JOURNAL OF APPLIED PHYSICS 106, 123906 共2009兲 Dynamic compact model of thermally assisted switching magnetic tunnel junctions M. El Baraji,1,2 V. Javerliac,1 W. Guo,2 G. Prenat,2,a兲 and B. Dieny2 1 CROCUS Technology, 5 place Robert Schuman, 38000 Grenoble Cedex, France SPINTEC, UMR(8191), CEA/CNRS/UJF/Grenoble-INP//INAC, 17 rue des Martyrs, 38054 Grenoble Cedex, France 2 共Received 6 April 2009; accepted 12 October 2009; published online 21 December 2009兲 The general purpose of spin electronics is to take advantage of the electron’s spin in addition to its electrical charge to build innovative electronic devices. These devices combine magnetic materials which are used as spin polarizer or analyzer together with semiconductors or insulators, resulting in innovative hybrid CMOS/magnetic 共Complementary MOS兲 architectures. In particular, magnetic tunnel junctions 共MTJs兲 can be used for the design of magnetic random access memories 关S. Tehrani, Proc. IEEE 91, 703 共2003兲兴, magnetic field programmable gate arrays 关Y. Guillement, International Journal of Reconfigurable Computing, 2008兴, low-power application specific integrated circuits 关S. Matsunaga, Appl. Phys. Express 1, 091301 共2008兲兴, and rf oscillators. The thermally assisted switching 共TAS兲 technology requires heating the MTJ before writing it by means of an external field. It reduces the overall power consumption, solves the data writing selectivity issues, and improves the thermal stability of the written information for high density applications. The design of hybrid architectures requires a MTJ compact model, which can be used in standard electrical simulators of the industry. As a result, complete simulations of CMOS/MTJ hybrid circuits can be performed before experimental realization and testing. This article presents a highly accurate model of the MTJ based on the TAS technology. It is compatible with the Spectre electrical simulator of Cadence design suite. © 2009 American Institute of Physics. 关doi:10.1063/1.3259373兴 I. INTRODUCTION The recent development of spintronics allowed the emergence of a new kind of nonvolatile memory, called magnetic random access memories 共MRAMs兲. They combine intrinsic nonvolatility, high-speed, high-density, low-power consumption, hardness to radiations, and endurance. The emergence of new magnetic tunnel junction 共MTJ兲 write schemes 共based on thermal assistance and/or spin transfer兲 allows performance improvement and much better prospects in terms of scalability, speed, and power consumption in comparison with the first generation of MRAM. As a result, MRAM could become in the long term a universal memory, combining the advantages of flash memories 共nonvolatile but power consuming, slow to write, and subject to aging兲, static random access memories 共fast but volatile and with a large footprint兲, and dynamic random access memories 共volatile兲.1–3 Many integrated circuit manufacturers are interested in this new technology 共IBM, Freescale, Hitachi, etc.兲 and startups specialized in MRAM are appearing 共CROCUS-Technology, Grandis, etc.兲. Today, several groups are also investigating the use of magnetic devices for other logic circuits 共reprogrammable logic or ASICs兲. The design of complex hybrid architectures requires tools to simulate the full architecture and to perform physical verifications 共design rule checking, extraction, layout versus schematic兲 of the circuit before fabricating it. In this paper, we present a compact model of the thermally assisted switching 共TAS兲 MTJ. The paper is organized as follows: we first introduce the basic element used in a兲 Electronic mail: guillaume.prenat@cea.fr 0021-8979/2009/106共12兲/123906/6/$25.00 magnetic logic, the MTJ. Then, a description of the compact model is given. Finally, simulation results of the model are presented. II. MTJS A. Description A MTJ 关Fig. 1共a兲兴 is a nanostructure composed of two ferromagnetic 共FM兲 layers separated by an insulator. The magnetization of one FM layer 共hard layer兲 is pinned and acts as a reference layer. The magnetization of the other FM layer 共storage layer兲 is free, and in the presence of a uniaxial magnetic anisotropy, can be switched between two stable directions, parallel 共P兲 or antiparallel 共AP兲 to the reference layer. Transitions between P and AP states present a hysteretic behavior. Depending on the relative orientation of the magnetizations, the stack’s resistance varies: the resistance of the AP state is larger 共typically by 50%–200%兲 than the resistance of the P state. The digital information is coded by the magnetic configuration of the stack. Reading information FIG. 1. 