable a-GST) and the model in [67] (drifted, doped a-GST) at various temperatures. I use Eswitch(300 K)

= 25.01 MV / m in this work and Eswitch(300 K) = 155 MV/m for the curves in [67]...... 53

Figure 29: (a-e) x-y and (f-j) x-z temperature cut planes while switching the 3D OTS in Figure

27a illustrate filamentary on-state conduction. (j) Current and (k) device voltage transients resulting from the applied Vapp used to generate the I-V in Figure 27c. Superscripts “-” and “+” refer to the time steps (1 ns increments) before and after the subscripted event. (Figure 27a: RLoad

= 1 kΩ, hOTS = 50 nm, rOTS = 100 nm, Vapp = 5 V / 60 ns triangular pulse)...... 54

Figure 30: The switching voltage increases with hOTS both (a) without and (b) with field dependent conductivity. Including field dependent conductivity reduces the switching field’s (c) magnitude and (d) sensitivity to hOTS. (Figure 27b: RLoad = 5 kΩ, hOTS = 25 to 100 nm, rOTS = 100 nm, Vapp =

5 V / 5 s triangular pulse)...... 55

Figure 31: (a) The switching current (voltage) decreases (increases) and (c) the switching power decreases with increasing Eth in (3). The device switches thermally before the field contribution

8 becomes significant for Eth > 1×10 V / m. As a result, further increases to Eth result in the same switching characteristics. (Figure 27b: RLoad = 5 kΩ, hOTS = 100 nm, rOTS = 100 nm, Vapp = 5 V / 5 s triangular pulse)...... 56

Figure 32: The (a) voltage, (b) current, and (c) power required to switch as the ambient temperature changes. Vswitch decreases monotonically, but Iswitch and Pswitch have more complex

xi relationships with Tambient. (Figure 27b: RLoad = 5 kΩ, hOTS = rOTS = 100 nm, Vapp = 5 V / 5 s triangular pulse)...... 57

Figure 33: Switching characteristics of OTS devices with varying radii, ramp times, and heights.

The simulated holding currents and voltages are weakly dependent on hOTS, but the switching voltage ~doubles as as hOTS doubles. (Geometry in Figure 27b) ...... 58

Figure 34: (a) Reset OTS+PCM, (b) isolated OTS, and (c) isolated reset PCM. (d) Pre-switching

I-V characteristics scaled by Vswitch and Iswitch in the OTS using the model in this work (spheres) and experimental data extracted from [58] (squares). I-V characteristics from this work show similar switching voltages but lower scaled switching currents. The schematic of the simulation setup is shown in Figure 26. The OTS+PCM reset animation for this simulation is available in supplementary material...... 59

xii

List of Tables

TABLE I ...... 8

TABLE II ...... 9

xiii

1. Introduction

Phase change memory (PCM) stores information in the atomic configuration of materials, such as the high electrical resistivity amorphous and low electrical resistivity crystalline phases of chalcogenides. In certain materials phase change is controllable, stable, and reversible, making

PCM a nonvolatile memory technology. Flash memory is presently the dominant nonvolatile memory technology, and PCM has faster write (~10 ns vs ~100 µs) and read (~10 ns vs ~10 µs) operations and better endurance (109 vs 105) than flash. PCM can be implemented in crossbar arrays, achieving a 4F2 packing factor compared to the 6F2 necessary for flash, and can be used for logic, offering ~50% area reduction (though higher write power) than comparable CMOS implementations [1].

Some of the properties that make PCM a good memory technology also pose interesting technical challenges: phase change can be achieved in nanoseconds in nanometer scale devices, but such devices cannot be imaged optically due to their small feature size. Electron microscopy can produce sub-nanometer resolution images, but traditional electron microscopy techniques are too slow to capture nanosecond transients. Nanosecond resolution transmission electron microscopy images can be obtained by applying short, high intensity electron probe beam pulses, and this technique has been used to extract nucleation and growth rates in the phase change material GeTe at previously inaccessible but technologically relevant temperatures [2], [3].

However, it has yet to be successfully applied to the phase change material Ge2Sb2Te5 with sufficient temporal and spatial resolution to extract these parameters due to its higher nucleation rate [2], [4].

In this work, I develop physics- and thermodynamics- based models for PCM materials and devices, then use these models in PCM simulations to try to better understand effects that are

1 difficult to observe. I adapt a finite element phase change model to capture thermodynamically consistent energy exchange during phase change, including heterogeneous melting (melting which begins at grain boundaries and material interfaces). I model electrical conductivity, thermal conductivity, specific heat, and the Seebeck coefficient in the common phase change material

Ge2Sb2Te5 as well as in TiN and SiO2, and I model thermal transport across interfaces between these materials. I use these models in finite element simulations of PCM devices including mushroom cells, pillar cells, ovonic threshold switches, and a PCM cell with an ovonic threshold switch as an access device.

2. Finite Element Framework

Throughout this work I use finite element simulations to analyze phase change memory devices. The finite element phase change model I use (and update throughout this work to capture new phenomenon associated with phase change) was published as a two part paper, the second of which I coauthored [5], [6]. The first part details an effective media phase change framework, with any point in the geometry described as being some percentage amorphous and some percentage crystalline [5]. The second part extends the model to capture discrete grains and grain boundaries using stochastic nucleation, growth, and amorphization [6]. I describe the model published in [6] here, and detail my additional developments throught the rest of this work.

The finite element phase change model solves for the crystal density field-vector CD =

(CD1, CD2, CD3):

푑퐶퐷 𝑖 = 푁푢푐푙푒푎푡𝑖표푛 + 퐺푟표푤푡ℎ + 퐴푚표푟푝ℎ𝑖푧푎푡𝑖표푛 , (1) 푑푡 𝑖 𝑖 𝑖 where t is time, Nucleation models the stochastic formation of crystalline nuclei within amorphous material, Growth models the growth of crystals within amorphous material, and Amorphization models the crystalline to amorphous phase transition above the melting temperature Tmelt. The phase of the material is defined by the sum of components: CD = Σ(CDi) = 0 for the amorphous or

1 for the crystalline phase. Grain orientation is given by the distribution of CDi values, with grain boundaries defined where |∇CD⃗⃗⃗⃗⃗ | > 5×10-3 nm-1.

Nucleation randomly generates nuclei at a temperature dependent rate:

푁푢푐푙푒푎푡𝑖표푛𝑖 = 훼푛푢푐 × (1 − 퐶퐷) × (푟푛푛푢푐 < 푝푛푢푐) × 푟푛𝑖 (2)

-1 where αnuc = 0.8 ns is a constant, pnuc is the steady state nucleation rate found in [7], and rnnuc and rni are functions which return random numbers uniformly distributed between 0 and 1 over a

3 square grid that updates at regular time intervals. rnnuc < pnuc determines if a nucleus will appear at a given space and time, and rni gives the values of CDi used for that grain.

Growth has a stability term (stbl) which creates stability points for CD at 0 and 1 and a diffusivity term (diff) which grows grains outwards:

퐶퐷 퐺푟표푤푡ℎ = 푠푡푏푙 × 𝑖 + ∇(푑𝑖푓푓)∇퐶퐷 𝑖 퐶퐷 𝑖 푠푡푏푙 = 훼푠푡푏푙 × 푣(푇) × 푝푤푠푡푏푙(퐶퐷) (3) 푑𝑖푓푓 =훼푑𝑖푓푓 × 푣(푇) × 푝푤푑𝑖푓푓(퐶퐷)

-1 where αstbl = 0.8 nm and αdiff = 0.2 nm are constants, pwstbl and pwdiff are piecewise control functions (Figure 1), and v is the phase change velocity. A valley in pwstbl prevents small perturbations above 0 from triggering crystallization (Figure 1a inset). Nucleation or templated growth from an adjacent crystal is required to escape this valley.

Amorphization brings each CDi (and thus CD) to 0 when T > Tmelt:

퐴푚표푟푝ℎ𝑖푧푎푡𝑖표푛𝑖 = 훼푎푚표푟푝ℎ × (푇 > 푇푚푒푙푡) × 퐶퐷𝑖 (4)

-1 1 where αamorph = -1 ns is a constant(a) and (T0.05 > Tmelt) is a step function (0 to 1 over a 1 K window

(CD) 0.00 centered at Tmelt). stbl -0.05 pw 0 0.00 0.05 1

(b)(a) 0.05 diff

(CD) 0.00 pw stbl -0.05 pw 0 0.00 0.05 1

(b)(c)

diff

(1-CD)

pw stbl

pw 0 1 (c)0.0 0.5 1.0

CD (1-CD)

Figure 1: Control functions forstbl (a) stbl and (b) diff. The stability well near 0 in (a) prevents small perturbations from triggering crystallization, requiringpw 0 nucleation or templated growth to reach the stability point at CD = 1. 0.0 4 0.5 1.0 CD This phase change model uses the steady state nucleation and growth rates calculated in

Burr et al.[7], sacrificing accuracy for speed compared to the non-steady state atomistic cellular automata simulations in that work. Later in this work (Chapter 3.3) I derive a different growth function based on the thermodynamics of phase change.

In electro-thermal device simulations, (1) is coupled with electric current

∇ ⋅ 퐽 = ∇ ⋅ (−𝜎∇푉 − 𝜎푆∇푇) = 0 (5) and heat transfer physics

푑푇 푑 푐 − ∇ ⋅ (푘∇푇) = −∇푉 ⋅ 퐽 − ∇ ⋅ (퐽푆푇) + 푞 , (6) 푚 푝 푑푡 퐻 where J is current density, σ is electric conductivity, V is electric potential, S is the Seebeck coefficient, dm is mass density, cp is specific heat, k is thermal conductivity, and qH accounts for the latent heat of phase change. The thermoelectric current in (5) is only appropriate when the gradient in S is negligible. A more general expression is

∇ ⋅ 퐽 = ∇ ⋅ (−𝜎∇푉 − 𝜎∇(S푇)) = 0. (7)

I use (5), the default physics used for the thermoelectric effect in COMSOL, throughout this work.

A more detailed explanation of thermoelectrics is give in Chapter 4.4.

In [6], qH is only used for the latent heat of crystallization (Δhcrys)

푑‖퐶퐷⃗⃗⃗⃗⃗ ‖ 푞 = 1 ⋅ Δℎ ⋅ 푑 , (8) 퐻 푑푡 푐푟푦푠 푚 and the latent heat of fusion is captured by an abrupt increase in cp near Tmelt. In Chapter 3.2, I derive a thermodynamic model which couples the specific heats of the amorphous and crystalline phases to both the latent heats of crystallization and fusion, which are treated as a unified latent heat of phase change. This model more appropriately captures the temperature and phase dependence of specific heat and the latent heat of phase change, and it conserves energy over multiple phase change cycles. Equation (8) does not conserve energy over multiple phase change

5 cycles: Δhcrys = 3.9 kJ/mol is released during crystallization, but that energy is never “re-absorbed” and thus 3.9 kJ/mol is added to the system with every crystallization event.

In the rest of this chapter I describe my contributions to [6], which include using adaptive meshing to more efficiently anneal large geometries, modeling the evolution of grains over time in fully annealed geometries, and analyzing the effects of stochastic nucleation on device behavior.

Except where otherwise stated, material parameters used in the following simulations are modeled as described in Chapter 4.

2.1 Adaptive Meshing

Mesh elements of ~1 nm are required to model phase change in GST at nucleation sites, at grain boundaries (expected to be ~0.5 nm [8]), and at crystalline/amorphous interfaces. However, coarser meshing can be applied in other areas where phase is expected to have a small spatial gradient and time derivative (Figure 2c). The required mesh size at a point in space changes over the course of a simulation, so I apply adaptive meshing to reduce simulation times without significantly impacting accuracy. First, a quick but inaccurate solution is calculated using a coarse

(~2.5 nm) mesh over a short time period. This solution is used to find areas where a user defined function (error) is high, and mesh density is increased in these locations. Selecting

푒푟푟표푟 = |훻퐶퐷⃗⃗⃗⃗⃗ | (9) tightens the mesh at nucleation sites, grain boundaries, and crystalline/amorphous interfaces while keeping the original coarse mesh within bulk amorphous or crystalline areas. A new mesh is periodically calculated from the original coarse mesh using (9) to adjust for expanding grains and newly generated nuclei. Adaptive meshing yields similar final grain maps as static meshing (Figure

2) while reducing computation time by ~ 45 % in 5 simulations which use different random seeds used for stochastic nucleation (TABLE I). Adaptive meshing tends to be unstable during electrothermal device operation simulations due to large thermal and field gradients even within bulk crystalline and amorphous areas. However, it is useful during annealing simulations to quickly analyze the effects of varying nucleation and growth rates, geometries, and thermal profiles or to quickly produce a large number of different grain maps to be used as initial conditions in device simulations, e.g. in [9] where a large number of grain maps in large geometries were used to analyze statistical effects of the latent heat of crystallization at different temperatures.

Figure 2: GST anneal with close-ups showing mesh elements. Adaptive (a,c) and static (b,d) meshing techniques produce similar results, but adaptive meshing reduces computation time (Table I).

TABLE I COMPARISON OF MESHING TECHNIQUES Static Adaptive Random Percent Meshing Meshing Seed Difference Time (s) Time (s)

1 3864 2321 39.9% 2 4644 2396 48.4% 3 5852 2494 57.4% 4 3708 2497 32.7% 5 4253 2311 45.7%

2.2 Dynamic Grains

Crystal grains are expected to evolve over time even in fully crystalline GST, especially at

elevated temperatures [10]. Grain boundary migration after crystallization is driven by decreasing

free energy. This free energy is highest at grain boundary defects, grain boundaries with high

curvature, and complex grain intersections [11]. Larger grains tend to grow while smaller grains

shrink, reducing grain boundary area and approaching a more stable configuration. I capture grain

boundary migration by periodically amorphizing grain boundaries and then recrystallizing (Figure

3). After an amorphization-crystallization cycle, the radii of curvature of crystal grains tend to

increase and small grains disappear (Figure 3b,c). Multiple cycles result in a steadily increasing

average grain size (TABLE II) (grain size calculated with ImageJ [12]). The grain sizes found in

these simulations are on order with the 15 nm – 25 nm fcc grains found experimentally with X-

Ray diffraction [10].

Figure 3: Grain maps after an initial anneal (a), during boundary amorphization (b), and after regrowth (c).

