Copyright

by

Urmimala Roy

2015

The Dissertation Committee for Urmimala Roycerties that this is the approved version of the following dissertation:

Towards high-density low-power spin-transfer-torque

random access memory

Committee:

Sanjay K. Banerjee, Supervisor

Leonard F. Register, Co-Supervisor

Emanuel Tutuc

S. V. Sreenivasan

Maxim Tsoi

Towards high-density low-power spin-transfer-torque

random access memory

by

Urmimala Roy, B.E.; M.Tech.

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulllment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

August 2015

To my family

Acknowledgments

I would like to thank my academic advisor Prof. Sanjay K. Banerjee for

his guidance and support. It has been an exceptional opportunity to be able

to work under his supervision for the last ve years. I sincerely thank him for

accepting me in his group. He has always given me complete freedom to choose

my research directions, shaping my identity as an independent researcher. I

take this opportunity to thank my co-supervisor Prof. Leonard F. Register for

his insightful advice every time I stumbled upon a problem. Not only did I

learn from his enormous knowledge of semiconductor physics, discussions with

him helped me learn how a research problem should be handled. It has been a

privilege to work with Prof. Register. I will remain indebted to Prof. Banerjee

and Prof. Register for their endless help and support.

I thank Prof. Maxim Tsoi for his guidance in the spin-transfer-torque

experiments, and my dissertation research in general. I also thank Prof. Tsoi

for serving on my PhD committee and his valuable suggestions on this disser-

tation. I thank Prof. Emanuel Tutuc and Prof. S. V. Sreenivasan for being in

my PhD commitee and for their valuable advice.

During my ve years at UT Austin I have had the privilege of working

with a number of extremely talented students. I take this opportunity to thank

Fahmida Ferdousi and Heidi Seinige for the excellent collaboration in my rst

v

year at UT. Fahmida introduced me to most of the tools I have learned to use in

the clean room. In the second half of my PhD, when I started working more on

the modeling and simulation, I have been fortunate to collaborate with Tanmoy

Pramanik and Rik Dey, on the multi-bit spin-transfer-torque random access

memory (STTRAM) and the topological insulator based STTRAM projects.

Working with Tanmoy and Rik has been a wonderful experience to say the

least. Thanks to them for being so patient with me, when I would stop by one

of their desks to talk about a long calculation I was in the middle of, and they

would help me as much as they could. I can not probably thank Tanmoy and

Rik enough.

I thank other past and present graduate students and post-docs at

the Microelectronics Research Center, Nima Asoudegi, Dax Crum, William

Hsu, Mustafa Jamil, Seyoung Kim, Emmanuel Onyegam, Yujia Zhai, Dhar-

mendar Reddy, Michael Ramon, Jason Mantey, Chris Corbet, Xuehao Mou,

Jiwon Chang, Hema Chandra Prakash Movva, Tanuj Trivedi, Amritesh Rai,

Amithraj Valsaraj, Sayan Saha, Donghyi Koh, Sk. Fahad Chowdhury, Ky-

ounghwan Kim, Milo Holt, Maruthi Yogesh, Atresh Sanne, Sushant Sonde,

Domingo Ferrer, Samaresh Guchhait, Bahniman Ghosh, Sarmita Majumdar,

Priyamvada Jadaun and Nitin Prasad, for their warm friendship and incred-

ible help. Special thanks to Anupam Roy for advice on this dissertation and

the discussions on Bengali literature and Indian politics on our way back from

PRC.

I express my sincere gratitude to Jean Toll, Christine Wood and Melanie

vi

Gullick for helping so much in every possible way. I thank Ricardo Garcia and

Johny Johnson for all their help during my brief stint in the clean room.

I had the opportunity to work as an intern in the Process Modeling and

Applications Group at Intel Corp. during the summer of 2014. I would like to

thank David L. Kencke for his guidance and mentoring. I would also like to

thank Cory E. Weber, Charles Kuo and Angik Sarkar, for their help making

my Intel experience such a wonderful one.

I would like to thank my friends at Austin, Arindam Sanyal, Avik

Ray, Somsubhra Barik, Ayan Acharya, Debarati Kundu, Abhik Ranjan Bhat-

tacharya and Rudra Narayan Chatterjee, Proma Bhattacharya and those out-

side Austin, Srimoyee Sen and Ankur Guha Roy for their company. Last ve

years would not be such a nice experience if they were not here. I take this

opportunity to thank Avhishek Chatterjee for being extremely patient and

considerate and an amazing friend. It has been a pleasure to know him. For

the last one and half years, I have been blessed with another family, besides

my own. I thank my parents-in-law, my sister-in-law Sudipa, her husband

Dimitri and my little niece Tinni for being in my life.

My heartfelt thanks to my sisterDidibhai and her husbandDebmalya da

for their love, tireless support and encouragement. They both being successful

researchers, their experience and advice helped me take decisions at every step

of my academic career so far. My husband Subhendu, who has been a friend

and a classmate ever since I joined IIT Bombay and a fellow PhD student at

UT Austin for the last four years, has always been a pillar of support. I thank

vii

him for being there for me always.

Lastly, I thank my parents Maa and Bapi for having nurtured in me a

deep sense of the value of education and for their love and trust.

viii

Towards high-density low-power spin-transfer-torque

random access memory

Publication No.

Urmimala Roy, Ph.D.

The University of Texas at Austin, 2015

Supervisors: Sanjay K. Banerjee and Leonard F. Register

In this work, we investigate the prospects for spin-transfer-torque ran-

dom access memory (STTRAM) as the new generation low-power high-density

non-volatile memory. Possible means to lower the switching current and in-

crease the packing density of STTRAM are proposed. In an STTRAM cell, the

logical value of the memory bit is stored as orientation of magnetic moment

in its ferromagnetic free layer. The bit typically consists of two thin lm

ferromagnets (FM) separated by an insulating tunnel barrier as in a magnetic

tunnel junction (MTJ) structure. One of the two FM layers has xed magne-

tization direction, while the other layer is free to be switched. We rst study

STT-assisted switching in spin valve structures with in-plane, perpendicular,

and canted magnetizations in free and (or) reference layers using point con-

tact measurements to explore the use of non-collinear magnetizations in free

and xed FM to reduce both switching current and time. Next, we consider

ix

the possibility of storing two memory bits within a single MTJ with a cross-

shaped free layer that could still be addressed by one selection transistor. We

provide a detailed discussion of the switching dynamics and associated regions

of reliable switching currents, in addition to illustrating the eects of varying

device geometry on the latter. Moving on from the standard MTJ structure,

we then consider the possible use of topological insulators, as opposed to the

xed FM, as the spin-polarizer layer. It has been established that spin and

momentum are locked helically in the surface states of a three-dimensional

topological insulator (TI). Suggestions of possible use of the TI spin-polarized

surface states in spintronic devices to induce reversal of a magnet have been

made using theoretical and experimental studies. Here, we simulate mag-

netization reversal of a metallic nanomagnet by an underlying TI. The TI,

thanks to the spin-momentum helical locking of the surface states, causes a

spin-polarized current injection to the FM above it. We study the eciency

of the spin injection as a function of varying transparency of the TI-FM inter-

face. The transport in the TI and the FM is assumed to be diusive at room

temperature for the assumed resistance values. Finally, we take into account

random thermal uctuations leading to write error rate (WER) in STTRAM

write operation and use Fokker-Planck method and stochastic Landau-Lifshitz-

Gilbert-Slonczewski equation to model WER in STTRAM. We conclude with

possible future research directions.

x

Table of Contents

Acknowledgments v

Abstract ix

List of Tables xiv

List of Figures xv

Chapter 1. Introduction 1

1.1 The memory hierarchy . . . . . . . . . . . . . . . . . . . . . . 21.2 STTRAM as the potential universal memory . . . . . . . . . . 51.3 Basic principles of STTRAM operation . . . . . . . . . . . . . 8

1.3.1 Write mechanism: Spin-transfer-torque . . . . . . . . . . 81.3.2 Read mechanism: Tunnel magnetoresistance . . . . . . . 91.3.3 Current driven switching modes . . . . . . . . . . . . . 11

1.3.3.1 Thermal regime . . . . . . . . . . . . . . . . . . 111.3.3.2 Precessional regime . . . . . . . . . . . . . . . . 111.3.3.3 Dynamic regime . . . . . . . . . . . . . . . . . . 11

1.4 Recent progress in STTRAM . . . . . . . . . . . . . . . . . . . 121.5 Current challenges for STTRAM . . . . . . . . . . . . . . . . . 131.6 Overview of this dissertation . . . . . . . . . . . . . . . . . . . 17

Chapter 2. Perpendicular and canted magnetic anisotropy forSTTRAM 19

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Fabrication and characterization of in-plane, canted, and PMA

spin valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Spin valve fabrication . . . . . . . . . . . . . . . . . . . 222.2.2 Characterization by point-contact measurements . . . . 23

2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

xi

Chapter 3. Spin-transfer-torque switching of a ferromagneticcross towards multi-bit STTRAM 27

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Equilibrium magnetizations of the cross . . . . . . . . . 293.2.2 Magnetization dynamics under STT . . . . . . . . . . . 32

3.2.2.1 State 1 to State 2 switching . . . . . . . . . 353.2.2.2 State 1 to State 3 switching . . . . . . . . . 353.2.2.3 State 1 to State 4 switching . . . . . . . . . 37

3.2.3 Thermal stability . . . . . . . . . . . . . . . . . . . . . . 403.3 Packing density considerations . . . . . . . . . . . . . . . . . . 423.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 4. Spin-transfer-torque switching of a metallic nano-magnet due to an underlying topological insulator 45

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Transport model . . . . . . . . . . . . . . . . . . . . . . 484.2.2 Self-consistent transport and magnetization dynamics cal-

culation . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 5. Write error rate in STTRAM 72

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.1.1 Challenges for WER simulation using micromagnetic eects 74

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.1 Fokker-Planck calculation of WER within the macrospin

approximation . . . . . . . . . . . . . . . . . . . . . . . 755.2.2 WER simulation including micromagnetic eects . . . . 76

5.2.2.1 Method . . . . . . . . . . . . . . . . . . . . . . 765.2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

xii

Chapter 6. Conclusion 83

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . 86

Bibliography 88

Vita 101

xiii

List of Tables

2.1 Slope of STT for samples I-III. . . . . . . . . . . . . . . . . . . 25

4.1 Parameters assumed for transport in the TI-FM bilayer . . . . 64

xiv

List of Figures

1.1 Memory hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Comparison of STTRAMwith other existing and emerging mem-

ory technologies, created following Ref. [1]. . . . . . . . . . . . 61.3 STTRAM cell with 1 MTJ and 1 selection transistor. . . . . . 71.4 STT in a FM-nonmagnet-FM trilayer stack . . . . . . . . . . . 81.5 Magnetoresistance (MR) in a FM-NM-FM stack. . . . . . . . 10

2.1 (a) is a typical magnetoresistance plot of sample II with a mag-netic eld applied perpendicular to the sample plane and a dcbias current of -1.3 mA. (b-d) are STT slope from point contactmeasurements for sample I-III respectively, with open symbolsfor up-sweep and closed symbols for down-sweep. This gurehas been reproduced from Ref. [2] with permissions. . . . . . . 23

3.1 Cross-shaped MTJ stack . . . . . . . . . . . . . . . . . . . . . 283.2 Equilibrium magnetizations; (a) State 1; (b) State 2; (c)

State 3; (d) State 4; l1/w is referred to as AR of short arm,l2/w is referred to as AR of long arm. This gure has beenreproduced from Ref. [3] with permissions. . . . . . . . . . . . 31

3.3 (a) 1 → 2 switching with J = 5 × 107A/cm2 (b) Pre-switchingoscillations with dierent J values; inset: linear t of 0 crossingtime for dierent values of short branch AR (l1/w) and longbranch AR (l2/w). l1, l2 and w are dened in Figure 3.2. Thisgure has been reproduced from Ref. [3] with permissions. . . 36

3.4 1 → 3 switching with J = 3.8×108A/cm2. This gure has beenreproduced from Ref. [3] with permissions. . . . . . . . . . . . 38

3.5 1 → 4 switching with two current pulses of opposite polaritiesand dierent magnitudes. This gure has been reproduced fromRef. [3] with permissions. . . . . . . . . . . . . . . . . . . . . . 39

3.6 Switching regions to State 2 and State 3 starting from State1 with increasing amplitude of current pulse in steps of J =5 × 106 A/cm2 for dierent designs of cross. This gure hasbeen reproduced from Ref. [3] with permissions. . . . . . . . 40

xv

4.1 (a) Schematic of the TI-FM bilayer, x is the transport, z is theout-of-plane direction. (b) The lled circle denotes the shiftedFermi circle in presence of a current supported by the surfacestates. The arrows denote the spin vectors of the surface statesbecause of spin-momentum helical locking. This gure has beenreproduced from Ref. [4] with permissions. . . . . . . . . . . . 54

4.2 Transport in the bilayer for xed FM magnetization along +xand an applied voltage of 0.2 V for (a) small coupling (normal-ized interlayer conductivity = 1/100), (b) moderate coupling(normalized interlayer conductivity = 1/9), and (c) Large cou-pling (normalized interlayer conductivity = 1). (d) Resistanceof the bilayer vs. interlayer conductance. Inset in (d): TI re-sistivity as a function of the normalized interlayer conductance.This gure has been reproduced from Ref. [4] with permissions. 61

4.3 Variation in device metrics with varying interlayer coupling,with the FM magnetized along the +x-direction and an appliedvoltage of 0.2 V. (a) spin torque ratio (spin Hall angle) obtainedwith respect to nominal TI conductivity. (b) spin torque ratiowith respect to TI conductivity self-consistently decreased bythe process of exerting STT. (c) Spin injection eciency, (d)Spin-transfer-torque (STT) via injection across the interface,and eld-like torque (FLT) via exchange coupling between thespin-polarized electrons in the TI and the nanomagnet. Thisgure has been reproduced from Ref. [4] with permissions. . . 69

4.4 (a) Switching of the FM from +y to -y for an applied voltageof -0.2 V, so that current is in -x-direction. (b) Switching ofthe FM from -y to +y for an applied voltage of 0.2 V, so thatcurrent is in +x-direction. (c) Switching time ts (dened in thetext) and write energy per write operation vs. applied voltageacross the bilayer, for the normalized interlayer conductivity of1. The TI fails to switch the FM below a critical voltage as canbe seen from the plot of inverse of switching time with voltagein the inset of (c). (d) Switching time ts (dened in the text)and write energy per write operation vs. normalized interlayerconductivity, for an applied voltage of 0.2 V. Similarly, the TIfails to switch the FM below a critical interlayer conductivity,as can be seen from the plot of inverse of switching time withcoupling in the inset of (d). This gure has been reproducedfrom Ref. [4] with permissions. . . . . . . . . . . . . . . . . . . 70

4.5 a) bilayer resistance, b) Spin injection eciency and c) Switch-ing time and write energy per write operation for articially in-creased FM in-plane resistivity and varying interlayer couplingd) Write energy as a function of the FM resistivity divided bythe FM thickness. This gure has been reproduced from Ref. [4]with permissions. . . . . . . . . . . . . . . . . . . . . . . . . . 71

xvi

5.1 Write error from micromagnetic simulations. 20 trials are shown,each simulated with a random thermal seed for the random ther-mal magnetic eld. A current pulse is applied at 5 ns, startingfrom uniform magnetization in +z at 0 ns. The current pulse isturned of at 11 ns, the trials which are above mz=0 at the timethe current pulse is turned o, go back to the initial magneti-zation of mz=1 and cause an error. . . . . . . . . . . . . . . . 73

5.2 WER from Fokker-Planck method, for diameter D=40 nm andD=80 nm, for normalized current i = I/Ic0 of 2 and 6, as afunction of normalized time t/t0. This gure has been reusedfrom Ref. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3 WER from Fokker-Planck method, for diameter D=40 nm andD=80 nm, for normalized current i = I/Ic0 of 2 and 6, as afunction of physical time. This gure has been reused fromRef. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4 WER from macrospin (Fokker-Planck) calculations comparedwith micromagnetic calculations for D=40 nm. This gure hasbeen reused from Ref. [5]. . . . . . . . . . . . . . . . . . . . . 80

5.5 WER from macrospin (Fokker-Planck) calculations comparedwith micromagnetic calculations for D=80 nm. This gure hasbeen reused from Ref. [5]. . . . . . . . . . . . . . . . . . . . . 81

5.6 Scaling of WER from micromagnetic simulations for D=20 nmto 80 nm as a function of normalized time t/t0 for normalizedcurrents i=2 and i=6. This gure has been reused from Ref. [5]. 82

xvii

Chapter 1

Introduction

Spintronics, the commonly used name for spin-transport-electronics,

employs electron spin in solid state devices and has been an area of great

fundamental and practical interest, owing to exciting physics and the promise

of applications in technology. Each electron carries a unit of charge (-e) as

well as a unit of spin angular momentum h2, or equivalently, a xed magnetic

moment. The spin angular momentum can only have two values along any

direction, ± h2, and is normally labeled as up or down spin. In normal

metals, e.g. Cu, electrons of both the spins are randomly oriented and the

density of states of the two spins are the same at the Fermi level. In certain

metals such as Fe, Co or Ni, and many other alloys of these, due to huge

internal exchange interactions, the atomic magnetic moments align parallel to

each other. The densities of states for up and down spin electrons at the Fermi

level are dierent and the two spin bands are shifted in energy. Fe, Co or Ni are

so-called ferromagnetic metals. Many properties in these materials, e.g. the

electrical conductivity, are thus spin-dependent as a result of the spin-splitting

in the energy bands.