共Color online兲 Schematics of MTJ: schematic magnetoresistance response 共a兲 and reading operation 共b兲. 106, 123906-1 © 2009 American Institute of Physics 123906-2 J. Appl. Phys. 106, 123906 共2009兲 El Baraji et al. FIG. 3. 共Color online兲 Schematic of TAS writing operation. FIG. 2. 共Color online兲 Schematic of FIMS writing operation. consists in measuring its effective resistance 关Fig. 1共b兲兴. Writing a MTJ can be performed by different methods, corresponding to different MTJ generations. B. Field induced magnetic switching MTJ In this first generation, the MTJ’s magnetic state is written by means of a magnetic field generated by current lines 共Fig. 2兲. If the current densities are large enough, the generated magnetic field switches the magnetization of the storage layer. In this technology, the selection of a MTJ to be written requires activating two current lines perpendicular to each other: only the cell at the intersection of the two lines is selected to be written. Moreover, the thermal stability of the magnetic orientation in the storage layer in each junction at equilibrium is determined by the MTJ’s aspect ratio 共see Sec. III C 3兲. This writing approach has been extensively studied between 1996 and 2000 but it was then realized that the mechanism of write selectivity provided by the combination of two perpendicular pulses of magnetic fields does not provide enough margin in regards to the switching field distribution from cell to cell. Half selected bit could switch yielding an excessive soft error rate. This selectivity issues was latter solved by freescale by the introduction of the so-called toggle write scheme in which the soft layer consists of a sandwich formed by two antiferromagnetically coupled FM layers.4 However, the approach remains hardly scalable due to the large current needed to produce the required write magnetic fields. The limitation in scaling down comes from the electromigration limit in the bit and word lines. C. TAS MTJ The second generation5 relies on the strong temperature dependence of the switching field. In this approach, an antiferromagnetic layer is used to pin the magnetization of the soft layer by exchange bias. A current pulse of a few nanoseconds applied through the stack heats the junction 共Joule heating兲 above a critical temperature 共blocking temperature TB typically of the order of 180– 250 ° C兲, freeing the storage layer magnetization. The latter can then be switched with a single relatively weak pulse of magnetic field 共Fig. 3兲. In this technique, the write selectivity is based on the combination of heating and applying an external magnetic field. Thanks to this approach, the thermal stability of the written information is excellent down to very small cell size 共 艋 20 nm兲, only the heated MTJ being sensitive to the magnetic field. In this technology, only one current line is required, resulting in a better integration density and the writing field required is weaker, typically half to one-third of that required in field induced magnetic switching 共FIMS兲 technology. Typically, a field of 30 Oe can be used 共at least 70 Oe in FIMS兲 and the writing time is imposed by the heating and cooling durations 共around 10 ns兲. Moreover, the magnetization of the storage layer could potentially be set in any direction of the plane, allowing to code several bits per junction 共multilevel storage兲. However, as in standard FIMS, this particular implementation would require two sets of perpendicular conducting lines to be able to generate any in-plane orientation of the magnetic field. III. COMPACT MODELING A. State of the art Several MTJ’s compact models have already been published in the literature. For example, a FIMS MTJ model written in HSPICE language6 and a spin transfer torque 共STT兲 MTJ model written in VERILOGA language7 were proposed. Both take into account thermal effects, but they only describe the static behavior of the magnetization. Another compact model of the MTJ, applicable for the FIMS and STT writing schemes, was also proposed.8 It is written in VHDL-AMS language and describes the transient behavior of the magnetization. However, the thermal dependence of the parameters is not taken into account. In all these models, the device temperature is fixed and does not vary dynamically during the simulation. B. Implementation The present model is based on the initial static and transient model of the FIMS MTJ:9 