TABLE II DYNAMIC GRAIN GROWTH Average Max Min Percent Cycles Radius Radius Radius Increase (nm) (nm) (nm) 0 17.5 ± 13.5 - 26.8 2.62 1 17.9 ± 13.8 2.7 27.4 2.79 2 19.6 ± 13.8 12.3 27.5 2.68 3 20.8 ± 14.2 18.9 27.5 2.48

2.3 Cycle-to-Cycle Variations

Stochastic nucleation and growth dynamics cause variations in percolation paths and grain maps during device operation. Hence, the time and power necessary to cycle a device and the set and reset resistances can vary. Cycle-to-cycle resistance variations in both set and reset states are presented in Figure 4. The coefficient of variation (CV) for all reset states after the first cycle is

~0.5%. The first reset state resistance is ~46% below the others due to nucleation near the TiN/GST interface which reduces the effective amorphous length of the reset state. Grain boundary resistances contribute proportionally more to the total resistance in the set state than the reset state; hence the set state is more sensitive to grain map changes leading to a ~4% cycle-to-cycle CV.

Figure 4: Grain maps for set (a, c) and reset (b, d) states in a 10 nm mushroom cell. Cycle-to-cycle resistance for set (blue) and reset (red) states over 10 cycles is shown in (e). Voltage and max temperature within the GST for the first cycle are shown in (f). 10

3. Thermodynamics of Phase Change

3.1 Thermodynamic Description of Phases

A phase is an atomic configuration associated with a minimum of Gibb’s free energy (G) at some temperature T (Figure 5). The global minimum of G corresponds to the stable phase at T, and local minima are metastable phases. Small rearrangements from a metastable phase result in a higher G and are thus thermodynamically damped: they are undone at a higher rate than they occur.

However, a large enough rearrangement (or large enough series of small rearrangements) may result in the formation of the stable phase or another metastable phase. This phase change requires an energy at least equal to the height of the “hill” between phases. Metastable-to-stable phase transitions will eventually occur as long as there is enough energy to overcome the energy barriers between phases. The time scale associated with phase transitions increases exponentially with

Amorphous Glass Increasing Gibb’s Free Free Energy Increasing Gibb’s

liquid fcc hcp Atomic Configuration

Figure 5: Schematic illustration showing metastable (liquid and fcc) and stable (hcp) phases in GST at some temperature below the melting point. The amorphous glass is not at a minimum and so is not a true thermodynamic phase, but atoms tend not to re-configure because of the high viscosity [13]. 11 increasing 1/T: The amorphous-to-crystalline transition in GST occurs over decades (~108 seconds) at 300 K but nanoseconds at higher temperatures (~550 to 750 K).

GST has three technologically relevant thermodynamic phases: face centered cubic (fcc) crystalline, hexagonal close packed (hcp) crystalline, and liquid (l). hcp-GST is stable at temperatures below Tmelt and l-GST above, while fcc is metastable at temperatures both below and above Tmelt. l-GST tends to transition to the metastable fcc-GST upon cooling below Tmelt, which then eventually transitions to hcp-GST. The fcc-GST to hcp-GST transition is slow enough that fcc-GST is more often encountered as the crystalline phase in GST PCM devices; hence, I use fcc-

GST material parameters for the crystalline phase of GST in the models and simulations in this work.

I model the liquid, undercooled liquid, and amorphous “phases” of GST as a single phase with continuous material parameters in this work but will briefly describe their thermodynamic differences here. Atoms do not instantly rearrange into a crystalline configuration upon cooling below Tmelt; the material remains in the now metastable liquid phase (often described as an

“undercooled liquid”) until, after some time, a cluster of atoms rearranges into a crystalline nucleus and begins to grow [13]. The exact atomic configuration of the metastable liquid phase is a function of temperature, so atoms must rearrange into a new, slightly different configuration with small temperature changes. The time required for this rearrangement increases as temperature decreases because the viscosity (η) increases with decreasing T. In practical cooling experiments, the rearrangement time eventually becomes so large that either a crystal forms before the metastable liquid configuration can be reached or the viscosity becomes so large that effectively no atomic motion occurs: the atoms have “frozen” in a configuration which closely corresponds to the metastable liquid configuration of a higher temperature. The material is said to transition from a

12 metastable undercooled liquid to an amorphous glass when this freeze-out occurs, and the temperature at which freeze-out occurs is the glass transition temperature Tglass. Tglass depends on the experimental cooling rate: slower cooling allows more time for the atoms to rearrange and thus a lower Tglass [13].

Though referred to as the amorphous phase throughout this work, I note here that amorphous GST is neither stable nor metastable and is not a true thermodynamic phase: small rearrangements of atoms can result in a lower G and are thus thermodynamically favorable (Figure

5), but generally do not occur because η is too large. Over time random fluctuations do occur due to thermal energy in the system, and the amorphous glass slowly approaches the (temperature dependent) atomic configuration of the metastable liquid phase. This “structural relaxation” is often associated with resistance drift, where the resistance of the amorphous phase can increase by an order of magnitude over ~1 year even at 300 K [14]. However, the observed resistance drift at cryogenic temperatures where structural relaxation is expected to be negligible [15] and the sensitivity of resistance to photo-absorption at these temperature [15], [16] suggest that structural relaxation may be inadequate to model resistance drift at all temperatures and that charge trapping within the amorphous material or in sub-critical fcc-nuclei embedded in the amorphous material may play an important role in explaining this phenomenon.

3.2 Coupling the Latent Heat of Phase Change and Specific Heat

The simulations in Chapter 3 [6] used the same specific heat (cp(T)) for crystalline, amorphous, and molten GST, sharply increasing cp(T) near the melting temperature (Tmelt to Tmelt

+ 10 K) to account for the latent heat of fusion (Δhf). Reference [6] additionally incorporated the latent heat of crystallization using the value measured at the glass transition temperature

(Δhcrys(Tglass)) in [17]. However, the specific heats of amorphous and crystalline phases are expected to diverge for T > Tglass, and materials can crystallize at any temperature where Tglass <

T < Tmelt. Since Δhcrys(Tglass)  Δhf [17], models which incorporate these latent heats separately without coupling to specific heat do not conserve energy in amorphization-crystallization cycles.

In this section, I model both the latent heat of crystallization and the latent heat of fusion as the latent heat of phase change Δha,c(T). I ensure that energy is conserved in amorphization- crystallization cycles by coupling Δha,c(T) to cp.

Enthalpy (h(T)) is the amount of heat stored in a material. h cannot be directly measured, but calorimetry can be used to measure its temperature derivative cp(T) [18]:

dℎ(푇) 푐 (푇) = . (10) 푝 d푇 cp is the amount of energy necessary to raise the temperature of a substance by 1 K, and it is temperature dependent. The enthalpy difference between the amorphous and crystalline phases,

Δha,c(T), is the heat released or absorbed during phase change and can also be measured via calorimetry:

Δℎ푎,푐(푇) = ℎ푎(푇) − ℎ푐(푇). (11)

Equations (10) and (11) can be used to model of h(T) and cp(T) for crystalline and amorphous GST.

Integrating (10) between arbitrary temperatures T1 and T2 gives the change in enthalpy between the two temperatures:

푇2 ℎ(푇2) − ℎ(푇1) = ∫ 푐푝(푇)푑푇. (12) 푇1

The change in Δha,c(T) between these temperatures can be found with (11):

Δℎ푎,푐(푇2) − Δℎ푎,푐(푇1) = (ℎ푎(푇2) − ℎ푎(푇1)) − (ℎ푐(푇2) − ℎ푐(푇1)). (13)

Rearranging (13) gives

Δℎ푎,푐(푇1) = Δℎ푎,푐(푇2) − [(ℎ푎(푇2) − ℎ푎(푇1)) − (ℎ푐(푇2) − ℎ푐(푇1))], (14) into which can be substituted (12):

푇2 Δℎ푎,푐(푇1) = Δℎ푎,푐(푇2) − ∫ (푐푝,푎(푇) − 푐푝,푐(푇)) 푑푇. (15) 푇1

Experimental Δha,c(T) values are typically reported as heat absorbed during melt (the latent heat of fusion Δhf), with the temperature assumed to be Tmelt, or the heat released during crystallization

(the latent heat of crystallization Δhcrys) of a slowly heated amorphous sample with the crystallizaiton temperature near Tglass. Values ranging from Δhf ≈ 68 ~ 129 J/g [17], [19]–[23] and

Δhcrys ≈ 35 J/g [17], [19], [23] have been reported and used for parameter modeling in GST, with the wide variance in Δhf appearing to be due to the accidental misuse of the latent heat of fusion for a different phase change material. Here, I use the reported GST values Δhf = 128.9 J/g [17] and

Δhcrys = 34.2 J/g [17] for Δha,c(Tmelt) and Δha,c(Tglass). Substituting Δhf = Δha,c(Tmelt) for Δha,c(T2) in

(15):

푇푚푒푙푡 Δℎ푎,푐(푇1) = Δℎ푓 − ∫ (푐푝,푎(푇) − 푐푝,푐(푇)) 푑푇. (16) 푇1

Equation (16) can be used to calculate Δha,c(T) given models for cp,a(T) and cp,c(T). I assume the amorphous and crystalline phases have the same specific heat below the glass transition (cp,c(T) = cp,a(T) for T < Tglass), which is typical for glass formers [13]. I model cp,c(T) with a quadratic function fit to GST measurements performed from 300 K to 650 K [24] (Figure 6a), with cp,c(T) constant for T > 700 K (maximum of the quadratic). I then model cp,a(T) for T > Tglass as a constant

15 such that the specific heat difference between phases accounts for the difference in the latent heats of phase change at Tglass and Tmelt:

푇푚푒푙푡 Δℎ푓 − Δh푐푟푦푠(푇푔푙푎푠푠) = ∫ (푐푝,푎(푇) − 푐푝,푐(푇)) 푑푇. (17) 푇𝑔푙푎푠푠

I use a cubic function over a 10 K window centered at Tglass to smoothly connect the two cp,a(T) segments. Having defined cp,c(T) and cp,a(T), I then calculate Δha,c(T) using (16) (Figure 6b).

I implement Δha,c(T) in simulations by modifying (8):

푑‖퐶퐷⃗⃗⃗⃗⃗ ‖ 푞 = 1 ⋅ Δℎ (푇) ⋅ 푑 , (18) 퐻 푑푡 푎,푐 푚 leading to local temperature variations during phase change (Figure 7). I fix the initial and boundary temperatures (Tinit, Tbnd), allowing heat exchange at the boundaries of the simulated area but not out-of-plane. I use the steady state nucleation rate Iss(T) and growth velocity vg(T) from

[7]. Iss(T) peaks at ~600 K while vg(T) increases until ~720 K (Figure 8). Heating at crystal growth fronts increases vg as long as T < 720 K, which in turn increases d||퐶퐷⃗⃗⃗⃗⃗ ||1/dt, releasing more heat

(a) 400 C ) p,a H (T) - H (T )

-1 a,c crys glass

K -1 C

200 [2002[26] Kalb] p,c

(J kg

p C

0 (b)

Hf

100 )

-1 Ha,c(T)

H(J g 50 

Hcrys(Tglass) 0 300 400 500 600 700 800 T (K)

Figure 6: (a) cp,a(T) and cp,c(T) models and measured values [24]. The shaded area is the enthalpy difference between phases minus Δhcrys(Tglass). (b) Enthalpy difference between phases.

16 and creating positive feedback (Figure 7a). Heating crystalline-amorphous interfaces above 720 K decreases vg because the growth velocity begins to decrease with temperature, creating negative feedback. Conversely, heat absorbed during amorphization cools the crystalline-amorphous interface and slows the amorphization process (Figure 7b).

Grain distributions after annealing are a function of both Iss(T) and vg(T): higher Iss leads to more, smaller grains while higher vg leads to fewer, larger grains (Figure 8). In 300 nm x 300 nm simulations with a 20 nm out-of-plane depth (Figure 9), I obtain the smallest grains when annealing with Tinit = Tbnd ≈ 550 K: the increase in Iss(T) from 550 K to 600 K is outweighed by the increase in vg(T) (Figure 8, Figure 9). This is true even when Δha,c(T) is not accounted for, but including Δha,c(T) results in fewer grains at or above Tinit = Tbnd ≈ 550 K as local temperatures increase up to ~180 K above Tbnd (Figure 9). Δha,c(Tglass) is released when crystallizing at any temperature using the model I co-developed in [6], which underestimates Δha,c(T) above Tglass and thus affects grain distributions to a lesser extent than the model presented here (Figure 9).

c-GST a-GST

= = 600K

bnd

= =

init T (a) 600 T (K) 630 100 nm

c-GST a-GST

= = 900K

bnd

= =

init T (b) 870 T (K) 900

Figure 7: Simulated (a) crystal growth and (b) amorphization with initial and boundary temperatures (Tinit, Tbnd) fixed at (a) 600 K and (b) 900 K. Out-of-plane depth is 20 nm. 17

0 30 10 v (T) g 10-2 20 vg,max 10-4

-6

(nm) 10 avg

r 10 I (T) ss 10-8 Iss,max

400 500 600 700 800 900

T (K)

Figure 8: Average grain radius (blue markers, left axis) depends on Iss(T) and vg(T) [7] (solid and dashed lines, scaled by maximum values, right axis). Radii are from 300 nm x 300 nm simulations using the model in this work, see Figure 9 for simulation details. 18

CD1 T (K) 0 1 650 850

(a) (b) This work

6,7

et et al

Woods [2017Woods]

(e) (f)

= 0 =

H Q

100 nm

Amorphous 100 100 700 K 700 K (g) 650 K 600 K 650 K 550 K 500 K 550 K

10 (nm) r r (nm) r

This work [2017Woods Woods] et al.6,7

QH = 0 1 1 0 500 1000 1500 0 500 Rank1000 1500 Rank

Figure 9: Snapshots of simulated grain (left column) and temperature (right column) maps when annealing with Tinit

= Tbnd = 650 K comparing (a,b) this work, where QH gives rise to ~180 K increase in local temperature, to (c,d) Woods et al.[6] and (e,f) neglecting the latent heat of phase change. (g) shows grains from 5 simulations at each temperature and QH model ranked by size. Out-of-plane depth is 20 nm. Grain analysis is performed with ImageJ[12]. 19

The relative contribution of the latent heat of phase change depends on device structure, thermal boundary conditions, operational speed, and operational voltages. I simulate reset and set operations in nanoscale PCM cells to compare the presented model to that in [6]. There is no substantial change in the voltage or current necessary to reset a 45 nm x 50 nm confined cell (45 nm out-of-plane depth), and the temperature profiles while melting are similar using either model

(Figure 10). However, the maximum GST temperature hovers near Tmelt for ~2 ns when cooling at the end of reset using the previous model due to the spike in cp(T) from Tmelt to Tmelt + 10 K, which releases Δhf as the liquid cools below Tmelt and thus models a liquid-to-solid phase transition, even when crystallization does not occur. The model presented here treats the molten phase as a continuum of the undercooled liquid and amorphous phases and does not release this latent heat of fusion. The model presented here predicts a higher cp,a(T) for Tmelt > T > Tglass, and thus a-GST cools more slowly in this temperature range and stays near the peak nucleation temperature for a longer duration (Figure 10b). This increases the nucleation probability while quenching at the end of reset (pre-nucleation), which can reduce set time and increase device-to-device and cycle-to- cycle variability [25]–[28] (Figure 11). In 30 reset simulations of a 10 nm mushroom cell (Figure

11), the presented model shows more nuclei in 10 cases and fewer nuclei in 2 compared to the model in [6]. These nuclei assist set by providing crystallization sites and establishing a preferential current path (Figure 11). Pre-nucleation cannot be leveraged to directly reduce set time and voltage in conventional memory implementations, where a uniform waveform should set all cells, due to its stochastic nature. However, randomness introduced by pre-nucleation makes PCM cells suitable for hardware security applications (e.g. true random number generation, physical unclonable functions, and physical obfuscated keys [29]–[32]).

