UNIVERSITY OF CALIFORNIA

Los Angeles

A Magnetic Tunnel Junction Compact

Model for STT-RAM and MeRAM

A project report submitted in partial satisfaction

of the requirements for the degree

Master of Science in Electrical Engineering

by

Daniel Scott Matic

2013

This project was performed under the direction of Professor Kang Wang

and in collaboration with fellow M.S. student Dheeraj Srinivasan

© Copyright by

Daniel Scott Matic

2013

i

TABLE OF CONTENTS

1. Introduction ..................................................................................................................... 1

1.1 Motivation for STT-RAM and MeRAM ........................................................................ 1

1.2 Report Outline ............................................................................................................ 4

2. The Magnetic Tunnel Junction ................................................................................... 5

2.1 Magnetic Anisotropy and Device Structure ................................................................. 5

2.2 Resistance Hysteresis and TMR ................................................................................ 6

2.3 Switching Regimes ..................................................................................................... 8

3. The Compact Model ..................................................................................................... 10

3.1 The Landau-Lifshitz-Gilbert Equation ...................................................................... 10

3.2 Spin and Field-Like Torque ...................................................................................... 12

3.3 External and Demagnetization Fields ....................................................................... 13

3.4 Voltage Controlled Magnetic Anisotropy .................................................................. 14

3.5 Thermal Noise ........................................................................................................... 16

3.6 Heun’s Method .......................................................................................................... 16

4. Simulating MTJ Switching ......................................................................................... 18

4.1 STT Switching ............................................................................................................ 19

4.2 VCMA Field-Assisted Switching ............................................................................... 19

4.3 VCMA + STT Thermally Activated Switching ........................................................... 21

4.4 VCMA Precessional Switching ................................................................................. 22

ii

Conclusion ........................................................................................................................... 25

References............................................................................................................................ 27

Appendix .............................................................................................................................. 30

iii

LIST OF FIGURES

1.1 SEM of a 400nm MTJ and the Stable MTJ States .......................................................... 3

2.1 In-Plane MTJ and Perpendicular MTJ ............................................................................ 6

2.2 Energy Wells and R-H Loop ............................................................................................. 7

2.3 Switching Regimes for an STT-RAM MTJ ....................................................................... 8

3.1 MTJ Illustration for Compact Model .............................................................................. 10

3.2 Precessional and Damping Motion ................................................................................. 11

3.3 Spin and Field-Like Torque Motion ............................................................................... 13

3.4 VCMA Experimental Results and Visual Aid .................................................................. 15

4.1 Typical STT-RAM Cell .................................................................................................... 19

4.2 VCMA Field Assisted RP RAP and RAP RP ............................................................... 20

4.3 VCMA + STT RP RAP and RAP RP ............................................................................. 21

4.4 Illustration of Precessional Switching ........................................................................... 23

A.1 STT Switching Simulation Results ................................................................................ 30

A.2 Close-Up of STT Switching ............................................................................................. 31

A.3 VCMA Field-Assisted Switching Simulation Results..................................................... 32

A.4 VCMA + STT Thermally Activated Switching Results (RP RAP, RP) .......................... 33

A.5 VCMA + STT Thermally Activated Switching Results (RAP RP, RAP) ......................... 34

A.6 Parrallel State VCMA Precessional Switching Simulation Results ................................ 35

A.7 Anti-Parallel State VCMA Precessional Switching Simulation Results ......................... 36

iv

LIST OF TABLES

1.1 A Comparison of Traditional and Magnetic Memory Technologies................................ 2

4.1 Universal MTJ Simulation Parameters .......................................................................... 18

4.2 Parameters for STT Switching Simulation ..................................................................... 19

4.3 Parameters for VCMA Field-Assisted Switching Simulation ........................................ 20

4.4 Parameters for VCMA + STT Thermally Activated Switching Simulation .................... 22

4.5 Parameters for VCMA Precessional Switching Simulation ........................................... 24

v

ABSTRACT OF THE PROJECT REPORT

A Magnetic Tunnel Junction Compact

Model for STT-RAM and MeRAM

by

Daniel Scott Matic

Master of Science in Electrical Engineering

University of California, Los Angeles, 2013

Professor Kang Wang, Advisor

This report presents a compact model of a magnetic tunnel junction (MTJ) for use in the

design and simulation of spin-transfer torque (STT) and magnetoelectric random access memory

(MeRAM). In STT-RAM, the magnetic state of an MTJ is switched by applying a spin-polarized

current through the junction. In MeRAM, the MTJ state is manipulated primarily by a voltage-

induced electric field. Both of these competing state-of-the-art non-volatile magnetic memory

technologies offer significant area and energy advantages over traditional SRAM, DRAM, and

Flash memories. In this work, the physics behind an MTJ is explored to develop a Verilog-A

model that captures the switching dynamics and accounts for the following phenomena: spin-

torque, field-like torque, external magnetic fields, shape anisotropy, the voltage-controlled

magnetic anisotropy (VCMA) effect, and thermal noise. Simulations were performed in Cadence

to confirm correct switching operation of an STT-RAM MTJ and junctions designed for MeRAM

switching in both the precessional and thermally activated regimes.

