Compact Modeling of STT-RAM and MeRAM A Verilog-A model of Magnetic Tunnel Junction Behavioral Dynamics

Transcription

1 UNIVERSITY OF CALIFORNIA, LOS ANGELES Compact Modeling of STT-RAM and MeRAM A Verilog-A model of Magnetic Tunnel Junction Behavioral Dynamics Dheeraj Srinivasan 3/8/2013 +This work was done under the advisement of Prof. Kang Wang and in collaboration with fellow M.S. student Daniel Matic.

2 Outline 1. Abstract 2. Magnetic Tunnel Junctions 2.1. Introduction History Basic Operational Principles 2.2. Current-controlled Magnetic Tunnel Junctions Introduction Switching Mechanism Advantages and Disadvantages Conclusion 2.3. Voltage-controlled Magnetic Tunnel Junctions Introduction Switching Mechanism VCMA Field Assisted Switching VCMA + STT thermally activated switching Precessional Switching Advantages and Disadvantages Conclusion 3. Modeling of MTJ Characteristics 3.1. Behavioral Dynamics Landau-Lifshitz Equation Effective Magnetic Field Demagnetization Field Anisotropy VCMA Effects Thermal Noise Variation STT Current and Field-like Torque MTJ Resistance Heun Method 3.2. Conclusion and Parameters 4. Model Verification 4.1. STT-only switching 4.2. VCMA Field Assisted Switching 4.3. VCMA + STT thermally activated switching 4.4. Precessional Switching 5. Conclusion 6. References Appendix A 2

3 1. Abstract In recent years, Spin-Transfer Torque Magnetoresistive Random Access Memory (STT- RAM) has been found to be very power inefficient, and when integrated with CMOS, area inefficient. The advantages of MRAM remain that they have the density of DRAM, the speed of SRAM, and the non-volatility of Flash memory [3]. In this work, a highly-accurate physics based model for a current (STT) and voltage-controlled device (VCMA) is presented along with different switching mechanisms. The voltage-controlled device aims to improve on the power and speed performance of STT-RAM. The behavioral dynamics of the device are modeled in Verilog-A, through the use of the Landau-Lifshitz equation (LLE) [11]. The model is then carefully tuned to match the components of the fabricated device, which include Anisotropy, Field-like torque and most importantly, thermal fluctuations. In addition, to improve the accuracy and performance of the model, a Second-order Runge-Kutta or Heun Method is employed to solve the LLE. 3

4 2. Magnetic Tunnel Junctions This section will briefly review the physics of Magnetic Tunnel Junctions (MTJs) and provide insight into established methods of operation and discuss the pros and cons of each. Early and current work on these devices will also be highlighted Introduction Ever since the technology boom in the early 2000s, there has been a strong effort to scale down the size of complementary metal-oxide semiconductors (CMOS), in order to achieve higher performance and smaller area. In recent years, this rapid scaling has led to a plateau, where experts predict that scaling beyond 22nm CMOS will yield little upside, due to high leakage and other short-channel effects. Due to these effects, cells that were once stable are now susceptible to power-loss and other volatile effects. In order to keep developing products that continue to improve performance, either as a replacement or a supplement to existing CMOS, there was an investment into magnetic non-volatile memories [1] History of MTJs Investigation into the properties of MTJs began around the mid-1970s when M. Jullière discovered the relative change in resistance between two states, based on spin polarization, and summarized his findings in a conductance model [2]. Interest in MTJs reached its rising point, in the early-1990s when T. Miyazaki found magnetoresistance variations at room temperature. Since that time, MTJs have been optimized to improve the resistance difference between the two states. Currently, MTJs that have very high TMRs, on the order of 100%, are in use and research into this area has expanded [3] Basic Operational Principles An MTJ, the simplest of spintronic devices, consists of two ferromagnets separated by a thin tunnel barrier. This basic structure is shown in Fig.1.1. One of the ferromagnetic layers, which has a fixed magnetic orientation determined during fabrication, is called the fixed layer of the device. The other ferromagnetic layer, whose orientation is determined by the effective magnetic field, is called the free layer. The device is characterized by its resistance which changes based on the orientation of the two magnetization directions of the two ferromagnets. The magnetization direction is dependent on the spin-polarized tunneling between the minority and majority states [4]. This change in resistance is classified by the tunnel magnetoresistance ratio or TMR. 4