(a) Vapp(t) GST TiN Vdd This work 6,7 45 nm [2017Woods Woods] et al. Si SiO2 QH = 0 1500 1350

(b) A)

1000  1300

I ( I A)  500 1250 I ( I 5 6 7 8 9 10 time (ns) 0

900 950 (K) (c) max 900

800 T

700 5 6 7 8 9 10

time (ns) (K) 600 I (T) / I > 0.5

max ss ss,max T 500 1.7 ns 400 3.1 ns 300 5 10 15 20 25 time (ns)

Figure 10: (a) Reset of a confined cell (45 nm out-of-plane depth) with a square pulse Vapp(t), amplitude Vreset = Vdd = 6 V. (b) Current and (c) temperature in the GST lag slightly when modeling the latent heat of fusion as in this section (yellow spheres) or as in [6] (blue spheres) compared to when the latent heat is neglected (solid line). Woods et al. [6] releases the latent heat of fusion when cooling, giving a relatively flat maximum temperature from 8 - 10 ns (c). The model in this section uses a higher cp,a(T) for Tmelt > T > Tglass, resulting in slower cooling and more time spent near the peak nucleation temperature (highlighted region in (c)).

Vapp(t) CD1 0 1 Amorphous Molten Vdd

(a) 0 ns (b) 80 ns (c) 130 ns (d) 200 ns This Work

(e) TiN (f) (g) (h)

6,7

et al

[2017Woods] Woods 20 nm SiO2

(i) 10-5 This work 6,7 Read Reset Woods[2017 Woods] et al. Set Read

-7

I (A) 10 Read 10-9 0 25 50 75 100 125 150 175 200 225 time (ns)

Figure 11: Reading and writing a PCM mushroom cell (10 nm out-of-plane depth) using (a-d) the models in this section and (e-h) the models in [6]. The models in this section use a higher cp,a(T) for Tmelt > T > Tglass, resulting in slower cooling and more probable nucleation after reset (c,g). This “pre-nucleation” reduces set time, as evidenced by the third read operation: a 40 ns pulse sets only the pre-nucleated device (d,h,i). Pre-nucleation decreases set time; however, it is a stochastic process. Read, reset, and set are performed with square Vapp(T) pulses with magnitudes Vread = 0.1 V, Vreset = Vdd = 2 V, and Vset = 1.3 V.

3.3 Heterogeneous Melting

Solids tend to melt heterogeneously: the liquid phase initially forms at high energy sites such as grain boundaries and material interfaces. While many materials heat ~20% above Tmelt before the liquid phase forms within the bulk solid, heterogeneous melting may occur below Tmelt

[33]. In this section, I model and analyze the impacts of heterogeneous melting in PCM devices.

PCM retention, endurance, and speed depend on the physics underlying crystallization and melting. I calculate temperature and grain size dependent phase change velocities in GST based on kinetic and thermodynamic parameters. I then show in simulations that heterogeneous melting can help account for the experimentally demonstrated PCM performance improvement with decreasing grain size [34], [35].

Tmelt is the temperature at which the Gibbs free energy difference between bulk liquid and crystalline phases (Δglc) is zero. However, melting becomes thermodynamically favorable below

Tmelt at crystal interfaces. The Gibbs free energy of a spherical crystal surrounded by liquid (ΔGcrys) is calculated by classical nucleation theory as

4 훥퐺 = − 𝜋푟3훥푔 + 4𝜋푟2훾 , (19) 푐푟푦푠 3 푙푐 푙푐 where r is the crystal radius and γlc is the energy penalty at a liquid-crystal interface (Figure 12a).

(19) has extrema at r = 0 and r = rc, the critical radius:

2훾푙푐 푟푐 = . (20) 훥푔푙푐

Crystals with r < rc are subcritical and can reduce ΔGcrys by shrinking, i.e. melting. rc increases with T: Δglc decreases with increasing T, crossing 0 at Tmelt. γlc is difficult to measure in GST and often used as a fitting parameter; however, γlc increases with T in metals as well as in the

2 semiconductors Si and Ge [36]–[40]. In this work, I use γlc = 75 mJ/m , a temperature-independent value which allows classical nucleation theory to accurately model nucleation and growth in GST

2 (a) 2 (b) 10 4r lc

G

crys 101

(nm)

-4/3r3g lc 100 Increasing Energy Increasing r c 400 600 800 Increasing Radius T (K)

Figure 12: (a) General behavior of volume (dashes), surface area (dash-dots), and total (solid) contributions to the

Gibbs free energy of a crystal. (b) rc increases with T, diverging to infinity at Tmelt. over a wide temperature range[41]. 1 nm and 10 nm radius GST grains become subcritical at ~640

K and ~840 K, respectively, with the parameters in this section (Figure 12b).

Grain boundary melting may be thermodynamically favorable below Tmelt (pre-melting) even for supercritical grains (r > rc) if

2훾푙푐 + 푤훥푔푙푐 < 훾푐푐, (21) where w is the width of liquid formed and γcc is the crystalline-crystalline interface (grain boundary) energy penalty. Not all grain boundaries pre-melt at the same temperature since γcc depends on the misorientation Φ between grains, as calculated via phase field crystal simulations

[42] and shown experimentally with colloidal crystals [43]. While (21) presents a thermodynamic pathway for only a finite w, a supercritical grain may become subcritical while pre-melting and consequently melt entirely.

Regardless of the underlying mechanism, melting is a transient process and requires kinetic and thermodynamic parameters to model. I model phase change velocity (v) as:

훥푔 푣 = 푣 (1 − exp (− )) (22) 푘𝑖푛푒푡𝑖푐 푅푇 where vkinetic is the kinetic upper limit, R is the gas constant, and Δg is the non-negative thermodynamic driving force. (22) is appropriate for atomically rough interfaces [44], predicts a

24 smooth derivative of v as crystallization transitions to melt at Δg = 0 [44], and has been used to model crystallization dynamics in glass formers including GST [45], [46].

I model vkinetic for GST as in [46]: the temperature dependence is determined from ultrafast digital scanning calorimetry [46], and vkinetic(900 K) is calculated from the liquid viscosity given by molecular dynamics simulations (η(900 K) ≈ 1.1 mPa s) [47].

I calculate Δg from Δglc, which is thermodynamically related to the differences in enthalpy

(Δhlc) and entropy (Δslc) between phases:

Δ푔푙푐(푇) = Δℎ푙푐(푇) − 푇Δ푠푙푐(푇). (23)

Δhlc and Δslc can be calculated from the difference in specific heat between phases (Δcp,lc):

( ) 푇 ( ′) ′ Δℎ푙푐 푇 = ∫0 Δ푐푝,푙푐 T dT (24a)

푇 ′ = Δℎ푙푐(푇푥) + ∫ Δ푐푝,푙푐(푇 )푑푇′ (24b) 푇푥

푇 Δ푐 (T′) Δ푠 (푇) = ∫ 푝,푙푐 dT′ (25a) 푙푐 0 T′

′ 푇 Δ푐푝,푙푐(푇 ) ′ = Δ푠푙푐(푇푦) + ∫ 푑푇 (25b) 푇푦 푇′ where Tx and Ty are temperatures at which Δhlc and Δslc are known. I treat the amorphous and liquid phases as a single material with a sharp change in thermodynamic parameters at the glass transition temperature (Tglass = 431 K[17]), neglecting the dependence of these parameters and Tglass on thermal history [13]. I use Δhlc(916 K) = 128.9 J/g [17] and Δslc(Tmelt) = Δhlc(Tmelt)/Tmelt as fixed values, and then calculate Δhlc(T), Δslc(T), and Δglc(T). I use Tmelt = 858 K here, consistent with resistivity measurements on GST thin films [48]. I calculate phase change velocities using Δglc for the upper limit (corresponding to a grain with an infinite radius v∞) and Δgcrys for crystals with radius r (vr):

푚퐺푆푇 Δ푔푐푟푦푠 = Δ퐺푐푟푦푠 × (26) 푉표푙푟×푑푚

25 where m is molar mass, Vol is volume, and dm is mass density. I also use mGST and dm to convert

Δglc from J/g to J/mol before using it in (22). v is a continuous function which changes sign at the size dependent critical temperature (i.e. when r = rc, Figure 12b), implemented using Δgcl = -Δglc.

Smaller grains are expected to melt faster or crystallize slower. Hence; using v∞, as in the simulations below, gives a lower bound for melting and an upper bound for crystallization velocities.

I now adapt the finite element phase change framework to capture heterogeneous melting.

Beginning with this section, I use a 2-vector instead of a 3-vector CD, which is sufficient to capture

2 2 grain boundaries while reducing the number of equations solved. I use CD = √퐶퐷1 + 퐶퐷2 instead of CD = Σ(CDi), fixing all crystalline vectors at an equal length in CD-space and ensuring that

their lengths approach zero (they melt) at the same rate (Figure 14b). Nucleation now generates )

-1 (a) 400 c 1 1 K p,l Δ푐푝,푙푐푑푇 = 4 -1 4 1 200

cp,c

(J kg p c 0 (b) h 100 lc

50 slc g

0 lc Energy (J/g) Energy (c) v [20] 2 kinetic [2012 Orava] [2012 Burr] v 0 [25] v7.5nm v v

-2 2.5 nm Velocity (m/s) Velocity 400 600 800 Tmelt T (K)

Figure 13: (a) The integral of Δcp,lc accounts for the difference in the latent heats of fusion and crystallization (Δhlc(916

K) = 128.9 J/g and Δhlc(431 K) = 34.2 J/g, respectively[17]). (b) Calculated thermodynamic parameters. (c) Phase change velocities for bulk, 7.5 nm (dash-dots), and 2.5 nm (dashes) radius grains and that calculated by Burr et. al [7].

Negative values denote melting. vkinetic stabilizes with the liquid viscosity at higher temperatures; vkinetic(T > 2000 K) ≈ 60 m/s. 26

(a) CD3 (b) CD2 퐶퐷 = 퐶퐷 = 1 퐶퐷 = 퐶퐷 = 1 1 𝑖 1 2

1 θCD CD CD1 2 0 1 1

CD 퐶퐷 = 퐶퐷 2 1 2 𝑖 Figure 14: Crystalline vectors defined by (a) a 3-vector with CD = Σ(CDi) as in Woods et. al [6] and (b) a 2-vector with CD = ||CD||2 as used here. The cut plane and dotted line show allowed crystalline values. nuclei with a random angle (10° ≤ θCD ≤ 80°) instead of a random CDi distribution. θCD can be mapped to a physical grain orientation (θphys) with a function that depends on the dimensionality and crystalline structure of interest, e.g. to a range of 0° ≤ θphys < 90° for 2-D simple cubic structures or 0° ≤ θphys < 60° for 2-D hexagonal structures. I define grain boundaries wherever there is a high gradient in θCD (|훻θCD| > 2.9 °/nm).

I update v(T) from the velocity curve given in [7] to v∞(T). The two curves are qualitatively similar for T < Tmelt, but v∞(T) has a higher peak velocity and is also defined for T > Tmelt (Figure

13c).

I initiate melting at grain boundaries and material interfaces by modifying Amorphization:

퐴푚표푟푝ℎ𝑖푧푎푡𝑖표푛𝑖 = 훼푚푒푙푡 × 푣(푇) × (푇 > 푇푚푒푙푡) × 퐶퐷𝑖 × 퐻푀, (27) where HM is 1 at heterogeneous melting sites (grain boundaries and material interfaces) and 0 elsewhere. (T > Tmelt) is implemented as a step function which goes from 0 to 1 over a 1 K window centered at Tmelt. I modify stbl to the vector stbl such that there is a stability valley at CD ⪅ 1 when

T > Tmelt to maintain well-defined grains during melt (Figure 15c):

푠푡푏푙𝑖 = si n(퐶퐷𝑖) × 훼푠푡푏푙 × 푣(푇) (푇 < 푇 ) × 푝푤 (si n(퐶퐷 ) × 퐶퐷) . (28) × ( 푚푒푙푡 푠푡푏푙 𝑖 ) +(푇 > 푇푚푒푙푡) × 푝푤푠푡푏푙(1 − (si n(퐶퐷𝑖) × 퐶퐷))

(T < Tmelt) is implemented as a step function which goes from 1 to 0 over a 1 K window centered at Tmelt. I use the sign function (-1/+1/0 for negative/positive/0 arguments, respectively) in (28) to

1 (a) 0.05

(CD) 0.00 stbl -0.05 pw 0 0.00 0.05 1

(b)

diff pw

0 1

(c)

(1-CD) stbl

pw 0 0.0 0.5 1.0 CD

2 2 properly call control functions with the non-negative CD = √퐶퐷1 + 퐶퐷2 if CDi becomes negative due to numerical errors. Lastly, I set αdiff = -0.2 nm in (3) for T > Tmelt so diffusion works appropriately when v(T) is negative. I can control the temperature at which melting begins (Tmelt), the time required for a liquid layer to form between grains (via αmelt), and the melting rate [v(T), T

> Tmelt] with this framework. As in [6], this model can extend to 3-D by re-calibrating constants and solving for nucleation in discrete cubes instead of squares.

To demonstrate this updated model, I first simulate heterogeneous melting of a polycrystalline 2-D 200 nm square (20 nm out of plane depth) by setting T = 900 K (Figure 16). A liquid layer forms at grain boundaries and around the perimeter in ~1 ns, then grows at v∞(900 K)

≈ 1.3 nm/ns.

Next, I simulate electrical cycling of a 30 nm wide confined cell (30 nm out of plane depth).