it takes into account the static and dynamic behaviors of the magnetization, the electrical characteristics of the MTJ depending on its magnetic state, the temperature dependence of the parameters, and the dynamics of the temperature, allowing TAS modeling. The physical equations describing all these phenomena are implemented in an electrical equivalent model written in C language,9,10 compiled with the compiled model interface of Cadence. The resulting library is compatible with Spectre and Ultrasim simulators. Figure 4共a兲 shows a very simple memory cell based on TAS technology, in a Cadence schematic. It includes a MTJ and a selection MOS 共Metal Oxide Semiconductor兲 transistor. The MTJ symbol has two input nodes, bl0 and bl1, representing the ends of the stack which is modeled by its resistance, magnetoresistance, and capaci- 123906-3 J. Appl. Phys. 106, 123906 共2009兲 El Baraji et al. approximation for the device we use, we can write the magnetocrystalline energy and the corresponding equivalent field as in Eqs. 共1兲 and 共2兲, where ␪msl is the angle between the magnetic moment and the easy axis eជy, Ku is the uniaxial anisotropy constant, and M s is the magnetization saturation value, ជ ·H ជ k, Ek = Ku sin2 ␪msl ⇔ Ek = − M 共1兲 with (b) (a) FIG. 4. 共Color online兲 TAS MTJ model implementation: MTJ model symbol used with a selection transistor in a Cadence simulation schematic 共a兲 and corner file used with the model 共b兲. tance. Two other nodes, fl0 and fl1, represent the ends of the writing line which is used to generate the magnetic field. If the writing technology requires additional lines 共multilevel storage, for example兲, two additional nodes per line are made available. For development ease, internal nodes can be accessible, representing, for example, the temperature or the magnetic state of the junction. The generic physical models need to be supplied with input parameters. Some of them are geometrical parameters 共for example, the size or the shape of the MTJ兲. They can be set by the user to perform simulations. The other parameters are technological and depend on the manufacturer. The model library is thus used with a corner file which contains all these technological informations. Figure 4共b兲 illustrates the use of such a corner file: libsmtj.so is the library containing the physical model of the MTJ, and the corner file “CROCUS” contains the parameters associated with the technology used by the company CROCUSTechnology, for example. Together, these files describe the behavior of the MTJs prepared with the technology of CROCUS-Technology. The C language approach for describing the model is very efficient in terms of flexibility in comparison with standard high level description languages such as VERILOGA and VHDL-AMS. The result is a library specific to Cadence, but the code can be easily migrated to other platforms, the principle remaining the same. ជk= H Ku m eជ . ␮0 M s y y 共2兲 3. Demagnetizing field Another source of anisotropy in magnetic nanostructures is due to their shape via the magnetostatic energy. From a general standpoint, in a magnetic nanostructure of ellipsoidal shape, the magnetization always tends to align with the longest dimension of the ellipsoid. This effect is described ជ d = −关N兴 M ជ, through a demagnetizing field expressed by H where 关N兴 is the tensor of demagnetization coefficients and ជ the magnetization. The tensor 关N兴 depends on the shape of M the device.11–13 In a sample of ellipsoidal shape, the larger the aspect ratio of the ellipsoid, the larger the shape anisotropy and correlatively the larger the thermal stability of the magnetization. Magnetic flat layers patterned in an elliptical shape are usually described as ellipsoidal. The demagnetizing field and associated demagnetizing energy are then written 关Eq. 共3兲兴 as 冤 nx 0 0 ជ Hd = − M s 0 n y 0 0 0 nz 冥冤 冥 mx ␮ ជ. my ⇔ Ed = − 0 Hជd · M 2 mz 共3兲 4. Zeeman energy The Zeemann energy results from the interaction beជ and the external applied field tween the magnetic moment M ជ a. Its expression is given in Eq. 共4兲, where ␪mha is the angle H ជ and H ជ a. between M ជa·M ជ = − ␮0HaM s cos ␪mha . E z = − ␮ 0H 共4兲 C. Magnetic materials description 1. Macrospin approximation To develop these models, we always proceed under the macrospin approximation, considering that the magnetization of the FM layer is homogeneous and can be described by a single macroscopic magnetic moment. This approximation is all the more relevant that the sample is small. 