1

CHAPTER 1

Introduction

STT-RAM and MeRAM are two new and promising magnetic memory technologies that

offer significant advantages over existing memories. STT-RAM works on the principle of the spin-

transfer torque effect whereas MeRAM operates by exploiting the VCMA effect. This chapter

offers a brief motivation for STT-RAM and MeRAM, provides more detail on the benefits of each

technique, and outlines the rest of the report.

1.1 Motivation for STT-RAM and MeRAM

The three most mature and prevalent memory technologies are static RAM (SRAM),

dynamic RAM (DRAM), and Flash memory. They all have different fundamental principles of

operation, and thus each excels in a couple of performance metrics, but falls short in others.

Consequently, the memory hierarchy of a typical system must integrate all three types for optimal

performance, area, and cost. SRAM works by using cross-coupled CMOS inverters to drive the

cell output lines to the desired logical levels. This approach, which leverages positive feedback,

makes SRAM the fastest of the aforementioned technologies with read and write speeds in the

GHz range. Unfortunately, the speed comes at the cost of very low cell density (each SRAM cell

requires six transistors), and leakage current is becoming an increasing problem with transistor

scaling. DRAM, on the other hand, has a higher density because each cell only requires a single

transistor and a capacitor to store charge (the presence or absence of which indicates a logical one

or zero). However, DRAM requires a periodic refresh cycle to prevent data loss from capacitor

leakage, and it is fabricated using a separate process from that of standard CMOS. Flash is the

2

only non-volatile memory of the three and it works by tunneling electrons onto the gate of a

floating gate transistor. The presence or absence of electrons on the floating gate alters the

transistor’s threshold voltage, and this can be detected as a logical one or zero. Flash has the

smallest area and a very fast read speed, but the write operation requires high internal voltages

(to create hot tunneling electrons) and is very slow. Lastly, even though SRAM and DRAM are

volatile, their memory endurance is very high compared to the limited read and write cycle

lifetime of Flash.

STT-RAM and MeRAM each have the potential to become a true “universal” memory

providing the speed of SRAM, the density of DRAM, and the non-volatility of Flash. This kind of

memory would significantly increase performance and decrease cost and power consumption in

systems that integrate multiple application specific memories. Even among these two candidates,

MeRAM promises to be substantially more area and energy efficient than STT-RAM. This opens

up the additional exciting possibility of using MeRAM to create non-volatile logic because the

switching energy per bit is competitive with modern CMOS.

Table 1.1: A Comparison of Traditional and Magnetic Memory Technologies [1]

In general, magnetic memory works by storing binary information in the magnetic

moment of a ferromagnetic material. The basic magnetic memory storage element is called a

magnetic tunnel junction (MTJ) and its structure consists of two ferromagnetic layers separated

by a thin nonconductive tunneling barrier. An SEM picture of an actual 400nm MTJ nanopillar

is shown on the next page.

3

Figure 1.1: SEM of a 400nm MTJ from [2] (Left) and the MTJ Stable States (Right)

One of the ferromagnetic layers (called the fixed layer) has a permanent magnetization and the

other layer (called the free layer) has a magnetization that is free to change. The MTJ has two

stable states: the parallel state in which the free and fixed layer magnetic moments align, and the

anti-parallel state in which the free layer magnetization is in the opposite direction to that of the

fixed layer. The parallel configuration results in a low MTJ resistance (denoted as RP) and the

anti-parallel configuration yields a high MTJ resistance (denoted as RAP). In a well-engineered

MTJ, the ratio of these two resistances is large enough to read RP as a logical zero and RAP as a

logical one or vice versa. The difference between magnetic memory technologies is the method

used to switch the magnetization of the free layer. In STT-RAM, the state is switched by applying

enough current through the MTJ. In MeRAM, the MTJ state is switched primarily by applying a

voltage across the junction.

The main goal of this project was to design a compact Verilog-A model to aid the design of

STT-RAM and MeRAM. Such a model is necessary to correctly simulate MTJ switching dynamics,

meet memory control circuit timing, predict temperature effects, and to measure switching

probabilities. Without a model that is compatible with today’s modern circuit simulation tools, it

is impossible to design an actual working circuit.

Parallel

Anti-Parallel

4

1.2 Report Outline

Chapter 2 reviews the concept of magnetic anisotropy, delves into more detail about the

MTJ device structure, discusses resistance hysteresis, and concludes by describing the switching

regimes of an MTJ. Chapter 3 presents detailed information about the model itself, each of the

physical phenomena it captures, and the numerical solving method used. Chapter 4 presents

intuitive descriptions of the different ways to switch an MTJ in both STT-RAM and MeRAM. The

report concludes with a few words on some worthwhile future improvements to the model and all

Cadence simulation results are found in the Appendix.

5

CHAPTER 2

The Magnetic Tunnel Junction

The purpose of this chapter is to provide more detail about the most basic building block

of magnetic memory, the MTJ. Additional details about the structure of MTJs used in both STT-

RAM and MeRAM will be discussed. Resistance hysteresis and tunnel magnetoresistance ratio

will be explained and the concept of magnetic anisotropy will be reviewed so the reader can better

understand the VCMA effect presented in Chapter 3. Lastly, the most basic form of the Landau-

Lifshitz-Gilbert (LLG) equation is introduced to examine the precessional and thermally activated

switching regimes of an MTJ.