I use a regular array of equally sized grains for the initial crystallinity conditions rather than a random grain map so that I can explicitly define grain size (Figure 17a). I vary the durations of

(a) t0 (b) t0 + 1 ns (c) t0 + 3 ns TK = 900

10° 90° θCD Liquid 200 nm Figure 16: (a) A polycrystalline GST square is brought to 900 K at t0. (b) Liquid forms at grain boundaries in 1 ns and (c) melts at 1.3 nm/ns. rectangular current pulses with 0.1 ns rise and fall times for reset and set of devices with 7.5 nm,

2 15 nm, and 25 nm radius grains. The devices achieve a reset resistance increase (R/RInitial) of >10 in 4.2 ns when r = 7.5 nm but require 5.2 ns to reach this contrast when r = 15 nm or 25 nm (Figure

17b, Figure 18).

I use devices reset to similar R/RInitial for set comparisons. 7.5 nm grains set more quickly due to templated growth from quenched-in crystals when R/RInitial ≈ 20 (Figure 17c, Figure 19).

However, fewer quenched-in crystals remain after stronger resets, increasing the resistance ratio and set times: set is achieved through templated growth from crystalline fronts at the top and

τ = (a) (b) τReset = (c) Set 200 nm Iapp Initial 4.1 ns 4.2 ns 5.2 ns Reset 1 ns 2 ns

SiO2

TiN

50nm r = 7.5nm 50 nm

30 nm

TiN r = 15nm

Tbnd = 300 K r θCD

80° r =25nm 10° Amorphous Figure 17: (a) Schematic illustration of geometry and initial and boundary conditions used in simulations. Iapp is a square pulse with 0.1 ns rise and fall times and a magnitude of 500 μA for reset or 50 μA for set. (b) Crystal maps of initial conditions and after reset with 3 different pulse durations (τReset) for 7.5 nm, 15 nm, and 25 nm grains. The amorphous area (and hence reset resistance) increases with τReset. The Initial column is used for RInitial in Figure 18 and Figure 19. A ~20x increase in cell resistance requires a 4.1 ns reset pulse for 7.5 nm grains and a 4.2 ns reset pulse for 15 nm or 25 nm grains (dashed borders). (c) Quenched-in crystals embedded in the amorphous GST in the r = 7.5 nm case grow simultaneously, allowing for shorter set times. 29 bottom contacts. Reference [34] experimentally showed a trend of decreased reset and set times as radii decreased from 8.5 nm to 5 nm but achieved sub-nanosecond reset while requiring ~50 ns for set. The longer reset and shorter set times in these simulations are consistent with a v∞ that underestimates melting velocities but overestimates crystallization velocities, resulting in less melting during reset, more crystallization while cooling after reset, and faster crystallization during set.

103

102 Initial IReset = 500 A 101 r = 7.5 nm R / R R / + r = 15 nm r = 25 nm 100 3 4 5 6 7 8 9 10  (ns) Reset Figure 18: ‘Reset’ resistance after a square current pulse (500 μA magnitude and τReset duration). Smaller grains allow for faster reset, while larger grains have a wider pulse duration window for intermediate values. RInitial corresponds to the cells in the Initial column in Figure 17b.

2 10 Set = 50 A r = 7.5 nm + r = 15 nm

Initial r = 25 nm

101 R / R / R

100 0 1 2 3 4  (ns) Set

Figure 19: ‘Set’ resistance after a square current pulse (50 μA magnitude and τSet duration) beginning with ~20x (dashed lines) and ~100x (solid lines) reset resistances for each grain size (Figure 18). Smaller grains allow for faster set when the initial reset resistance is low. Set time becomes less dependent on grain size as fewer quenched-in crystals remain in the amorphous region and reset resistance increases; crystallization is achieved through templated growth from the same number of similarly spaced crystal-amorphous interfaces. RInitial corresponds to the cells in the Initial column in Figure 17b. 30

4. Material Parameters

4.1 Measuring Material Parameters

Phase change material measurements are often made on thin films and nanoscale devices. Thin films allow precise control of temperatures and temperature gradients, and phase can be observed non-destructively via optical microscopy and atomic force microscopy (AFM). However, heating and cooling rates on thin films are limited to ~104 K s-1 [46], so it is difficult or impossible to perform measurements on metastable phases at temperatures where metastable-metastable (e.g. undercooled liquid-to-fcc) or metastable-stable (e.g. fcc-to-hcp) phase changes occur in < 100 µs.

Thin film measurements of “as-deposited” amorphous GST are often used as a starting point for characterizing the melt quenched amorphous phase. As-deposited a-GST is obtained by depositing

Ge, Sb, and Te in the correct ratio (2-2-5 for GST) at temperatures low enough that these atoms do not crystallize. The result shows no long range order, similar to the liquid and amorphous phases, but has a lower density than l-GST or melt-quenched a-GST and is expected to have a different electrical conductivity (σ), thermal conductivity (κ), specific heat (cp), and Seebeck coefficient (S). As-deposited a-GST serves as a useful starting point for estimating l-GST and melt- quenched a-GST parameters at low temperatures because it can be obtained at room temperature as a thin film, but care must be taken when using these parameters for the melt-quenched phase in models and simulations.

Nanoscale devices allow faster heating and cooling rates (~109 K s-1) than thin films and thus can be used to obtain metastable l-GST and melt-quenched a-GST (for T > Tglass and T < Tglass, respectively). These fast heating rates are obtained by electrical pulses, while cooling occurs via thermalization of the device with its surroundings, making it difficult to precisely determine transient temperatures without knowledge of the temperature dependent σ, κ, S, and cp of the phase

31 change material of interest, the same parameters of nearby materials (the metal contacts and insulating dielectric used in the device), and the temperature dependent thermal transport across material interfaces. The initial phase of a device can be determined by room temperature conductivity measurements. Melting is assumed to occur in initially crystalline devices when electrical conductivity rapidly increases at some current level, an assumption that is validated by the large decrease in electrical conductivity of the device after the pulse signifying that a crystalline-liquid-amorphous transition has occurred. Metastable l-GST can be obtained at known temperatures in nanoscale devices by electrically melting them on a heated chuck and allowing a short time for thermalization (~ 1 µs), but this technique till precludes measurements of l-GST at temperatures where the phase change time scale is less than 1 µs (above ~600 K) [49]. In addition to uncertainty in temperature during transient heating and cooling, phase cannot be easily and non- destructively observed via microscopy in nanoscale devices: dimensions < ~400 nm cannot be observed optically, encapsulated devices do not show height differences after phase change (the density difference between phases is realized as stress/strain rather than a change in volume) and thus cannot be measured via AFM, and scanning and transmission electron microscope (SEM and

TEM) imaging must be delicately performed to avoid disturbing the original phase.

The uncertainty of transient temperatures and phase in nanoscale devices and the difficulty of measuring metastable phases at all temperatures relevant to device operation in both thin films and nanoscale devices highlights the need for physically motivated parameter models for phase change memory materials. In the rest of this chapter I describe models for material parameters in fcc-GST

(hereafter c-GST), l-GST and amorphous GST (hereafter treated as the single phase a-GST), TiN, and SiO2. I also model thermal transport across interfaces of these materials.

4.2 Electrical Conductivity

Electrical conductivity in amorphous chalcogenides is temperature and field dependent. I model σ in a-GST (σaGST) as the sum of independent electrical field (σaGST,E) and temperature

(σaGST,T) dependent contributions, capped at the thermal contribution at the semiconductor-to-metal transition at T = 930 K

𝜎푎퐺푆푇(푇, 퐸̂) = min (𝜎푎퐺푆푇,푇(푇) + 𝜎푎퐺푆푇,퐸(퐸̂), 𝜎푎퐺푆푇,푇( 0 K)), (29) as described in Chapter 5.1. I also briefly detail the models here (Figure 20).

Reference [49] melts GST nanowires with an electrical pulse on a heated chuck. By varying the chuck temperature and allowing ~1 µs for thermalization after melt, [49] is able to obtain metastable liquid or amorphous GST between 300 and 600 K and measure σ. Measurements show

σ ∝ exp(T), suggesting that σaGST,T can be modeled as

퐸푎퐺푆푇(푇) 𝜎푎퐺푆푇,푇 = 𝜎푎퐺푆푇,푇0 × exp ( ), (30) 푘퐵푇 where σaGST,T0 is a constant, EaGST(T) is a quadratic function describing the depth of the effective trap level at T, and kB is the Boltzmann constant. Reference [50] (which I co-authored) fits equation

(30) to the measured data and finds (30) is well approximated by

𝜎푎퐺푆푇,푇 = 𝜎푎퐺푆푇,푇0 × exp(훼푎퐺푆푇,푇 × 푇), (31)

-1 -1 -1 where σaGST,T0 = 2.85 Ω m and αaGST,T = 2.02 K . Extrapolating this curve intercepts the thin film molten conductivity measured at Tmelt = 858 K [51], further validating this model (Figure 20).

Amorphous chalcogenides show an exponentially increasing current with increasing voltage, even at current levels too low for significant temperature increases. I model σaGST,E as

휎 (300 K) 𝜎 (퐸⃗ ) = 푇 exp(퐶 ⋅ |퐸⃗ |) (32) 퐸 100 1

T (K)

400 600 800 Tmelt 1000 106 T (K)  (930 K)

400 ) 600 800 T 1000 400 6004 T800 Tmelt 1000 106 TT (K) (K) -1 10 T m [2015 Adnane] 400400 600600 800800 (930T-1 T K) 10001000 400 ) 600 800 (930T melt K) 2 1000 ) T melt 6 4   -1  6 -1 10 T 10 10 ( [2010 Endo]

10 E



m m [2015 Adnane] -1 [2015 Adnane]  (930 -1 K) 0 )  T(930 K) [2005 Kato]

)  (930 K) ) 4 T 2  10

-1 4 

4   -1  T -1 1010 10 T

( [2010 Endo] [2013 Cil] 10 (  [2010[2015 Endo]Dirisaglik]

m E



-2 -1 [2015 Adnane] -1 [2015 Adnane] -1 0 22 10 [2005 Kato]  10  10  10 0 25 50 E 75 (  th

( [2010 Endo] (  E [2010[2015 Endo] Dirisaglik] [2013 Cil]

 E



0 -2 E (MV/m) 100 10 [2005 Kato]sigma_a 10 E [20150 Dirisaglik] 25[2013 Faruk Cil] metastable50 thth 75 -2 1010-2 sigma_a L'HaceneE (MV/m) Thin Film 0 25 50 EE 75 0 25 Faruk metastable50 thth 200575 Kato sigma_a sigma_a EE (MV/m) (MV/m)L'Hacene Thin Film sigma [2015 Faruk Faruk Dirisaglik metastable metastable] [2005 2005 Kato] Kato [2013 50 nm Cil] [2015 L'Hacene L'Hacene Adnane Thin Thin] Film Film [2010 sigma Endo] 20 nm 2005 2005 Kato Kato 50 nm Figure 20: Thermal and electric field contributions to electrical conductivity in a -sigma_a_T(930GST. Measured dataK) on thin films sigma sigma 20 nm (spheres above Tmelt) and nanowires (spheres below Tmelt and circles above Tmelt) included for reference. 50 50 nm nm sigma_a_T(930 K) -7 where C1 = 2.42×10 m/V is 20 20 chosen nm nm such that σE(Eth) = σT(Tmelt) × 10%. I use Eth = 56 MV/m, the sigma_a_T(930 sigma_a_T(930 K) K) switching field measured in as-deposited a-GST [52], and Tmelt = 858 K [51] and find that the switching field obtained (~28 MV/m) is similar to that measured for melt-quenched a-GST (~25

MV/m [53]) (Chapter 5).

fcc-GST is metastable and eventually transitions to hcp-GST when T < Tmelt. The fcc-hcp transition is slow enough for T < 640 K to allow conductivity measurements on fcc-GST; however, fcc-GST is polycrystalline, and the contributions of grain boundaries are difficult to extract from the effects of monocrystalline fcc-GST. Reference [51] anneals as-deposited a-GST thin films at progressively higher temperatures (Tanneal) (e.g. from 300 to 450 to 300 K, then 300 to 475 to 300

K, …) and finds the conductivity continuously increases when annealed above ~430 K due to the amorphous-to-fcc transition and again when annealed above ~640 K due to the fcc-to-hcp transition. Electrical conductivity in fcc-GST becomes less temperature dependent with increasing

Tanneal for 430 K < Tanneal < 600 K, while x-ray diffraction measurements on thin films show fcc grain sizes becoming larger with increasing Tanneal in this range [54]. This suggests that the

34 temperature dependence of fcc-GST is largely due to grain boundaries, and in this work I model the temperature dependence of σcGST as arising only from grain boundaries:

퐸푎,퐺퐵 𝜎푐퐺푆푇 = 𝜎푐퐺푆푇,0 (1 + 퐺퐵 (exp ( ) − 1)), (33) 푘퐵푇 which returns

𝜎푐퐺푆푇,0, 퐺퐵 = 0 𝜎푐퐺푆푇 = { 퐸푎,퐺퐵 } (34) 𝜎푐퐺푆푇,0 exp ( ) , 퐺퐵 = 1 푘퐵푇

-1 -1 where σcGST,0 = 1.45 Ω m is the temperature independent monocrystalline fcc conductivity, GB is 1 at grain boundaries and 0 elsewhere, Ea,GB is the grain boundary activation energy, and kB is the Boltzmann constant. In this work I use the Ea,GB = 30 meV at all grain boundaries. This value was extracted using thin film conductivity measurements from [51] and expected grain size for a given annealing temperature from [54]. It represents an average grain boundary, but in general

Ea,GB is expected to depend on the misorientation (Φ) between grains. Equation (33) can be modified to account for Φ dependent grain boundary conductivity by making Ea,GB a function of

Φ. If Φ = 0 corresponds to a continuous crystal, and Ea,GB increases with increasing Φ and has an intercept Ea,GB(Φ=0) = 0, then (33) returns σcGST = σcGST,0 for grains with the same orientation and progressively lower conductivities with increasing Φ: higher mis-alignments would give a lower conductivity, as expected due to more defects and thus increased scattering between grains with high Φ. σcGST,0 intercepts σaGST,T at T ~ 720 K. fcc-GST measurements are unavailable at these temperatures due to the fcc-hcp transition, but I treat c-GST as never being less conductive than a-

GST by modifying (33) to

퐸푎,퐺퐵 𝜎푐퐺푆푇 = max (𝜎푐퐺푆푇,0 (1 + 퐺퐵 (푒푥푝 ( ) − 1)) , 𝜎푎퐺푆푇) (35) 푘퐵푇 in my simulations.

I assume electrical conductivity in SiO2 is negligible and use σSiO2 = 0, as given for that material by COMSOL [55]. I use temperature-dependent conductivity values for TiN measured in [56].