5 (1.1) where and are the resistances for the anti-parallel (AP) or high resistance and parallel (P) or low resistance states between the two ferromagnets, respectively. Anti-parallel Parallel Anti-parallel Parallel (a) (b) Figure 1.1: MTJ Device layers in (a) In-plane and (b) Out-of-plane configurations In current MTJs, there are two main different types of magnetic switching regimes due to spin-transfer-torque and voltage effects: precessional and thermally activated. Precessional switching is known to take effect on a nanosecond time scale, whereas thermally activated switching happens in larger scales. As shown in Fig.1.2, the transition between the thermally activated and precessional regimes is in the range of 1-10ns. Figure 1.2: MTJ Switching Regimes [3] 5

6 The dynamics of the MTJ during precessional switching operation are governed by the following Landau-Lifshitz-Gilbert equation: (1.2) This equation, including other MTJ dynamics will be covered more extensively in Section 3 of this report. While looking into the thermally activated regime, it is possible to find a relation between the switching current (, the thermal stability ( of the device and the pulse duration ( that is applied. It can be modeled by the following equation [3]: (1.3) In this equation, is the natural time constant, and is the critical switching current of the system. As can be seen from Fig.1.2, the current shows a logarithmic relation to increasing pulse width with respect to the thermal stability. Another basic concept which defines the properties of an MTJ device is its resistance. The Jullière model, as mentioned in the introduction, divides conductance into two components:, the conductance due to the direct elastic tunneling and, the conductance that results from imperfections in the insulating layer (which in this case is the MgO layer). The total conductance as a function of the angle, the angle between the moment and the parallel unit vector, is given by [2]: (1.4) where and are spin-polarization factors for the ferromagnetic free and fixed layers. It can be inferred from the above equation that when it corresponds to the parallel state and for, to the anti-parallel state. For the purposes of the experiment, it is assumed that a perfect insulating layer is used, i.e.. Throughout this paper and in MTJ characteristics, the anisotropy is used to explain the physics of operation and aid in the voltage-induced switching. So this calls for a detailed explanation of magnetic anisotropy itself. Anisotropy can be classified into three different parts: magnetocrystalline, shape, and magnetoelastic. Magnetocrystalline anisotropy is where the atomic structure of the crystal itself produces preferential directions for the magnetization vector. Shape anisotropy is introduced due to the fact of a particle not being a perfect sphere causing an unequal demagnetization field in different directions. Finally, magnetoelastic anisotropy is where tension affects the magnetic behavior of the cell [5]. 6

7 2.2. Current-controlled Magnetic Tunnel Junctions Introduction Current-controlled MTJs or more commonly known as STT-RAMs are the most-popular type of spin-based devices. In 1996, the discovery of the effects of spin-transfer-torque by Slonczewski, revolutionized the way in which MTJs were written. Slonczewski theorized that by injecting a spin-polarized current into an MTJ, the magnetic orientation of the ferromagnets could be controlled by the angular momentum created by the current [1, 16]. Soon after verifying the effect through experiments, the STT component was integrated into the Landau-Lifshitz equation that governs the dynamics of MTJs Switching Mechanism As stated earlier, in STT switching, a current passed either from the fixed layer to the free layer, or vice versa determines the final state of the MTJ. When a current passes from the free layer to fixed layer, it switches the MTJ into the parallel state and flowing in the opposite direction switches it into the anti-parallel state ( Shown in Fig.1.3 as the current flows from the free layer to the fixed layer, electrons are polarized in the direction of the fixed layer and therefore the spin-polarization of the free layer is changed to that of the fixed layer, resulting in the parallel state. And, when current flows from the fixed layer to the free layer, the electrons flow against the current, and are therefore first polarized in the fixed layer s orientation and this orientation is reflected back on the free layer, resulting in the antiparallel state. In order to switch the state, the current supplied has to be larger than the critical switching current of the device. (a) (b) Figure 1.3: STT Writing Mechanism (a) from AP P and (b) from P AP [1] 7

