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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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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