Ferromagnetism in Fe, Co, Ni and their alloys has been known for a

1

long time. The recognition that the interaction of the electron spin angular

momentum with the solid can be manipulated for device applications, just

as well as that of its charge, only came recently. The discovery of the gi-

ant magnetoresistance by Albert Fert and Peter Grünberg [6, 7] (2007 Nobel

prize) marks an important milestone in the eld of spintronics and soon led

to several practical applications in magnetic data storage and read heads.

With advancements in nanoscale fabrication techniques the discovery of the

tunnel magnetoresistance (TMR) in a magnetic tunnel junction (MTJ) fol-

lowed [812] and found widespread applications in magnetic random access

memory (MRAM). Magnetization in the patterned ferromagnetic thin lms in

an MTJ traditionally has been controlled using magnetic elds. This disser-

tation is, however, focused on using a spin-polarized current in magnetization

switching for potential applications in random access memories.

1.1 The memory hierarchy

There are dierent levels of memory in a computer depending on the

proximity to the central processing unit (CPU). For example, CPU registers

store the data most frequently used by the CPU. The CPU registers can be

accessed very quickly (access time ∼ns). The cache memories, which also oer

fast access, are in the next level. Next, the main memory has larger size,

but typically would take longer to communicate with the CPU. The largest

information storage happens in a solid state drive (SSD) or in a hard disk drive

(HDD). The SSD (ash memory) oers faster access than the HDD. Access

2

CPU

Registers

Local secondary storage

(SSD, HDD)

Cache

(SRAM)

Sp

eed

an

d c

ost

Den

sity

Main memory

(DRAM)

Remote secondary storage

(tapes)

Figure 1.1: Memory hierarchy

time to HDDs are very large (∼ms). The division of memory usage according

to information storage density and speed of access from the CPU is referred

to as the memory hierarchy in computer architecture (shown schematically

in Figure 1.1).

Accordingly, dierent memory technologies are used in dierent levels

of the memory hierarchy depending on their access speeds and bit packing den-

sities. For example, the most commonly used memory technology for the cache

memory is static random access memory (SRAM), which is faster than the dy-

namic random access memory (DRAM) used in the main memory. However,

an SRAM cell occupies too large an area (∼100F2, where F is the minimum

3

feature size of the technology node). DRAM, on the other hand, occupies

smaller area per cell (∼6F2) and also oers fast access (access time ∼10-100

ns). With semiconductor technology approaching the so-called end of the

road-map it is getting harder to scale the commonly used semiconductor de-

vice based memories to smaller technology nodes. For example, increasing

leakage current of the SRAM cell has become a concern. For the DRAM

technology, as the cell is typically realized using a transistor and a capacitor,

realizing a certain capacitance in a certain maximum allowed footprint has

become challenging. It has been predicted that DRAM will be dicult to be

scaled further than the 20 nm node [13,14].

Moreover, the traditionally used magnetic-eld-assisted random access

memory (MRAM) is not scalable to smaller bits. In the traditional MRAM, a

smaller bit would require a larger magnetic eld for information to be recorded

in it, in turn requiring a larger write current [15]. This would increase the

power consumption by a large extent. Moreover, the commercially available

MRAM is slower than SRAM and less dense than DRAM [16].

In this scenario, the memory industry, therefore, is looking for alter-

natives, and preferably one that can be used in all the levels of the memory

hierarchy. With the theoretical predictions [17,18] and experimental observa-

tion [19] of the so-called spin-transfer-torque (STT), it was readily recognized

that using a spin-polarized current (SPC) to do the write operation instead of

a magnetic eld in a magnetic bit would greatly enhance the scalability of non-

volatile information storage. The counterpart of MRAM, in which the write

4

mechanism would be STT, is referred to as the spin-transfer-torque random

access memory (STTRAM).

1.2 STTRAM as the potential universal memory

In addition to the scalability, the access speed of STTRAM is expected

to be really high (access time ∼2-30 ns [13]), as has been demonstrated already.

Moreover, STTRAM promises a cell size of 6F2 or, possibly even smaller than

that [15]. With only few additional masks, STTRAM fabrication is compat-

ible with CMOS fabrication steps [20]. Hence, the widespread hope is that

STTRAM might be a candidate for the so-called universal memory, becom-

ing a potential replacement for memories in more than one of the levels in the

memory hierarchy, if not all of them [21].

A comparison of STTRAM with other emerging and existing mem-

ory technologies such as SRAM, DRAM, NAND and NOR ash, phase change

memory (PCM), magnetoelectric RAM (MeRAM) and ferroelectric RAM (FeRAM)

can be found in work by Yoda et al. [16] and has been shown schematically in

Figure 1.2, following previous work [1].

In a single STTRAM cell, a bit of information is stored as the mag-

netization direction of a patterned nanoscale thin lm ferromagnet (FM). In-

formation stored as the magnetization direction lends an inherent non-volatile

nature to this memory technology. Although information is stored as the

magnetization direction of a single FM known as the free FM, both read and

write mechanisms require a FM-nonmagnet-FM trilayer stack (schematically

5

1 ns 1 µs 1 ms

1 pJ

1 nJ

1 fJ

Write time

Wri

te e

ne

rgy

MeRAM

STTRAM

RRAM FeRAM

PCRAM

NOR NAND

Figure 1.2: Comparison of STTRAM with other existing and emerging mem-ory technologies, created following Ref. [1].

shown in Figure 1.3). The other FM in the trilayer stack is known as the

xed or the reference layer. The nonmagnetic layer is generally a tunnel

barrier, thus allowing reading of the bit using the TMR. The trilayer stack is

then called a magnetic tunnel junction (MTJ). Also, in an STTRAM cell, the

MTJ is accessed through a selection transistor.

There are two types of STTRAM bits currently being considered ac-

cording to the orientation of the free layer with respect to the plane of the

thin lm. If the easy axis is in the plane of the thin lm, the bit is referred

to as an `in-plane' bit. For an in-plane bit, the easy axis is engineered using

6

Bit Line

Source line

Word line

free layer

Tunnel barrier

fixed layer

(polarizer)

Figure 1.3: STTRAM cell with 1 MTJ and 1 selection transistor.

shape anisotropy of the magnet, so the free layer magnet is often a thin lm

ellipse. This way, the easy axis is along the longer of the two axes, i.e., the

major axis of the ellipse. The other type of MTJ is where the easy axis of

the free layer magnet is perpendicular to the plane of the thin lm. The easy

axis can be in a perpendicular direction owing to the so-called perpendicular

magnetic anisotropy (PMA) of the magnetic material. As will be discussed in

the subsequent sections in more detail, perpendicular STTRAM bits possess

7

inherently better performance compared to the in-plane bits owing to the dif-

ferent energy landscape. A comparative study using simulations can be found

in a previous study [22]. The international technology roadmap for semicon-

ductors (ITRS) predicts use of perpendicular materials for STTRAM from

the year 2017 and onwards [23]. In this chapter, rst the write mechanism of

STTRAM memory bit, which is the phenomenon of STT, will be described

along with the read mechanism of tunnel magnetoresistance (TMR). Then,

we briey discuss the recent experimental demonstrations of STTRAM. The

challenges for STTRAM to become a main stream memory technology will be

discussed along with some of the solutions currently being considered.

1.3 Basic principles of STTRAM operation

1.3.1 Write mechanism: Spin-transfer-torque

Fixed FM Free FM

Nonmagnetic

spacer

STT

Figure 1.4: STT in a FM-nonmagnet-FM trilayer stack

8

When a SPC traverses a ferromagnet (FM), the transverse component

of the spin angular momentum of the SPC is absorbed by the FM. STT arises

because of rapid dephasing, usually within a few atoms from the non-magnetic

layer into the FM, if the incident spin is not aligned with the magnetization

direction of the FM. As a result, the FM experiences a torque, which is known

as the spin-transfer-torque (STT). When the STT is large enough to overcome

the damping inherent in the FM, the STT might lead to magnetization reversal

of the FM [17, 18, 24]. The SPC is usually generated using spin-ltering by

another ferromagnet, the xed FM in the MTJ stack (shown schematically

in Figure 1.4).

1.3.2 Read mechanism: Tunnel magnetoresistance

Apart from the all electrical write mechanism, STTRAM takes advan-

tage of the MTJ stack for an electrical read out of the information written in

the free FM. The read-out is usually done by measuring the resistance of the

stack. The percentage change in resistance when the magnetizations in the

free and the xed FM are parallel (P) or antiparallel (AP) to each other is

referred to as magnetoresistance (MR) ratio (≡ (RAP − RP)/RP). With the

non-magnetic spacer (NM) between the two FM electrodes of the MTJ being

a thin dielectric tunnel barrier, the magnetoresistance arises because of spin-

dependent tunneling between the electrodes. The TMR eect was predicted

rst by Julliere [10] and experimentally observed independently by Moodera

et al. [9] and Miyazaki et al. [12]. In a ferromagnet, the density of states of

9

the two spin bands (up and down) are separated in energy. As a result, one

of the two spins can travel across the FM-NM-FM stack, only if the two FM

electrodes are parallely magnetized. On the other hand, if the the two FM

electrode magnetizations are antiparallel, neither of the two spins can make to

the other electrode from the NM. The simplest interpreation of phenomenon

of MR has been shown schematically in Figure 1.5. An MR ratio of more

than 100% has been predicted theoretically [25] and also has been experimen-

tally demonstrated [11] for single crystal Fe/MgO/Fe MTJs. The TMR of the

MTJ stack decides the read margin as well as the read speed of the STTRAM

bit [20].

NM FM FM

Parallel Antiparallel

FM FM NM

EF EF EF

N(E) N(E)

Figure 1.5: Magnetoresistance (MR) in a FM-NM-FM stack.

10

1.3.3 Current driven switching modes

Depending on the length of the write pulse there can be three distinct

modes of STTRAM switching [26]:

1.3.3.1 Thermal regime

If the current amplitude is small compared to the critical switching

current Ic0 and a long pulse is being used, the switching happens mainly by

thermal activation over the energy barrier. This regime is referred to as the

thermal regime. In the thermal regime, the initial (prior to the application

of the write pulse) probability distribution of the free layer magnetization is

irrelevant, whereas the temperature, and hence thermal uctuations during

the reversal is important [26].

1.3.3.2 Precessional regime

In the precessional regime, the pulse length is short and the pulse am-

plitude is several times higher than Ic0. In the precessional regime, the thermal

uctuation during the reversal does not aect the switching distribution, only

the initial thermal distribution of the magnetization is important [26].

1.3.3.3 Dynamic regime

The dynamic regime is the region in the pulse length-pulse amplitude

plane where neither of the above two cases are true and both initial thermal

distribution and the thermal uctuations during the reversal might play im-

11

portant role. The dynamic regime is important from practical point of view

since STTRAM is most likely to be operated in this regime for applications of

interest.

The stochastic nature of the switching will be discussed in more detail

in Chapter 5.

1.4 Recent progress in STTRAM

Considerable amount of eort has been put in the last decade to im-

prove the performance of STTRAM and meet the specications in the ITRS.

A comparison of performance achieved by dierent groups can be found in

the review by Yoda et al. [16]. Historically, the in-plane magnetized MTJs

matured technologically before the perpendicular MTJs. However the critical

switching current was as high as hundreds of micro-amperes, thereby ending

up requiring a very large selection transistor and reducing the packing den-

sity. A major advancement was made by Toshiba in proposing perpendicular

bits [27]. Even before the demonstration of STT switching in perpendicularly

magnetized MTJ by Toshiba [27], it was being noted by other groups that using

PMA, instead of shape anisotropy and in-plane magnetization in the free layer

might oer better switching current for the same anisotropy eld and studies

on spin-valves were carried out [28]. However, a spin-valve(with a metal as

the non-magnetic spacer instead of a dielectric tunnel barrier) suers from a

small magnetoresistance ratio(∼1%), which is small compared to the require-

ment of electrical read out. An ecient read-out needs TMR ∼150% [16, 23].

12

In [27], a buer layer of CoFeB was used to lessen mismatch between the MgO

tunnel barrier and the perpendicularly magnetized TbCoFe, though a TMR

ratio of only 15% was seen. Further improvements in the perpendicular MTJ

technology followed and led to a demonstration of as high as 200% TMR [29].

The other concerns towards making commercially viable STTRAM is

the write speed and the write energy, though both are related to the write

current. Writing by 1 ns long current pulse and write energy as small as 0.05

pJ/bit has been demonstrated by Toshiba [30]. Though perpendicular bits

are considered to be the solution for the problem of high switching current,

problems with materials with PMA include [31]:

• Crystal match with the MgO tunnel barrier for high spin injection e-

ciency and high TMR.

• The material has to have low enough damping constant(<0.01).

Apart from high TMR, small write current, write energy and write

time, a suciently high endurance of the cell has to be ensured. Endurance

of the bit refers to the maximum number of write cycles a cell can withstand

before suering from a damage [31].

1.5 Current challenges for STTRAM

Inspite of having a potential for scalability to smaller technology nodes

unlike its magnetic-eld-assisted counterpart, STTRAM faces a number of se-

rious challenges before it can be made commercially viable, as will be discussed

13

in more detail in the following paragraphs. The main obstacle is due to the high

write energy, stemming from the high write current density required for reli-

able switching of the bit [1,13]. Switching energy from recent demonstrations

are high, ∼100 fJ (Ref. [1] and references therein). The high switching energy

compared to the switching energy of a CMOS transistor switch might prohibit

the use of STTRAM in application that require frequent read and write oper-

ations [1]. The switching current density (Jc0) is the minimum charge current

density needed to be injected to the nanomagnet to overcome the damping

in the system. A model to calculate Jc0 was proposed by J. Sun [32]. The

model assumed a monodomain magnetic body subject to a magnetization dy-

namics given by Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation which

is driven out of equilibrium by a spin-polarized current injected perpendicular

to its plane. Using this model, switching current for monodomain in-plane

magnetized body can be written as [22,32]:

Jc0 =1

η

2eα

hMst(2πMs +HK) (1.1)

where, η is spin-transfer eciency, α is the damping factor,Ms is the saturation

magnetization characteristic of the material, HK is the anisotropy eld, t is

the thickness of the free layer. The energy barrier to be crossed while being

thermally switched, and hence the thermal stability factor (∆) of the bit is

eectively decided by the anisotropy eld HK. Hence, Eq. (1.1) essentially

dictates the Jc0-∆ trade-o [22]. A ∆ of ∼60 is required for ∼10 years of data

14

retention. The trade-o between switching current and thermal stability of a

bit is discussed in more detail while motivating Chapter 2.

In case of perpendicularly magnetized bit, due to absence of the 2πMs

term in Eq. (1.1), Jc0 is expected to be smaller for same HK and hence, same

thermal stability [22,28]. Work described in Chapter 2 is motivated by this.

Another problem introduced by the high values of Jc0 is the large area

of the selection transistors which will be required to drive the high current,

which in turn limits the maximum packing density achievable [1]. A possible

route to larger packing density can be the ability to address two bits stored in

a single nanomagnet, which is still addressed by a single selection transistor. A

multi-level cell might enable scaling down below the 6F2 cell area [20]. Work

described in Chapter 3 is motivated by search for a possible route towards

multi-bit STTRAM.

STTRAM is expected to possess innite endurance thanks to the fact

that there is no inherent wear-out mechanism involved with ipping the mag-

netization back and forth using STT. In contrast, competing emerging mem-

ory technologies such as the phase-change memory (PCM) and the resistive

random access memory (RRAM) suer from wear-out as the writing involves

physical displacement of the atoms [13]. However, wear-out in an STTRAM

does take place due to large write current crossing the tunnel barrier and di-

electric break down of the tunnel barrier after a certain number of operations

is a concern. A possible solution that has been proposed is to use a dierent

device geometry than the two terminal MTJ structure where the read and

15

write paths are the same. In the so-called spin Hall geometry, a heavy metal

such as Ta or Pt is used to inject spin to the free FM for the write opera-

tion [33,34]. In the spin Hall geometry the read and write paths are separate,

thereby saving the tunnel barrier in the MTJ read path from wear-out due

to large write current being sent through it. However, the three terminal ge-

ometry might require larger area per cell compared to the two terminal MTJ.

Also, a material with a high spin Hall angle is needed as the spin polarizer, so

that the write current density is small enough for practical applications. With

discovery of topological insulators (TI) with spin-momentum locked surface

states [35], a TI has become a promising candidate for the spin Hall mate-

rial. SPC injection to a FM using a TI has been reported experimentally [36]

and low-temperature magnetization switching has also been demonstrated [37].