2. Magnetocristalline anisotropy The interaction between the magnetic moment and the crystalline lattice is responsible for a privileged direction 共that we will denote as eជy兲, in which the magnetization will spontaneously align itself without external field. In a first D. Tunnel magnetoresistance The tunnel magnetoresistance 共TMR兲 is the relative variation of the stack resistance according to the relative magnetization of the two layers. The value of the conductance is given by Eq. 共5兲 共Julliere model14兲, where P1 and P2 are the polarizations of the first and second electrode, respectively, ␪slhl is the angle between the magnetizations of the two electrodes, and GT is the tunnel conductance, which depends on the bias voltage 共V兲 of the junction and on its temperature 共T兲, G共␪,T,V兲 = GT共T,V兲共1 + P1 P2 cos ␪slhl兲. 共5兲 123906-4 J. Appl. Phys. 106, 123906 共2009兲 El Baraji et al. Pi共T兲 = P0i共T兲共1 − ␣iT3/2兲. 1. Temperature dependence of the tunnel conductance With the steadily growing speed and density of electronic systems, their thermal characterization is becoming a key task in thermal engineering. MTJs are themselves subjected to Joule heating. Since their magnetic and electrical properties depend on the temperature, a complete set of temperature dependent parameters is required to achieve an accurate temperature simulation. The dependance of GT upon the temperature is given by Eq. 共6兲 共Stratton model15兲, where the constant ␭ is given by ␭ = 共␲toxk / ប兲冑2me / e, where tox is the oxide thickness, k the Boltzmann constant, ប the reduced Plank constant, me the electron mass, and e its charge, GT共T,V兲 = GT共0,V兲 ␭T . sin共␭T兲 共6兲 In this model, GT represents the elastic conduction of the electrons through the tunnel barrier. To describe the thermal variations in the conductance, a second term Gie of inelastic conductance has to be added. This term depends on the temperature according to Eq. 共7兲, where ␶n is a material dependent constant and ␤ depends on the number of states occupied by the electrons when traversing the tunnel barrier. For a second order system, ␤ = 34 , Gie共T兲 = ␶nT␤ . 共7兲 The conductance also depends on the bias voltage applied across the MTJ stack, that will be referred as “polarization voltage.” This variation follows Eq. 共8兲 共Brinkmann model16兲, where ␤ = e冑2metoxd␾ / 24ប␾3/2 and ␦ 2 2 = e metox / 12ប␾, where ␾ is the height of the tunnel barrier and d␾ the barrier asymmetry. GT共0 , 0兲 is the tunnel conductance at 0 V, 0 K. It is given by Eq. 共9兲 共Simmons model17兲, GT共0,V兲 = GT共0,0兲共1 − 2␤V + 3␦V2兲, GT共0,0兲 = k0k1A 冑␾ 2tox 冑 e−k1tox ␾ . 共8兲 共9兲 with k0 = e2 / 2␲h, k1 = 4␲冑2mee / h, and A the surface of the junction. From the previous equations, we can evaluate the value of the TMR 共Moodera model18兲, which is defined by TMR = 共GP − GAP兲 / GAP, where GP and GAP are the conductances in parallel and antiparallel configurations, respectively. From Eq. 共5兲, we can write Eq. 共10兲, 2GT共V,T兲P1 P2 , GT共V,T兲共1 − P1共T兲P2共T兲兲 + Gie共T兲 TMR共T,V兲 = TMR共T,0兲 . V2 1+ 2 Vh 共12兲 E. Heat propagation The equations given in Sec. III D allow calculating the temperature dependence of different parameters such as M s and performing simulations taking into account the working temperature. However, the principle of the TAS technology is to very rapidly 共within a few nanoseconds兲 heat the junction to unpin the magnetization of the soft layer when writing and then quickly quench it down to insure its thermal stability in the newly written state. Then, the dynamic temperature variations have to be taken into account in this model. This temperature variations are described by the heat transfer equation 关Eq. 共13兲兴, where T is the stack temperature, ␳ is the volumic density of the material, c is the MTJ specific heat, and ␭th is the thermal conductivity, 共10兲 where the temperature dependence of the polarization is given by Eq. 共11兲,19 where ai is a material dependent constant, 共13兲 F. Dynamic behavior The dynamic behavior of a normalized magnetic moment m ជ under an effective magnetic field Hជeff is described by the Landau–Lifshitz–Gilbert model,21 Eq. 共14兲. The precession torque is responsible for a precession motion around the effective magnetic field. The precession frequency 共called Larmor frequency兲 is given by Eq. 共15兲. The damping torque is responsible for the relaxation of the magnetization toward the magnetic field direction, dm dm ជ ជ ជ , = − ␥ 0m ជ ⫻H ជ⫻ eff + ␣m dt dt Precession term T0 = 3. Expression of the TMR TMR共T,V兲 = The TMR value also decreases with the voltage V applied to the junction,20 following Eq. 