8 Advantages and Disadvantages Though STT switching provides fast switching speeds and a relatively simple operating mechanism, it also does not bare well under technology scaling. Given the fact that STT current required to switch a device scales inversely with transistor size, and the current trend towards smaller transistors, the power consumed in an STT device will become increasingly large. Also, in order to provide these currents, transistors will have to be scaled up causing an area and power tradeoff. Despite these effects, STT-RAMs remain widely developed and used and there are currently various studies into optimizing the switching current and energy. Recently, a paper published by the DRL group at UCLA, suggests alternate writing mechanisms for STT-RAM in order to reduce the power dissipation of the MTJ, and thereby render it compatible with modern day CMOS technology [6, 21] Conclusion In the above section, a current-controlled switching mechanism was presented and its pros and cons were analyzed. STT-RAM remains the leader in magnetic memory in industry but there needs to be more research and optimization in order to make it continually compatible with shrinking CMOS transistors. STT current required can be reduced by introducing perpendicular anisotropy [20], but this still requires further research. STT provides a simple, efficient method to store data and has a lot of scope for development in the future. 8

9 2.3. Voltage-Controlled Magnetic Tunnel Junctions Introduction Voltage-controlled MTJ s or Voltage-controlled magnetic anisotropy MTJs (VCMA MTJs) are devices that are controlled and operated using voltages, magnetic field and inherent properties of the ferromagnets. As discussed in previous sections, current-controlled devices are currently very area and power inefficient. In order to fill the void for a device with lower power and comparable performance, people have started to research different methods of controlling the MTJ; one such way is through voltage. By using voltage to actively control the device there is no a constant flow of charge, thereby reducing the power consumption. Another benefit to voltage-controlled devices is that recent experiments show the possibility of using similar fabrication processes as that for STT-RAM [7] Switching Mechanism The interfaces of the magnetic films with the non-magnetic oxides are susceptible to electric field variations. Particularly, the interface of oxides such as MgO with metallic ferromagnets has shown to have a large perpendicular magnetic anisotropy (PMA). This PMA effect has also been proven to be enhanced due to the effect of voltage [7]. Due to this PMA effect, the in-plane magnetization vectors are pushed out-of-plane and create the necessary effect to switch the states. With voltage-controlled MTJs there are various switching mechanisms that can be utilized; three such mechanisms are discussed below VCMA Field-Assisted Switching One of the mechanisms currently being used extensively is known as VCMA Field-assisted switching. This mechanism utilizes an external magnetic field which decides the final orientation of the magnetic moments. In the case of an initial in-plane magnetization of the free and fixed layers, the switching of the states is achieved by tuning the interfacial perpendicular anisotropy as the voltage pulse is applied. When the pulse is applied, the magnetic moment in the free layer departs from the in-plane orientation, and settles to a meta-stable perpendicular state (out-of-plane). When the pulse is removed, the orientation of the applied magnetic field, either favoring anti-parallel or parallel, forces the moment to settle to one of the energy-favorable states [7, 17]. This is depicted in Fig.1.4. In the similar case of out-of-plane free and fixed layers, the voltage pulse reduces the interfacial perpendicular anisotropy and moves the moment to an in-plane meta-stable state. Then, the field is applied in the direction and causes the moment to become parallel or anti-parallel with respect to the fixed layer. 9

10 Even though VCMA field-assisted switching provides an efficient and concrete mechanism, it requires a bipolar external field to support the switching. If there is a need for a mechanism that reduces the field and provides faster operation, precessional switching should be used. (a) Figure 1.4: VCMA Field-Assisted Switching - In-plane MTJ (a) from P AP (b) from AP P [7] VCMA + STT Thermally-Activated switching (b) Another switching mechanism that is used in the thermally activated regime is VCMA+STT switching. As discussed in previous sections, STT switching alone is no longer a practical mechanism due to the power and area inefficiencies that are introduced, and unlike the VCMA field-assisted switching, this mechanism requires a relatively small unidirectional bias field. But, the main attraction to this method of switching is its ability to work with unipolar voltages [8, 9]. First, a small effective magnetic bias field is introduced in the direction opposite to that of the STT current flow. This is done to ensure that there are forces in both orientations for correct switching. After this bias field is introduced, the interfacial anisotropy of the device is calibrated to a value close to the compensation point. For the purposes of these experiments, the field was created to favor the parallel state, and the current the anti-parallel state. For anti-parallel to parallel switching, a relatively small positive voltage is applied, which moves the magnetic moment closer to the compensation point, and introduces a relative small STT current. As the moment is at the compensation point, and the force of the STT current is fairly weak, the magnetic bias field forces the moment to the energy favorable state it favors, the parallel orientation. In order for the moment to remain in the anti-parallel state, a large positive voltage is applied, and STT current overcomes the effect of the field [8, 9]. Similarly for switching from the parallel to the anti-parallel state, a large voltage is applied, which moves the magnetic moment past the compensation point, at which time the strong STT current forces the moment to the anti-parallel state. In order to keep the state at parallel, 10

