Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices
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Chapter 2 Fundamentals of Spintronics
This chapter provides an overview of the fundamental concepts and phenomena in spintronics that are relevant to the research presented in this dissertation. It covers the basic principles of the electron’s spin, magnetoresistance,
MTJs, STT, and SOT.
2.1 The Electron’s Spin
Spintronics refers to the field of research that focuses on harnessing the electron’s spin degree of freedom to develop novel electronic devices. The electron’s spin can be manipulated using various techniques, allowing for the creation
of devices that have the potential to be faster and more energy efficient than traditional charge-based devices [13]. Since the electron is a point-like particle, its spin is a purely quantum mechanical property without a classical
analogue, and should be treated as an intrinsic angular momentum carried by the electron. The quantum mechanical description of the electron’s spin was introduced by Wolfgang Pauli in 1927 [14], and is essential for the study of
spin-related phenomena. The length of the electron spin is given by
\(\seteqnumber{0}{2.}{0}\)
\begin{equation}
S = \sqrt {s(s+1)}\hbar = \frac {\sqrt {3}}{2}\hbar ,
\end{equation}
where \(s = 1/2\) is the spin quantum number of the electron and \(\hbar = h/(2\pi )\) is the reduced Planck constant. The spin angular momentum of electrons is described by the quantum mechanical spin operator
\(\seteqnumber{0}{2.}{1}\)
\begin{equation}
\bm {\hat {S}} = \frac {\hbar }{2}\bm {\hat {\sigma }},
\end{equation}
where \(\bm {\hat {\sigma }} = (\hat {\sigma }_x, \hat {\sigma }_y, \hat {\sigma }_z)\) is the vector of Pauli matrices, and a hat \(\hat {\,}\) denotes a quantum operator.
The electron carries a magnetic moment proportional to its spin, which can interact with magnetic fields. In the presence of a magnetic field, the spin vector is projected onto the direction of the magnetic field, e.g., along \(z\),
yielding
\(\seteqnumber{0}{2.}{2}\)
\begin{equation}
\hat {S}_z = \bm {e_z}\cdot \bm {\hat {S}} = \frac {\hbar }{2}\hat {\sigma }_z,
\end{equation}
where \(\bm {e_z}\) is the unit vector along the \(z\) direction. The projection direction is referred to as the quantization axis, as the possible values of the projected spin form a discrete set of values. The eigenvalues of \(\hat
{S}_z\) provide the possible physical values of the spin projected along the \(z\) direction, and are given by
\(\seteqnumber{0}{2.}{3}\)
\begin{equation}
S_z = m_s\hbar ,
\end{equation}
where \(m_s = {\mathbin {\textpm }} 1/2\) is the spin magnetic quantum number. These two possible orientations of the spin vector in a magnetic field are usually called "spin-up" (\(m_s = +1/2\)) and "spin-down" (\(m_s =
-1/2\)). The associated magnetic moment is given by
\(\seteqnumber{0}{2.}{4}\)
\begin{equation}
\mu _{z} = -g_s\frac {\mu _B}{\hbar } S_z,
\end{equation}
where \(\mu _B = e\hbar /(2m_e)\) is the bohr magneton, \(e\) is the elementary charge, \(m_e\) is the electron mass, and \(g_s\) is the electron spin g-factor. For a free electron \(g_s \approx 2.0023\), thus \(\mu _z
\approx \mp \mu _B\). Note that the sign of the magnetic moment is opposite to that of the spin due to the negative charge of the electron.
In materials, the electron’s spin can interact with neighboring spins through exchange interactions, leading to collective magnetic phenomena such as ferromagnetism and antiferromagnetism [15]. In NM materials, the band
structure is typically spin-degenerate; however, in FM materials, the exchange interaction between the magnetization and the conduction electrons leads to a splitting of the spin-up and spin-down conduction bands. Thus, when an
electric field is applied, the spin splitting results in a current with a net spin direction (i.e., spin polarization) [15]. This spin polarization of the electronic states is a key feature exploited in spintronic devices. To avoid confusion
later in the text, from here on, the conduction band with magnetic moment parallel to the magnetization (spin antiparallel) will be referred to as the majority spin band and will be denoted by \(\uparrow \), and the conduction
band with magnetic moment antiparallel to the magnetization (spin parallel) is referred to as the minority spin band and will be denoted by \(\downarrow \).
In addition to the intrinsic spin magnetic moment, the electron also has an orbital magnetic moment associated with its orbital angular momentum in the atomic nuclei and crystal field potential. Using the Dirac equation, a
relativistic quantum theory of the electron, it can be shown that an interaction between the electron’s spin and its orbital angular momentum, known as the SOC, appears. The electrons experience the SOC as an effective
momentum-dependent field that couples to their spin. In systems with broken inversion symmetry, this leads to a SOC that is odd in the electron’s momentum, which is the basis for several important phenomena in spintronics. In
the last two decades, the study of phenomena originating in the SOC has attracted significant interest due to their potential applications; consequently, a subfield of spintronics has emerged, referred to as spin-orbitronics [16].