Work described in Chapter 4 is motivated by eciency of STT switching of a

metallic nanomagnet(as opposed to an insulating one) by SPC injection by an

underlying TI.

The write time for an STTRAM bit for a given write pulse amplitude

is a stochastic quantity. Alternatively, the probability that a bit is written,

i.e., the magnetization of the free layer ips when a write pulse is applied for

a given duration can, in general, be less than 1. The fact that the switching

time is a random quantity, requires the circuit designers to have a write pulse

much longer than the average write time to ensure a certain safety margin.

The probability that a bit will not be written after a write pulse is applied is

referred to as the write error rate (WER). The WER should be below 10−9 if

16

there is an error correction code (ECC) and below 10−18 in absence of any ECC,

for correct operation as a working memory [31]. The necessity to understand

the reasons behind write error and improve WER motivates Chapter 5.

1.6 Overview of this dissertation

The goal of this dissertation is to investigate possible ways of decreasing

the switching current and increasing the packing density in STTRAM memory

[25].

• Chapter 2 focuses on use of perpendicular and canted magnetic anisotropies

for ecient STT switching in spin-valve structures [2].

• Chapter 3 focuses on micromagnetic simulations of STT switching of a

ferromagnetic cross [3]. We investigate possibility of using a ferromag-

netic cross for a multi-bit STTRAM.

• Chapter 4 describes modeling study of a STTRAM bit using a TI as the

spin polarizer layer in a spin Hall geometry [4].

• Chapter 5 briey describes the eects of including thermal uctuations

into STTRAM modeling. We describe our ongoing work on thermal

uctuation induced error in STTRAM write operation using Fokker-

Planck calculations and micromagnetic simulations including random

thermal magnetic eld.

17

• I will summarize and conclude this dissertation in Chapter 6 with a

discussion on possible future research directions.

18

Chapter 2

Perpendicular and canted magnetic anisotropy

for STTRAM

In this chapter we try to exploit magneto-crystalline anisotropy for

ecient STT switching using spin-valve experiments. The results have been

published in Ref. [2].1

2.1 Motivation

In a spin valve, anisotropy axes of the two ferromagnetic layers can be

in the plane of the sample, perpendicular to the plane or making an angle to

either. Materials with perpendicular magnetic anisotropy (PMA) are being

considered widely because of a promise of larger retention owing to higher

coercive eld and lower switching current due to lower damping constants [28].

Reduction in switching current with PMA materials have been reported [38

1The results in this chapter has been published in Ref. [2] (Journal of Applied Physics,vol. 111, no. 7, p. 07C913, 2012). The paper is titled Spin-transfer-torque switching inspin valve structures with perpendicular, canted, and in-plane magnetic anisotropies" by U.Roy, H. Seinige, F. Ferdousi, J. Mantey, M. Tsoi, and S. K. Banerjee. The contributions forthis paper is as follows. The spin valve samples were prepared by U. Roy and F. Ferdousiusing sputtering and e-beam evaporation processes optimized by F. Ferdousi. Point contactmeasurements were carried out by H. Seinige under the supervision of M. Tsoi. J. Manteyhelped to set-up experiments. H. Seinige and U. Roy prepared the manuscript. All authorsreviewed and commented on the manuscript.

19

40]. As briey discussed in Chapter 1, the trade-o between thermal stability

factor ∆(= Eb/kBT ) (where Eb is the energy barrier between the minima of

the energy landscape, kB is Boltzmann's constant and T is temperature) and

the zero-temperature critical switching current density Jc0 is inherently better

in case of magnets with PMA, compared to a magnet with easy axis in the

plane of the thin lm of the magnet due to shape anisotropy. The critical

switching current for an in-plane bit is given by Eq. 2.1:

Jc0 =1

η

2eα

hMst(2πMs +HK) (2.1)

whereas, for bits with PMA, the equivalent equation for Jc0 is:

Jc0 =1

η

2eα

hMst(HK) (2.2)

As also briey discussed in Chapter 1, the absence of the term 2πMs gives rise

to the fact that for the same HK, Jc0 for a PMA bit will be smaller than an

in-plane bit. The thermal stability factor of the bit, ∆ is given by:

∆ =Eb

kBT=KeffV

kBT=µ0HKMsV

2kBT(2.3)

In Eq. 2.3, Eb is the energy barrier the bit has to cross while switching between

the minima of its energy landscape. Eb, and hence∆ is decided by the eective

20

anisotropy energy per unit volume Keff . The eective anisotropy eld HK is

related to Keff by HK = 2Keff/µ0Ms. µ0 is the permeability of vacuum and

Ms is the saturation magnetization of the FM per unit volume.

Due to this better thermal stability versus switching current trade-o,

PMA bits are preferred over in-plane for the scaled down technology nodes

for STTRAM. However, if both the free layer and xed layer magnetizations

are perfectly perpendicular to the plane of the thin lm, STT will be zero as

the STT vector goes as m × (m × mp), where m is the unit vector along the

free layer magnetization and mp is the unit vector along the xed layer mag-

netization direction. Hence, the STT becomes zero if mp and m are perfectly

collinear. The STT induced switching gets started when mp and m become

slightly non-collinear due to either thermal uctuations or spatially varying

spin-texture within the free layer. However, relying on thermal uctuation

to cause non-collinear magnetizations in the xed and free layer might cause

widely stochastic nature and large range of the switching delay [32,41]. To cir-

cumvent this problem, designing non-collinear easy axes of the free and xed

layer has been proposed [41]. Precessional switching for non-collinear magne-

tization in free and reference layers leading to reduction in switching current

density and time has been studied using simulations [42].

In this work, we study STT-assisted switching in spin valve structures

with in-plane, PMA, and canted magnetizations in free and (or) reference

layers using point contact measurements to investigate the use of non-collinear

magnetizations in reducing switching current and time.

21

2.2 Fabrication and characterization of in-plane, canted,

and PMA spin valves

2.2.1 Spin valve fabrication

To fabricate the spin valve samples, p-Si (100) wafers were piranha

cleaned and wet-oxidized to grow about 400 nm thick layer of silicon dioxide. A

seed layer consisting of TaN 50/Ta 6/Pd 25 (thicknesses in nm) was deposited

on oxidized Si wafer to grow a subsequent [111] oriented stack. TaN/Ta was

deposited using DC magnetron sputtering. A 25 nm thick Pd seed layer on

TaN/Ta was deposited using electron-beam evaporation at a base pressure

of 5e-6 Torr. For the PMA and canted spin valves, Co/Pt and(or) Co/Ni

multilayer, also deposited using electron-beam evaporation were used as the

free and reference layers. For the in-plane sample, Co was used for the both

layers. The detailed sample compositions starting after the seed layers were

as follows: Co 5/Cu 4/Co 2/Cap (thicknesses in nm) for sample I; Co 0.7/Pt

1.6/[Co 0.7/Pt 1.7]3/Co 0.7/Ni 0.7/Co 0.4/Cu 4/Co 0.4/Ni 0.6/Co 0.2/Pt

0.6/Cap for sample II; [Co 0.4/Pt 1.2]4/Co 0.2/Ni 0.4/Co 0.2/Cu 4/Co 0.2/Ni

0.4/Co 0.2/Pt 0.6/Cap for sample III. For the canted samples the amount of

magnetization's tilt from the normal to the sample plane was quantied using

two vibrating sample magnetometry (VSM) measurements with magnetic eld

in the plane of the sample and perpendicular to the sample plane following

previous report [43]. The amount of canting of the reference layers were found

to be about 45 for sample II and about 12 for sample III. Sample preparation

(for samples II and III) and characterization was done in collaboration with

22

Fahmida Ferdousi and have been discussed in more detail elsewhere [44].

2.2.2 Characterization by point-contact measurements

-10 0 10

-5

0

5

I(mA)

B(mT)

d)

-60 -40 -20 0 20 40 60

-2

0

2

I(mA)

B(mT)

c)

-2 -1 0 1 2

-4

0

4

I(mA)

B(mT)

b)

-150 0 150

15.10

15.15

R()

B(mT)

a)

Figure 2.1: (a) is a typical magnetoresistance plot of sample II with a magneticeld applied perpendicular to the sample plane and a dc bias current of -1.3mA. (b-d) are STT slope from point contact measurements for sample I-IIIrespectively, with open symbols for up-sweep and closed symbols for down-sweep. This gure has been reproduced from Ref. [2] with permissions.

Point-contact technique [19, 45, 46] was used to perform experimental

studies of STT in the spin valve samples. Point-contact measurements were

performed by Heidi Seinige in UT-Physics and was supervised by Prof. Maxim

Tsoi. Point contacts to all three samples were created between electrochem-

ically sharpened Cu tips and sample lms using a standard mechanical sys-

23

tem [19, 45]. At room temperature magnetoresistances of such point contacts

are measured while applying dierent values of dc bias current to the contact.

A typical magnetoresistance curve is shown in Figure 2.1a) for sample II and

magnetic eld applied perpendicular to the sample plane. At large negative

(positive) values of magnetic eld, magnetizations of the free and reference

layers are parallel, so the resistance is low. When the eld is swept to a small

positive (negative) value, the free layer switches rst so the resistance rises

when the layer magnetizations become antiparallel. Further increase of the

magnetic eld results in a resistance decrease when the reference layer also

switches and the magnetizations become parallel again. In the following we

refer to the measurements of magnetoresistance in magnetic eld sweeping up

(from negative to positive) and down (from positive to negative) as up-sweep

and down-sweep, respectively. For PMA and canted samples, the magnetic

eld was applied in perpendicular direction to the sample plane whereas for

the in-plane spin valve the applied magnetic eld was in the plane of the sam-

ple. The magnetic eld at which the free layer switches its orientation was

observed to vary with the applied dc bias current. We tentatively attribute

these variations to the eect of STT and use them to evaluate the eciency

of STT in our experiments.

Figures 2.1(b)-2.1(d) show the free layer switching-eld variations with

the dc bias current in samples I-III, respectively, for both the up-sweep (open

symbols) and downsweeps (solid symbols). The least-squares linear ts to the

data are also shown and their slopes were used to compare the eciencies of

24

Table 2.1: Slope of STT for samples I-III.

Sample Up-sweep Down-sweep Contact Estimated Estimated currentslope (T/A) slope (T/A) resistance (Ω) contact density

size (nm) for I=1mA (A/m2)Sample I -0.281 -0.441 11.8 15.2 1.3× 1012

Sample II 13.7 -10.9 15.1 13.2 1.8× 1012

Sample III 1.82 -6.75 12.0 15.0 1.4× 1012

the STT in dierent samples. In sample III we observed little, if any, eect

of the positive bias on the switching eld, so only negative currents were used

to evaluate the eciency of the STT (see the linear ts in Figure 2.1(d)).

Table 2.1 summarizes these observations. The current was found to be most

ecient in aecting the switching-eld in sample II with a 45 tilt. In contrast,

the in-plane sample showed a very weak dependence of the switching-eld on

current. The signs of the STT slopes are related to the presence of dierent

canted phases in the respective spin congurations, as was also seen in some

other reports [28].

2.3 Conclusions

We have performed a comparative study of STT-switching in samples

with in-plane, canted, and perpendicular magnetic anisotropies. STT was seen

to be more ecient in perpendicular/canted samples, both from micromag-

netic simulations (discussed in Ref. [44, 46]) and point contact experiments,

compared to an in-plane spin valve. In the samples with out-of-the-plane

anisotropy, point contact measurements showed larger eect of STT in the

25

sample with 45 tilt compared to the sample with 12 tilt. The in-plane ge-

ometry showed less eciency of STT to switch magnetization in both micro-

magnetic simulations and experiments. The experimental observations agree

qualitatively with the micromagnetic simulations, though exact quantitative

comparison of the two could not be carried out. The reason is, a dc bias current

and an external magnetic eld were used for the point contact measurements

whereas the simulations were carried out for a pulsed current. Also, the mag-

netization volume probed in point contact experiments can be inuenced by

magnetic congurations and dynamics of neighboring domains. The latter

would call for simulations in much larger volumes compared to the dimensions

of spin valves simulated in the present work.

26

Chapter 3

Spin-transfer-torque switching of a

ferromagnetic cross towards multi-bit STTRAM

We propose a multi-bit STTRAM using micromagnetic simulations of a

ferromagnetic cross. The results in this chapter has been published in Ref. [3].1

3.1 Motivation

Currently, the switching current densities (106-107 A/cm2) required to

switch the bit using the STT eect is high. In the most widely used scheme

an STTRAM cell comprises of a MTJ connected to an access transistor (1T-

1MTJ). The access transistor has to be wide enough to support the write

current of the MTJ. Currently, the switching current densities being very high,

the area of the selection transistor becomes really large and might be as high

as 50F2 [47]. Thus the area of the transistor becomes the bottleneck for the

1The results have been published in the Journal of Applied Physics (vol. 113, no. 22,pp. 223904, 2013). The paper is titled Micromagnetic study of spin-transfer-torque switch-ing of a ferromagnetic cross towards multi-state spin-transfer-torque based random accessmemory" by U. Roy, T. Pramanik, M. Tsoi, L. F. Register, and S. K. Banerjee. The con-tributions for this paper is as follows. U. Roy and T. Pramanik ran the micromagneticsimulations. U. Roy developed basic code for a single simulation. T. Pramanik batch pro-cessed the simulations for wide range of device dimensions and current density. U. Roywrote the manuscript. M. Tsoi, L. F. Register and S. K. Banerjee supervised the work andreviewed and commented on the manuscript.

27

Figure 3.1: Cross-shaped MTJ stack

area per bit and in turn, the packing density of the technology [1]. To improve

the packing density, for example, alternative access schemes other than the

1T-1MTJ scheme have been considered [47]. Another route towards improving

packing density could be the use of a multi-bit cell. Multi-bit STTRAM device

has been previously proposed using multiple MTJs connected in series [48]

or two dierent domains in the free layer switching at two dierent current

densities [49]. In this chapter, we explore the possibility of storing two memory

bits within a single MTJ with a cross-shaped free layer that could still be

addressed by one selection transistor. The studied STTRAM structure is

similar to device proposed in a recent patent [50]. However, in this work we

provide a detailed discussion of the switching dynamics and associated regions

of reliable switching currents. We also study the eects of varying device

28

geometry on the regions of reliable switching current. A schematic picture of

the imagined cross-shaped MTJ stack has been shown in Figure 3.1.

With two long branches in the free layer of the considered MTJ struc-

tures, there are an associated four valleys in the energy landscape. The angular

positions of the free layer magnetization in these stable positions with respect

to the xed layer are designed so that the resistance of the MTJ has distinct

values when the free layer magnetization resides in each of these valleys. Using

micromagnetic simulations, we show switching between energy valleys using

spin polarized current. We engineer the energy barrier between valleys by

controlling the dimension of the branches.

First, we simulate the equilibrium magnetizations of the cross-shaped

free layer. The dimensions of the cross are adjusted such that energy barrier

between any two energy minima is at least ∼60 kBT (kB is Boltzmann's con-

stant and T is absolute temperature). Then switching scheme between the

four energy minima of the cross shaped free layer has been proposed by using

varying amplitude of current pulse applied to the free layer.

3.2 Results

3.2.1 Equilibrium magnetizations of the cross

Equilibrium spin texture of the cross has been shown in Figure 3.2. As

can be seen in Figure 3.2, there are four equilibrium magnetizations, equiv-

alently, two easy axes, owing to the 4 elongated arms. The energy barrier

between any two minima would be the same for a symmetric cross. Here,

29

however, we have considered the cross to be asymmetric. That is, the short

arms are each 40 nm long along the longer axis and 20 nm long along the

shorter axis. The long arms are each 60 nm longer along the long axis and 20

nm long along the shorter axis. The thickness in the out-of-the-plane direction

is 2 nm. The magnetization is saturated along a direction making a very small

angle with the branches applying a large enough magnetic eld, and then al-

lowed to relax back to zero magnetic eld. The magnetization relaxes to the

spin-textures shown in Figure 3.2.(a)-(d). As the magnetization relaxes to

the congurations in Figure 3.2.(a)-(d) with no external magnetic eld, these

correspond to the minima of the energy landscape. Here we simulate only the

free layer thereby ignoring any dipolar eld from the xed FM. This can be

achieved in a real MTJ device using a synthetic antiferromagnet as the xed

layer [51].

The assumed material parameters are saturation magnetization Ms =

1400 × 103 A/m and uniform exchange constant A = 30 × 10−12 J/m. These

material parameters are typical for cobalt [52]. We neglect any crystalline

anisotropy so as to be able to engineer the easy axes of the free magnet using

shape anisotropy only. The shape anisotropy of the branches and hence the

barrier height between the minima are eectively controlled by the aspect

ratio (AR) of the branches for a given volume. The shape anisotropy of a FM

depends on its demagnetization tensor. The demagnetization tensor can be

determined for an ellipsoid of a given AR from Osborn's work [53]. This does

not hold true for any geometry other than an ellipsoid.