11 a small voltage is applied and the STT current is unable to overcome the magnetic bias field. As can be seen from the above description, the two different states flip based on different methods and different voltages but the voltages remain unipolar. By using unipolar voltages for switching, it is ensured that power is minimized and only one kind of voltage generator is needed. Even though this mechanism offers up a new form of switching, it leaves questions unanswered as to how the bias field is picked and how this physically affects the energy states of the electrons in the MTJ. To understand this, it is important to begin with the concept of an R-H loop. For an MTJ, an R-H loop is used to denote the fields at which the orientation of the moment is stable and unstable. In order to set the correct bias field for this type of switching, an initial R-H loop is constructed, where the field to switch the states without any applied voltage is measured, both AP P and P AP [18]. As measured in the devices, this field is measured to be, shown in Fig.1.5(a). Next, a small positive voltage (in this case ) is applied and the R-H loop is re-measured. Since there is a voltage pushing the moment closer to the compensation point, it requires a smaller field to flip orientations:, shown in Fig.1.5(b). Finally when a large voltage is applied (, it is seen that the R-H loop further contracts to values of, shown in Fig.1.5(c). By looking at the R-H loop it is easy to observe the method of switching being proposed here. If a bias field, is chosen such that it is greater than the value of the R-H loop at but less than that at, it is possible to switch from the anti-parallel to the parallel state. If is set to be around and the MTJ is initialized to the antiparallel state, the only energy favorable state available for the electrons to go to would be at the parallel state. But this applies to the AP P switching, because if the state is initialized to parallel, there would be no way to switch it to the opposite state. V = 0V R AP V = 0.2V R AP V = 1V R AP R P R P R P -165 Oe 165 Oe -75 Oe 75 Oe -25 Oe 25 Oe (a) (b) (c) Figure 1.5: Resistance-Hysteresis loops for varying voltages and their scaling trend 11

12 This problem is solved by the use of an STT current. The STT current acting on the R-H loop shifts the P AP transition point to the left of the indicated. By doing so, when the MTJ is initialized in the parallel state, it goes to the energy-stable anti-parallel state, as shown in Fig.1.6. An important point to note is that if the anisotropy constant is picked, such that the moment is at the compensation point, the R-H loop completely collapses. So for the above experiments, the anisotropy constant needs to be carefully picked close to the compensation point, so that the value of the required can be reduced [18]. The region in which, if initialized, the states are stable is referred to as the coercivity of the device. Simulation results for this switching are shown in Section 4. H bias R AP AP P R P R AP P AP R P Figure 1.6: R-H Loop under H bias and STT current Even though VCMA + STT switching provides an efficient and concrete mechanism, it requires careful tuning of the parameters and fields. If there is a need for a mechanism that reduces the field and provides faster operation, precessional switching should be used, since this mechanism is only valid in the thermally-activated regime Precessional Switching -300 Oe After looking at two switching mechanisms in the thermally-activated regime, it is seen that though they are robust, their speed is in the range of tens of nanoseconds. For applications that require switching on the order of 1ns, precessional switching devices can be used [10]. Precessional switching is done via the use of short voltage pulses that are applied in order to 12

13 favor a particular state. After initializing the moment to one of the two energy-favorable states, the interfacial perpendicular anisotropy constant is set such that the moment is brought out-of-plane or in-plane by applying about, in the in-plane and out-of-plane junction cases respectively. When the moment approaches the meta-stable state, thermal fluctuations that are inherent to the device cause an oscillatory pattern around this region. As the moment oscillates, and when it is in the preferred direction, the voltage pulse is dropped, causing the moment to relax to the chosen stable state. For a perpendicular junction, due to the moment s affinity to remain for long times at the energy-favorable state, a field in the direction was added to offset the field at an angle that was not perfectly. By doing so, it was possible to ensure relatively fast precessional switching (100% switching probability) and reduce thermal fluctuations. Precessional switching in the three-dimensional spherical domain is depicted in Fig.1.7, and it is juxtaposed with the damping of the device. m z m z m y m x m y m x (a) (b) Figure 1.7: In a perpendicular junction, (a) damping and (b) precessional switching [10] Advantages and Disadvantages Compared to STT switching, different forms of VCMA switching provide more energy efficient operation. Though STT switching is fast, VCMA switching performance in the precessional regime is comparable. The advantage of VCMA Field induced switching is that it does not require an STT current to switch, as it depends on the magnetic field bias. With VCMA + STT switching, the duo of STT current and a bias magnetic field is used in order to provide unipolar switching capabilities. Depending on the desired magnetic field, the designer can choose between these two mechanisms that are in the thermally-activated regime. If fast switching is required, precessional switching must be used, though the fact that it heavily depends on the width of the input pulse may pose a risk. 13