2.1 Magnetic Anisotropy and Device Structure

A basic understanding of magnetic anisotropy is essential to describe the MTJ device

structure and switching behavior. Magnetic anisotropy is defined as the dependence of magnetic

properties on a preferred direction. There are several kinds of anisotropy: magnetocrystalline,

shape, magnetoelastic, and exchange anisotropy. These arise from the structure of the crystal

lattice, the physical shape of the ferromagnet, any strain in the material, and from interactions

between ferromagnets and antiferromagnets, respectively [3]. For our purposes, it is sufficient to

understand that any combination of these anisotropy sources determines the free layer

ferromagnet’s energetically favorable direction of spontaneous magnetization, called the easy

axis. The opposite (i.e. positive and negative) directions along the easy axis are equivalent,

resulting in two stable magnetization states. In the case of an MTJ, these two states are the

parallel and anti-parallel configurations.

6

As mentioned briefly in Chapter 1, the most basic structure of an MTJ is two ferromagnetic

layers separated by an insulating tunneling oxide, where the magnetic moment of one layer is

fixed and the other can change freely. Figure 2.1 below shows two different types of MTJs, one

with an in-plane easy axis and the other that exhibits dominant perpendicular anisotropy, as well

as example material stacks for each from [2].

Figure 2.1: In-Plane MTJ (Left) and Perpendicular MTJ (Right)

As you can see, many different material layers of varying thicknesses (numbers are in nm) are

required to fabricate an MTJ. The three layers highlighted in red are the CoFeB ferromagnetic

layers and the MgO tunneling oxide. MeRAM uses perpendicular MTJs and the early generations

of STT-RAM MTJs were in-plane junctions (although newer versions are now also perpendicular

[4]). For the sake of demonstrating the versatility of the compact model, STT switching will be

simulated with an in-plane junction and MeRAM switching with a perpendicular MTJ.

2.2 Resistance Hysteresis and TMR

Magnetic anisotropy gives rise to an easy axis along which there exists two stable states

for the free layer magnetization. When no external magnetic field is applied (H = 0), these states

can be visualized as two energy wells separated by a potential barrier as shown below in Figure

2.2 on the left. For a certain magnitude of external magnetic field, called the coercivity or coercive

field 𝐻𝐶, the depth of the present state’s energy well decreases such that the opposite becomes

more energetically favorable, and the MTJ free layer switches states. The MTJ remains in this

Parallel

Anti-Parallel

Parallel

Anti-Parallel

7

new state until a coercive field in the opposite direction is applied. If the resistance is plotted

versus applied external field under zero bias voltage, an R-H loop like the one shown below is

observed which clearly indicates resistive hysteresis.

Figure 2.2: Energy Wells [2] (Left) and R-H Loop (Right)

The coercivity can be thought of as a measure of the potential barrier height or as the strength of

the magnetic anisotropy along the easy axis.

The parallel state results in a low resistance (RP) and the anti-parallel state yields a high

resistance (RAP). The physics behind the resistance change are beyond the scope of this report,

but one of the most important figures of merit for an MTJ is its tunnel magnetoresistance ratio

(TMR) which is defined mathematically below:

𝑇𝑀𝑅 =𝑅𝐴𝑃 − 𝑅𝑃

𝑅𝑃

It is desirable to have a high TMR (100% and above) because this allows for larger sensing margins

and lower read error rates in memory. With the CoFeB-MgO material system, TMRs of 604%

have been obtained at room temperature [5] but typical values are between 50 – 150%.

+HC

RP

RAP

-HC

RP RAP

RAP RP

8

2.3 Switching Regimes

The switching behavior of an MTJ is generally divided into two regimes: precessional and

thermally activated switching. Figure 2.3 below shows this distinction for an STT-RAM MTJ.

Figure 2.3: Switching Regimes for an STT-RAM MTJ [6]

Precessional switching occurs when more than the critical switching current flows through an

STT-RAM MTJ or when the perpendicular anisotropy in an MeRAM MTJ has been reduced below

a critical amount by the VCMA effect. The free layer magnetic moment switches states on a

nanosecond or sub-nanosecond time scale [7] and the dynamics are well described by the Landau-

Lifshitz-Gilbert (LLG) equation [8]:

𝑑�⃗⃗�

𝑑𝑡= −𝛾′(�⃗⃗� × �⃗⃗� 𝑒𝑓𝑓) − 𝛼𝛾′�⃗⃗� × (�⃗⃗� × �⃗⃗� 𝑒𝑓𝑓)

9

This equation will be discussed in more detail (including additional terms) in Chapter 3.

Thermally activated switching occurs when the STT current density is below the critical value or

when the perpendicular anisotropy in an MeRAM MTJ is only slightly reduced. In this case,

thermal noise agitation may be strong enough to jolt the free layer magnetization into the opposite

state over a timescale of tens of nanoseconds or longer [9].

10

CHAPTER 3

The Compact Model

This chapter examines every detail of the compact Verilog-A MTJ model and discusses

some of the physics behind each of the phenomena the model captures: spin and field-like torque,

external and demagnetization fields, the VCMA effect, and thermal noise.

3.1 The Landau-Lifshitz-Gilbert Equation

The full version of the Landau-Lifshitz-Gilbert (LLG) equation is presented below in S-I units and

models the basic MTJ structure shown in Figure 3.1.