共12兲, where Vh is defined as the voltage at which the TMR amplitude has decreased to half of its low bias value 关TMR共T , Vh兲 = TMR共T , 0兲 / 2兴. ␦ 2T c ␳ ␦ T . = ␦x2 ␭th ␦t 2. Dependence of the conductance on bias voltage 共11兲 2␲ . ␥␮0Heff 共14兲 Damping term 共15兲 G. Static behavior The static behavior of the FIMS or TAS MTJ is described by the Stoner–Wolfarth model:22 it consists in finding the equilibrium state of the magnetization under an applied field by minimizing the total energy of the system, the latter being given by the sum of the contributions given in Sec. III C: E = Ek + Ed + Ez. The minimization of the energy consists in finding the value of ␪ such as ⳵E / ⳵␪ = 0 and ⳵2E / ⳵␪2 艌 0. 123906-5 J. Appl. Phys. 106, 123906 共2009兲 El Baraji et al. FIG. 5. 共Color online兲 Simulation results for the memory cell: heating current through the stack 共a兲, time dependance of the temperature 共b兲, magnetic field generation current 共c兲, and bias voltage accross the junction 共d兲. IV. SIMULATION RESULTS In this section, we present simulation results under Cadence for the very simple memory cell of Fig. 4共a兲. The heating current through the junction is generated by using the selection transistor, activated by the voltage source applied on its gate. The magnetic field is generated by sending a current through the field generation line by means of a bidirectional current source. Simulation results are presented in Fig. 5. Figure 5共a兲 shows the heating current through the junction, Fig. 5共b兲 shows the resulting time dependence of its temperature, and Fig. 5共c兲 shows the bidirectional field generation current 共negative here兲 through fl0 and fl1. The blocking temperature, that enables the switching of the magnetization of the MTJ storage layer, is 150 ° C. When the blocking temperature is reached, a magnetic field is applied to switch the magnetization of the junction: Figure 5共d兲 represents the voltage across the MTJ. For a constant heating current, the voltage drops when applying the magnetic field, illustrating the switching of the magnetic state and the associated resistance change. In the insert, we can see the damped precessions around the effective magnetic field during switching. At the switching time, the resistance of the stack changes as well as the power transmitted for a given heating current. This results in a change in the temperature characteristic of the junction in Fig. 5共b兲. We can also notice that even before and after switching, for a constant current through the junction, the voltage is not perfectly constant: this is due to the increase in temperature resulting in parameter variations, in particular, a decrease in the resistance. V. CONCLUSION A very flexible, complete, and accurate compact model of the TAS MTJ was presented. This model is compiled in C language, compatible with the Spectre simulator of the standard Cadence design suite. It can be easily migrated to other simulators. It has already been used in the design of various demonstrators in the framework of French research projects. This model could be used in the design of spintronics circuits, to describe the electrical behavior of the MTJ as the Berkeley short-channel IGFET model does for the CMOS 共Complementary MOS兲 transistor. Further improvements of the model require additional development of CAD tools to integrate the model in a full magnetic design kit for the standard design suites. Corner files must be edited containing accurate data about the dispersion of the parameters, derived from characterized samples produced by a given technology. The model should also allow Monte Carlo or noise simulations. The ac and noise modules should be added to the model 共for rf oscillators design兲. Furthermore, the model must be steadily improved according to the progress in knowledge of MTJ’s physics. However, already in its present form, it constitutes a very important tool for the design of complex hybrid CMOS/magnetic architectures. ACKNOWLEDGMENTS The work and results reported were obtained with research funding from the French National Research Agency 共ANR兲 under PNANO program, contract CILO-MAG 共06NANO-066兲. The views expressed are solely those of the authors, and the other Contractors and/or the European Community and/or ANR cannot be held liable for any use that may be made of the information contained herein. 1 S. Tehrani, J. M. Slaughter, M. Deherrera, B. N. Engel, N. 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