14 Conclusion In the above section, a voltage-controlled switching mechanism was presented and its pros and cons were analyzed. VCMA switching offers researchers with a new method that avoids the power and area downside that STT-RAM has, while using properties inherent to the fabricated device itself. Unlike STT-RAM which has a simple one-step switching, VCMA provides different mechanisms that are tailored to different applications based on the required speed and performance. Voltage-controlled switching provides designers with the ability to build more devices using MTJs. 14

15 3. Modeling of MTJ Characteristics Modern day device models help in simulating and verifying the functionality of a component to avoid failures prior to the fabrication phase. The obstacle that prevents the design and production of new memory components featuring voltage-controlled devices is a lack of a compact MTJ model that accurately models actual device behavior after fabrication. It is vital these effects are included in such a model to ensure performance and yield for new memory. The following section details such a model for a voltage-controlled device in Verilog-A. This model takes into account: device thermal noise, current, voltage and magnetic field effects Behavioral Dynamics Landau-Lifshitz Equation The precessional switching dynamics of the magnetic moment ( ) in the presence of a field, is defined in Section 2 of this report via a Landau-Lifshitz Equation (LLE). But in the case of voltage-controlled devices the equation is expanded to account for spin-transfertorque, field-like torque and the magnetic field is changed to include voltage, anisotropy, and thermal effects [11]. Thus, the normalized LLE for a perpendicular junction is given by: (1.4) where, is the absolute value of the gyromagnetic ratio ( ) divided by where is the damping factor of the MTJ device, is a unit vector in the direction of the magnetization vector, is Planck s constant, is the relative permeability of the free layer, is the charge of an electron, is the area of the MTJ device, is the thickness of the free layer, and and are constants of scaling in the field-like torque. The spinpolarization constant, for simplification is assumed to be independent of temperature and is related to TMR by: (1.5) where TMR is the same as defined in Section 2 of this document. The current density,, used in Eq.1.4, is approximated to be fairy linear and is calculated from the applied voltage, device resistance and area. 15

16 Figure 1.8: Basic MTJ Structure with LLE components Effective Magnetic Field The effective magnetic field for a perpendicular junction is given by: (1.6) where corresponds to the externally applied magnetic field, is the demagnetization field, is the interfacial perpendicular anisotropy (magnetocrystalline) field, is the field due to voltage effects, and is the field resulting from thermal noise in the system Demagnetization Field The demagnetization field, also known as shape anisotropy, results from the non-spherical nature of the free layer, resulting in unequal fields and a pre-determined easy axis of operation; in this case, where corresponds to a demagnetization tensor vector in all directions. By assuming the free layer to be a relatively flat ellipsoid, the demagnetization tensor constants can be calculated by Osborn [12]: (1.7) 16

17 where and are the complete elliptic integrals of the first and second kind, calculated in MATLAB, with the argument of: (1.8) It is important to note that demagnetization field, as state above only depends on the device dimensions and not the magnetocrystalline anisotropy of the device, i.e. PMA Anisotropy As described in Section 2, there are three different types of anisotropy. In the previous part, shape anisotropy or demagnetization was covered, and in this part, the magnetocrystalline anisotropy that results from the atomic structure of the interfacial junction is studied [5, 19]. For voltage control to be effective, it is vital to use atomic crystal properties to tune the device. Currently, the junction between the free layer and insulating layer is tuned to provide an optimal anisotropy that allows operation at small unipolar voltages. This is done by introducing a magnetic field that is solely based on the junction, and keeps the moment in the perpendicular state. In the effective magnetic field the anisotropy term is: (1.9) where corresponds to the thickness of the free layer, is the relative permeability of the free layer, is the magnitude of the magnetization vector, is the z-component of the magnetization unit vector and corresponds to the anisotropy constant. Note from the above equation that the anisotropy is only in the direction, meaning it reinforces the perpendicular field. The anisotropy term plays a large role in the precessional and VCMA + STT switching as detailed in section 2. One important point to address is how the anisotropy can tune the magnetic vector to the compensation point. The compensation point is the point at which the magnetization orientation switches from out-of-plane to in-plane solely due to the factor of the anisotropy. Thus, the closer that this pushes the orientation to the compensation point, the faster the switching dynamics VCMA effects Another term that contributes to the effective field is that of VCMA. The integral part of this MTJ device is the voltage-control capability. By introducing this voltage-induced magnetic field term, it is possible to modulate the amount of effective magnetic field in the system. As explained in Section 2, the voltage term works hand-in-hand with the anisotropy term in order to employ switching. In the effective magnetic field, the VCMA term is: 17