Figure 3.1: MTJ Illustration for Compact Model (Modified from [6])

𝑑�⃗⃗�

𝑑𝑡= −𝛾′(�⃗⃗� × �⃗⃗� 𝑒𝑓𝑓) − 𝛼𝛾′�⃗⃗� × (�⃗⃗� × �⃗⃗� 𝑒𝑓𝑓) +

𝛾′ℏ𝑃𝐽

𝜇0𝑞𝑡𝑓𝑙𝑀𝑠

[�⃗⃗� × (�⃗⃗� × 𝑝 ) + (𝛽1 + 𝛽2 ∙ 𝐴 ∙ 𝐽)(�⃗⃗� × 𝑝 )]

�⃗⃗� 𝑒𝑓𝑓 = �⃗⃗� 𝑒𝑥𝑡 + �⃗⃗� 𝑑𝑒𝑚 + �⃗⃗� 𝑎𝑛𝑖 − �⃗⃗� 𝑉𝐶𝑀𝐴 + �⃗⃗� 𝑡ℎ [𝐴 𝑚⁄ ]

11

𝛾′ =𝛾

1+𝛼2 [𝑚 (𝐴 ∙ 𝑠)⁄ ] 𝑃 = √𝑇𝑀𝑅

𝑇𝑀𝑅+2

The LLG equation describes the motion of the free layer magnetization unit vector �⃗⃗� in the

presence of an effective magnetic field �⃗⃗� 𝑒𝑓𝑓. The third and fourth terms in the differential

equation account for spin and field-like torque, and all other effects are modelled in the five

components of �⃗⃗� 𝑒𝑓𝑓. 𝑝 is the unit vector of the fixed layer magnetization, 𝜇0 is the permeability

of free space [H/m], 𝛾 is the gyromagnetic ratio 𝛾𝑒𝜇0 [m/(A∙s)], and 𝛼 is the material-dependent

Gilbert damping constant.

The action of the first two terms in the LLG equation can be visualized with the help of

Figure 3.2 (shown for an in-plane junction).

Figure 3.2: Precessional and Damping Motion

The first term is responsible for precessional motion, the circular rotation around the unit sphere

at a given value along the x-axis. The second term creates a damping torque that forces �⃗⃗� to align

with �⃗⃗� 𝑒𝑓𝑓. The combination of these effects results in the magenta trajectory.

The motion of �⃗⃗� over time by itself is not useful for a circuit simulation. However, its

motion relative to 𝑝 (the fixed layer moment) determines the conductance of the MTJ according

to the Jullière model [10]:

Precessional motion

Damping motion

12

𝐺(𝜃) = 𝐺𝑇(1 + 𝑃2 cos 𝜃) + 𝐺𝑆𝐼 [1/Ω]

𝑃 is the spin-polarization factor, the percentage of electrons whose intrinsic angular momentum

(spin) align with magnetization direction as current flows through the MTJ. It is assumed that 𝑃

is the same for both the fixed and free layers. 𝐺𝑇 is the direct elastic tunneling conductance and

𝐺𝑆𝐼 is additional conductance resulting from imperfections in the MgO. 𝜃 is the angle between �⃗⃗�

and 𝑝 . The cosine of this angle is nothing more than the x-component of �⃗⃗� for an in-plane

junction, but it is equal to the z-component of �⃗⃗� in a perpendicular MTJ.

3.2 Spin and Field-Like Torque

The spin-transfer torque effect was discovered by J.C. Slonczewski in 1996 [11]. Prior to

this discovery, magnetic memory used Oersted-field switching in which the MTJ state was toggled

with a magnetic field generated by relatively large currents flowing in nearby interconnects [12].

Not only is this approach power hungry, but the switching current actually increases as MTJ

device dimensions are reduced. For STT-RAM, however, the critical switching current decreases

along with downward scaling. Recent experiments have also suggested an additional field-like

torque that can explain the absence of pre-switching oscillations, but its mathematical form is still

controversial [13]. The spin and field-like torque terms in the LLG equation are reproduced

below:

𝑑�⃗⃗�

𝑑𝑡= ⋯

𝛾′ℏ𝑃𝐽

𝜇0𝑞𝑡𝑓𝑙𝑀𝑠

[�⃗⃗� × (�⃗⃗� × 𝑝 ) + (𝛽1 + 𝛽2 ∙ 𝐴 ∙ 𝐽)(�⃗⃗� × 𝑝 )]

ℏ is the normalized Planck constant [J/s], 𝑃 is the same spin-polarization factor, 𝐽 is the current

density in the MTJ [A/m2], 𝑞 is the charge of an electron [C], 𝑡𝑓𝑙 is the thickness of the free layer,

𝑀𝑠 is the saturation magnetization [A/m], 𝛽1 and 𝛽2 [1/A] are the field-like torque constants, and

𝐴 is the MTJ area. Slonczewski explained that by passing a current through a magnetically

13

polarized layer one produces a spin-polarized current. If such a current is directed into another

magnetic layer, the angular momentum can be transferred to that layer, changing its orientation.

Figure 3.3: Spin and Field-Like Torque Motion

Figure 3.3 above depicts the effect of both spin and field-like torque on the motion of the free layer

magnetization �⃗⃗� . All this adds up to the following: sufficient current flowing from the fixed (free)

to the free (fixed) layer writes the MTJ into the anti-parallel (parallel) state [14].

