Recent outstanding improvements in the development of computer memories have been possible thanks to the down-scaling of semiconductor devices. This, however, has increased the stand-by power consumption of traditional
volatile components, such as static and dynamic random-access memory (SRAM and DRAM, respectively), especially due to the presence of leakage currents [1]. By introducing
nonvolatile memory components, stand-by power consumption can be mitigated, as they do not require refreshing of the memory bits. While being a valuable candidate, nonvolatile flash memories have poor endurance and are also
becoming increasingly complex and expensive to downscale for embedded application [2]. Moreover, the price for a gigabit of traditional flash memories does not follow the
down-scaling of the technology node, as it ceased to decrease. These factors prompt emerging memories entering the market to replace NOR flash, SRAM, and DRAM for stand-alone and embedded applications. Spin-transfer
torque magnetoresistive random-access memory (STT-MRAM) is a nonvolatile memory which possesses a simple structure and whose fabrication is compatible with CMOS processing. Instead of relying on charge for storing
information, MRAM is based on the electron’s spin. In contrast to traditional flash memory, STT-MRAM presents good speed and high endurance. This makes it particularly attractive for both, stand-alone as well as embedded
applications, for example, in Systems-on-Chip, where STT-MRAM is poised to replace SRAM and flash memory [2]–[11].
The core of modern STT-MRAM cells consists of a magnetic tunnel junction (MTJ), a sandwich of two ferromagnetic (FM) layers separated by a tunnel barrier (TB), see Fig. 1.1. The
FM layers are referred to as reference layer (RL) and free layer (FL). The RL is fixed either by proper choice of materials, or by antiferromagnetic coupling to an additional ferromagnetic pinning layer, while the magnetization of the
FL can be reversed. The use of MTJs with perpendicular magnetic anisotropy (PMA) permits to obtain better thermal stability, better scalability, and a lower switching current [12].
Figure 1.1: MTJ structure with the inclusion of NM contacts (light blue). The structure is composed of a reference layer (red), a tunnel barrier (green) and a free layer (yellow). The RL magnetization is fixed, while the magnetization
in the FL is free to move. The figure is adapted from [13].
When the magnetization vectors are in the parallel (P) state, the electrical resistance is lower than in the anti-parallel (AP) state, providing a way to store binary information. Switching between these two stable configurations can
be achieved by letting an electric current flow through the structure. Electrons tunneling from the fixed RL become spin-polarized, generating a spin current. When entering the FL, the non-equilibrium spin accumulation acts on
the magnetization via the exchange interaction. When the magnetization vectors are not aligned, the transverse spin current components are quickly absorbed, generating the spin-transfer torque through conservation of angular
momentum [14], [15]. If the current is sufficiently strong, the magnetization of the free layer can be switched between
the two stable P and AP configurations, relative to the RL.
The development of accurate simulation tools is essential in helping the design of efficient devices. Modeling of STT switching, which allows to describe the writing process of an STT-MRAM cell, requires a solution of the
time-dependent Landau-Lifshitz-Gilbert (LLG) equation with the inclusion of a term describing the torque acting on the magnetization. Such a task can be performed by assuming a Slonczewski-like torque approach [16]. In micromagnetic modeling of STT switching, the typical simplified approach is to assume that the current density is position- and time-independent [17]. In circuits, however, it is often the voltage, rather than the current density, that remains fixed during switching. The resistance of the tunnel junction depends on the relative
magnetization alignment of the free and the reference layer, so the current through the structure is not constant during the process. Moreover, as the magnetization of the FL is not uniform at switching, but depends on the
position, so does the local tunneling conductance. The assumption of a constant current density is violated, especially in advanced MTJs with a tunnel magnetoresistance (TMR) ratio, characterizing the difference between P and
AP resistance, of about 200% and higher [18]. It is thus important to investigate the switching behavior under fixed bias voltage and the effects of non-uniform currents,
which is the focus of the first part of this thesis.