4.3 Thermal Conductivity

Heat is carried throughout a material by electrons and phonons. Thermal conductivity can be modeled as the sum of electron (κel) and phonon (κph) contributions:

휅 = 휅푒푙 + 휅푝ℎ. (36)

The electronic contribution is related to the electrical conductivity via the Wiedemann Franz Law:

휅푒푙 = 퐿𝜎푇 (37) where L = 2.44×10-8 W Ω K-2 is the Lorentz number. I treat the phonon contribution in a-GST and c-GST as degrading as the number of broken bonds in these materials increases, using σ as a measure of the number of broken bonds as in [57]: more broken bonds result in more free carriers and thus a higher σ. Assuming κph becomes 0 at the semiconductor-to-metal transition gives

휎(푇) 휅푝ℎ = 휅푝ℎ,0 (1 − ), (38) 휎푎퐺푆푇,푇(930 K) where κph,0 is a constant calculated as

휎(푇) 1 휅 = (휅 ( 00 K) − 휅 ( 00 K)) × (1 − ) . (39) 푝ℎ,0 푚푒푎푠푢푟푒푑 푒푙 휎(930 K)

κmeasured(300 K) is the total thermal conductivity measured at T = 300 K. I use κaGST,measured(300 K)

-1 -1 -1 -1 = 0.25 W m K [58] and κcGST,measured(300 K) = 0.45 W m K [58] in this work.

I model thermal conductivity in TiN using measured data [59] and in SiO2 as the temperature independent 1.4 W m K-1 given for that material by COMSOL [55].

4.4 Seebeck Coefficient

Reference [56] measures the Seebeck coefficient in as-deposited a-GST thin films during a sequence of anneals to progressively higher temperatures, as described in Section 4.2. These measurements show S is positive in both as-deposited amorphous and fcc-GST, suggesting that holes dominate thermoelectric effects in both phases. S in as-deposited a-GST decreases linearly with increasing T for 300 K < T < 400 K, at which point the thin films begin to crystallize. Thin film measurements above the melting point also show a linearly decreasing S but with a different slope [56]. The thin films tend to break down after melting, making repeated measurements of S in the molten liquid difficult to obtain, but based on the data given in [56] I use

1 0 푆푎퐺푆푇 × 푇 + 푆푎퐺푆푇 푇 < 858 푆푎퐺푆푇 = { 1 0 } (40) 푆푙퐺푆푇 × 푇 + 푆푙퐺푆푇 푇 ≥ 858 where Si are constants of the ith order linear fitting terms for the subscripted phases (aGST for as- deposited amorphous and lGST for the molten liquid).

The Seebeck coefficient of fcc-GST (ScGST) increases linearly with increasing T and has a y-intercept of zero, as expected in materials at temperatures where conduction is unipolar. As with electrical conductivity, I assume ScGST ≤ SaGST, but here use an inverse-sum-of-inverses to achieve a smooth transition between the two curves:

1 푆푐퐺푆푇 = 1 (41) 1 − − 1 4 1 4 − [(푆푐퐺푆푇×푇) +(푆푎퐺푆푇) 4] where S1 corresponds to the first order linear fitting term of measurements on the subscripted phase.

I assume thermoelectric effects within SiO2 are negligible due to the small number of carriers

-14 -1 and set SSiO2 = 10 V K . I use measured values from 300 K to 900 K [56] for STiN.

4.4.1 Seebeek Coefficient from Band Diagram

The Seebeck coefficient in semiconductors can be calculated from their energy band diagrams. In this section, I present an alternative model for SaGST based on the band diagram calculated in [60]. I also modify ScGST to continue linearly increase until 30 K below the semiconductor-to-melt transition, at which point it decreases to SaGST over a 30 K window (from

900 to 930 K). This results in a Seebeck differential between phases at Tmelt and consequently significant thermoelectric effects at crystalline-liquid interfaces during reset. The simulations in this section on a double mushroom geometry were performed by Noah Del Coro, an undergraduate student I mentored [61].

The Peltier coefficient describes the energy carried by the average electron or hole (Πe or

Πh) and is equal to the sum of their potential (band edge to fermi level) and kinetic (3/2kBT) energies:

3 Π푒 = − ((퐸푐 − 퐸퐹) + 푘퐵푇) 2 (42) 3 Π = (퐸 − 퐸 ) + 푘 푇 ℎ 퐹 푣 2 퐵 where Ec and Ev are the conduction and valence band edges, respectively, EF is the fermi level, and kB is the Boltzmann constant (Figure 21c). Thermoelectric effects result in an open circuit (Voc) voltage under a fixed temperature gradient (ΔT), expressed by the Seebeck coefficients

푉 Π S = 표푐 = 푒 푒 Δ푇 푇 푉 Π . (43) S = 표푐 = ℎ ℎ Δ푇 푇

Both electrons and holes contribute to thermoelectric effects under bipolar conduction, resulting in a measured S that is a function of Se and Sh and can be modeled by weighting these contributions

∗ ∗ by carrier concentrations (n and p) and their effective mass ratio (푚ℎ/푚푒) [62]:

∗ ∗ 푝 (푚ℎ/푚푒)푛 푆 = 푆ℎ ∗ ∗ + 푆푒 ∗ ∗ . (44) 푝+(푚ℎ/푚푒)푛 푝+(푚ℎ/푚푒)푛 39

I calculate Se and Sh using the effective trap level for metastable aGST calculated in [60] as EF

(Figure 21). Sh is close in both magnitude and slope to measurements of as-deposited a-GST from

300-400 K (Figure 21d), consistent with the unipolar conduction assumed when deriving that band gap. Using Sh as the Seebeck coefficient for a-GST will be an over-estimate once bipolar conduction begins. However, the unipolar assumption in [60] correctly describes resistivity even at Tmelt ~ 858 K, so unipolar conductivity may be an appropriate approximation even near melt due

101 (a) [2015 Dirisaglik]

-1 m)

 10

 E -E ( F v   -3 k T B  10 0e aGST

-5 10 [2017 Adnane]

 = -(E - E + 3 k T) (b) e c F 2 B 0.5 Ec 0.0 E

E (eV) E F E -0.5 v  = E - E + 3 k T h F v 2 B 1.5 (c)

− 1.0 e

E (eV) E 0.5  h 0.0 -S (d) 3 e + [2006 Baily] [2017 Adnane] 2 Sh

1 S(mV/K)

Sc 0

0 200 400 600 800 TMelt T (K) Figure 21: (a) Resistivity measurements on metastable a-GST nanowires [49] show an exponential dependence with T that intercepts thin film measurements at Tmelt = 858 K [48]. (b) [60] used (a) to calculate a temperature dependent band diagram for metastable aGST. I use that diagram to calculate (c) Peltier and (d) Seebeck coefficients for electrons and holes, finding close agreement with S measurements on as-deposited a-GST thin films (spheres [48] and crosses [63]).

∗ ∗ ∗ ∗ to either a low population of electrons or a low 푚ℎ/푚푒. Room temperature values of 푚ℎ/푚푒 include from 0.65 in Ge, 0.69 in Si, and 7.9 in GaAs [64].

Simulations of a double mushroom phase change memory device (Figure 22) illustrate thermoelectric effects using the models presented in this section. This geometry shows polarity dependent heating or cooling in the top and bottom GST layers, as evidenced by the size of the

Liquid VGate θ I crystal Vdd Amorphous 10° 80° TiN 0 ns (a) GST (f)

SiO2 100 nm

(b) (g) 24 ns

43 ns

(d) (i) 90 ns

2 2 (e) 100 100(j) 80 80

60 V 60 (V)

1 V (V) 1 A)

Gate Gate A)



 Gate

I 40 Gate I 40

I (

I ( V 20 V 20 0 0 0 0 20 30 40 50 20 30 40 50 time (ns) time (ns) With Thermoelectrics Without Thermoelectrics Figure 22 Snapshots of a transient reset simulation (a-d) with and (f-i) without modeling thermoelectric effects. Thermoelectrics result in more heating in the top and less heating in the bottom phase change layer, as evidenced by the size of the amorphous areas after quenching. Vdd = 1.9 V for both devices, resulting in a reset-to-initial resistance ratio of 703x with and 1040x without thermoelectrics. The device resistance is lower during melt without thermoelectrics, resulting in a slightly higher current (e,j). 41 amorphous areas in those layers. Thermoelectric heating within the bulk material is on the same order of magnitude as Joule heating, while the thermoelectric heating at boundaries is even larger

(Figure 23).

2 (a) 표푢푙푒 = 퐽 (b) 푇퐸 = −∇ ⋅ 퐽푆푇 푛푡푒푟푓푎푐푒 = − 퐽 푆𝑖푇𝑖 − 푆 푇 /(1 nm)

3 ) 100

)

3 m

m (c)  μ 0 10 nm

10 nm Fig. 2c

Q (W/ -100

Heat (W/ 0 2 4 6 8 10 -3 x (nm) Figure 23 Heat at the peak of reset (Figure 22c) due to (a) Joule, (b) thermoelectric, and (c) interfacial thermoelectric contributions. Interfacial heating is calculated assuming the heat at material interfaces (W/μm2) is released within 1 nm from the interface. There is significant Joule heating at the lower GST/heater interface (~10s of W/μm3), but this is countered by thermoelectric cooling, so the GST near that interface remains crystalline. Heating and cooling at liquid-crystal interfaces due to the difference in Sh and Sc at Tmelt can be seen in (b): the magnitude of S is small, but

QTE is large due to the abrupt change in the material, and hence a high S gradient.

4.5 Specific Heat

Specific heat is a measure of how much energy is required to increase temperature in a material. The specific heat of fcc-GST (cp,fcc-GST) has been measured from 300 to 600 K and is fit well by a quadratic function that has a maximum at T ~ 700 K [24]. Room temperature measurements of the specific heat of amorphous GST (cp,a-GST) are very similar to cp,fcc-GST, as expected in amorphous materials below the glass transition temperature. The amorphous phase must have a larger specific heat than the fcc phase between Tglass and Tmelt to account for the difference in energy released upon crystallization (the latent heat of crystallization) near Tglass and the energy absorbed upon melting (the latent heat of fusion) at Tmelt (Figure 24):

2 2 1 0 푐푝,푐퐺푆푇 × 푇 + 푐푝,푐퐺푆푇 × 푇 + 푐푝,푐퐺푆푇 푇 < 00 K 푐푝,푐퐺푆푇 = { 푐표푛푠푡 } (45) 푐푝,푐퐺푆푇 푇 ≥ 00 K

푐푝,푐퐺푆푇 푇 < 푇푔푙푎푠푠 푐푝,푎퐺푆푇 = { 푐표푛푠푡 } (46 푐푝,푎퐺푆푇 푇 ≥ 푇푔푙푎푠푠

i th const where cp is a fitting constant of the i order fitting term and cp is a constant used as the specific heat above a certain temperature for the subscripted phase. This model appropriately conserves energy due to the latent heat of phase change over multiple crystallization-amorphization cycles, as described in Chapter 3.2.

-1 -1 I use the temperature independent 730 J kg K for SiO2 (given for that material by

COMSOL [55]) and 784 J kg-1 K-1 for TiN ([65]).

14200

14000 cp,a

400

)

-1

K -1

[2002 Kalb]

cp,c

(J kg

p c 200 cp,c and cp,a in [2017 Woods]

0 300 400 500 600 700 800 900 1000

T (K) Figure 24: Specific heat in a-GST (blue) and c-GST (red) developed for this work, as well as measured data from [24]. The model in [6] used the same specific heat for a-GST and c-GST at all temperatures, with a spike near the melting point capturing the latent heat of fusion both upon heating and cooling (green).

4.6 Thermal Boundary Conductance

Poor coupling between electrons and phonons across material interfaces leads to steep

2 thermal gradients in nanoscale devices, characterized by a thermal boundary resistance RTh (m K

-1 -1 GW ) or thermal boundary conductance GTh = RTh [66]–[68]. In this section I use the variable G to refer to thermal boundary conductance; G refers to the Gibb’s free energy in the rest of this work. I model GTh between two materials Mat1 and Mat2 (퐺푀푎푡1/푀푎푡2) as the sum of electron

푒푙 푝ℎ (퐺푀푎푡1/푀푎푡2) and phonon (퐺푀푎푡1/푀푎푡2) contributions:

푒푙 푝ℎ 퐺푀푎푡1 푀푎푡2 = 퐺푀푎푡1/푀푎푡2 + 퐺푀푎푡1/푀푎푡2. (47)

푒푙 퐺푀푎푡1/푀푎푡2 is expected to depend on σ, and I treat it as negligible at insulator interfaces (SiO2).

푒푙 5 Additionally, the room temperature 퐺푀푎푡1/푀푎푡2 is expected be ~10 higher at c-GST/metal interfaces than a-GST/metal interfaces; however, room temperature measurements of Gc-GST/TiN and Ga-GST/TiN differ by only one order of magnitude [67]. Reference [67] suggests a Schottky

푒푙 barrier at GST-TiN interfaces is responsible for the apparently minor 퐺퐺푆푇/푇𝑖푁, so that thermal

푒푙 conductance across the interface is achieved primarily through phonons. If true, 퐺퐺푆푇/푇𝑖푁 should vary significantly with temperature and field and may be significant during PCM device operation.

Experimental values of 퐺푎 퐺푆푇/푇𝑖푁 at elevated temperatures are unavailable, but experimental

푒푙 values of 퐺푐 퐺푆푇/푇𝑖푁 show a linear increase with increasing T [66]. Here, I model 퐺퐺푆푇/푇𝑖푁 by adding a boundary condition at GST/TiN boundaries that treats thermal conductance across those interfaces as if there were an extra layer of GST with a certain thickness and thermal conductivity.

I use the room temperature cGST/TiN depletion zone width as the layer thickness, which I

19 -3 calculate as Wd ~ 7 nm by assuming the number of free carriers in GST (n) is 1.6⋅10 cm , the c-

GST/TiN offset voltage is 0.3 V, and the relative permittivity of c-GST (εr) is 16. I calculate the

45 layer thermal conductivity using the phase-dependent electrical conductivity and the Wiedemann

Franz Law. The resulting electron contribution to thermal boundary conductance is negligible at a-GST/TiN interfaces at 300 K, but accounts for ~35% of the total conduction at c-GST/TiN interfaces. The electron contribution dominates at elevated temperatures in both phases (Figure

25b).