3.3 External and Demagnetization Fields

External and demagnetization fields, �⃗⃗� 𝑒𝑥𝑡 and �⃗⃗� 𝑑𝑒𝑚, are the first two terms of the effective

magnetic field, �⃗⃗� 𝑒𝑓𝑓, in the LLG equation. In this model, any bias field necessary for experiments

can be programmed directly into �⃗⃗� 𝑒𝑥𝑡. Bias fields are required for some types of switching as will

be explained in Chapter 4. The demagnetization field in an MTJ originates from shape anisotropy

and is of the following form:

�⃗⃗� 𝑑𝑒𝑚 = −𝑀𝑠(�⃗⃗� ∙ �⃗⃗� ) [𝐴 𝑚⁄ ]

Its name and the negative sign indicate that this field reduces the total magnetic moment. The

free layer is assumed to be a very flat ellipsoid so the components of �⃗⃗� can be calculated using the

formulas in [15]:

Precessional motion

Damping motion

Spin-Torque

Field-Like Torque

p

14

𝑁𝑥 =𝑡𝑓𝑙

𝐿(1 − 𝑒2)1/2

𝐹 − 𝐸

𝑒2

𝑁𝑦 =𝑡𝑓𝑙

𝐿

𝐹 − (1 − 𝑒2)𝐸

𝑒2(1 − 𝑒2)1/2

𝑁𝑧 = 1 −𝑡𝑓𝑙

𝐿

𝐸

(1 − 𝑒2)1/2

F and E are the complete elliptic integrals of the first and second kind with the argument:

𝑒 = (1 −𝑊2

𝐿2 )

1/2

where W and L are the width and length of the MTJ. The demagnetization field components of

all MTJs used in this project are weak in the x and y-directions (see Figure 3.1 for the coordinate

system). This explains why the stable states for an in-plane MTJ are indeed in the x-y plane. For

perpendicular MTJs, however, the magnetocrystalline anisotropy is engineered to be stronger

than the shape anisotropy, leading to an easy axis that is in the z-direction.

3.4 Voltage Controlled Magnetic Anisotropy

It has been observed that the interface between an MgO tunneling oxide and a CoFeB

ferromagnetic layer can exhibit strong perpendicular magnetic anisotropy (PMA) [16-18].

Furthermore, it has been discovered that this PMA is sensitive to a voltage applied across the

MgO-CoFeB junction [19-21]. Using a voltage to modulate the PMA of an MTJ is called the voltage

controlled magnetic anisotropy (VCMA) effect and it is the basic principle of operation behind

MeRAM. As mentioned in Chapter 1, MeRAM has the potential to be significantly more energy

and area efficient compared with STT-RAM because fundamentally, no current flow is required

for the VCMA effect. The intrinsic PMA of the MTJ (�⃗⃗� 𝑎𝑛𝑖) and the VCMA effect (�⃗⃗� 𝑉𝐶𝑀𝐴) are the

third and fourth components of �⃗⃗� 𝑒𝑓𝑓 in the LLG equation.

15

�⃗⃗� 𝑎𝑛𝑖 =2𝐾𝑖

𝑡𝑓𝑙𝜇0𝑀𝑠𝑚𝑧�̂� [𝐴 𝑚⁄ ]

�⃗⃗� 𝑉𝐶𝑀𝐴 =2𝜉𝑉

𝜇0𝑀𝑠𝑑𝑀𝑔𝑂𝑡𝑓𝑙𝑚𝑧�̂� [𝐴 𝑚⁄ ]

𝐾𝑖 is the PMA constant [J/m2], 𝜉 is the VCMA constant [J/(V∙m)], 𝑉 is the voltage across the MTJ,

and 𝑑𝑀𝑔𝑂 is the thickness of the oxide layer. Notice that both terms are in the z-direction (out-of-

plane in Figure 3.1), the strength of each is proportional to 𝑚𝑧, and the VCMA term subtracts from

the PMA term in �⃗⃗� 𝑒𝑓𝑓. This means that applying a positive (negative) voltage across the MTJ

reduces (increases) its PMA, and thus reduces (increases) its coercivity. This is consistent with

experimental results from [22] shown below in Figure 3.4 on the left.

Figure 3.4: VCMA Experimental Results (Left) [22] and Visual Aid (Right) [23]

A nice visual aid of the VCMA effect is provided by [23], which illustrates that a voltage

modifies the energy barrier height relative to both states as opposed to a magnetic field, which

alters it in only one direction (see Figure 2.2). From [20], a VCMA constant as high as 37 fJ/(V∙m)

has been experimentally observed. Applying negative voltages across the MTJ, strengthening the

PMA and the stability of its present state, could be used to avoid any potential read disturbance

in MeRAM.

16

3.5 Thermal Noise

The last and final component of �⃗⃗� 𝑒𝑓𝑓 in the LLG equation is the thermal noise term �⃗⃗� 𝑡ℎ.

Thermal noise creates random fluctuations in the free layer magnetization and therefore in the

MTJ resistance.