30

Figure 3.2: Equilibrium magnetizations; (a) State 1; (b) State 2; (c) State3; (d) State 4; l1/w is referred to as AR of short arm, l2/w is referredto as AR of long arm. This gure has been reproduced from Ref. [3] withpermissions.

31

3.2.2 Magnetization dynamics under STT

The resistance of the MTJ stack is a function of the cosine of the

angle between the free layer and xed layer magnetizations. So we propose

to pin the xed layer magnetization at around 22.5 with respect to the short

branch. This serves to provide four dierent cosine values of the equilibrium

magnetization directions of the central part of the cross with respect to the

xed layer magnetization direction and, hence, four dierent resistance values.

Non-collinear magnetizations in free and xed layers of a MTJ or spin-valve

have been previously used [42,54,55].

Micromagnetic simulator OOMMF [56] is used to simulate magnetiza-

tion dynamics of the free layer as a spin-polarized current pulse is applied to

it. The Landau-Lifshitz-Gilbert equation is solved with an STT term added,

as has been implemented in OOMMF:

dm

dt= −|γ|m×Heff+α(m× dm

dt)+ |γ|βϵ(m×mp×m)−|γ|βϵ′(m×mp) (3.1)

where,

γ = gyromagnetic ratio

α= damping constant

m = reduced magnetization

Heff = spatially varying eective magnetic eld

β = | hµ0e

| JtMs

t = free layer thickness

Ms = saturation magnetization

32

J = charge current density

mp = (unit) polarization direction of the spin-polarized current

ϵ = PΛ2

(Λ2+1)+(Λ2−1)(m.mp)

P = spin polarization

ϵ' = secondary STT term (eld-like-torque)

In Eq. (3.1), ϵ gives the dependence of the spin-polarization on the

time-varying free layer magnetization direction m. We assume Λ to be equal

to 1, following previous studies [42] for simplicity. We assume P to be 0.4. We

set ϵ' to 0.06, considering eld-like-torque to be 30% of the STT term in Eq.

(3.1) [42]. α is assumed to be 0.01. mp is taken to be 0.92, 0.382, 0. This

represents the fact that the xed layer FM is assumed to be pinned making

22.5 angle with the short branch, with its magnetization vector lying in the

x-y plane. We assume the xed layer magnetization is pinned by coupling

to an adjacent anti-ferromagnetic layer although the STT acts both on the

free layer and the xed layer. We imagine that the whole MTJ stack has

a cross-shaped area and the shape and lateral size of the pinned (reference)

layer is the same as the shape and size of the free layer. We assume uniform

spin-polarized current is being injected into the free FM over its entire area.

However, with the assumption that the only information from the xed layer

to the free layer is via the spin-polarized charge injected into the latter, the

exact shape and size of the xed layer are not important for our conclusions,

as long as it encompasses the entire area of the free layer and the xed layer

33

magnetization is pinned at a certain angle with respect to the cross.

The magnetization dynamics of the cross is simulated for a negative

polarity pulse of amplitude varying from 1 × 107A/cm2 to 5 × 108 A/cm2 in

steps of 5×106 A/cm2 to investigate the eect of STT on the magnetization of

the ferromagnetic cross. Cell size for micromagnetic simulation is taken to be 1

nm in both directions in plane of the cross, while in out-of-the-plane direction

the cell height is equal to the thickness of the FM lm, i.e., 2 nm. A smaller

cell size in x-y plane does not make any observable dierence in the dynamics.

Eect of thermal uctuations has been neglected for simplicity. For all of

the dynamic simulations in presence of an injected spin-polarized current, the

initial spin distribution is the steady state magnetization of State 1 described

in the previous section. For a current pulse, the minimum current value at

which switching occurs depends on the pulse-width used. The pulse-width has

to be larger than the duration of the pre-switching oscillations. By denition,

critical switching current density for STT switching is the current density

for which small-amplitude oscillations start to grow with time [32]. The 0

crossing time (ts) for the x-component of spatially averaged magnetization

(< mx >) for dierent amplitudes of the negative going current pulse could

be t by [22,32,57]:

1/ts ∝ (J − Jc0) (3.2)

The curve is extrapolated to nd out critical switching current Jc0, below

which the precession induced by STT is damped by the system. Here, after

starting from State 1", as the system energy increases with time for a J>Jc0,

34

the system may settle down to either State 2 or State 3 after crossing the

energy barrier between the stable states.

3.2.2.1 State 1 to State 2 switching

For a low range of current, the system crosses the energy barrier to

switch only the short branch and the long branch does not switch. This be-

haviour is because of the smaller eective shape anisotropy eld for the short

branch, and asymmetry of the spin polarization direction (mp) of the injected

current between the branches. If J is greater than Jc0 (and not too large as is

discussed later) switching from State 1 to State 2 (1→2 switching) takes

place. Therefore, henceforth we will refer to Jc0 as J1→2. If the current pulse

remains on after 1→2 switching takes place, the system attains a steady state

magnetization distribution even in the presence of the current (after 6 ns in

Figure 3.3-(a)). This result corresponds to the scenario of Eq. (3.1) with dmdt

set to 0. When the current is turned o, the magnetization is damped to the

closest equilibrium magnetization conguration. As shown in Figure 3.3, the

system is damped to State 2 after the current pulse is switched o at around

10 ns.

3.2.2.2 State 1 to State 3 switching

If the current density value is even higher, the long branch also switches

and the system settles to State 3, until the maximum current used in the

present work (5 × 108 A/cm2). This process happens through a systematic

35

0.0 0.2 0.4 0.6 0.8 1.0

-0.2

0.0

0.2

0.4

0.6

<mX>

Time (ns)

increasing J

32 40 48 56

0.3

0.6

0.9

l1/w l2/w 2 3 2.25 3 2.5 3 2 2.5

1/t s(1/ns

)

J (MA/cm2) (b)

0 2 4 6 8 10 12-0.8

-0.4

0.0

0.4

0.8

J (MA/cm

2 )

<mx,y,z>

Time (ns)

<mx> <m

y> <m

z>

-40

-20

0 J

(a)

Figure 3.3: (a) 1 → 2 switching with J = 5 × 107A/cm2 (b) Pre-switchingoscillations with dierent J values; inset: linear t of 0 crossing time for dier-ent values of short branch AR (l1/w) and long branch AR (l2/w). l1, l2 and ware dened in Figure 3.2. This gure has been reproduced from Ref. [3] withpermissions.

36

sequential switching of the short branch rst and then the long branch. A

typical switching from State 1 to State 3(1→3 switching) is shown in Fig-

ure 3.4 for J = 3.8× 108A/cm2. Henceforth we refer to the minimum current

density for reliable 1→3 switching as J1→3.

3.2.2.3 State 1 to State 4 switching

State 4 (Figure 3.2-(d)) cannot be reached by the same polarity of

current pulse as could be used to switch to State 2 and State 3 from State

1. A negative polarity of pulse on State 1 always tries to switch both the

branches and J1→3 > J1→2. However, using 2 dierent pulses of opposite

polarities consecutively, State 4 can be reached from State 1 (Figure 3.5).

Figure 3.6 summarizes simulation results of switching for dierent de-

signs of cross. All the designs have a central square region of 20 nm by 20

nm, with varying lengths of the short and long arms. What we refer to as

a grey region in which switching is almost random function of the current

density is not shown, or rather is shown as breaks in switching current densi-

ties, for clarity. These results demonstrate that J1→3 becomes smaller with a

smaller AR of long branch for a given width, as can again be understood in

terms of the eective shape anisotropy eld of the long branch. Figure 3.6 also

provides some insight into design of STT-devices with a biaxial energy land-

scape (including but perhaps not limited to the cross-shaped free layer devices

under consideration here). For a given length of the short arm, if the long

arm is made closer in length to the short arm, although J1→3 becomes smaller,

37

0.0 0.5 4 6 8

-0.8

-0.4

0.0

0.4

0.8

J (M

A/c

m2 )

<mx,

y,z>

Time (ns)

<mx>

<my>

<mz>

-300

-200

-100

0

J

Figure 3.4: 1 → 3 switching with J = 3.8 × 108A/cm2. This gure has beenreproduced from Ref. [3] with permissions.

38

0 1 2 3 4 5 6

-0.8

-0.4

0.0

0.4

0.8

J (M

A/c

m2 )

<mx,

y,z>

Time (ns)

<mx> <my> <mz>

-400

-200

0

J

Figure 3.5: 1 → 4 switching with two current pulses of opposite polaritiesand dierent magnitudes. This gure has been reproduced from Ref. [3] withpermissions.

39

1 2 3 4 5

Switching to ``State 3" Switching to ``State 2" No switching

l1/w=2, l

2/w=2

l1/w=2.5, l

2/w=3

l1/w=2, l

2/w=2.5

l1/w=2.25, l

2/w=3

Current Density (x108A/cm2)

l1/w=2, l

2/w=3

Figure 3.6: Switching regions to State 2 and State 3 starting from State1 with increasing amplitude of current pulse in steps of J = 5 × 106 A/cm2

for dierent designs of cross. This gure has been reproduced from Ref. [3]with permissions.

monotonic switching regime to State 2 is observed in an increasingly narrow

window, and eectively can be considered to be a part of the grey region for

symmetric structure for equal short and long arms. Simulation for dierent

designs of cross and for dierent values of pulse amplitude was performed by

Tanmoy Pramanik by batch processing of input codes developed by me.

3.2.3 Thermal stability

As has been discussed in Chapters 1 and 2, thermal stability of a

STTRAM bit is an important consideration. The thermal stability require-

40

ment often sets a trade-o between maximum storage density achievable and

the requirement for data retention, where the latter requires an energy barrier

of about 60kBT (kB is Boltzmann's constant and T is absolute temperature)

for 10 year retention [22]. In general, a magnet might not switch under ther-

mal uctuation as a mono-domain. That is, spins might not be parallel to

each other while switching. In that case, there will be another contribution

to the energy barrier, the exchange energy [58]. The total barrier will depend

on the path the system takes in the phase-space while thermally switching.

To estimate the thermal stability of the bit, we follow a method similar to

that reported for stability calculations for an elliptic magnet [58]. Starting

from State 1, a large magnetic eld (500 mT) is applied along the y axis

to saturate the magnetization along the hard axis of the short branch. The

dierence of the demagnetization energy in these two states is estimated to be

the energy barrier due to shape anisotropy for the system to go from State

1 to State 2 for in-plane rotation under thermal uctuation. That is, we

ignore the contribution due to exchange energy while switching. For the cross

with a 20 nm by 20 nm center, and branches with ARs of 2 and 3, this en-

ergy is approximately equal to 62.7kBT at T=300 K. As the short branch has

a smaller energy barrier, this energy barrier limits the bit's stability against

in-plane rotation of magnetization under thermal activation. We have not dis-

cussed simulation results on smaller cross sizes than this (i.e. smaller center

sizes while keeping the aspect ratios the same) as these are not expected to be

thermally stable according to the 60kBT energy barrier benchmark.

41

3.3 Packing density considerations

The maximum current density simulated for the STT-induced magne-

tization switching of the ferromagnetic cross is 5×108 A/cm2, which translates

to a current range of ∼20 mA for the switching from State 1 to State 3.

This is however very large compared to the target write current values for

STTRAM in the future technology nodes [23]. Better understanding of the

magnetization dynamics and reason behind the chaotic magnetization dynam-

ics observed for a certain range of the applied current might require analysis

of the spin-wave modes excited by STT [59]. Wiser engineering of the shape

might lead to lesser chaotic nature of the dynamics, hopefully leading to a

smaller grey region. As commented before, the high write current predicted

arises due to:

1. The inherent worse trade-o between thermal stability and write cur-

rent density for in-plane bits, as also described in detail in Chapter 2.

The multi-bit scheme proposed here is inherently dependent on shape

anisotropy and hence needs in-plane magnets. Thus the proposed scheme

suers from the worse-trade o between thermal stability and write cur-

rent density compared to perpendicular materials. Thus even the critical

switching current density to State 2 (∼107 A/cm2) is also very high

compared to expectations in the international technology roadmap for

semiconductors [23].

2. Secondly, the grey region between regions of reliable switching to State

42

2 and State 3 pushes the reliable switching regime to State 3 to high

current densities. This eect might get worse once the eect of thermal

uctuations is taken into account. Eects of including thermal eects will

be discussed for an uniaxial energy landscape in more detail in Chapter

5. Presumably, thermal eects, as will be important in any real system,

will have two fold eects on the switching of the cross considered here.

Firstly the grey region might become wider than has been predicted in

Figure 3.6. Secondly, thermal uctuations give rise to wide range of

switching time even for a bit with uniaxial energy landscape, thereby

requiring the current density to be applied for reliable write operation

to be higher than the average current density of switching.

In the light of the considerations listed above, the proposed multi-bit scheme

might not lead to a considerable improvement in packing density, if at all. How-

ever, the study here shows the possibility of using shape of cross to engineer a

biaxial energy landscape and multi-bit storage for non-volatile memories.

3.4 Conclusion

We have studied STT switching of a ferromagnet with an energy land-

scape that is eectively biaxial, in contrast with conventional MTJs with ellip-

tic cross-section where the shape denes an eective uniaxial energy landscape.

Here by the combined eect of the spin polarization direction and the ARs of

the arms of the cross we demonstrated that by introducing asymmetry, mono-

tonic regions of switching with increasing spin-polarized current density can

43

be achieved. The results potentially could be used towards storing more than

one memory bit in a single MTJ-like structure for improved storage density.

44

Chapter 4

Spin-transfer-torque switching of a metallic

nanomagnet due to an underlying topological

insulator

An alternative being considered for the two-terminal MTJs is the spin

Hall geometry, where the read and write current of the bit would ow through

dierent paths. The three terminal geometry would save the thin tunnel barrier

in the MTJ from the damage due to high write current owing through it

during write operations and the MTJ stack can only be used for the read-

out [13]. In this chapter we consider a topological insulator as the polarizer in

a spin Hall device.1

1The results have been published in the Journal of Applied Physics (vol. 117, no. 16,pp. 163906, 2015). The paper is titled Magnetization switching of a metallic nanomagnetvia current-induced surface spin-polarization of an underlying topological insulator" by U.Roy, R. Dey, T. Pramanik, B. Ghosh, L. F. Register, and S. K. Banerjee. The contributionfor this paper is as follows. U. Roy developed the necessary equations and implemented thenumerical solution for the transport model and magnetization dynamics with helps from R.Dey and T. Pramanik. U. Roy wrote the manuscript. L. F. Register provided importantsuggestions and insights into the transport model. All authors reviewed and commented onthe manuscript.

45

4.1 Motivation

Topological insulators (TIs) having Dirac-cone-like band structures pos-

sess spin-momentum helically-locked surface states [35], i.e., spin and wave-

vector, k′ are locked. With the discovery of TIs, it has been recognized mag-

netization switching using the surface states of a TI can be a promising route

towards lower current and lower power non-volatile memory [36,37,60,61]. A

charge current supported by the TI surface states gives rise to spin accumula-

tion at the TI surface in proportion to the TI surface current density because

of the spin-momentum helical locking [36,61].

Theoretical studies [62,63] of TI-ferromagnet (FM) bilayer systems con-

sidered the FM experiencing a torque due to exchange coupling between the

the FM moment and TI surface moment in presence of a charge current on the

TI layer. In an earlier work, magnetization switching of a magnet using a TI

was calculated using a non-equilibrium Green's function approach for trans-

port in the TI [61]. These theoretical treatments are particularly appropriate

for insulating ferromagnets such as EuS [64], GdN [65] and Yttrium iron gar-

net (YIG) on the TI. However, of these, only YIG, has a Curie temperature

above room temperature, making it appropriate for broad device applications.

More technologically developed are the ferromagnetic metals as used

in conventional spin-transfer-torque memories. In addition, as in conventional

STTRAM, these materials allow for STT via injection of spin-polarized charge

carrier injection into the FM. However, in a recent experimental study of spin-

transfer ferromagnetic resonance in permalloy by spin injection from surface

46

states of a Bi2Se3 TI, it was recognized that charge shunting will be important

[36]. Shunting of the current to the ferromagnetic metal reduces spin injection

eciency for a given total current. In addition, in theoretical modeling that

accompanied the latter experimental results, the charge current in the TI was

assumed to be constant with position along the TI-FM bilayer channel [36].

While appropriate for that experimental analysis, this approximation will not

hold when the length of the FM in the transport direction is short, as for the

nanomagnet dimensions relevant to STTRAM applications.

In this scenario, in this chapter we consider the applicability of FMs

to TI-based STTRAM. We consider a thermally-stable, conducting nanomag-

net subject to spin-polarized current injection (and extraction) from (to) the

surface states of a TI. We calculate parallel transport in the TI and the FM,

and focus on the eciency of switching as a function of transport between

the TI and the FM across the interface and within each layer. Although we

do not model it here, we are aware that a FM lm on the TI surface can

aect the band structure of the surface Dirac electrons, opening a gap at the

Dirac point or shifting the Dirac point in momentum space [66]. However, the

basic conclusions of our analysis should hold to the extent that signicant spin-

momentum helicity is maintained within the energy range of interest about the

Fermi level. Transport in the TI, as well as in the FM, is treated as diusive

(for reasons to be detailed momentarily) at room temperature on the scales

of interest, such that it can be captured by drift-diusion (DD) simulation.