18 (2.0) where corresponds to the thickness of the free layer, is the thickness of the insulating layer, is the relative permeability of the free layer, is the magnitude of the magnetization vector, is the z-component of the magnetization unit vector and corresponds to the VCMA constant. Again, it is noted that the VCMA field component is also in the direction. But by looking at Eq.1.6, it is possible to come to the conclusion that since the VCMA term subtracts from the anisotropy term, and brings it closer to the compensation point, the of the anisotropy term needs to be larger so that when the voltage is applied, the net field in the direction is reduced to be closer to the compensation point. In brevity, the VCMA term attempts to bring the moment closer to the in-plane direction to allow for easier switching Thermal Noise Variation In a fabricated MTJ device, there always exists a form of random noise under which the system operates. In order to have an effective model of the actual device, it is imperative to include thermal noise fluctuations and observe voltage and resistance behavior. By assuming a random Gaussian noise, with mean zero and variance of one ( ), it is possible to model the random noise. In the effective magnetic field, the thermal noise variation term is: (2.1) where corresponds to Boltzmann s constant, is the temperature of the cell in Kelvin, is the Gilbert damping constant, is the relative permeability of the free layer, is the magnitude of the magnetization vector, is the absolute value of the gyromagnetic ratio ( ) divided by, is the volume of the MTJ device and corresponds to the time-step of computation. The coefficient vector is a Gaussian random variable and is defined as: (2.2) where denotes the seed of the random variable that is automatically generated on-the-fly in Verilog-A to provide different values for. An important point to note is that thermal noise field affects all of the axes equally and thus is not specified in a particular direction. 18

19 STT Current and Field-like Torque The STT current plays a vital role in VCMA switching and its dynamics are modeled by Slonczewski s term [16] in Eq.1.4. The STT current does not contribute to the effective magnetic field, as is in the case of the voltage, but rather balances it based on the applied voltage and device state resistance. In addition to STT, there is another component that needs to be taken into account, called Field-like torque (FLT). The existence of a field-like torque was predicted for metal spin valves but shown to be less that STT, allowing for approximation. But in MTJs, it is predicted that both the torques could have equal magnitude and that the field-like torque would have a quadratic dependence on the voltage. Though some measurements report contradictory signs of the field-like term in different cases, for the experiments conducted in this paper, it is assumed that field-like torque assists STT. As research continues into the effects of FLT, one paper reports that field-like torque can possibly explain the absence of pre-switching oscillations and the decaying of oscillations of the resistance after switching [13] MTJ Resistance As discussed before, the MTJ has two resistance states: anti-parallel and parallel, and they are related to each other via the TMR. Also suggested earlier was Jullière s model for the conductance of an MTJ. Assuming for this particular case that, the equation is further simplified to: By using the relation between and, it is possible to approximate: (2.3) (2.4) Another key point to make is how to extract the direct elastic tunneling component of Eq.2.3 by using parameters such as and. From earlier, it is known that when it corresponds to the parallel state and for, to the anti-parallel state, so: (2.5) From the above equations and the fact that the angle between the magnetization vector and the parallel component vector, is nothing but the -th component of the unit magnetization vector, Eq.2.3 can be rewritten as: (2.6) 19