�⃗⃗� 𝑡ℎ = 𝜎 √2𝑘𝐵𝑇𝛼

𝜇0𝑀𝑠𝛾′𝒱Δ𝑡

[𝐴 𝑚⁄ ]

𝑘𝐵 is the Boltzmann constant [J/K], 𝑇 is the temperature [K], 𝒱 is the volume of the free layer,

and Δ𝑡 is the simulation time step. 𝜎 is a unit vector whose x, y, and z components are independent

Gaussian random variables with mean = 0 and standard deviation = 1. These components are

produced using Verilog-A built-in random number generator functions. One of the major

consequences of thermal noise is that the switching behavior becomes probabilistic [24-25]. This

aspect of the model is essential for measuring switching probabilities and for simulating any

switching behavior in the thermally activated regime.

3.6 Heun’s Method

P. Horley et al. performed an investigation of numerical simulation techniques for solving

the LLG equation [26]. They concluded that a second order approach is required to obtain a

correct solution and that Heun’s method is a reasonable compromise between accuracy and

computation time. Given an ordinary differential equation of the form 𝑦′(𝑡) = 𝑓(𝑡, 𝑦(𝑡)) with

𝑦(𝑡0) = 𝑦0, the Fundamental Theorem of Calculus tells us the following:

𝑦(𝑡𝑖+1) = 𝑦(𝑡) + ∫ 𝑦′(𝑢)𝑡𝑖+1

𝑡𝑖

𝑑𝑢

The simplest way of numerically approximating the solution is Euler’s method:

𝑦𝑖+1 = 𝑦𝑖 + ℎ𝑓(𝑡𝑖, 𝑦𝑖)

17

where ℎ is the computation time step. This amounts to approximating the integral as a Riemann

sum, in other words it estimates the area under the function 𝑦′(𝑡) with a series of rectangles.

Heun’s method is more accurate because it approximates the integral using the Trapezoidal Rule.

�̃�𝑖+1 = 𝑦𝑖 + ℎ𝑓(𝑡𝑖, 𝑦𝑖)

𝑦𝑖+1 = 𝑦𝑖 +ℎ

2[𝑓(𝑡𝑖, 𝑦𝑖) + 𝑓(𝑡𝑖+1, �̃�𝑖+1)]

The real benefit here is that the accuracy of Heun’s method increases quadratically with a decrease

in the time step whereas the accuracy of Euler’s method only increases linearly [27]. For all these

reasons, the compact MTJ model uses Heun’s method to solve the LLG equation.

18

CHAPTER 4

Simulating MTJ Switching

This chapter takes an in-depth look at four different means of switching an MTJ: STT

current-induced switching, VCMA field-assisted switching, VCMA + STT thermally activated

switching, and high speed VCMA precessional switching. The model parameters used in each

Cadence simulation are tabulated and the results are shown in the Appendix. Below is a table of

parameters that are common across all the simulations.

Table 4.1: Universal MTJ Simulation Parameters

𝑳 𝑾 𝒅𝑴𝒈𝑶 𝒕𝒇𝒍 𝑻𝑴𝑹

170 nm 60 nm 1.1 nm 1.1 nm 100%

𝑻 𝜶 𝜸 𝑴𝒔 𝑮𝑺𝑰

300 K 0.02 221276 m/(A∙s) 1.2 × 106 A/m 0

𝑵𝒙 𝑵𝒚 𝑵𝒛 𝜷𝟏 𝜷𝟐

0.0045 0.0152 0.9803 0 0

Unfortunately, time constraints prevented any simulations demonstrating the field-like torque

effect, so it is not included in any of the results even though the functionality exists. Optimizing

𝛽1 and 𝛽2 should be the first order of business for improving the model.

19

4.1 STT Switching

As discussed in Chapter 3, passing a current through a magnetically polarized metal

creates a spin-polarized current that can transfer its polarization to another magnet via the spin-

transfer torque effect. Delivering a positive current greater than the critical switching current

from the fixed (free) to the free (fixed) layer switches the MTJ into the anti-parallel (parallel)

state. Table 4.2 shows the model parameters unique to the STT simulation and Figure 4.1 shows

a typical STT-RAM cell complete with an access transistor and a bit-line to pulse current through

the MTJ.

Table 4.2: Parameters for STT Switching Simulation

𝑴𝑻𝑱 𝑻𝒚𝒑𝒆 𝑹𝑷 𝑲𝒊 𝝃

In-plane 700 Ω 0 0

𝐾𝑖 is set to zero to eliminate any PMA to ensure the

easy axis is in-plane, and 𝜉 is set to zero to disable the

VCMA effect. Assuming VDD = 1V, a low resistance

MTJ (RAP = 1.4 KΩ) is required because a fairly high

current of 700 µA is necessary to flip states. The

simulation results are shown in pages 30 and 31 in the Appendix. The close-up waveforms show

evidence of the magnetic moment �⃗⃗� precessing before the spin-torque becomes dominant and

flips the state.