We take into account the contributions to the magnetization dynamics of the

47

torque from transport of spin-polarized carriers between the TI and the FM,

as well as the torque from exchange coupling between the TI surface electrons

and the FM. We calculate switching time and write energy per write operation

as a function of the ease of transport across the TI-FM interface. Switching

energies and times can compare favorably to conventional spin-torque mem-

ory schemes for substantial interlayer conductivity. Nevertheless, we nd that

shunting of current from the TI to metallic nanomagnet can substantially limit

eciency for high FM conductivity. Exacerbating the problem, spin-transfer-

torque from the TI eectively increases the TI resistivity. Thus, the eective

conductivity of the FM layer becomes a critical design consideration for TI-

based STTRAMs. The results have been published in Ref. [4].

4.2 Results

4.2.1 Transport model

In the energy range of interest, we model the energy dispersion in the

TI to be linear, with spin and momentum helically locked, and the energy

dispersion in FM to be parabolic for the up and down spin bands, which are

separated by an exchange splitting energy.

For transport in the TI in absence of any coupling between the TI and

the FM, the mean free path for momentum relaxation l0 for the assumed mo-

mentum relaxation rate and Fermi velocity (as specied in Table 4.1) is ∼15

nm. However, following a previous study [67], we note that there will be an ad-

ditional contribution to the TI in-plane current due to out-of-plane tunneling

48

of carriers between the TI and the spin up and spin down energy bands of the

FM that eectively reduces the conductivity in the TI. Essentially, the rate of

spin loss to the FM in units of the intrinsic spin angular momentum of elec-

trons h/2 must be matched by a comparable rate of electron momentum loss

for these helically-locked spin-momentum states of the TI. Therefore, when the

average spin-transfer length that a charge carrier travels in the TI to provide a

spin angular momentum of h/2 to the nanomagnet, ls,||, becomes comparable

to or smaller than l0, then the net mean free path, l−1 ≈ l−10 + l−1

s,||, and associ-

ated eective TI conductivity are reduced. Here, we have simulated only the

eective conductivity reduction that falls out naturally from the spin-weighted

interlayer electron transport, and not that due to the exchange interaction.

For the maximum interlayer coupling considered here, ls,|| is ∼1.4 nm,

l becomes ∼1.3 nm, and the TI resistance goes up over an order of magnitude.

Therefore, absent coupling between the TI an FM where l0 is about half the

channel length, the assumption of diusive transport is reasonable for our

purposes if imperfect. As the interlayer coupling that we rely upon for STT

increases, the diusive approximation becomes increasingly accurate. Within

the DD approximation, the net spin is then proportional to the current ow

for these spin-helically locked states.

For transport in the metallic FM, with the characteristic time scale for

the local moments in the FM orientation (order of nanoseconds) being much

larger than the time scale involved in the dynamics of conduction electrons

(order of picoseconds), the transport in the TI-FM bilayer can be obtained for

49

a given FM magnetization direction within a (quasi-) steady-state approxima-

tion [68]. Moreover, phase coherence between spin up and spin down states,

i.e., spin transverse to the FM magnetization direction, is absorbed by the

FM within atomic distances [69], which are small on the scale of interest here.

Accordingly, spin accumulation in the FM adiabatically follows the axis of FM

magnetization, which allows us to model in-plane spin transport in the FM in

terms of spin up and spin down channels.

We have assumed spatial homogeneity within the FM for each spin-

dependent quasi-chemical potential in the surface-normal (z) direction because

of the small layer thickness of the FM. Within the remaining nominal direc-

tion of transport (x), we explicitly consider transport for each (inter-coupled)

spin-degree of freedom within the FM within a DD approximation. For large

momentum relaxation lengths and associated large conductivities within the

FM, the quasi-chemical potential for each spin channel in the metal becomes

essentially constant (variation of a few meV at most) along the channel, mak-

ing the detailed transport model within the metal a moot issue. Only for

substantially reduced conductivities with correspondingly reduced momentum

relaxation lengths, for which a diusive transport model is justied, are sig-

nicant variations in the Fermi levels within the FM observed.

The rapid decay of coherence between spin up and spin down compo-

nents, also allows the use of a Golden Rule-like approach to transport across

the TI-FM interface following Ref. [67]really across a short distance near the

interfacebetween the spin-momentum helical-locked surface states of the TI

50

and the spin up and spin down states of the FM, with momentum (but not

spin) randomization between initial and nal states.

Following Ref. [67], three coupled DD equations are solved, one in the

TI and two for the FM, subdivided by spin. Here, we have assumed homogene-

ity in the TI-FM bilayer system along the width (y direction) for simplicity, so

that we need not consider transport in that direction. Using the quasi-static

spin-dependent eective carrier densities in the FM, dened by nσ ≡ DOSσµσ

(σ =↑, ↓) and eective electron density in the TI, nTI ≡ DOSTIµTI, where DOS

indicate the density of states and the µ indicate the quasi-chemical potentials,

the drift-diusion equations in terms of the particle currents take the form

(written in 3D for generality although only solved in 1D here),

jσ = −Dσ∇nσ (4.1)

in the FM, and,

jTI = −DTI∇nTI + jt (4.2)

in the TI. The Dσ and DTI are the spin-dependent diusivities in the FM, and

the diusivity in the TI, respectively. The jt in Eq. (4.2) is the additional term

due to out-of-plane tunneling of carriers between the FM and the TI previously

mentioned and discussed further below. The variations in the eective carrier

densities represent variations in the quasi-chemical potentials, due to actual

changes in carrier densities and/or eld-induced drift. These eective carrier

densities also are subject to the steady-state current continuity equations [67]

Dσ∇2nσ − (nσ

τσ− nσ

τσ)− iσ = 0 (4.3)

51

for each spin channel in the FM, and,

−∇ · jTI + (i↑ + i↓) = 0 (4.4)

in the TI. In Eq. (4.3), the τσ are the spin-ip times for the FM material, and

the overbars indicate the opposite spin channel. The iσ are spin-dependent

interlayer currents. They are calculated here, after Ref. [67], using Fermi's

Golden Rule,

iσ =2π

h

∑k′,k,σ

| ⟨ψk′|H|ψk,σ⟩ |2

×(fF,k,σ − fT,k′)δ(ϵk,σ − ϵk′).

(4.5)

Here, h is the reduced Planck's constant, k,k′=wave vectors in the FM and

in the TI, respectively, ψk,σ, ψk′ are the energy eigenfunctions corresponding

to energy eigenvalues ϵk,σ in the FM and ϵk′ in the TI, respectively. fF,k,σ

and fT,k′ are the occupation probabilities of the states labelled by k and σ in

the FM and k′ in the TI, respectively, according to the Fermi-Dirac distribu-

tion. H is the coupling Hamiltonian between the TI and the FM states. The

value of the coupling energy H used for the interlayer transport calculation

will depend on the materials and detailed nature and quality of the TI-FM in-

terface. Moreover, we established a mapping between the coupling energy and

the conductance across the interface between the TI surface charge layer and

the nanomagnet center, the latter being more intuitive for analysis of trans-

port.However, we note that, the out-of-the-plane current is not proportional

to the voltage drop across the interface, because of the shift in the Fermi circle

of the TI in presence of a TI surface current. So, in general, this mapping

52

is only approximate. We do not try to justify a particular value of coupling

energy, or even the Golden Rule, per se; rather the coupling strength is simply

used to vary the interlayer conductance.

As mentioned earlier, there will be an additional component jt in the

in-plane TI current due to out-of-plane tunneling of carriers between the TI

and the FM, in addition to the drift and diusion currents that are together

captured by spatial gradient in nTI and, hence, the quasi-chemical potential

µTI [67]. This additional contribution, consistent with Eq. (4.5) is [67],

jt =2π

h

∑k′,k,σ

vk′| ⟨ψk′|H|ψk,σ⟩ |2

×(fF,k,σ − fT,k′)δ(ϵk,σ − ϵk′)

(4.6)

with only the x component surviving under the conditions assumed in this

work. Here, vk′(= ∂ϵk′/h∂k′) is the velocity in the TI state labelled by the

wave vector k′. (Because of spin-momentum helical locking, each carrier in-

jected into the TI with resulting in-plane velocity vk′ , will also carry a unit

of spin angular momentum oriented perpendicular to vk′ with a helicity as

shown in Figure 4.1(b), the associated STT can be calculated self-consistently,

as done in Section III.)

We limit the interlayer transport by matching the resistance between

the TI sheet charge layer and the center of the FM over a 1 nm distance along

the layers to the resistance within the TI sheet charge layer to go 1 nm, which

still allows for rapid shunting to the metal for high in-plane FM conductivities.

That is, we match the minimum interlayer resistivity (units of Ω-m2) divided

by a 1 nm length along the TI, to the self-consistently obtained eective TI

53

Figure 4.1: (a) Schematic of the TI-FM bilayer, x is the transport, z is theout-of-plane direction. (b) The lled circle denotes the shifted Fermi circle inpresence of a current supported by the surface states. The arrows denote thespin vectors of the surface states because of spin-momentum helical locking.This gure has been reproduced from Ref. [4] with permissions.

sheet resistivity (units of Ω) for this minimum interlayer resistivity times a

1 nm length along the TI, which is 25.5 µΩ-m (corresponding to a TI sheet

resistivity of 25.5 µΩ-m ÷ 1 nm = 25.5 kΩ, and an interlayer resistivity of 25.5

µΩ-m × 1 nm = 25.5 fΩ-m2). (The associated interlayer coupling potential is

∼0.1 eV.) We use this minimum interlayer resistivity (maximum interlayer con-

ductivity) as a reference for normalizing larger interlayer resistivities (smaller

interlayer conductivities).

Parameters assumed for the DD model of the TI-FM bilayer are shown

in Table 4.1, except as otherwise noted. The parameters assumed for the TI

are reasonably consistent with previous reports [35, 36] and the parameters

for the FM are representative of a generic ferromagnet. The patterned FM

is taken to be 30 nm along the nominal transport direction (x direction) as

already noted, 100 nm normal to the transport direction in the TI-FM plane

54

(y), and 2 nm thick normal to the TI-FM plane (z), with the magnetic easy

axis along the FM's long axis. The axes conventions are shown in Figure 4.1.

In Figures 4.2 and 4.3, we present the results of the DD transport simulations

for a xed FM magnetization direction.

As can be seen in Figures 4.2(a)-(c), increased coupling between the

TI and the FM states leads to larger shunting of the TI surface current by

the FM. The shunting is increased both due to the increased interlayer con-

ductance and the self-consistent reduction in TI intralayer conductance. The

inset of Figure 4.2(d) shows the linear increase in eective TI intralayer resis-

tivity as a function of the interlayer conductance. Note that this resistivity

quickly becomes dominated by the eects of the interlayer transport. Never-

theless, Figure 4.2(d) shows that, except to a small degree initially, the overall

resistance of the bilayer decreases due to shunting from the TI through the

metallic nanomagnet. Also, as predicted by analytical calculations [67], the

resistance of the bilayer is a parabolic function of the y component of the FM

magnetization. However, the relative variation is only approximately 4% of

the average.

In terms of the ability to torque the nanomagnet, a commonly used

metric is the spin Hall angle" or spin torque ratio", dened as [36],

θ|| = (2e

h)σs,||σTI

= (2e

h)Js,||JTI,3D

=tTI

ls,||. (4.7)

For convenience, we have explicitly included here the latter (far right-hand

side) representation in terms of the eective (or assumed) thickness of the

55

TI charge layer, tTI, relative to the previously discussed average spin-transfer

distance within the TI, ls,||. σs,|| ≡ Js,||/E is the TI version of the spin Hall con-

ductivity, where Js,|| is the component of spin current density injected across

the interface into the nanomagnet, which is transverse to the FM magnetiza-

tion direction and E is the longitudinal electric eld. σTI ≡ JTI,3D/E is the 3D

conductivity of the TI, where JTI,3D is the eective charge current density in

the TI per unit area, JTI,2D/tTI, where JTI,2D is current density per unit width.

e is the electronic charge, and h is the reduced Planck's constant. To illustrate

σs,|| here, we consider the y-component of the spin current density injected

across the interface, Js,||, with the FM magnetized along the +x direction, and

an applied voltage of 0.2 V dropped (nonuniformly) along the bilayer. To allow

more ready comparison to prior experiments [36], we use the value of tTI that

was used in that work, 8 nm. Figure 4.3(a) shows variation of spin torque ratio

with varying normalized interlayer coupling obtained in terms of the nominal

TI conductivity as in Ref. [36]. Because the TI conductivity changes due to

interlayer current, in Figures 4.3(a) and 4.3(b), respectively, we have shown

the spin torque ratio as a function of interlayer coupling with respect to the TI

conductivity absent eects of STT for reference, and with respect to TI con-

ductivity reduced because of STT. In Figure 4.3(b), what is shown is the local

value obtained in the center of the nanomagnet, but the result varies very little

(<1%) with position. In contrast, Figure 4.3(a) shows the ratio of the average

value of Js,|| divided by the average value of the electric eld, to the xed TI

conductivity in the absence of the interlayer transport (nominal conductiv-

56

ity), as calculated in Ref. [36]. For comparison, the reported experimentally

determined spin torque ratio from Ref. [36] with (not coincidentally) similar

parameters is in the range of 2 to 3.5 [36], which lies between the results of

Figures 4.3(a) and 4.3(b). The comparison is imperfect in either case. Regions

near the beginning and end of the FM dominate the torque for this nanoscale

system (to be elaborated on shortly), presumably unlike for the much larger

system considered in Ref. [36], and the current ow is actually much less in

the TI layer within the center in the simulated system than would be expected

based on the nominal TI conductivity, as assumed in Ref. [36]. However, con-

sistent with Eq. (4.7) here, the experimental spin torque ratios reported in

Ref. [36] suggest that the spin-transfer length may have been signicantly less

than nominal mean free path in the TI and, thus, that STT may have been

the dominant eective source of scattering in the TI. Therefore, those reported

spin torque ratios are arguably conservative estimates, qualitatively consistent

with the dierence between Figures 4.3(a) and 4.3(b).

From an application point of view, a perhaps more important gure of

merit may be the spin injection eciency, the amount of spin angular momen-

tum provided in units of h/2 per electron injected into the bilayer system,

γs = (2e

h)Is,||Ibilayer

. (4.8)

Is,|| is the y-component of the total spin current injected across the entire

TI-FM interface area, with, for the results of Figure 4.3(c) below, the FM

magnetized in the +x direction. In the denominator of Eq. (4.8), Ibilayer is the

57

total charge current drawn by the TI-FM bilayer. Specically, Figure 4.3(c)

shows the spin injection eciency plotted as a function of normalized inter-

layer conductance for an applied voltage of 0.2 V. γs rst increases rapidly

with the interlayer conductance. However, once shunting becomes signicant

the injection eciency tends toward saturation, if still at a value greater than

unity. Essentially, as alluded to above, as the shunting increases, one reaches

a pattern where injected electrons enter the FM at one end from the excess

of +x directed electrons, providing a -y directed spin in these spin helically

locked states, and is extracted at the other end into the dearth of -x directed

electrons, extracting a +y directed spin, which is equivalent to injecting an-

other -y directed spin. However, little if any spin injection occurs between the

ends because transport in the TI is otherwise avoided.

As previously noted, we consider sources of torque on the nanomagnet,

that due to the physically-injected spin current density across the interface that

we simply label STT" consistent with the mechanism STT used in conven-

tional STT memory, and torque on the FM because of the exchange coupling

between the TI surface moment and the FM that we refer to as the eld-like

torque" (FLT). This latter component acts like an eective magnetic eld act-

ing on the magnet. With the magnetization and the spin injection across the

interface in-plane initially, STT component produces an in-plane (∥) torque,

while the FLT produces a plane-normal (⊥) torque. However, with an out-of-

plane magnetization component, the torque direction becomes mixed. How-

ever, for charge current ow in the x-direction, both remain nonzero as long as

58

the magnetization orientation of the FM is not entirely along the y-direction.

The direction of the STT vector τ|| is given by m × (m × Is) where, m is

the time-varying FM magnetization direction and Is is the spin polarization

direction of the injected current across TI-FM interface. The direction of FLT

τ⊥ is given by y ×m, where y is the unit vector along y. Ultimately, both

terms vanish only when the magnetization is parallel to the TI surface spin

orientation, which is along the y-direction.