푝ℎ 푝ℎ I model phonon contributions at a-GST interfaces (퐺푎퐺푆푇 푆𝑖푂2 and 퐺푎퐺푆푇 푇𝑖푁) as constant using values measured at room temperature. I model phonon contributions at c-GST interfaces

푝ℎ 푝ℎ (퐺푐퐺푆푇 푆𝑖푂2 and 퐺푐퐺푆푇 푇𝑖푁) as constant at low and intermediate temperatures, but assume they degrade to their corresponding a-GST values as T increases from 720 K (where σcGST = σaGST) to

푝ℎ TMelt. Room temperature 퐺푀푎푡1/푀푎푡2 values are extracted from [66]–[68], with GST-TiN values chosen such that the calculated electron contribution and the phonon contribution add to the total thermal boundary conductance measured at room temperature:

푒푙 푝ℎ 퐺푐퐺푆푇 푇𝑖푁( 00 ) = 퐺푐퐺푆푇 푇𝑖푁( 00 ) + 퐺푐퐺푆푇 푇𝑖푁( 00 ). (48)

(a)

) -1 [1] K 30 -2 [2010 Reifenberg] [2013 Lee] GST-SiO 20 2 [2010 Lee]

G m (MW [1] 10 a

(b) ) -1 60

K GST-TiN

-2 [2] 40

a G m (MW [3] 0 300 400 500 600 700 800 900 T (K) Figure 25: Thermal boundary conductance at (a) GST-SiO2 and (b) GST-TiN interfaces due to phonons (dashed), electrons (dotted), and the sum of both (solid) for c-GST (red) and a-GST (blue). Measured data from [66] (triangles), [58] (spheres), and [68] (stars) are included for reference. 46

5. Ovonic Threshold Switching

PCM is CMOS back-end-of-line compatible, allowing memory integration on-chip with

CMOS circuitry to eliminate latency from off-chip memory access [70]. PCM can be implemented

as a crossbar array, allowing high device density (4F2) in multiple memory layers and efficient

neuromorphic computing [71]. Crossbars consist of perpendicular word and bit lines with memory

elements sandwiched between these lines at the cross-points (Figure 26). Each word or bit line is

connected to Vdd or ground through a transistor, and cross-points can be randomly accessed by

activating their corresponding word and bit lines. Non-selected devices can form undesirable

current sneak paths between selected and non-selected lines; hence access devices with non-linear

6 current-voltage (I-V) characteristics, high Ion/Ioff ratios (Ion/Ioff ~10 for a 1000x1000 device array),

300 K VDevice

I RLoad

Vapp Top Electrode OTS Middle aGST Electrode (a) TiN TiN(e) PCM cGST 45 nm Bottom SiO 2 Electrode (b) (f)

300 K

Melting Figure 26: A schematic illustration of a cross-point cell with an Ovonic threshold switch in series with a phase change (c) memory element. The shown cell(g) structure is used for 2D analysis to compare modeling results to the experimental results in [69].

47 Homogeneous Homogeneous Heterogeneous Melting

(d) (h)

Crystal Orientation Liquid 10 nm Amorphous 2 and high drive capabilities to write (Iwrite ~MA/cm ) are needed at each cross-point [72]. Ovonic threshold switches (OTS) made from amorphous chalcogenides are one such access device.

Amorphous chalcogenides are highly resistive under low electric fields and exhibit

“threshold switching” to a highly conductive on-state at a threshold voltage (Vth). Once switched, these materials remain in the on-state while a minimum holding current or voltage (Ihold or Vhold) is maintained (Figure 27) [73]. Crystallization dynamics of these materials determine whether they are more suitable for an OTS or a PCM. PCM materials include various stoichiometries of Ag-In-

Sb-Te and Ge-Sb-Te, with typical crystallization times on the order of 10 ns [74]. OTS materials remain in their amorphous phase during normal operation and are often characterized by the number of switching cycles they withstand before failure (through, e.g., material damage or

8 crystallization): > 600 for GeTe6 [75], > 10 for AsTeGeSiN [76], and unknown for As-doped Se-

Ge-Si, a material used in a commercial OTS+PCM crossbar array [77].

Vapp (t) 2D - (a) 3D (b)

I RLoad Rotational Device

V rSiO2 1 µm

hOTS = 50 nm

hTiN = 1 µm

rOTS = 100 nm

SiO TiN aGST (c) 2

00 2 1010 2 3D (~2 days) iii 2D Rotational (~5 min) iv ii 1010-3-3

10-6-6 v 10 0 1

00 0 1

I (mA) I (mA) I i -1-1 2 3D3D (~2 days) -2-2 2D2D -RotationalRot. (~5 minutes) (~5 min)

-1.5-1.5 -1.0-1 -0.5-0.5 0.00 0.5 1.01 1.5 1 VVdevice (V) (V) Device Figure 27: (a) 3D and (b) 2D-rotational OTS geometries used in this work. T = 300 K is used as the initial condition and as the boundary condition at the top and bottom of the TiN contacts. (c) When a 3 V / 60 ns triangular pulse is applied at Vapp, the device (i) is highly resistive until Vswitch ~ 1.75 V, (ii) switches on in ~10 ps, (iii - iv) remains on until the voltage and current drop below Vhold ~ 0.4 V and Ihold ~ 0.25 mA, and (v) returns to the high resistance off- state. Symmetric switching behavior is observed when0 a negative ramped pulse is applied. The 2D-rotational and 3D

simulations give similar results, as expected for rotationally symmetric filamentary switching. Inset in (c) is the first

quadrant in logarithmic scale. (mA) I

49 -1

-2

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

VDevice (V) There is still debate on the mechanism(s) underlying threshold switching despite many studies investigating this phenomenon [53], [69], [73], [78]–[85]. Vth scales linearly with device thickness, suggesting a field-based mechanism at a threshold Eth [82]. Theoretical arguments and the presence of crystalline filaments in failed devices suggest that on-state conduction is filamentary [80], [82].

Current-field (I-퐸⃗ ) measurements on amorphous chalcogenides typically show an ohmic

1 regime at low 퐸⃗ , an intermediate regime where ln(퐼) ∝ 퐸⃗ or 퐸⃗ 2, and a high field regime where I increases at a super-exponential rate with 퐸⃗ [86]. Reference [86] reviews conduction mechanisms in amorphous materials and shows that multiple mechanisms can fit measured data through the tuning of parameters which are otherwise difficult to validate (e.g. trap-to-trap distance, effective carrier mass, and carrier mobility). Models for these mechanisms have been proposed which define carrier concentrations (n) and mobilities (μ) as functions of 퐸⃗ and temperature (T) such that all three I-퐸⃗ regimes are captured with an electrical conductivity 𝜎 = 푞푛휇, but such techniques are computationally expensive in addition to relying on multiple unknown fitting parameters.

Reference [53] proposes a field-based switching model where carrier concentrations rapidly increase once trap states near the Fermi band are filled and fits the model to amorphous (a-)

Ge2Sb2Te5 (GST) measurements. References [78], [85] propose a field-assisted thermal model based on multiple trap barrier lowering and fit the model to a-GeTe and doped a-GST measurements. References [81], [87] ascribe switching to crystalline filaments which form under high 퐸⃗ and fit the model to a-GST using relaxation oscillations. They suggest that these filaments become unstable at low 퐸⃗ in OTS materials but remain stable in PCM materials.

Here, I model conductivity in a-GST as the sum of T and 퐸⃗ dependent terms (σa = σT + σE).

This model does not require a computationally expensive evaluation of the (density of states ×

Fermi function) integral at every T, 퐸⃗ combination where conductivity is needed; hence, it is appropriate for transient finite element simulations with dynamic T and 퐸⃗ . While this model trades accuracy for ease of computation, simulations show that it (i) can be tuned to fit a wide range of switching fields, (ii) captures the appropriate changes in threshold switching as I systematically vary ambient conditions, geometries, and the rise and fall times of applied pulses, and (iii) can reproduce the behavior of a series PCM+OTS device when used with a finite element phase change model [5], [6], [9], [88].

5.1 Computational Model

I use a-GST material parameters as in [6], [88] to simulate an OTS material. Of particular interest to this work is σa, which I model as in [65] and cap at σmax = σT(930 K) (Figure 28):

𝜎푎(푇, 퐸⃗ ) = min( 𝜎푇(푇) + 𝜎퐸(퐸⃗ ) , 𝜎푚푎푥 ), (49) which is equivalent to assuming that free carriers are excited to a band edge via independent thermal and electrical processes:

𝜎푎(푇, 퐸⃗ ) = min( 푞(푛푇 + 푛퐸)휇 , 𝜎푚푎푥 ), (50) where nT and nE are carriers excited via thermal or electrical processes and 휇 is the free carrier mobility.

I fit σT to low-퐸⃗ measurements of metastable a-GST wires [49] and molten GST thin films

[56], as described in [50] (Figure 28a). Measurements of liquid GST show a semiconductor-to- metal transition near 930 K [89], with σ becoming practically independent of T. I therefore limit

5 -1 -1 σa(T, 퐸⃗ ) to σT(930 K) = 4.1×10 [Ω m ], which is in line with the highest conductivities measured in molten GST [89]–[91](Figure 28a).

σE is assumed to be an exponential which contributes 1% of σT(300 K) at zero field and

10% of σT(Tmelt) at Eth:

휎 (300 𝐾) 𝜎 (퐸⃗ ) = 푇 exp(|퐸⃗ | ⋅ 퐶 ) (51) 퐸 100 1

-7 where C1 = 2.42×10 m/V is chosen such that σE(Eth) = σT(Tmelt) × 10%. I use Eth = 56 MV/m, the breakdown field measured in as-deposited a-GST [52], and Tmelt = 858 K [56] (Figure 28a).

I include σ-퐸⃗ curves at various temperatures calculated using the models in this work (for metastable a-GST) and the models in [78] (for drifted, doped a-GST) for comparison (Figure 28b).

σT dominates at low fields, while σE begins to dominate at higher and higher fields with increasing

T (K)

400 600 800 Tmelt 1000 6 (a) 10  (930 K) [28] ) 4 T  [30]

-1 10 T [31] m -1 2 [26]  10

( E



100 [32] 10-2 0 25 50 Eth 75 E (MV/m) 4 (b) 10 700 K 3 700 K 10

) 2

-1 10 m

-1 101 [27] 

( 0

 [11] 10 300 K 300 K This Work 10-1 0.0 0.5 1.0 1.5 E/E (300 K) switch Figure 28: (a) T and 퐸⃗ dependent contributions to electrical conductivity in (1); temperature is uncertain for T > Tmelt. (b) Conductivity-Field behavior using the model in this work (metastable a-GST) and the model in [78] (drifted, doped a-GST) at various temperatures. I use Eswitch(300 K) = 25.01 MV / m in this work and Eswitch(300 K) = 155 MV/m for the curves in [78].

5.2 Simulations

I first simulate switching in 3D and 2D-rotational geometries by applying a 3V / 60 ns triangular pulse (Figure 27a,b). Results show filamentary switching with current confined to the central portion of the device (Figure 29a-j), with practically identical I-V characteristics for 3D and 2D-rotational simulations (Figure 27c). 3D simulations show some instability of the filament location (slightly off-center in Figure 29d,i) and filament migration over time.

I next evaluate the impact of σE by simulating switching with σE = 0 (Figure 30a) and compare the results with σE defined as in (3) (Figure 30b). Vswitch and Iswitch are the values at which

- + - + (a) tswitch (b) tswitch (c) tpeak (d) thold (e) thold

(f) (g) (h) (i) (j)

200 nm 300 T (K) 950 2 (k)

I (mA) I 0 (l) V 4 app 2

V V(V) Device 0 t t 0 switch 50 tpeak 100 hold time (ns)

Figure 29: (a-e) x-y and (f-j) x-z temperature cut planes while switching the 3D OTS in Figure 27a illustrate filamentary on-state conduction. (j) Current and (k) device voltage transients resulting from the applied Vapp used to generate the I-V in Figure 27c. Superscripts “-” and “+” refer to the time steps (1 ns increments) before and after the subscripted event. (Figure 27a: RLoad = 1 kΩ, hOTS = 50 nm, rOTS = 100 nm, Vapp = 5 V / 60 ns triangular pulse). 54

VDevice begins to decrease. Threshold switching occurs even with σE = 0 due to thermal runaway.

However, defining σE as in (3) gives a switching field (Eswitch = Vswitch / hOTS) that is smaller and less dependent on hOTS (Figure 30c). Some hOTS dependence is still observed due to changing thermal conditions.

Reference [52] reports switching fields from 8.1 MV/m (as-deposited a-Ge15Sb85) to 94

MV/m (as-deposited 4 nm thick a-Sb). I examine the tunability of this model by varying Eth in (3) from 5.6 to 560 MV/m (Figure 31). Results show Eswitch varying from 5 to 42.5 MV/m. Eswitch =

(a)  = 0 (b)  as in (3) 800 E E increasing h

600 OTS A)  400 increasing h

OTS I ( 200

0 0 1 2 3 4 0 1 2 3 4 V (V) VDevice (V) Device

60 (c) E = 0

40 (MV/m) 20  as in (3) Switch E E 0 1.00 E = 0

0.75 (d)

25 25 nm Switch

/E 0.50

E as in (3)

Switch 0.25 E 0.00 25 50 75 100 h (nm) OTS Figure 30: The switching voltage increases with hOTS both (a) without and (b) with field dependent conductivity.

Including field dependent conductivity reduces the switching field’s (c) magnitude and (d) sensitivity to hOTS. (Figure

27b: RLoad = 5 kΩ, hOTS = 25 to 100 nm, rOTS = 100 nm, Vapp = 5 V / 5 s triangular pulse). 55

25.01 MV/m when Eth = 56 MV/m, similar to the Eswitch = 28.75 MV/m measured in [53] for melt- quenched a-GST. σE becomes negligible compared to σT when Eth > 200 MV/m even for high fields: the σT used in this work precludes Eswitch > 42.5 MV/m; a reduced σT is required for higher switching fields. Iswitch decreases by ~100x as Vswitch increases, resulting in a decrease in switching power (Pswitch) from ~100 to 20 μW (Figure 31c).

Vswitch has been shown to decrease with increasing ambient temperature (Tambient), while the temperature behavior of Iswitch and Pswitch are less clear [78]. I simulate switching while varying

Tambient (the initial temperature and the fixed top and bottom TiN boundary temperatures, Figure

27b) from 300 to 400 K (Figure 32). Vswitch decreases as expected (Figure 32a). Iswitch at first decreases and then increases with increasing Tambient, while Pswitch monotonously decreases in the

Tambient range simulated but is beginning to flatten with increasing T by 400 K.