4.2 VCMA Field-Assisted Switching

Recall that when a positive voltage is applied across an MTJ built for use in MeRAM, the

VCMA effect reduces the coercivity, equivalently lowering the potential barrier between the two

Figure 4.1: Typical STT-RAM Cell [1]

20

stable states. This phenomena can be exploited to write an MTJ with the help of a small external

bias field. The process is best described pictorially using R-H loops:

Figure 4.2: VCMA Field-Assisted RP RAP (Left) and RAP RP (Right)

In the far left picture, the MTJ is stable in the RP state in the presence of a small positive bias field

at V = 0 V. However, the application of 1 V decreases the coercivity by enough such that the RP

state becomes unstable and the MTJ must switch to RAP. VCMA field-assisted switching is

possible with an 𝐻𝑏𝑖𝑎𝑠 that satisfies |𝐻𝐶0| > |𝐻𝑏𝑖𝑎𝑠| > |𝐻𝐶1| where 𝐻𝐶0 and 𝐻𝐶1 are the coercivities

at 0 and 1 V, respectively. This was the first kind of electric field-induced MTJ switching

demonstrated [22, 28]. A major practical limitation is the fact that the polarity of 𝐻𝑏𝑖𝑎𝑠 must be

reversed to switch in the opposite direction. Table 4.3 contains all the parameters specific to the

VCMA field-assisted switching simulation, and the results are found on page 32 in the Appendix.

Table 4.3: Parameters for VCMA Field-Assisted Switching Simulation

𝑴𝑻𝑱 𝑻𝒚𝒑𝒆 𝑹𝑷 𝑲𝒊

Perpendicular 100 KΩ 1.0056364 × 10-3 J/m2

𝝃 𝑯𝒃𝒊𝒂𝒔 𝑺𝒘𝒊𝒕𝒄𝒉𝒊𝒏𝒈 𝑽𝒐𝒍𝒕𝒂𝒈𝒆

37 fJ/(V∙m) ±50 Oe �̂� 1 V

This simulation requires thermal noise because the switching is thermally activated. The MTJ

switches from RP to RAP and vice versa in approximately 60 ns.

For the sake of simplicity, the STT term in the LLG equation was set to zero in this

simulation for two reasons. First, in principle this type of switching does not require any current

flow. Secondly, STT current shifts both of the 1 V R-H loops in Figure 4.2 horizontally to the left

+Hbias +Hbias V = 0V V = 1V V = 0V -Hbias -Hbias V = 1V

21

because positive current favors the anti-parallel state according to the current sign convention in

Figure 3.1. Switching in both directions is certainly still possible but would require asymmetric

𝐻𝑏𝑖𝑎𝑠 values. Including the STT effect in this case unnecessarily complicates the simulation and

adds little to no additional understanding.

4.2 VCMA + STT Thermally Activated Switching

VCMA field-assisted switching is unattractive because it requires both a positive and

negative bias field for reversible switching. It turns out that by incorporating the VCMA and STT

effects simultaneously, it is possible to achieve fully reversible switching using a single bias field

and only positive voltages. Figure 4.3 provides an intuitive graphic explanation of how this is

accomplished.

Figure 4.3: VCMA + STT RP RAP (Left) and RAP RP (Right)

The MTJ is stable in the RP state in the presence of a small negative bias field at V = 0V (①). The

application of a low voltage, 0.2 V in our simulation, does not switch the MTJ because the bias

field keeps it stable in the parallel configuration (②). The small STT current shifts the R-H loop

to the left (positive current favors the RAP state) but only slightly. However, 1.1 V induces a large

enough current to switch the MTJ via the STT effect, which collapses and translates the R-H loop

① ②

③

④

⑤

⑦

⑥

⑧

22

to the left beyond the value of −𝐻𝑏𝑖𝑎𝑠 (③-④). If the 1.1 V is removed, the MTJ remains stable in

the RAP state because the coercivity returns to its original value at V = oV (⑤). Applying a low

voltage in the anti-parallel state, though, reduces the coercivity by enough such that the bias field

pushes the magnetization into the parallel state (⑥). The MTJ will remain in the RAP mode in

the presence of 1.1 V because the STT current is large enough to force it to stay there (⑦-⑧).

One may still question the usefulness of this scheme because it appears to require a negative

external bias field. In fact, this bias field can be intentionally engineered into the magnetic field

of the fixed layer eliminating the need for any external stimuli besides a switching voltage.

The parameters specific to this simulation are shown in Table 4.4 and the results are found

on pages 33 and 34 in the Appendix.

Table 4.4: Parameters for VCMA + STT Thermally Activated Switching Simulation

𝑴𝑻𝑱 𝑻𝒚𝒑𝒆 𝑹𝑷 𝑲𝒊

Perpendicular 27.5 KΩ 0.985091 × 10-3 J/m2

𝝃 𝑯𝒃𝒊𝒂𝒔 𝑺𝒘𝒊𝒕𝒄𝒉𝒊𝒏𝒈 𝑽𝒐𝒍𝒕𝒂𝒈𝒆𝒔

37 fJ/(V∙m) -100 Oe �̂� 0.2, 1.1 V

The results show that the MTJ switches states in approximately 60 ns. The RP RAP current at

1.1 V is 40 µA and the RAP RP current at 0.2 V is 3.6 uA. This adds up to an average switching

current of only 21.8 uA, much smaller than that required for the STT-RAM MTJ.