With both the charge current ow in the TI and the magnetization

orientation of the FM in the +x-direction as for Figure 4.3(d), the magnitude

of STT per unit moment of the FM, τ||, is estimated as [36],

τ|| =Is,||

MsVFM. (4.9)

Here, Ms and VFM are the saturation magnetization per unit volume and the

volume of the FM, respectively. Is,|| is the magnitude of the y-component of

the total spin current injected across the interface into the nanomagnet. The

normalized magnitude of the FLT per unit moment, τ⊥, is estimated as,

τ⊥ =HexsTI

µBM2D

. (4.10)

In Eq. (4.10), Hex is the eective exchange energy, µB is Bohr-magneton,

sTI = JTI,2D/ev is number of TI surface spins per unit area, v is the Fermi

velocity in the TI, andM2D is areal density of the FM moments. The quantity

Hex/M2D eectively determines the strength of exchange coupling between the

FM moment and the spins at the TI surface. We calculate M2D as MstFM/µB,

59

where tFM is the thickness of the FM. Both Hex and the interlayer coupling

used in the Golden Rule calculation should be proportional to the extent of

wave function overlap across the interface. As an illustrative example, Hex is

taken to be the same as the (variable) coupling energy between the TI and the

FM used to calculate the interlayer transport. However, the STT contribution

is proportional to the square of the interlayer coupling, while the FLT varies

only linearly with this coupling. The maximum value of Hex (∼0.1 eV as for

the STT term) when coupled to the volume of a single surface ferromagnetic

moment, Ms/µB ≈ 8 Å3, produces a value of Hex/M2D of ∼0.4 eV-Å2, which

is conservative if still reasonably consistent with previously used theoretical

values [60, 70] in the range of 2-13 eV-Å2.

Figure 4.3(d) shows the magnitudes of the STT and FLT for an ap-

plied voltage of 0.2 V and the FM magnetized in +x-direction, as a function

of the normalized interlayer transparency. Experimental work suggests that

the torques may be comparable [36], but little should be read into the relative

strengths of these processes as simulated here given the above uncertainties.

However, while the STT term mirrors the quasi-saturation behavior of the spin

injection eciency as expected, after an initial rise, the FLT actually decreases

with increasing interlayer transparency. This dierence lies in the linear de-

pendence of the FLT on the coupling energy, vs. the quadratic dependence of

STT on the coupling energy. As a result, the shunting of current out of the

TI with respect to increasing interlayer transparency increases faster than the

FLT for a given current within the TI. Also, as previously noted, we did not

60

consider the FLT contributions to the TI conductivity. To do so presumably

would have further increased the shunting and further decreased both STT

and FLT somewhat.

0 5 10 15 20 25 300

100

200

300

FM current TI current

Cur

rent

(A)

Position (nm)

0.0 0.2 0.4 0.6 0.8 1.00.5

0.6

0.7

Bi

-laye

r res

ista

nce

(k)

Normalized interlayer conductance0 5 10 15 20 25 300

100

200

300

400

Cur

rent

(A)

Position (nm)

FM current TI current

0 5 10 15 20 25 300

100

200

300

Cur

rent

(A)

Position (nm)

FM current TI current

a) b)

c) d)0.0 0.2 0.4 0.6 0.8 1.0

0

5

10

15

20

25

TI re

sist

ivity

(k)

Normalized interlayer conductance

Figure 4.2: Transport in the bilayer for xed FM magnetization along +xand an applied voltage of 0.2 V for (a) small coupling (normalized interlayerconductivity = 1/100), (b) moderate coupling (normalized interlayer conduc-tivity = 1/9), and (c) Large coupling (normalized interlayer conductivity =1). (d) Resistance of the bilayer vs. interlayer conductance. Inset in (d): TIresistivity as a function of the normalized interlayer conductance. This gurehas been reproduced from Ref. [4] with permissions.

61

4.2.2 Self-consistent transport and magnetization dynamics calcu-

lation

A saturation magnetization per unit volume Ms of 1,100 kA/m and a

damping constant α of 0.01 are assumed to simulate the magnetization dynam-

ics of the FM. The STT vector is calculated from the component of the injected

(or extracted) spin current vector that is transverse to the FM magnetization

direction. We solve for the magnetization dynamics of the FM self-consistently

with the transport model, using the Landau-Lifshitz-Gilbert (LLG) equation

with the added STT and FLT terms:

dm

dt= −γm× (Heff + τ⊥y) + α(m× dm

dt)

−γ(m× ((Is/MsVFM)×m)).

(4.11)

Here, m is the unit vector along the FM magnetization direction, γ is the

magnitude of the gyromagnetic ratio, Heff is the eective magnetic eld due

to the shape anisotropy of the FM, which is added with the eective magnetic

eld leading to the FLT, τ⊥y. The negative sign in front of the STT term in

Eq. (4.11), (m× ((Is/MsVFM)×m) ≡ (Is − (Is ·m)m)/MsVFM), results from

the magnetic moment of the incoming electrons being in the opposite direction

to the electron spin angular momentum. [24] The total injected spin current

vector Is is obtained from,

Is =2π

h

∑k′,k,σ

h

2sk′| ⟨ψk′|H|ψk,σ⟩ |2

×(fF,k,σ − fT,k′)δ(ϵk,σ − ϵk′)

(4.12)

Here we have simply replaced the velocity vk′ in Eq. (4.6) with the spin

angular momentum (h/2)sk′ , where sk′ is the unit vector for the spin direction

62

in the TI.

We model the FM as a macrospin, thereby neglecting spatially varying

spin-texture (micromagnetic eects) in the FM. Taking micromagnetic eects

into account could aect our quantitative results. However, for simplicity and

to focus on identifying trends and providing intuitive understanding, we em-

ploy the macrospin approximation. The FM has a thermal stability factor

of 66 kBT at room temperature (where kB is Boltzmann's constant and T is

absolute temperature), as estimated using demagnetization factors calculated

using the model proposed by Aharoni, [71] to ensure thermal stability at room

temperature as required for non-volatile memory applications. The results of

self-consistent calculation of the DD transport equations and magnetization

dynamics are shown in Figures 4.4(a) and 4.4(b) for two opposite polarities of

the applied voltage, where mx, my, mz are x, y and z components of the FM

magnetization. The switching time is dened here as the time taken for my to

cross zero from the time the write pulse is turned on. An initial magnetiza-

tion direction unit vector of [0.0141, -0.9999, 0] is assumed for magnetization

switching from -y to +y, and similarly an initial magnetization direction unit

vector of [0.0141, 0.9999, 0] is assumed for switching from +y to -y. For

magnetization strictly along the +y or -y directions, both STT and FLT are

zero as noted earlier; a small o-axis component of magnetization is required

to initiate the switching process, as would normally be provided by thermal

uctuation.

Write energy per write operation, as has been shown in Figures 4.4(c)

63

Table 4.1: Parameters assumed for transport in the TI-FM bilayerParameter Value

2D resistivity of TI 2.19×103 ΩMomentum relaxation time in the TI 3.89× 10−14 sFermi Level of the TI w.r.t Dirac point 0.4 eV

Fermi velocity of the TI 4× 105 m/sFermi level of the FM

w.r.t band minima of spin down band 2.5 eVExchange splitting of the

of FM between up and down spin bands 2.15 eVSpin down to

spin up ip time for the FM 1 psMinimum resistivity of the FM 8.9× 10−8 Ω-m

Conductivity polarization of the FM -0.45Density of states polarization of the FM -0.45Eective mass of electron in the FM 0.8 × 9.1 × 10−31 kg

and 4.4(d), is calculated assuming the write pulse is long enough so that the

magnetization, for example for switching from -y to +y, settles above 0.9999.

For a given coupling strength between the TI and the FM, there is a minimum

value of the applied voltage along the TI-nanomagnet layer below which the

STT is unable to ip the FM, as illustrated by the plot of inverse switching

time vs. voltage in the inset of Figure 4.4(c). For a given voltage across the

bilayer, there is a minimum critical value of the interlayer coupling below which

the STT is unable to ip the FM, as illustrated by the plot of inverse switching

time with normalized interlayer coupling in the inset of Figure 4.4(d).

As previously discussed, the spin injection eciency tends to saturate

with increasing interlayer conductivity due to the shunting of current out of

64

the TI layer through the metallic nanomagnet layer. If such shunting could be

avoided, injection eciency and device performance for strong interlayer con-

ductivity could be substantially improved, as has been illustrated in Figure

4.5 by decreasing the in-plane conductivity within the ferromagnet. Figures

4.5(a) and 4.5(b) show the bilayer resistance and spin injection eciency as

a function of interlayer conductance, respectively, for an applied voltage of

0.2 V, with the FM magnetized in the +x-direction, and FM conductivity re-

duced by a factor of 10,000 compared to that of iron that was used in Table

4.1. Figure 4.5(c) shows corresponding write energies and switching times.

The spin injection eciency, increases monotonically, instead of saturating.

The switching time values remain the same as the switching times for normal

FM conductivity, but the write energy per write operation decreases by ap-

proximately one order of magnitude compared to Figure 4.4(d). Essentially,

the same amount of torque is provided for a given voltage drop, even if the

voltage drop is distributed somewhat dierently along the bilayer, but the cur-

rent shunting, which decreases the overall resistance and, thus, increases the

current and power consumption without providing any additional torque, is

avoided. Figure 4.5(d) shows write energy as a function of the FM sheet resis-

tivity, the FM resistivity divided by its thickness. The left most data point in

Figure 4.5(d) corresponds to the bulk resistivity of iron, 8.9 × 10−8 Ω-m. In

these simulations, substantial improvements are achieved only when the FM

sheet resistivity becomes comparable to the nominal resistivity of the TI sur-

face charge layer, and near saturation when the FM sheet resistivity reaches

65

the self-consistently reduced conductivity of the TI charge layer. Thus, the

eective conductivity of the FM layer relative to that of the TI layer becomes

a critical design consideration for TI-based STTRAM.

Achieving sheet resistivities in the metal comparable to those of the TI

is a non-trivial issue, and one that we do not attempt to resolve. We, however,

do note that thin metallic layers are already limited by the interactions with

the surface when the layer thickness falls below the mean free path of the

electron in the metal. In this regard, our use of a bulk resistivity for iron

represents a pessimistic limit even for iron. Starting with high resistivity FM

materials would help, of course. In Ref. [36], the 3D conductivity for an 8

nm thick permalloy was measured to be about an order of magnitude less

than the that of the iron reference here, and to be a decreasing function of

thickness, so that it presumably would be less for the 2 nm thickness considered

here. These changes in FM resistivity would get us into the transition region in

Figure 4.5(d). Deliberately producing a rough surface or otherwise deliberately

reducing the thin lm conductivity could further help. We also note that our

simulation tool, at least explicitly, does not take into account the eects of

interlayer transport on the FM conductivity. Much as for the TI layer in

the presence of the FM layer, the FM resistivity for very thin lms may also

be substantially larger than in its absence. Use of an insulating thin lm

FM such as YIG to eliminate shunting and relying on FLT might be ideal,

provided sucient exchange coupling, and should be explored. However, YIG

is less technologically developed. A lower saturation magnetization could aect

66

scalability, subject to thermal stability. Also, use of an insulating FM could

aect the read mechanism, not otherwise discussed here. These latter topics

are subjects for further exploration.

In the context of STTRAM scaling, the bilayer has a resistance of ∼522

Ω for normal FM resistivity and maximum interlayer coupling (Figure 4.2(d)),

so that an applied voltage of 0.2 V produces a current of ∼380 µA. For in-

creased FM resistivity due to interlayer transport with maximum interlayer

coupling, the bilayer has a resistance of ∼7.25 kΩ (Figure 4.5(a)). Therefore,

an applied voltage of 0.2 V corresponds to a current of 27.6 µA. Here, we

have calculated the write energy, which is eectively decided by the current

drawn by the bilayer. (Also, we believe, making the magnet partially perpen-

dicular, similar to [10], will lead to better write performance for the TI-based

devices as well). Therefore, comparing with the projections in the Interna-

tional Technology Roadmap for semiconductors, [23] the TI-FM STTRAM bit

could enable downscaling of the transistors and, thus, STTRAM memory if

the high-resistive FM limit can be approached directly or eectively.

4.3 Conclusion

We have considered a thermally-stable, conducting nanomagnet sub-

ject to spin-polarized current injection and exchange coupling from the spin-

helically locked surface states of a TI to evaluate possible non-volatile STTRAM

applications. We calculated switching time and energy consumed per write

operation using self-consistent transport, spin-torque, and magnetization dy-

67

namics calculations. The calculated switching energies and times compare

favorably to conventional spin-torque memory schemes for substantial inter-

layer conductivity. Nevertheless, we nd that shunting of current from the TI

to metallic nanomagnet can substantially limit eciency. Exacerbating the

problem, spin-transfer-torque from the TI eectively increases the TI resistiv-

ity. We show that for optimum performance, the sheet resistivity of the FM

layer should be comparable to or larger than that of the TI surface layer. Thus,

the eective conductivity of the FM layer becomes a critical design consider-

ation for possible TI-based non-volatile STTRAMs. The bulk FM resistivity

values used in the high conductivity limits are certainly exaggerations for thin

lms with carriers subject to surface scattering. However, to achieve eective

sheet conductivities in the FM as low as that of the TI will require creative

solutions if metallic FMs are to be used. Finally, we emphasize that our focus

has been on identifying essential device physics, trends, and design considera-

tions. For various reasons including the use of perhaps conservative interlayer

coupling and exchange energies, the quantitative results presented here, even

in the limit of high FM resistivity, should not be considered to represent the

limit of performance for TI/metallic FM based STTRAM.

68

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

To

rque

(T)

Normalized interlayer conductance

STT FLT

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

Spin

inje

ctio

n ef

ficie

ncy

Normalized interlayer conductance

0.0 0.2 0.4 0.6 0.8 1.00

2

4

Spin

torq

ue ra

tio w

.r.t.

mod

ified

TI c

ondu

ctiv

ity

Normalized interlayer conductance0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

Spin

torq

ue ra

tio w

.r.t.

nom

inal

TI c

ondu

ctiv

ity

Normalized interlayer conductance

a) b)

c) d)

Figure 4.3: Variation in device metrics with varying interlayer coupling, withthe FM magnetized along the +x-direction and an applied voltage of 0.2 V. (a)spin torque ratio (spin Hall angle) obtained with respect to nominal TI conduc-tivity. (b) spin torque ratio with respect to TI conductivity self-consistentlydecreased by the process of exerting STT. (c) Spin injection eciency, (d)Spin-transfer-torque (STT) via injection across the interface, and eld-liketorque (FLT) via exchange coupling between the spin-polarized electrons inthe TI and the nanomagnet. This gure has been reproduced from Ref. [4]with permissions.

69

0.05 0.10 0.15 0.200

3

6

9W

rite

ener

gy (p

J)

t s (ns)

Voltage (V)

0.05

0.10

0.1 0.2

0.5

1.0

t s -1 (n

s-1)

Voltage (V)

0.0 0.5 1.0 1.5 2.0-1.0

-0.5

0.0

0.5

1.0

Mag

netiz

atio

n

Time (ns)

mx

my

mz

0.0 0.4 0.80

4

8

12

t s (ns)

Normalized interlayer conductance0.0

0.2

0.4

0.6

0.8

W

rite

ener

gy (p

J)d)c)

b)

0.0 0.4 0.80.0

0.5

1.0

t s-1 (n

s)-1

Normalized interlayer conductance

0.0 0.5 1.0 1.5 2.0-1.0

-0.5

0.0

0.5

1.0

Mag

netiz

atio

n

Time (ns)

mx my mz

a)

Figure 4.4: (a) Switching of the FM from +y to -y for an applied voltage of-0.2 V, so that current is in -x-direction. (b) Switching of the FM from -yto +y for an applied voltage of 0.2 V, so that current is in +x-direction. (c)Switching time ts (dened in the text) and write energy per write operation vs.applied voltage across the bilayer, for the normalized interlayer conductivityof 1. The TI fails to switch the FM below a critical voltage as can be seenfrom the plot of inverse of switching time with voltage in the inset of (c). (d)Switching time ts (dened in the text) and write energy per write operation vs.normalized interlayer conductivity, for an applied voltage of 0.2 V. Similarly,the TI fails to switch the FM below a critical interlayer conductivity, as canbe seen from the plot of inverse of switching time with coupling in the insetof (d). This gure has been reproduced from Ref. [4] with permissions.

70

0.1 1 10 1000.00

0.02

0.04

0.06

0.08

0.10

0.12B. TI resistivity for maximum interlayer coupling

B

Writ

e en

ergy

(pJ)

resistivity/thickness of the FM (k )

A

A. nominal TI resistivity

0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

Writ

e en

ergy

(fJ)

t s (ns)

Normalized interlayer conductance

1

10

100

d)c)

b)

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

Bi-la

yer r

esis

tanc

e (k

)

Normalized interlayer conductance0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

Spin

inje

ctio

n ef

ficie

ncy

Normalized interlayer conductance

a)

Figure 4.5: a) bilayer resistance, b) Spin injection eciency and c) Switchingtime and write energy per write operation for articially increased FM in-planeresistivity and varying interlayer coupling d) Write energy as a function of theFM resistivity divided by the FM thickness. This gure has been reproducedfrom Ref. [4] with permissions.