Next, I systematically vary rOTS, hOTS, and the rise time (τrise) of Vapp in the geometry shown in Figure 27b (Figure 33). Results agree with expected OTS behavior: Vswitch approximately

2 4 10 (a)

A) 3

 (V) ( 2

1 Switch

Switch 10

I 1 V 0 100 (b)

W) 

( 50

Switch 25 P 0 107 108 EEth (V/m) (V/m) Threshold Figure 31: (a) The switching current (voltage) decreases (increases) and (c) the switching power decreases with 8 increasing Eth in (3). The device switches thermally before the field contribution becomes significant for Eth > 1×10

V / m. As a result, further increases to Eth result in the same switching characteristics. (Figure 27b: RLoad = 5 kΩ, hOTS

= 100 nm, rOTS = 100 nm, Vapp = 5 V / 5 s triangular pulse). 56 doubles as hOTS doubles [73], Vswitch decreases with increasing τrise, approaching a minimum value

[83], Ihold and Vhold are only weakly dependent on hOTS [82], and switching characteristics are only weakly dependent on rOTS due to filamentary conduction in the on-state.

3 (a)

(V) 2

[2016[11] Gallo] Switch

V 1 This Work 0

15 (b) A)

 10 ( 5

Switch I 0 (c)

30 W)

 20 (

10 Switch P 0 300 320 340 360 380 400 T (K) Ambient Figure 32: The (a) voltage, (b) current, and (c) power required to switch as the ambient temperature changes. Vswitch decreases monotonically, but Iswitch and Pswitch have more complex relationships with Tambient. (Figure 27b: RLoad = 5 kΩ, hOTS = rOTS = 100 nm, Vapp = 5 V / 5 s triangular pulse). 57

5 V / 10 ns 5 V / 100 ns 5 V / 1 s 800

600 A)

 400

= 25 nm = I (

OTS 200 r 0 800

600 A)

 400 = 50 nm =

I ( 200

OTS r 0 h = 100 nm 800 OTS h = 50 nm 600 OTS

hOTS = 25 nm A)

 400 = 100 nm =

I ( 200 OTS r 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 V (V) V (V) V (V) Device Device Device Figure 33: Switching characteristics of OTS devices with varying radii, ramp times, and heights. The simulated holding currents and voltages are weakly dependent on hOTS, but the switching voltage ~doubles as as hOTS doubles. (Geometry in Figure 27b) Finally, I simulate an OTS and PCM in series (OTS+PCM, Figure 26) based on the devices fabricated and characterized in [69]. I use a 2D, 45 nm fixed out-of-plane-depth simulation instead of a 2D-rotational simulation to more appropriately model phase change dynamics in the PCM with (1). I use a 500 nm depth in the bit line to account for its large thermal mass (Figure 26), set

Tambient = 300 K as the initial and fixed TiN boundary temperatures, and reset the device with a 5

V / 5 ns square pulse (1 ns rise and fall times) at Vapp followed by 1 μs for thermalization (starting at Figure 26 and ending at Figure 34a). I then sweep Vapp from 0 to 2.5 V over 50 ns to characterize the OTS + reset PCM. I also simulate an isolated OTS and isolated reset PCM in the same way by replacing the PCM or OTS, respectively, with TiN (Figure 34c,d). I plot the I-V characteristics

before switching, dividing currents and voltages by the isolated OTS Iswitch and Vswitch values in

order to compare our results to those presented in [69] (Figure 34e). The results are similar, with

푂푇푆 the OTS limiting the current in the reset PCM until 푉퐷푒푣𝑖푐푒 / 푉푠푤𝑖푡푐ℎ > 2 (Figure 34d). This limits

the current during read in a reset cell while allowing high current in a set cell, creating a large read

margin. The smaller scaled currents for the PCM and OTS+PCM in our simulation could be due

to a difference in the amorphous volume in the reset PCM; the fixed out-of-plane depth in our

simulations, which cannot capture filaments smaller than 45 nm in depth and may thus

overestimate Iswitch in the OTS; or parameter differences between aGST and the (unreported) OTS

material used in [69]. (a) TiN (e)

aGST cGST TiN SiO2 (a) OTS+PCM (b) OTS (c) PCM

SiO2 150 nm (b) (f )

1 [2009 K a u]Melting OT S 1 [2009 Kau] OTSPCM T hi s W ork (d) 1 [5][2009 Kau] OTS OTS (c) This Work (g )+ OTS

PCMThis Work + OTS

Switch OT S PCM + OTS Switch PCM

PCM

OTS

Switch I / I I /

PCM

I / I / I

Homogeneous Homogeneous Heterogeneous Heterogeneous Melting 0 0 0 0 1 0 2 1 2 (d) 0OTS 1 (h ) 2 V / V OTS Switch V / V OTSSwitch Figure 34: (a) Reset OTS+PCM, (b) isolated OTS, and (c)V isolated / VSwitch reset PCM. (d) Pre-switching I-V characteristics

scaled by Vswitch and Iswitch in the OTS using the model in this work (spheres) and experimental data extracted from [69] (squares). I-V characteristics from this work show similar switching voltages but lower scaled switching currents. The schematic of the simulation setup is shown in Figure 26. The OTS+PCM reset animation for this simulation is available in supplementary material. Cr y st al Or i ent at i on L iqu id 59 10 n m A m orph ou s 6. Conclusion

I have developed a finite element phase change model to capture thermodynamically consistent phase change phenomena by coupling the phase-dependent specific heat to the latent heat of phase change and initiating the crystalline-to-liquid phase transition heterogeneously.

These phenomena can have significant effects on phase change memory device operation: the latent heat of crystallization results in the actual temperature during crystallization being higher than that predicted by only Joule and thermoelectric heating, and this latent heat increases with increasing temperature. Heterogeneous melting results in faster melt rates in materials with smaller grains because each grain boundary can serve as a melting site. Isolated “quenched-in nuclei” that remain after quenching can serve as growth sites during set, so smaller grains can also reduce set times if the correct electrical pulses are chosen for reset. Experimental data shows reduced write and erase speeds in PCM devices with reducing grain sizes, consistent with these predictions.

I have also developed models for fcc and amorphous Ge2Sb2Te5 parameters relevant to phase change memory device operation: the electrical conductivity, thermal conductivity, Seebeck coefficient, and thermal boundary conductance. These parameters are temperature dependent and can be used to predict material behavior at temperatures where measurements are unavailable. The bandgap used to model the temperature dependence of electrical conductivity in metastable amorphous Ge2Sb2Te5 captures electrical conductivity from 300 K to 858 K. The electric field dependence of electrical conductivity in amorphous Ge2Sb2Te5 results in threshold switching that shows the correct variations as geometries, ambient conditions, and transient pulsing conditions are varied. The bandgap model used for electrical conductivity is also used to calculate the Seebeck coefficient and shows that the Seebeck contribution from holes in metastable a-GST is similar in magnitude and slope to Seebeck measurements on as-deposited amorphous Ge2Sb2Te5.

Simulations using this Seebeck coefficient for amorphous Ge2Sb2Te5 and extrapolating room temperature Seebeck measurements of crystalline Ge2Sb2Te5 predict a Seebeck differential between the two phases at melt, resulting in thermoelectric heating and cooling at crystalline/amorphous boundaries during reset as demonstrated in simulations of a double mushroom phase change memory device. The phonon contribution from thermal conductivity is expected to decrease with increasing temperature, resulting in a thermal conductivity that is largely electronic at high temperatures. This prediction also suggests that the thermal boundary conduction at Ge2Sb2Te5/SiO2 interfaces decreases significantly at high temperatures, predicting better thermal isolation and lower reset currents near melt but also making quenching more difficult.

This finite element phase change framework and physics-based material parameter modeling presented in this work allow for simulations of reset, set, and read in phase change memory devices, threshold switches, and combinations of the two. These simulations can be used to analyze thermal crosstalk during device operation, the evolution of grain maps as devices are cycled, and the dependence of device operation on geometry, ambient conditions, and applied electrical pulses.

7. References

[1] N. Kan’an, H. Silva, and A. Gokirmak, “Phase change pipe for nonvolatile routing,” IEEE

J. Electron Devices Soc., vol. 4, no. 2, pp. 72–75, 2016.

[2] G. H. Campbell, J. T. McKeown, and M. K. Santala, “Time resolved electron microscopy

for in situ experiments,” Appl. Phys. Rev., vol. 1, no. 4, 2014.

[3] M. K. Santala, B. W. Reed, S. Raoux, T. Topuria, T. Lagrange, and G. H. Campbell,

“Irreversible reactions studied with nanosecond transmission electron microscopy movies:

Laser crystallization of phase change materials,” Appl. Phys. Lett., vol. 102, no. 17, 2013.

[4] M. K. Santala et al., “ Nanosecond in situ transmission electron microscope studies of the

reversible Ge 2 Sb 2 Te 5 crystalline ⇔ amorphous phase transformation ,” J. Appl. Phys.,

vol. 111, no. 2, p. 024309, 2012.

[5] Z. Woods and A. Gokirmak, “Modeling of Phase-Change Memory: Nucleation, Growth,

and Amorphization Dynamics During Set and Reset: Part I—Effective Media

Approximation,” IEEE Trans. Electron Devices, vol. 64, no. 11, pp. 4466–4471, Nov. 2017.

[6] Z. Woods, J. Scoggin, A. Cywar, L. Adnane, and A. Gokirmak, “Modeling of Phase-Change

Memory: Nucleation, Growth, and Amorphization Dynamics during Set and Reset: Part II

- Discrete Grains,” IEEE Trans. Electron Devices, vol. 64, no. 11, pp. 4472–4478, Nov.

2017.

[7] G. W. Burr et al., “Observation and modeling of polycrystalline grain formation in

Ge2Sb2Te5,” J. Appl. Phys., vol. 111, no. 10, p. 104308, May 2012.

[8] O. Cueto, V. Sousa, and G. Navarro, “Coupling the Phase-Field Method with an

electrothermal solver to simulate phase change mechanisms in PCRAM cells,” in 2015

International Conference on Simulation of Semiconductor Processes and Devices

(SISPAD), 2015.

[9] J. Scoggin, R. S. Khan, H. Silva, and A. Gokirmak, “Modeling and impacts of the latent

heat of phase change and specific heat for phase change materials,” Appl. Phys. Lett., vol.

112, no. 19, p. 193502, 2018.

[10] K. Cil, “Temperature dependent characterization and crystallization dynamics of

Ge2Sb2Te5 thin films and nanoscale structures,” Ph.D. dissertation. Dept. Elec. and Comp.

Eng., University of Connecticut, Storrs, CT, USA., 2015.

[11] X. Zhang, J. S. Chen, and S. Osher, “A multiple level set method for modeling grain

boundary evolution of polycrystalline materials,” Interact. Multiscale Mech., vol. 1, no. 2,

pp. 191–209, 2008.

[12] C. A. Schneider, W. S. Rasband, and K. W. Eliceiri, “NIH Image to ImageJ: 25 years of

image analysis,” Nat. Methods, vol. 9, no. 7, pp. 671–675, 2012.

[13] W. Kauzmann, “The Nature of the Glassy State and the Behavior of Liquids at Low

Temperatures,” Chem. Rev., vol. 43, no. 2, pp. 219–256, Oct. 1948.

[14] D. Ielmini, A. L. Lacaita, and D. Mantegazza, “Recovery and drift dynamics of resistance

and threshold voltages in phase-change memories,” IEEE Trans. Electron Devices, vol. 54,

no. 2, pp. 308–315, 2007.

[15] R. ~S. Khan, S. Muneer, N. Noor, H. Silva, and A. Gokirmak, “Evidence of Charge

Trapping Giving Rise to Resistance Drift of Metastable Amorphous

Ge$_{2}$Sb$_{2}$Te$_{5}$,” in APS Meeting Abstracts, 2019, p. K33.001.

[16] M. Kaes and M. Salinga, “Impact of defect occupation on conduction in amorphous

Ge2Sb2Te5,” Sci. Rep., vol. 6, no. August, pp. 1–12, 2016.

[17] J. A. Kalb, “Crystallization kinetics in antimony and tellurium alloys used for phase change

recording,” Ph.D. dissertation. I. Physikalisches Institut der Rheinisch-Westfälische

Technische Hochschule (RWTH) Aachen, 52056 Aachen, Germany, 2006.

[18] J. A. Kalb, M. Wuttig, and F. Spaepen, “Calorimetric measurements of structural relaxation

and glass transition temperatures in sputtered films of amorphous Te alloys used for phase

change recording,” J. Mater. Res., vol. 22, no. 03, pp. 748–754, Mar. 2007.

[19] C. Peng, L. Cheng, and M. Mansuripur, “Experimental and theoretical investigations of

laser-induced crystallization and amorphization in phase-change optical recording media

Experimental and theoretical investigations of laser-induced crystallization and

amorphization in phase-change optical r,” J. Appl. Phys., vol. 4183, no. 1997, 1997.

[20] S. Senkader and C. D. Wright, “Models for phase-change of Ge2Sb2Te5 in optical and

electrical memory devices,” J. Appl. Phys., vol. 95, no. 2, pp. 504–511, 2004.

[21] K. Sonoda, A. Sakai, M. Moniwa, K. Ishikawa, O. Tsuchiya, and Y. Inoue, “A compact

model of phase-change memory based on rate equations of crystallization and

amorphization,” IEEE Trans. Electron Devices, vol. 55, no. 7, pp. 1672–1681, 2008.

[22] J. Kalb, F. Spaepen, and M. Wuttig, “Calorimetric measurements of phase transformations

in thin films of amorphous Te alloys used for optical data storage,” J. Appl. Phys., vol. 93,

no. 5, pp. 2389–2393, 2003.

[23] T. Ohta, “Phase-change optical memory promotes the DVD optical disk,” J. Optoelectron.

Adv. Mater., vol. 3, no. 3, pp. 609–626, Sep. 2001.

[24] J. Kalb, “Stresses, viscous flow and crystallization kinetics in thin films of amorphous

chalcogenides used for optical data storage,” der Rheinisch-Westfalischen Technischen

Hochschule Aachen, 2002.

[25] U. Russo, D. Ielmini, A. Redaelli, and A. Lacaita, “Intrinsic data retention in nanoscaled

phase-change memories—Part I: Monte Carlo model for …,” IEEE Trans. Electron

Devices, vol. 53, no. 12, pp. 3032–3039, 2006.

[26] A. Redaelli, D. Ielmini, U. Russo, and A. L. Lacaita, “Intrinsic Data Retention in

Nanoscaled Phase-Change Memories—Part II: Statistical Analysis and Prediction of

Failure Time,” IEEE Trans. Electron Devices, vol. 53, no. 12, pp. 3040–3046, 2006.

[27] U. Russo, D. Ielmini, and A. L. Lacaita, “Analytical modeling of chalcogenide

crystallization for PCM data-retention extrapolation,” IEEE Trans. Electron Devices, vol.