4.3 VCMA Precessional Switching

As alluded to in Chapter 2, high speed switching is possible if the PMA of an MeRAM MTJ

is reduced below a critical amount via the VCMA effect. When this happens, the easy axis moves

from out-of-plane to in-plane and the free layer magnetic moment vector �⃗⃗� follows along the

magenta trajectory in Figure 3.1. This precessional motion results in an oscillation of the MTJ

23

resistance as it settles to the final value of 1/𝐺𝑇 given by 𝐺(𝜃) = 𝐺𝑇(1 + 𝑃2 cos 𝜃) + 𝐺𝑆𝐼 when 𝜃 =

90° and 𝐺𝑆𝐼 = 0. Ref. [29] suggests that if the voltage is removed when the resistance approaches

its minimum (maximum) value, i.e. when the z-component of �⃗⃗� is the most negative (positive),

the MTJ will relax to the parallel (anti-parallel) state. Figure 4.4 is an illustration of this switching

technique. The initial state and the trajectory of �⃗⃗� during the pulse are shown in red. The final

state and the motion of �⃗⃗� after the pulse is removed are shown in blue. Both vectors are tilted to

the left by a slight angle for visual clarity.

Figure 4.4: Illustration of Precessional Switching (Modified From [29])

The parameters used in the precessional switching simulation are shown in Table 4.5 and

the results are on pages 35 and 36 in the Appendix. Initial simulations showed that the exact

trajectory of �⃗⃗� is very sensitive to the initial conditions at the time the voltage pulse begins.

Thermal noise was causing large variations in the optimum time to switch off the voltage

(variations were large enough for a switching probability of only 80%). The inclusion of a negative

bias voltage of -100 Oe in the x-direction corrects this problem. For 100 simulations, the model

predicted switching to either state with pulse widths of 1.2 and 2.4 ns at 100% probability. Again,

this bias field is small enough that it could be engineered into the fixed layer.

τpulse = 1.2 ns τpulse = 2.4 ns τpulse = 3.6 ns τpulse = 4.8 ns τpulse = 5.2 ns

24

Table 4.5: Parameters for VCMA Precessional Switching Simulation

𝑴𝑻𝑱 𝑻𝒚𝒑𝒆 𝑹𝑷 𝑲𝒊

Perpendicular 100 KΩ 1.0056364 × 10-3 J/m2

𝝃 𝑯𝒃𝒊𝒂𝒔 𝑺𝒘𝒊𝒕𝒄𝒉𝒊𝒏𝒈 𝑽𝒐𝒍𝒕𝒂𝒈𝒆

37 fJ/(V∙m) -100 Oe 𝑥 1.2 V

VCMA-induced precessional switching is the most exciting switching process because it is both

very fast and very energy efficient because the VCMA effect does not require current flow.

25

Conclusion

The outcome of this project is a compact model of a magnetic tunnel junction for the

design and simulation of spin-transfer torque and magnetoelectric random access memory. Spin-

torque, field-like torque, external magnetic fields, shape anisotropy, the VCMA effect, and thermal

noise are all built-in. The model can easily simulate both in-plane and perpendicular MTJs

designed for use in STT-RAM and/or MeRAM, and it is compatible with any commercial

computer-aided circuit design software that can compile and run Verilog-A. Simulations in

Cadence confirm that the model captures the dynamics of STT switching, VCMA field-assisted

switching, VCMA + STT thermally activated switching, and VCMA precessional switching.

There are a number of opportunities for future work both in improving the model itself

and using it to provide critical data necessary for memory design. First, while the model has the

capability to calculate field-like torque, time constraints prevented any investigation into whether

or not it behaves as suggested in [13]. Reasonable values for 𝛽1 and 𝛽2 need to be found

empirically or from literature. Secondly, additional temperature dependencies need to be

accounted for. For example, the saturation magnetization 𝑀𝑠 and the spin-polarization factor 𝑃

are quantities that change with temperature. Also, the computation speed of the model may be

improved. Heun’s method was chosen as a good compromise between accuracy and simulation

time from the recommendation in [26]. In that study, however, the length of simulations using

different numerical methods was measured for the same time step. The accuracy of Heun’s

method increases quadratically with a decrease in time step compared to Euler’s method. The

accuracy of the fourth-order Runge-Kutta method increases quartically for the same time step

reduction. It should be possible to use a higher order method with an increased time step to see

an improvement in simulation time. Finally, the model enables one to measure switching

26

probabilities, assess thermal stability, and to perform a scaling study in order to see the effects of

reducing the MTJ dimensions. The model presented in this report can provide this invaluable

data to advance the design of future STT-RAM and MeRAM.

27

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29

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30

APPENDIX

Figure A.1: STT-Switching Simulation Results

31

Figure A.2: Close-Up of STT Switching

32

Figure A.3: VCMA Field-Assisted Switching Simulation Results

Voltage applied at 5 ns

33

Figure A.4: VCMA + STT Thermally Activated Switching Results (RP RAP, RP)

RP RAP at 1.1 V

RP RP at 0.2 V

Voltage pulsed from 10 – 70 ns

34

Figure A.5: VCMA + STT Thermally Activated Switching Results (RAP RP, RAP)

RAP RP at 0.2 V

RAP RAP at 1.1 V

Voltage pulsed from 10 – 70 ns

35

Figure A.6: Parallel State VCMA Precessional Switching Simulation Results

Voltage left on after 10 ns

τpulse

= 1.2 ns

τpulse

= 2.4 ns

36

Figure A.7: Anti-Parallel State VCMA Precessional Switching Simulation Results

Voltage left on after 10 ns

τpulse

= 1.2 ns

τpulse

= 2.4 ns