71

Chapter 5

Write error rate in STTRAM

The probability that a bit does not switch when a write current pulse

is applied is referred to as the write error rate (WER). For reliable operation

as a working memory the WER needs to be smaller than 10−9 in presence of

a error correction code (ECC) present in the chip or less than 10−18 if there

is no ECC [31]. The eect of thermal uctuations in the STT induced rever-

sal is two fold. First, thermal eects give rise to a probability distribution

associated with the initial magnetization of the bit, i.e., the magnetization of

the free layer when the write pulse is applied. Secondly, thermal uctuation

is equivalent to a random thermal magnetic eld acting on the magnetization

during the reversal. As has been discussed in Chapter 1 the two eects have

dierent relative importance depending on which regime the bit is being oper-

ated in, namely the precessional, the thermal or the dynamic regime [26]. As

a result of stochastic thermal eects, the switching of a bit is also a stochastic

phenomenon, as opposed to being deterministic.1

1Some of the results in this chapter has been presented in the 73rd Device ResearchConference, 2015, titled Write error rate in spin-transfer-torque random access memoryincluding micromagnetic eects by Urmimala Roy, David L. Kencke, Tanmoy Pramanik,Leonard F. Register, Sanjay K. Banerjee. U. Roy ran the micromagnetic simulations withhelps from D. L. Kencke and T. Pramanik. L. F. Register and S. K. Banerjee supervisedthe work. All authors reviewed and commented on the results.

72

In Figure 5.1, write error has been shown using stochastic micromag-

netic simulation. When the pulse is turned of at 11 ns, the trials which have

not nished switching, go back to their initial magnetization and cause write

error.

0 2 4 6 8 10 12 14 16 18−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (ns)

mz

Write error

Figure 5.1: Write error from micromagnetic simulations. 20 trials are shown,each simulated with a random thermal seed for the random thermal magneticeld. A current pulse is applied at 5 ns, starting from uniform magnetizationin +z at 0 ns. The current pulse is turned of at 11 ns, the trials which areabove mz=0 at the time the current pulse is turned o, go back to the initialmagnetization of mz=1 and cause an error.

73

5.1 Motivation

For a perpendicular bit, WER has been studied using Fokker-Planck

(FP) calculations [72] previously. WER has also been modeled using Landau-

Lifshitz-Gilbert (LLG) simulations for the magnetization dynamics including a

random thermal magnetic eld and an STT term, for an in-plane bit with and

without perpendicular magnetic anisotropy (PMA) [73]. These studies made

the assumption that the free layer magnetization can be approximated as a

single spin (the macrospin approximation), thereby neglecting micromagnetic

eects. Spatially varying spin-texture, however, gives rise to several important

eects such as sub-volume excitations [74]. Higher order spin-wave modes

might be related to experimentally observed branching of the WER [75,76].

5.1.1 Challenges for WER simulation using micromagnetic eects

Suppose a probability p of some event has to be estimated using Monte-

Carlo simulation. A model of the system is then simulated n times and at

each trial it is recorded if the event occurs or not [77]. Here the system under

consideration is inherently stochastic due to the random thermal uctuations

thereby requiring Monte-Carlo simulation. We simulate the STT switching of

a bit with a random train of thermal eld, so each time a trial is simulated the

magnetization follows a dierent trajectory, even for a same applied current

and hence, same STT.

In the simplest case the n trials are independent of each other and

condence interval and relative error of the simulated probability p with

74

respect to the target probability p are to be estimated. It can be shown that

the relative error (RE) will be given by: RE≈ z/(√n√p) if p << 1 [77] (z

will be 1.96 for a 95% condence interval). Hence, to ensure a small enough

relative error for a low probability event, a large sample size (a large value of

n) is required. For example, if a probability p=10−9 needs to be estimated

with a RE of < 10% a sample size greater than 3.84×1011 will be needed by

this approach [77]. For full micromagnetic simulations including a random

thermal eld, this is a prohibitively large computational burden.

5.2 Results

5.2.1 Fokker-Planck calculation of WER within the macrospin ap-

proximation

Here, we rst look at WER for a range of device dimensions within

the macrospin approximation. Fokker-Planck method for PMA bits has been

employed following a previous report [72]. We consider PMA devices with

circular cross-sections. The parameters considered are damping constant α of

0.01, PMA Ku = 8 × 105 J/m3, saturation magnetization Ms = 1100 × 103

A/m. Thickness of the free layer is kept constant at 1 nm while varying

the device diameter D. The spin polarization direction of the injected spin

current perpendicular to the plane of the free layer is assumed to be in z

(out-of-the-plane) direction with a spin polarization η of 0.4. The FP method

[72] uses the thermal stability factor ∆ as a parameter, also predicting little

dependence of the WER on ∆ if the physical time t is normalized by the

75

parameter t0 = (1 + α2)/αγµ0HK(with dimension of time) where, anisotropy

eld HK is given by the equation HK = 2Keff/µ0Ms and γ is the gyromagnetic

ratio. Keff is the eective PMA, Keff=Ku − 0.5× µ0M2s (Nzz −Nxx). Thermal

stability factor ∆ is given by ∆ = (V × Keff)/kBT . Here [Nxx Nyy Nzz]

are the demagnetization tensor calculated using a previous work [71]. Also,

the applied current I for the devices of varying diameter D is normalized by

Ic0=(4αe∆kBT )/ηh (e=electronic charge, h=Planck's constant). In Figures

5.2 and 5.3 respectively we show non-switched fraction for D=40 nm and 80

nm with ∆ of 36.4 and 104.6 respectively as a function of the normalized time

t/t0 and the physical time t. Figure 5.2 conrms previous prediction [72] of

little dependence of Pns on D (and hence on ∆), except for a small parallel

shift. Pns is calculated by the total probability that the bit has not crossed the

mz=0 point, starting from mz=1 (mz is the z component of the magnetization

normalized by Ms).

5.2.2 WER simulation including micromagnetic eects

5.2.2.1 Method

After looking at scaling of WER using FP calculations within the

macrospin approximation, we include micromagnetic eects in the WER cal-

culations. We use the micromagnetic simulator MUMAX3 [78] to simulate

non-switched fraction from 1000 simulated trials as a function of time. MU-

MAX3 is an open source micromagnetic simulation software which uses the

graphics processing unit (GPU) to do the computation. In addition to the de-

76

0 2 4 6 8 10 12

10−6

10−4

10−2

100

t/t0

Pns

D=80 nm, i=2D=40 nm, i=2D=80 nm, i=6D=40 nm, i=6

(a)

Figure 5.2: WER from Fokker-Planck method, for diameter D=40 nm andD=80 nm, for normalized current i = I/Ic0 of 2 and 6, as a function ofnormalized time t/t0. This gure has been reused from Ref. [5].

magnetizing, exchange and crystalline anisotropy elds, MUMAX3 is used in

this chapter to include the random thermal uctuation eld [78]. The random

thermal eld is given by [78,79]:

Btherm = η(step)

√2µ0αkBT

Bsatγδvδt(5.1)

In Equation 5.1, η is a random vector with a normal probability distribution,

δv is the cell volume and δt is the time step. A xed time step of 10−14 s is

used.

77

0 10 20 30 40

10−6

10−4

10−2

100

time (ns)

Pns

D=80 nm, i=2D=40 nm, i=2D=80 nm, i=6D=40 nm, i=6

Figure 5.3: WER from Fokker-Planck method, for diameter D=40 nm andD=80 nm, for normalized current i = I/Ic0 of 2 and 6, as a function ofphysical time. This gure has been reused from Ref. [5].

5.2.2.2 Results

In Figures 5.4 and 5.5, we plot Pns, the number of trials out of the 1000

trials in which the magnetization has not crossed themz=0 point, as a function

of time t for devices of D=40 nm and D=80 nm respectively. Before the write

pulse is applied, the perpendicular bit is allowed to relax for 5 ns starting from

a spatially uniform magnetization (mz=1). After the relaxation the magneti-

zation of the free layer has a distribution due to random thermal uctuation,

so as to ensure a physically meaningful initial condition when the write pulse

78

is applied. It has been veried that having more trials in the simulations than

1000 does not alter the Pns vs. time t plot until Pns∼10−2. Little should be

read from the lower than Pns∼10−2 tail of the plots that suers from a lack of

statistics. Also, to show the impact of including micromagnetic eects, instead

of assuming the free layer as a macrospin, we plot corresponding predictions

of WER from FP calculations. As can be seen from Figures 5.4 and 5.5,

macrospin calculations over-estimate the WER, because with micromagnetic

eects turned on, it is now much easier for the STT switching to get started.

The spatially varying spin-texture now allows the STT to be non-zero some-

where in the free layer, which starts the STT induced magnetization dynamics.

In Figure 5.6, the non-switched fractions from micromagnetic stochastic LLG

for D=20 nm, 40 nm and 80 nm are shown as function of t/t0.

The cell size for all micromagnetic simulations are 2.5 nm × 2.5 nm. As

can be seen from Figure 5.2 and Figure 5.6, micromagnetic simulations predict

a dierent scaling of WER as a function of normalized time t/t0 than Fokker-

Planck (macrospin) simulations. The arrows indicate increasing diameter of

the bits.

5.3 Conclusion

This chapter uses a sample size of 1000 for the WER simulations in-

cluding micromagnetic eects that might not be enough even for predicting a

Pns∼10−2. Simulation of the low probability tail (Pns∼10−6 or smaller) will

require a large eective sample size for Monte-Carlo type simulations. This

79

0 2 4 6 8 10 12 1410

−3

10−2

10−1

100

time (ns)

P ns

i=2, Micromagnetic

i=6, Micromagnetic

i=2, Fokker−Planck

i=6, Fokker−Planck

Figure 5.4: WER from macrospin (Fokker-Planck) calculations compared withmicromagnetic calculations for D=40 nm. This gure has been reused fromRef. [5].

calculation will require further research, possibly using strategies used for rare

event simulation using Monte-Carlo methods.

80

0 5 10 15 20 2510

−3

10−2

10−1

100

time (ns)

P ns

i=2, Micromagnetic

i=6, Micromagnetic

i=2, Fokker−Planck

i=6, Fokker−Planck

Figure 5.5: WER from macrospin (Fokker-Planck) calculations compared withmicromagnetic calculations for D=80 nm. This gure has been reused fromRef. [5].

81

0 0.5 1 1.5 2 2.5 3 3.5 4

10−2

10−1

100

t/t0

P ns

Increasing diameter(D=20 nm, 40 nm, 80 nm)

i=2i=6

Figure 5.6: Scaling of WER from micromagnetic simulations for D=20 nm to80 nm as a function of normalized time t/t0 for normalized currents i=2 andi=6. This gure has been reused from Ref. [5].

82

Chapter 6

Conclusion

6.1 Summary

In this dissertation, several routes towards lower write current and write

energy and increased packing density of STTRAM have been studied. In Chap-

ters 2, 3 and 4, we ignored the eects of thermal uctuations and focused on

identifying basic device physics to look for methods to improve, i.e., decrease

eectively the zero temperature critical current density Jc0 for switching a

magnet of given thermal stability. In Chapter 5, we described modeling meth-

ods to predict probability of not switching when a current pulse is applied,

including thermal eects. Major contributions include:

• Chapter 2 describes fabrication and characterization by point-contact

experiments of unpatterned thin lm spin valves with in-plane, perpen-

dicular and canted xed and free layers [2]. Non-collinear magnetizations

in the free and xed layers by having canted easy axis in either the free

or xed layer might provide a route towards shorter switching time for a

given write current or smaller write current for a given switching time.

However, as a result of having a spin polarization direction of the injected

current dierent than the easy axis of the free layer, this approach might

83

lead to worse write error rate performance. This issue will require further

work to be investigated in detail.

• Chapter 3 deals with proposal of a multi-bit STTRAM cell to potentially

improve packing density in the most commonly used 1T-1MTJ STTRAM

cell [3]. As has been discussed in Section 3.3, the high current density

for reliable switching predicted from micromagnetic simulations of STT

switching of a ferromagnetic cross might lead to negligible improvement

in packing density compared to single bit schemes. A possible solution

might be using canted magnetization in the free layer and use of voltage

controlled magnetic anisotropy (VCMA) [1].

• The possibility of using a topological insulator as the polarizer layer for

a three-terminal STTRAM bit in a spin Hall geometry is considered in

Chapter 4. However, using a three-terminal geometry rather than a two-

terminal MTJ, although it has advantages as pointed out in Chapter 4,

might require a larger cell area to be realized [13]. We nd that a signif-

icant improvement in write performance can be achieved in a TI based

STTRAM only if the free layer magnet eectively has comparable resis-

tivity (after converting to appropriate units, since the TI is eectively a

surface charge layer, whereas the magnet is a 3D conductor, even though

a thin lm) as the TI. Identifying suitable material systems for this pur-

pose will require further investigation. As pointed out briey in Chapter

4, another possible route towards improving performance might be using

84

partially perpendicular free layer, along the lines of previous works on

TI-based STTRAM [61]. Also, in Chapter 4, micromagnetic eects have

been ignored although there is non-uniform spin injection in the magnet.

• In Chapter 5, we model write error rate (WER) using Fokker-Planck cal-

culations within the macrospin approximation following previous works

[72]. We also simulate WER using micromagnetic simulations in order to

identify inuence of micromagnetic eects on the WER. Ideally a WER

as low as 10−9 needs to be simulated or experimentally predicted so as

to ensure a current density larger than that required for a WER smaller

than that is applied. However, a major challenge to predict WER ∼10−9

with suciently large condence (small enough condence interval and

relative error) is the large number of samples required (number of trials

to be simulated), which often introduces a prohibitively heavy computa-

tional burden. For example, in Chapter 5 we use a sample size of 1000

which might not be large enough even for predicting WER∼10−2. Sim-

ulating a suciently small WER to predict the current density needed

and also, to identify the reasons behind the extremely slow switching

cases (thereby being a bottleneck for the write pulse duration) will

require further research. Also, being able to simulate the rare events

with suciently high condence will allow us to investigate the anoma-

lous behaviour of WER, namely branching of the WER graph in larger

detail.

85

6.2 Future directions

In the light of discussion in the previous paragraphs, possible future

research directions can be listed:

• Multi-bit STTRAM using canted magnetization in the free layer can be

studied in order to reduce switching current from a perfectly in-plane

structure.

• The interlayer coupling energies for the TI-FM interlayer transport are

assumed to be constant and is assumed to be an adjustable parameter in

Chapter 4. A more realistic estimate will require experimental and(or)

theoretical (possibly using density functional theory) determination of

the interlayer coupling strength for material systems of interest.

• Identication of a suitable TI-FM bilayer material system that will oer

a suciently transparent interface for spin and charge injection from

the TI to the FM, while the FM remains weakly conducting to in-plane

transport in the plane of the FM will require extensive theoretical and

experimental investigation.

• In Chapter 4, micromagnetic eects have not been taken into account.

Micromagnetic eects might lead to interesting observations. Inclusion

of micromagnetic eects self-consistently with the transport model will

require further research.

86

• Determination of the low probability tail (Pns∼10−6 or less) demands

a large eective sample size for Monte-Carlo type simulations, as has

been explained in Chapter 5. This will require further research, possibly

employing strategies used for rare event simulation using Monte-Carlo

techniques.

87

Bibliography

[1] P. K. Amiri and K. L. Wang, Voltage-controlled magnetic anisotropy in

spintronic devices, SPIN, vol. 02, no. 03, p. 1240002, 2012.

[2] U. Roy, H. Seinige, F. Ferdousi, J. Mantey, M. Tsoi, and S. K. Baner-

jee, Spin-transfer-torque switching in spin valve structures with perpen-

dicular, canted, and in-plane magnetic anisotropies, Journal of Applied

Physics, vol. 111, no. 7, p. 07C913, 2012.

[3] U. Roy, T. Pramanik, M. Tsoi, L. F. Register, and S. K. Banerjee, Micro-

magnetic study of spin-transfer-torque switching of a ferromagnetic cross

towards multi-state spin-transfer-torque based random access memory,

Journal of Applied Physics, vol. 113, no. 22, p. 223904, 2013.

[4] U. Roy, R. Dey, T. Pramanik, B. Ghosh, L. F. Register, and S. K.

Banerjee, Magnetization switching of a metallic nanomagnet via current-

induced surface spin-polarization of an underlying topological insulator,

Journal of Applied Physics, vol. 117, no. 16, p. 163906, 2015.

[5] U. Roy, D. L. Kencke, T. Pramanik, L. F. Register, and S. K. Banerjee,

Write error rate in spin-transfer-torque random access memory including

micromagnetic eects, in Device Research Conference (DRC), 2015 73rd

Annual, pp. 147148, June 2015.

88

[6] M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petro, P. Etienne,

G. Creuzet, A. Friederich, and J. Chazelas, Giant magnetoresistance

of (001) Fe/(001) Cr magnetic superlattices, Physical Review Letters,

vol. 61, no. 21, p. 2472, 1988.

[7] G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Enhanced mag-

netoresistance in layered magnetic structures with antiferromagnetic in-

terlayer exchange, Physical Review B, vol. 39, no. 7, p. 4828, 1989.

[8] S. S. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant,

and S.-H. Yang, Giant tunnelling magnetoresistance at room temperature

with MgO (100) tunnel barriers, Nature Materials, vol. 3, no. 12, pp. 862

867, 2004.