54, no. 10, pp. 2769–2777, 2007.

[28] M. Rizzi et al., “Cell-to-cell and cycle-to-cycle retention statistics in phase-change memory

arrays,” IEEE Trans. Electron Devices, vol. 62, no. 7, pp. 2205–2211, 2015.

[29] M. Tehranipoor, D. Forte, G. S. Rose, and S. Bhunia, Eds., Security Opportunities in Nano

Devices and Emerging Technologies. CRC Press, 2017.

[30] K. Kursawe, A. R. Sadeghi, D. Schellekens, B. Škorić, and P. Tuyls, “Reconfigurable

physical unclonable functions - Enabling technology for tamper-resistant storage,” 2009

IEEE Int. Work. Hardware-Oriented Secur. Trust. HOST 2009, pp. 22–29, 2009.

[31] L. Zhang, Z. H. Kong, and C. H. Chang, “PCKGen: A Phase Change Memory based

cryptographic key generator,” Proc. - IEEE Int. Symp. Circuits Syst., vol. 1, pp. 1444–1447,

2013.

[32] L. Zhang, Z. H. Kong, C. H. Chang, A. Cabrini, and G. Torelli, “Exploiting process

variations and programming sensitivity of phase change memory for reconfigurable

physical unclonable functions,” IEEE Trans. Inf. Forensics Secur., vol. 9, no. 6, pp. 921–

932, 2014.

[33] Q. S. Mei and K. Lu, “Melting and superheating of crystalline solids: From bulk to

nanocrystals,” Prog. Mater. Sci., vol. 52, no. 8, pp. 1175–1262, 2007.

[34] W. J. Wang et al., “Engineering grains of Ge2Sb2Te5for realizing fast-speed, low-power,

and low-drift phase-change memories with further multilevel capabilities,” Tech. Dig. - Int.

Electron Devices Meet. IEDM, pp. 733–736, 2012.

[35] W. Wang et al., “Enabling universal memory by overcoming the contradictory speed and

stability nature of phase-change materials,” Sci. Rep., vol. 2, pp. 1–6, 2012.

[36] F. Spaepen, “A Structural Model for the Solid-Liquid interface in monatomic systems,”

Acta Metall., vol. 23, no. 6, pp. 729–743, 1975.

[37] F. Spaepen and R. B. Meyer, “The surface tension in a structural model for the solid-liquid

interface,” Scr. Metall., vol. 10, no. 1, pp. 37–43, 1976.

[38] F. Spaepen, “The temperature dependence of the crystal-melt interfacial tension: a simple

model,” Mater. Sci. Eng. A, vol. 178, no. 1–2, pp. 15–18, 1994.

[39] Z. Jian, K. Kuribayashi, and W. Jie, “Solid-liquid Interface Energy of Metals at Melting

Point and Undercooled State.,” Mater. Trans., vol. 43, no. 4, pp. 721–726, 2002.

[40] D. T. Wu, L. Gránásy, and F. Spaepen, “N ucleation and the Solid – Liquid Interfacial Free

Energy,” MRS Bull., vol. 29, no. 12, pp. 945–950, 2004.

[41] J. Orava and A. L. Greer, “Classical-nucleation-theory analysis of priming in chalcogenide

phase-change memory,” Acta Mater., vol. 139, pp. 226–235, 2017.

[42] J. Mellenthin, A. Karma, and M. Plapp, “Phase-field crystal study of grain-boundary

premelting,” Phys. Rev. B - Condens. Matter Mater. Phys., vol. 78, no. 18, pp. 1–22, 2008.

[43] A. M. Alsayed, M. F. Islam, J. Zhang, P. J. Collings, and A. G. Yodh, “Premelting at Defects

Within Bulk Colloidal Crystals,” Science (80-. )., vol. 309, no. 5738, pp. 1207–1211, 2005.

[44] D. R. Uhlmann and E. V. Uhlmann, “Crystal growth and melting in glass-forming systems,

a view from 1992,” in Nucleation and crystallization in liquids and glasses, 1993, pp. 109–

140.

[45] M. D. Ediger, P. Harrowell, and L. Yu, “Crystal growth kinetics exhibit a fragility-

dependent decoupling from viscosity,” J. Chem. Phys., vol. 128, no. 3, 2008.

[46] J. Orava, A. L. Greer, B. Gholipour, D. W. Hewak, and C. E. Smith, “Characterization of

supercooled liquid Ge2Sb2Te5and its crystallization by ultrafast-heating calorimetry,” Nat.

Mater., vol. 11, no. 4, pp. 279–283, 2012.

[47] J. Akola and R. O. Jones, “Structural phase transitions on the nanoscale: The crucial pattern

in the phase-change materials Ge2 Sb2 Te5 and GeTe,” Phys. Rev. B - Condens. Matter

Mater. Phys., vol. 76, no. 23, pp. 1–10, 2007.

[48] L. Adnane et al., “High temperature electrical resistivity and Seebeck coefficient of Ge 2 Sb

2 Te 5 thin films,” J. Appl. Phys., vol. 122, no. 12, p. 125104, Sep. 2017.

[49] F. Dirisaglik et al., “High speed, high temperature electrical characterization of phase

change materials: metastable phases, crystallization dynamics, and resistance drift,”

Nanoscale, vol. 7, no. 40, pp. 16625–16630, Oct. 2015.

[50] S. Muneer et al., “Activation energy of metastable amorphous Ge2Sb2Te5 from room

temperature to melt,” AIP Adv., vol. 8, no. 6, p. 065314, 2018.

[51] L. Adnane, F. Dirisaglik, M. Akbulut, and Y. Zhu, “High temperature seebeck coefficient

and electrical resistivity of Ge2Sb2Te5 thin films,” APS Meet., 2012.

[52] D. Krebs, S. Raoux, C. T. Rettner, G. W. Burr, M. Salinga, and M. Wuttig, “Threshold field

of phase change memory materials measured using phase change bridge devices,” Appl.

Phys. Lett., vol. 95, no. 8, pp. 1–4, 2009.

[53] D. Ielmini, “Threshold switching mechanism by high-field energy gain in the hopping

transport of chalcogenide glasses,” Phys. Rev. B - Condens. Matter Mater. Phys., vol. 78,

no. 3, pp. 1–8, Jul. 2008.

[54] K. Cil et al., “In-Situ XRD Measurements and Simulations to Determine Grain Sizes in

GeSbTe at Various Annealing Temperatures,” in Materials Research Society (MRS) 2015

Fall Meeting, 2015, no. KK3.21.

[55] COMSOL 5.3, “COMSOL Multiphysics.” COMSOL Inc., Burlington, MA, 2017.

[56] L. Adnane, N. Williams, H. Silva, and A. Gokirmak, “High temperature setup for

measurements of Seebeck coefficient and electrical resistivity of thin films using inductive

heating,” Rev. Sci. Instrum., vol. 86, no. 10, p. 105119, Oct. 2015.

[57] A. Faraclas et al., “Modeling of Thermoelectric Effects in Phase Change Memory Cells,”

IEEE Trans. Electron Devices, vol. 61, no. 2, pp. 372–378, Feb. 2014.

[58] J. Lee, E. Bozorg-Grayeli, S. Kim, M. Asheghi, H. S. Philip Wong, and K. E. Goodson,

“Phonon and electron transport through Ge2Sb2Te5films and interfaces bounded by

metals,” Appl. Phys. Lett., vol. 102, no. 19, pp. 1–6, 2013.

[59] J. F. Shackelford and W. Alexander, CRC materials science and engineering handbook.

CRC Press, 2001.

[60] S. Muneer et al., “Activation energy of metastable amorphous Ge2Sb2Te5 from room

temperature to melt,” AIP Adv., vol. 8, no. 6, p. 065314, 2018.

[61] N. Del Coro, J. Scoggin, and A. Gokirmak, “Thermoelectric Effects in Phase Change

Memory Cells - A Computational Analysis on Double Mushroom Cells,” in MRS Spring

Meeting, 2019.

[62] J. J. Gong et al., “Investigation of the bipolar effect in the thermoelectric material CaMg 2

Bi 2 using a first-principles study,” Phys. Chem. Chem. Phys., vol. 18, no. 24, pp. 16566–

16574, 2016.

[63] S. A. Baily, D. Emin, and H. Li, “Hall mobility of amorphous Ge2Sb2Te5,” Solid State

Commun., vol. 139, no. 4, pp. 161–164, 2006.

[64] R. F. Pierret, “Semiconductor Device Fundamentals,” New York, 1996.

[65] A. Faraclas, N. Williams, A. Gokirmak, and H. Silva, “Modeling of set and reset operations

of phase-change memory cells,” IEEE Electron Device Lett., vol. 32, no. 12, pp. 1737–

1739, Dec. 2011.

[66] J. P. J. P. Reifenberg et al., “Thermal Boundary Resistance Measurements for Phase-

Change Memory Devices,” IEEE Electron Device Lett., vol. 31, no. 1, pp. 56–58, 2010.

[67] J. Lee, E. Bozorg-Grayeli, S. Kim, M. Asheghi, H.-S. Philip Wong, and K. E. Goodson,

“Phonon and electron transport through Ge 2 Sb 2 Te 5 films and interfaces bounded by

metals,” Appl. Phys. Lett., vol. 102, no. 19, p. 191911, May 2013.

[68] J. Lee et al., “Decoupled thermal resistances of phase change material and their impact on

PCM devices,” in 2010 12th IEEE Intersociety Conference on Thermal and

Thermomechanical Phenomena in Electronic Systems, ITherm 2010, 2010, pp. 1–6.

[69] D. Kau et al., “A stackable cross point phase change memory,” in Technical Digest -

International Electron Devices Meeting, IEDM, 2009, pp. 1–4.

[70] H.-S. P. Wong et al., “Phase ChangeMemory,” Proc. IEEE, vol. 98, no. 12, pp. 2201–2227,

2010.

[71] A. Chen, Memory Selector Devices and Crossbar Array Design: A Modeling Based

Assessment, vol. 16, no. 4. Springer US, 2017, pp. 1186–1200.

[72] G. W. Burr et al., “Access devices for 3D crosspoint memory,” J. Vac. Sci. Technol. B,

Nanotechnol. Microelectron. Mater. Process. Meas. Phenom., vol. 32, no. 4, p. 040802,

2014.

[73] S. R. Ovshinsky, “Reversible electrical switching phenomena in disordered structures,”

Phys. Rev. Lett., vol. 21, no. 20, pp. 1450–1453, Nov. 1968.

[74] T. Matsunaga et al., “From local structure to nanosecond recrystallization dynamics in

AgInSbTe phase-change materials,” Nat. Mater., vol. 10, no. 2, pp. 129–134, 2011.

[75] M. Anbarasu, M. Wimmer, G. Bruns, M. Salinga, and M. Wuttig, “Nanosecond threshold

switching of GeTe 6 cells and their potential as selector devices,” Appl. Phys. Lett., vol.

100, no. 14, 2012.

[76] M. J. Lee et al., “Highly-scalable threshold switching select device based on chaclogenide

glasses for 3D nanoscaled memory arrays,” Tech. Dig. - Int. Electron Devices Meet. IEDM,

no. December, pp. 10–13, 2012.

[77] J. Choe, “Intel 3D XPoint Memory Die Removed from Intel OptaneTM PCM (Phase Change

Memory),” Tech Insights, 2017. [Online]. Available: https://www.techinsights.com/about-

techinsights/overview/blog/intel-3D-xpoint-memory-die-removed-from-intel-optane-

pcm/. [Accessed: 10-Dec-2018].

[78] M. Le Gallo, A. Athmanathan, D. Krebs, and A. Sebastian, “Evidence for thermally assisted

threshold switching behavior in nanoscale phase-change memory cells,” J. Appl. Phys., vol.

119, no. 2, 2016.

[79] R. W. Pryor and H. K. Henisch, “Nature of the on-state in chalcogenide glass threshold

switches,” J. Non. Cryst. Solids, vol. 7, no. 2, pp. 181–191, 1972.

[80] K. E. Petersen, “On state of amorphous threshold switches,” J. Appl. Phys., vol. 47, no.

1976, p. 256, 1976.

[81] M. Nardone, V. G. Karpov, D. C. S. S. Jackson, and I. V. Karpov, “A unified model of

nucleation switching,” Appl. Phys. Lett., vol. 94, no. 10, pp. 10–12, Mar. 2009.

[82] W. Czubatyj and S. J. Hudgens, “Invited paper: Thin-film Ovonic threshold switch: Its

operation and application in modern integrated circuits,” Electron. Mater. Lett., vol. 8, no.

2, pp. 157–167, 2012.

[83] M. Wimmer and M. Salinga, “The gradual nature of threshold switching,” New J. Phys.,

vol. 16, 2014.

[84] A. Athmanathan, D. Krebs, A. Sebastian, M. Le Gallo, H. Pozidis, and E. Eleftheriou, “A

finite-element thermoelectric model for phase-change memory devices,” Int. Conf. Simul.

Semicond. Process. Devices, SISPAD, vol. 2015–Octob, pp. 289–292, 2015.

[85] M. Le Gallo, M. Kaes, A. Sebastian, and D. Krebs, “Subthreshold electrical transport in

amorphous phase-change materials,” New J. Phys., vol. 17, no. 9, 2015.

[86] M. Nardone, M. Simon, I. V. Karpov, and V. G. Karpov, “Electrical conduction in

chalcogenide glasses of phase change memory,” J. Appl. Phys., vol. 112, no. 7, 2012.

[87] I. V. Karpov, M. Mitra, D. Kau, G. Spadini, Y. A. Kryukov, and V. G. Karpov, “Evidence

of field induced nucleation in phase change memory,” Appl. Phys. Lett., vol. 92, no. 17, pp.

4–6, 2008.

[88] J. Scoggin, Z. Woods, H. Silva, and A. Gokirmak, “Modeling heterogeneous melting in

phase change memory devices,” Appl. Phys. Lett., vol. 114, no. 4, pp. 1–6, Oct. 2019.

[89] R. Endo et al., “Electric resistivity measurements of Sb2Te3 and Ge2Sb2Te5 melts using

four-terminal method,” Jpn. J. Appl. Phys., vol. 49, no. 6, p. 5802, 2010.

[90] T. Kato and K. Tanaka, “Electronic Properties of Amorphous and Crystalline Ge 2 Sb 2 Te

5 Films,” Jpn. J. Appl. Phys., vol. 44, no. 10, pp. 7340–7344, 2005.

[91] K. Cil, F. Dirisaglik, and L. Adnane, “Electrical resistivity of liquid based on thin-film and

71 nanoscale device measurements,” on Electron Devices, 2013.