20 Heun Method In order to implement the LLE from Section 3, it is important to look into different mathematical methods that will provide the best closed-form approximation of dynamic magnetic behavior. First, the LLE was implemented using Euler s Method in Verilog-A. The Euler method is a simple and intuitive method that provides a solution relatively quickly. It uses the Riemann sum approximation for the integral (rectangular approximation). However, the accuracy of the Euler method only increases linearly with the step size. With this caveat, using the Euler method would require relatively small step sizes and long runtimes in order to provide an accurate result that would match actual dynamics. In order to improve runtime and accuracy of the results, it was important to look at other methods of implementation. The improved Euler method, also referred to as the second-order Runge-Kutta or Heun method, results in accuracy that scales quadratically with step size. Rather than the use of the Riemann sum rectangular approximation, it uses the Trapezoid rule. From fundamental calculus, it is known that the Trapezoid rule error decreases quadratically with the step size [14]. Using a basic mathematical curve and example it is possible to show the differences between the two methods. Assuming a continuous time approximation of a differential equation: (2.7) It is possible to approximate information about using information from. Thus, using Euler s Method, the approximation turns to: (2.8) where corresponds to the step size, and is the value of the function at the previous time, and is the y-intercept of the approximation. In order to implement Heun s method, which is second-order, information calculated by Euler s Method ( is used. By doing so, it is possible to arrive at an improved approximation: (2.9) 20

21 Figure 1.9: Depicts the Euler Method approximation (left) against Heun Method (right) [15] 3.2. Conclusion and Parameters In the above section, the dynamics of an MTJ are discussed in detail covering: the LLE, effective magnetic field, which includes demagnetization, anisotropy, VCMA, thermal noise, and also the preferred method of computation, Heun s Method. By including all these various effects in the Verilog-A compact model, it was possible to mimic fabricated device characteristics and predict behavior. The behavior of this model and each of the switching mechanisms were thoroughly tested using numerous test cases which are presented in the subsequent section. Presented below is a table that summarizes all of the physical and fitted MTJ parameters that were employed in the model. Geometric Parameters LLE Damping FLT Torque Constants Spin Polarization Magnetization Saturation Conductance and VCMA Demagnetization Tensor Table 1: Table of Physical and Fitted MTJ parameters 21

22 4. Model Verification In this section, results from different switching mechanisms are presented. The physics that drive these results are detailed in section 2 of this report STT-only switching In the following simulation, a pure STT operated device is tested by application of and observing the resistance variation over time. When is applied, the device favors the anti-parallel direction, indicating current is flowing from fixed to the free layer, and when is applied, the current direction is reversed, and flips to the parallel state. Here, the parallel resistance is set to and the anti-parallel resistance to. In addition to the regular switching plot, a zoomed-in version is also shown in Fig.A.1.1 and Fig.A.1.2 respectively VCMA Field-Assisted Switching For verifying VCMA Field-Assisted Switching, the with a anisotropy constant, was tuned to values of. With the moment initialized at either the anti-parallel or parallel state, and ensuring correct polarity for, the switching was observed. The parallel resistance was set to, and the anti-parallel resistance was. The result of this simulation is shown in Fig.A VCMA + STT Thermally-Activated Switching In VCMA + STT Thermally-Activated Switching, the was set to, and for AP P, was applied (to reduce the amount of STT current), and the switching is achieved via the bias field. To maintain the AP state, is applied in order for the STT current to overcome. In the case of P AP, is applied for the STT current to favor the anti-parallel state, and to remain at P, is applied to let the bias field favor the parallel state. In this scenario, due to the change in the coercivities, as shown in Section 2, the resistances were modified to and, maintaining 100%. The results are shown in Fig.A.1.4 and Fig.A Precessional Switching Finally to test Precessional switching, the was set to, which is close to the compensation point, and was applied. The pulses were timed, so that when the moments were in the correct region, the voltage would be removed. The pulse widths for the first and second peak/trough were and respectively.again, and. This is shown in Fig.A.1.6 and Fig.A

23 5. Conclusion In this paper, a compact model for STT-RAM and MeRAM was presented, and new switching mechanisms were described. Initially, the motivation for a new type of MTJ resulted from the power inefficiencies of STT-RAM. To combat this inefficiency, a voltagecontrolled device, that provided the customer with multiple switching capabilities, was designed and two regimes of operation were discussed: Precessional and Thermallyactivated. For thermally activated switching, VCMA Field-Assisted and VCMA + STT Thermally-Activated switching are presented and their pros and cons are analyzed. Finally, the model was built using behavioral dynamics of the device and tested thoroughly using a variety of test cases. This model can be used to simulate fabricated MTJ device behavior in a circuit or any other application. 23