[9] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, Large magne-

toresistance at room temperature in ferromagnetic thin lm tunnel junc-

tions, Physical Review Letters, vol. 74, pp. 32733276, Apr 1995.

[10] M. Julliere, Tunneling between ferromagnetic lms, Physics Letters A,

vol. 54, no. 3, pp. 225 226, 1975.

[11] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando, Giant

room-temperature magnetoresistance in single-crystal Fe/MgO/Fe mag-

netic tunnel junctions, Nature Materials, vol. 3, no. 12, pp. 868871,

2004.

89

[12] T. Miyazaki and N. Tezuka, Giant magnetic tunneling eect in Fe/Al2O3/Fe

junction, Journal of Magnetism and Magnetic Materials, vol. 139, no. 3,

pp. L231 L234, 1995.

[13] A. D. Kent and D. C. Worledge, A new spin on magnetic memories,

Nature Nanotechnology, vol. 10, no. 3, pp. 187191, 2015.

[14] J. Liu, B. Jaiyen, R. Veras, and O. Mutlu, Raidr: Retention-aware intel-

ligent dram refresh, in Computer Architecture (ISCA), 2012 39th Annual

International Symposium on, pp. 112, June 2012.

[15] D. Apalkov, A. Khvalkovskiy, S. Watts, V. Nikitin, X. Tang, D. Lottis,

K. Moon, X. Luo, E. Chen, A. Ong, A. Driskill-Smith, and M. Krounbi,

Spin-transfer torque magnetic random access memory (STT-MRAM),

Journal on Emerging Technologies in Computing Systems, vol. 9, pp. 13:1

13:35, May 2013.

[16] H. Yoda, S. Fujita, N. Shimomura, E. Kitagawa, K. Abe, K. Nomura,

H. Noguchi, and J. Ito, Progress of STT-MRAM technology and the

eect on normally-o computing systems, in IEEE International Electron

Devices Meeting (IEDM), 2012, pp. 113, IEEE, 2012.

[17] J. Slonczewski, Current-driven excitation of magnetic multilayers, Jour-

nal of Magnetism and Magnetic Materials, vol. 159, no. 1-2, pp. L1 L7,

1996.

90

[18] L. Berger, Emission of spin waves by a magnetic multilayer traversed by

a current, Physical Review B, vol. 54, pp. 93539358, Oct 1996.

[19] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and

P. Wyder, Excitation of a magnetic multilayer by an electric current,

Physical Review Letters, vol. 80, pp. 42814284, May 1998.

[20] A. Driskill-Smith, Latest advances and future prospects of STT-RAM,

2010.

[21] E. Y. Tsymbal and I. uti¢, eds., Handbook of Spin Transport and Mag-

netism. Chapman and Hall/CRC, 2012.

[22] D. Apalkov, S. Watts, A. Driskill-Smith, E. Chen, Z. Diao, and V. Nikitin,

Comparison of scaling of in-plane and perpendicular spin transfer switch-

ing technologies by micromagnetic simulation, IEEE Transactions on

Magnetics, vol. 46, no. 6, pp. 22402243, June.

[23] www.itrs.net, 2013.

[24] D. Ralph and M. Stiles, Spin transfer torques, Journal of Magnetism

and Magnetic Materials, vol. 320, no. 7, pp. 1190 1216, 2008.

[25] W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M. MacLaren, Spin-

dependent tunneling conductance of Fe|MgO|Fe sandwiches, Physical Re-

view B, vol. 63, p. 054416, Jan 2001.

91

[26] Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. Wang, and

Y. Huai, Spin-transfer torque switching in magnetic tunnel junctions and

spin-transfer torque random access memory, Journal of Physics: Con-

densed Matter, vol. 19, no. 16, p. 165209, 2007.

[27] M. Nakayama, T. Kai, N. Shimomura, M. Amano, E. Kitagawa, T. Na-

gase, M. Yoshikawa, T. Kishi, S. Ikegawa, and H. Yoda, Spin trans-

fer switching in TbCoFe/CoFeB/MgO/CoFeB/TbCoFe magnetic tunnel

junctions with perpendicular magnetic anisotropy, Journal of Applied

Physics, vol. 103, no. 7, pp. , 2008.

[28] S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and

E. E. Fullerton, Current-induced magnetization reversal in nanopillars

with perpendicular anisotropy, Nature Materials, vol. 5, p. 210, 2006.

[29] K. N. et al. in MMM-Intermag 2010, pp. EF07, IEEE, 2010.

[30] E. Kitagawa, S. Fujita, K. Nomura, H. Noguchi, K. Abe, K. Ikegami,

T. Daibou, Y. Kato, C. Kamata, S. Kashiwada, et al., Impact of ultra

low power and fast write operation of advanced perpendicular MTJ on

power reduction for high-performance mobile cpu, in IEEE International

Electron Devices Meeting (IEDM), 2012, pp. 294, IEEE, 2012.

[31] A. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R. Beach, A. Ong,

X. Tang, A. Driskill-Smith, W. Butler, P. Visscher, et al., Basic principles

of STT-MRAM cell operation in memory arrays, Journal of Physics D:

Applied Physics, vol. 46, no. 7, pp. 7400174020, 2013.

92

[32] J. Z. Sun, Spin-current interaction with a monodomain magnetic body:

A model study, Physical Review B, vol. 62, pp. 570578, Jul 2000.

[33] L. Liu, T. Moriyama, D. Ralph, and R. Buhrman, Spin-torque ferromag-

netic resonance induced by the spin hall eect, Physical Review Letters,

vol. 106, no. 3, p. 036601, 2011.

[34] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman,

Spin-torque switching with the giant spin hall eect of tantalum, Sci-

ence, vol. 336, no. 6081, pp. 555558, 2012.

[35] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topo-

logical insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single dirac cone

on the surface, Nature Physics, vol. 05, pp. 438 442, 2009.

[36] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun,

M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C.

Ralph, Spin-transfer torque generated by a topological insulator, Na-

ture, vol. 511, pp. 449451, 2014.

[37] Y. Fan, P. Upadhyaya, X. Kou, M. Lang, S. Takei, Z. Wang, J. Tang,

L. He, L.-T. Chang, M. Montazeri, G. Yu, W. Jiang, T. Nie, R. N.

Schwartz, Y. Tserkovnyak, and K. L.Wang, Magnetization switching

through giant spin-orbit torque in a magnetically doped topological insu-

lator heterostructure, Nature Materials, vol. 13, pp. 699 704, 2014.

93

[38] T. Kishi, H. Yoda, T. Kai, T. Nagase, E. Kitagawa, M. Yoshikawa,

K. Nishiyama, T. Daibou, M. Nagamine, M. Amano, S. Takahashi, M. Nakayama,

N. Shimomura, H. Aikawa, S. Ikegawa, S. Yuasa, K. Yakushiji, H. Kubota,

A. Fukushima, M. Oogane, T. Miyazaki, and K. Ando, Lower-current

and fast switching of a perpendicular TMR for high speed and high den-

sity spin-transfer-torque MRAM, IEEE International Electron Devices

Meeting, 2008.

[39] D. C. Worledge, G. Hu, P. L. Trouilloud, D. W. Abraham, S. Brown, and

M. C. Gaidis, Switching distributions and write reliability of perpendic-

ular spin torque MRAM, IEEE International Electron Devices Meeting,

2010.

[40] O. G. Heinonen and D. V. Dimitrov, Switching-current reduction in

perpendicular-anisotropy spin torque magnetic tunnel junctions, Jour-

nal of Applied Physics, vol. 108, p. 014305, 2010.

[41] A. Kent, B. Özyilmaz, and E. Del Barco, Spin-transfer-induced preces-

sional magnetization reversal, Applied Physics Letters, vol. 84, no. 19,

pp. 38973899, 2004.

[42] D. E. Nikonov, G. I. Bouriano, G. Rowlands, and I. N. Krivorotov,

Strategies and tolerances of spin transfer torque switching, Journal of

Applied Physics, vol. 107, no. 11, p. 113910, 2010.

[43] C. Zha and J. Åkerman, Study of pseudo spin valves based on l10 (111)-

oriented FePt and FePtCu xed layer with tilted magnetocrystalline anisotropy,

94

IEEE Transactions on Magnetics, vol. 45, no. 10, pp. 34913494, 2009.

[44] F. Ferdousi, Device Design and Process Integration of High Density Non-

volatile Memory Devices. PhD dissertation, University of Texas at Austin,

Department of Electrical and Computer Engineering, 2011.

[45] Z. Wei, Spin-Transfer-Torque Eect in Ferromagnets and Antiferromag-

nets. PhD dissertation, University of Texas at Austin, Department of

Physics, 2008.

[46] H. Seinige, Magnetic Switching and Magnetodynamics driven by Spin

Transfer Torque. MA dissertation, University of Texas at Austin, De-

partment of Physics, 2011.

[47] A. Chen, Feasibility analysis of high-density STTRAM designs with

crossbar or shared transistor structures, in 72nd Annual Device Research

Conference (DRC), 2014, pp. 295296, June 2014.

[48] T. Ishigaki, T. Kawahara, R. Takemura, K. Ono, K. Ito, H. Matsuoka,

and H. Ohno, A multi-level-cell spin-transfer torque memory with series-

stacked magnetotunnel junctions, in VLSI Technology (VLSIT), 2010

Symposium on, pp. 4748, IEEE, 2010.

[49] X. Lou, Z. Gao, D. V. Dimitrov, and M. X. Tang, Demonstration of

multilevel cell spin transfer switching in MgO magnetic tunnel junctions,

Applied Physics Letters, vol. 93, no. 24, p. 242502, 2008.

[50] D. V. Dimitrov, Z. Gao, and X. Wang. U.S. Patent No. 7834385.

95

[51] J. L. Leal and M. H. Kryder, Spin valves exchange biased by Co/Ru/Co

synthetic antiferromagnets, Journal of Applied Physics, vol. 83, no. 7,

pp. 37203723, 1998.

[52] Y. Zhou, J. Åkerman, and J. Z. Sun, Micromagnetic study of switching

boundary of a spin torque nanodevice, Applied Physics Letters, vol. 98,

no. 10, p. 102501, 2011.

[53] J. A. Osborn, Demagnetizing factors of the general ellipsoid, Phys. Rev.,

vol. 67, pp. 351357, Jun 1945.

[54] I. N. Krivorotov, D. V. Berkov, N. L. Gorn, N. C. Emley, J. C. Sankey,

D. C. Ralph, and R. A. Buhrman, Large-amplitude coherent spin waves

excited by spin-polarized current in nanoscale spin valves, Physical Re-

view B, vol. 76, p. 024418, Jul 2007.

[55] C.-Y. You, Reduced spin transfer torque switching current density with

non-collinear polarizer layer magnetization in magnetic multilayer sys-

tems, Applied Physics Letters, vol. 100, no. 25, p. 252413, 2012.

[56] M. J. Donahue and D. G. Porter, OOMMF user's guide. National Insti-

tute of Science and Technology Report No. NISTIR 6376 (unpublished),

1999.

[57] Z. Li and S. Zhang, Magnetization dynamics with a spin-transfer torque,

Physical Review B, vol. 68, p. 024404, Jul 2003.

96

[58] S. Homan, Y. Tserkovnyak, P. K. Amiri, and K. L. Wang, Magnetic bit

stability: Competition between domain-wall and monodomain switching,

Applied Physics Letters, vol. 100, no. 21, p. 212406, 2012.

[59] T. Pramanik, U. Roy, M. Tsoi, L. F. Register, and S. K. Banerjee, Micro-

magnetic simulations of spin-wave normal modes and the spin-transfer-

torque driven magnetization dynamics of a ferromagnetic cross, Journal

of Applied Physics, vol. 115, no. 17, pp. , 2014.

[60] I. Garate and M. Franz, Inverse spin-galvanic eect in the interface be-

tween a topological insulator and a ferromagnet, Physical Review Letters,

vol. 104, p. 146802, Apr 2010.

[61] Y. Lu and J. Guo, Quantum simulation of topological insulator based

spin transfer torque device, Applied Physics Letters, vol. 102, no. 7, pp. ,

2013.

[62] T. Yokoyama, J. Zang, and N. Nagaosa, Theoretical study of the dy-

namics of magnetization on the topological surface, Physical Review B,

vol. 81, p. 241410, Jun 2010.

[63] T. Yokoyama, Current-induced magnetization reversal on the surface of

a topological insulator, Physical Review B, vol. 84, p. 113407, Sep 2011.

[64] P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman,

and J. S. Moodera, Exchange-coupling-induced symmetry breaking in

97

topological insulators, Physical Review Letters, vol. 110, p. 186807, Apr

2013.

[65] A. Kandala, A. Richardella, D. W. Rench, D. M. Zhang, T. C. Flana-

gan, and N. Samarth, Growth and characterization of hybrid insulat-

ing ferromagnet-topological insulator heterostructure devices, Applied

Physics Letters, vol. 103, no. 20, pp. , 2013.

[66] L. A. Wray, S.-Y. Xu, Y. Xia, D. Hsieh, A. V. Fedorov, Y. S. Hor, R. J.

Cava, A. Bansil, H. Lin, and M. Z. Hasan, A topological insulator sur-

face under strong coulomb, magnetic and disorder perturbations, Nature

Physics, vol. 7, pp. 3237, 2011.

[67] T. Yokoyama and Y. Tserkovnyak, Spin diusion and magnetoresistance

in ferromagnet/topological-insulator junctions, Physical Review B, vol. 89,

p. 035408, Jan 2014.

[68] S. Zhang, P. M. Levy, and A. Fert, Mechanisms of spin-polarized current-

driven magnetization switching, Physical Review Letters, vol. 88, p. 236601,

May 2002.

[69] M. D. Stiles and A. Zangwill, Anatomy of spin-transfer torque, Physical

Review B, vol. 66, p. 014407, Jun 2002.

[70] J.-J. Zhu, D.-X. Yao, S.-C. Zhang, and K. Chang, Electrically control-

lable surface magnetism on the surface of topological insulators, Physical

Review Letters, vol. 106, p. 097201, Feb 2011.

98

[71] A. Aharoni, Demagnetizing factors for rectangular ferromagnetic prisms,

Journal of Applied Physics, vol. 83, no. 6, pp. 34323434, 1998.

[72] W. Butler, T. Mewes, C. Mewes, P. Visscher, W. Rippard, S. Russek, and

R. Heindl, Switching distributions for perpendicular spin-torque devices

within the macrospin approximation, IEEE Transactions on Magnetics,

vol. 48, pp. 46844700, Dec 2012.

[73] R. Heindl, W. H. Rippard, S. E. Russek, and A. B. Kos, Physical lim-

itations to ecient high-speed spin-torque switching in magnetic tunnel

junctions, Physical Review B, vol. 83, p. 054430, Feb 2011.

[74] J. Z. Sun, R. P. Robertazzi, J. Nowak, P. L. Trouilloud, G. Hu, D. W.

Abraham, M. C. Gaidis, S. L. Brown, E. J. O'Sullivan, W. J. Gallagher,

and D. C. Worledge, Eect of subvolume excitation and spin-torque ef-

ciency on magnetic switching, Physical Review B, vol. 84, p. 064413,

Aug 2011.

[75] T. Min, Q. Chen, R. Beach, G. Jan, C. Horng, W. Kula, T. Torng,

R. Tong, T. Zhong, D. Tang, P. Wang, M. min Chen, J. Sun, J. DeBrosse,

D. Worledge, T. Matt, and W. Gallagher, A study of write margin of

spin torque transfer magnetic random access memory technology, IEEE

Transactions on Magnetics, vol. 46, pp. 23222327, June 2010.

[76] E. R. Evarts, R. Heindl, W. H. Rippard, and M. R. Pufall, Correlation

of anomalous write error rates and ferromagnetic resonance spectrum in

99

spin-transfer-torque-magnetic-random-access-memory devices containing

in-plane free layers, Applied Physics Letters, vol. 104, no. 21, p. 212402,

2014.

[77] G. Rubino, B. Tun, et al., Rare event simulation using Monte Carlo

methods. Wiley Online Library, 2009.

[78] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez,

and B. Van Waeyenberge, The design and verication of MuMax3, AIP

Advances, vol. 4, no. 10, pp. , 2014.

[79] W. F. Brown, Thermal uctuations of a single-domain particle, Journal

of Applied Physics, vol. 34, no. 4, pp. 13191320, 1963.

100

Vita

Urmimala Roy received a Bachelor's degree in Electronics and Tele-

Communication from Jadavpur University, Kolkata in 2007 and Master's de-

gree in Microelectronics and VLSI from Indian Institute of Technology of Bom-

bay, Mumbai in 2009 in India. She started her doctoral studies at the Univer-

sity of Texas (UT) at Austin in Fall of 2010. Before joining doctoral studies,

she worked as a visiting researcher at University of Bologna in Bologna, Italy.

In the summer of 2014, she worked as a technical intern in the process modeling

group at Intel Corp.

Permanent address: urmimala@utexas.edu

This dissertation was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth's TEX Program.

101