Moreover, the implementation of a Slonczewski-like torque allows to approximately simulate the magnetization dynamics of a thin FL only. Recent STT-MRAM devices rely however on structures which are increasingly complex. In
order to boost the PMA provided by the interface between CoFeB and crystalline MgO, the FL is often capped with a second MgO layer [19]. Recently, more advanced structures
were proposed to boost the PMA even further, either by introducing more MgO layers in the FL or using the shape anisotropy of elongated FLs [20]–[22], while also improving scalability thanks to a reduced diameter. Accurate simulation tools can provide valuable support in the design of these ultra-scaled MRAM cells,
exemplified in Fig. 1.2. In order to model such devices, it is paramount to generalize the traditional Slonczewski approach to incorporate normal metal buffers or MgO barriers between
multiple CoFeB FL segments, as well as the barrier between RL and FL, and the torques coming from magnetization textures or domain walls that can be generated in elongated FLs. A more complete description of the process can
be achieved by computing the non-equilibrium spin accumulation across the whole structure. In a spin-valve structure with a non-magnetic spacer layer, this successfully accomplished by solving the spin and charge drift-diffusion
equations, both in a finite element (FE) [23], [24] and finite difference (FD) setting [25]. Inclusion of the MTJ properties is needed in order to describe modern MRAM devices in the drift-diffusion formalism, and to be able to apply it to switching simulations of
ultra-scaled MRAM cells. The second part of this thesis focuses on extending the drift-diffusion formalism, implemented in a FE solver based on open-source software, to account for transport properties of MTJs.
Figure 1.2: Model examples of elongated ultra-scaled MRAM cells. The NM contacts are in light blue, the RL is in red, the TBs are in green, and the FL segments are in yellow. The figure was published in [13].
1.1 Outline of the Thesis
This thesis is devoted to the employment and development of simulation tools capable of handling the different sources of torques acting in STT-MRAM devices, in order to be able to predict their switching behavior. The software
used to produce the results reported in this work is based on C++ libraries.
Chapter 2 gives a brief overview of the development and state of the art of modern STT-MRAM cells.
In Chapter 3, a summary of the derivation of the LLG equation for simulating the magnetization dynamics is reported, together with a description of the most prominent contributions to
the effective field therein. Moreover, simplified expressions for the spin-transfer torque acting in both spin-valves and MTJs are provided.
Chapter 4 focuses first on the computation of the current density redistribution in an MTJ with non-uniform magnetization. Then, the FD implementation of the LLG equation, employed by
an in-house solver, is described, and is updated with fixed current density, fixed total current, and fixed voltage approaches to the torque term.
Chapter 5 reports the results obtained by comparing switching simulations performed with the three different approaches to the torque term, showing how a correction to the current
value in the fixed current approaches is able to generate compatible results in all three models.
In Chapter 6, a brief description of the spin and charge drift-diffusion formalism, providing a more general expression for the torque, is reported. Moreover, an FE implementation of
the coupled LLG and drift-diffusion equations is described.
In Chapter 7, a way to deal with the TMR effect in the scope of the drift-diffusion formalism is derived. The implemented software is employed to study the torque dependence on
the system parameters, in order to reproduce the torque magnitude expected in MTJs.
In Chapter 8, an extension of the drift-diffusion formalism to account for the polarized tunneling spin-current is derived. The approach is shown to be capable of reproducing the
torque properties expected in MTJs. The updated equations, capable of predicting interactions between different sources of torque, are then applied to perform switching simulations in recently proposed ultra-scaled MRAM cells.
Moreover, by computing a solution to the drift-diffusion equations based on analytical expressions in the presence of ballistic corrections to the spin current, it is shown how a more complex oscillatory behavior of the torque can be
reproduced.
Finally, Chapter 9 reports a summary of the main results of the thesis.
The equations employed to derive a solution for the spin accumulation with analytical expressions are reported in Appendix A.
1.2 Research Setting
The research presented in this dissertation was conducted within the scope of the Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic (NOVOMEMLOG). The Christian Doppler Association promotes
the cooperation between research institutions and companies pursuing application-orientated basic research. For this laboratory, the cooperation was established between the Institute for Microelectronics at the TU Wien and
Silvaco Inc., a company developing and providing electronic device automation and software tools for Technology Computer-Aided Design (TCAD) .