24 6. References [1] F. Ren, Energy-Performance Characterization of CMOS/Magnetic Tunnel Junction (MTJ) Hybrid Logic Circuits, Ph.D. dissertation, Dept. Elect. Eng., UCLA, Los Angeles, CA, 2011, Chapters 1-2. [2] M. Jullière, Tunneling Between Ferromagnetic Films, Physics Letters A, vol. 54, no. 3, pp , [3] R. Dorrance, Modeling and Design of STT-MRAMs, Ph.D. dissertation, Dept. Elect. Eng., UCLA, Los Angeles, CA, 2011, Chapters 1-3. [4] S. Ikeda, J. Hayakawa, M. L. Young, F. Matsukura, Y. Ohno, T. Hanyu, and H. Ohno, Magnetic Tunnel junctions for spintronic memories and beyond, IEEE Trans. Electron Devices, vol. 54, no. 5, pp , May [5] M. McCaig, Permanent magnets in theory and practice, Pentech Press,1977. [6] K.L. Wang, J.G. Alzate, P. Khalili Amiri, Low-Power Non-Volatile Spintronic Memory: STT-RAM and Beyond, Journal of Physics D: Applied Physics, vol. 46, no. 7, January [7] P. Khalili Amiri, K.L. Wang, Voltage-Controlled Magnetic Anisotropy in Spintronic Devices, SPIN Magazine, vol. 2, no. 3, October [8] P. Khalili Amiri, P. Upadhyaya, J.G. Alzate, and K.L. Wang, Electric-Field-Induced Thermally Assisted Switching of Monodomain Magnetic Bits, Journal of Applied Physics, vol. 113, no. 1, January [9] W.G. Wang, C.L. Chen, Voltage-induced switching in magnetic tunnel junctions with perpendicular magnetic anisotropy, Journal of Applied Physics, vol. 46, no. 7, January [10] M. Dimian, Nonlinear Spin Dynamics and Ultrafast Precessional Switching, Ph.D. dissertation, Dept. Elect. Eng., University of Maryland, College Park, MD, 2005, Chapter 3. [11] T. Moriyama et al., Tunnel Magnetoresistance and Spin Torque Switching in MgObased Magnetic Tunnel Junctions with a Co/Ni Multilayer Electrode,, Applied Physics Letters, vol. 97, no. 7, p , [12] J.A. Osborn, Demagnetizing Factors of the General Ellipsoid, Phys. Rev., vol. 67, no , pp , June

25 [13] S. Garzon et al., Macrospin model to explain the absence of preswitching oscillations in magnetic tunnel junctions: Fieldlike spin-transfer torque, Physics Review Letters B, vol. 79, issue. 10, March [14] P. Horley et al., Numerical Simulations of Nano-Scale Magnetization Dynamics, Numerical Simulations of Physical and Engineering Processes, InTech, [15] P. Hewitt, The Euler-Heun Method, Class Notes EE214A, University of Toledo, [16] J.C. Slonczewski, Current-driven excitation of magnetic multilayers, Journal of Magnetism and Magnetic Materials, vol. 59, no. 1-2, pp. L1-L7, June [17] W.G. Wang et al., Electric-field-assisted switching in magnetic tunnel junctions, Nature Mater, 11, 64-68, [18] J.G. Alzate et al., Voltage-induced switching of magnetic tunnel junctions, Tech. Dig. IEEE Int. Electron Devices Meet, , [19] S. Ikeda et al., A perpendicular-anisotropy CoFeB-MgO magnetic tunnel junction, Nature Mater, 9, , [20] P. Khalili Amiri et al., Switching current reduction using perpendicular anisotropy in CoFeB-MgO magnetic tunnel junctions, Appl. Phys. Lett, vol. 98, no , [21] P. Khalili Amiri et al., Low write-energy magnetic tunnel junctions for high-speed spin-transfer-torque MRAM, IEEE Electron Devices Lett., vol. 32, pp ,

26 Appendix A Figure A.1.1. STT Current Switching 26

27 Figure A.1.2: STT Current Switching Zoomed-In Precessional Behavior 27

28 Figure A.1.3: VCMA Field-Assisted Switching 28

29 Figure A.1.4: AP P and AP AP VCMA + STT Switching 29

30 Figure A.1.5: P AP and P P VCMA + STT Switching 30

31 Figure A.1.6: Precessional Switching, initialized in the AP State 31

32 Figure A.1.7: Precessional Switching, initialized in the P State 32