\chapter{Electron Spin: Theoretical Foundations} \label{chap:spin}

\section{The Origin of Electron Spin}


% I could have opening paragraph on first ever electron-spin-based experiments, which date back to the 1880s (see wiki for eg)


Spin can generally be conceptualised as the rotation of a particle about a specific axis. However, elementary particles such as electrons are envisioned as point-particle-like, and cannot rotate in space in the conventional way that solids can. If an electron were to be visualised as a sphere of Lorentz radius $r$ rotating about its own axis, its velocity on the surface would be many times the speed of light in a vacuum. In addition, the simple concept of self-rotation about an axis doesn't explain simple spin features such as its quantised values. Hence, there is no clear-cut explanation as to how spin can emerge at a fundamental level. In the words of Richard Feynman; ``It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation" \cite{feynman1965feynman}.


\bigskip

Otto Stern and Walther Gerlach of the University of Hamburg are credited with the discovery of electron spin in the 1920s, albeit by an accidental discovery, as they had not even thought of the concept of spin before. They designed an experiment whereby silver was vapourised in an oven \cite{stern1922magnetische}, as in Figure \ref{fig:stern}. They  sought to measure the magnetic fields produced from electrons orbiting the silver atoms, knowing in advance that moving charges produce magnetic fields. What they hadn't predicted was the silver atoms would form a beam which, when passing through a magnetic field, exhibited two distinct angular momenta, with two discrete accumulation points displayed on the screen rather a continuous distribution of points. They assumed the two spots observed were due to quantised orbital angular momentum, which they had guessed would be $\pm \hbar$ (where $\hbar$ is the reduced Planck constant). However, in 1927 Ronald Fraser showed that the atoms had no orbital angular momentum, and suggested that the observed magnetic properties were due to electron spin \cite{fraser1927effective}, whereby the two possible spin states reflect the electron's alignment relative to an external magnetic field. The conclusion made by Goudsmit and Uhlenbeck \cite{uhlenbeck1925ersetzung} was electrons have an intrinsic angular momentum of $\pm \hbar/2$ which is comparable to the angular momentum of a classically spinning object, but they take only specific quantised values% of half a quantum and a magnetic moment of one Bohr magneton
. Their theory was largely strengthened by the realisation that the concept of electron spin explained the small energy shifts that produced the fine structure in spectral lines; something which had puzzled scientists for years. We can therefore conclude that while spin is sometimes visualised as the electron rotating around its own axis, a formal definition for spin at a fundamental level in quantum mechanics is as an intrinsic form of angular momentum, independent of classical rotation.


\bigskip

\begin{figure}[h!]
    \centering
    \includegraphics[width=0.85\linewidth]{Background/Figures/Stern.png}
    \caption{Experimental arrangement of the Stern-Gerlach apparatus.}
    \label{fig:stern}
\end{figure}

\bigskip

\section{The Quantum Mechanics of Spin}

% new quantum number s
Initially, the idea seemed controversial, as the notion of a spinning charged particle raised questions about relativistic effects, which were difficult to reconcile with classical physics. Quantum mechanics, as formulated by Schrödinger \cite{schrodinger1926schrodinger}, did not account for relativistic effects. Prior to the discovery of spin, quantum mechanics centred around Bohr's model of the hydrogen atom, which predicted that the electron orbitals were quantised. The principal quantum number $n$ determined the allowed orbital radius, the orbital quantum number $l$ determined the allowed shapes, while the magnetic quantum number $m$ determined its magnetic behaviour \cite{bandyopadhyay2008introduction}. A fourth quantum number, the spin quantum number $s$ (= $\frac{1}{2}$ for fermions), was introduced to represent the magnitude of this intrinsic angular momentum. The spin magnetic quantum number, $m_s$, describes its projection along a chosen axis (usually the $z$-axis). For an electron, $m_s$ can take two values: $+\frac{1}{2}$ (spin-up) or $-\frac{1}{2}$ (spin-down), as demonstrated by the two discrete parts of the beam in the Stern-Gerlach experiment.



% Pauli
Any observable quantity in quantum mechanics is associated with an operator, the eigenvalues for which are the expectation values of the physical quantity. There must consequently be a quantum mechanical operator for spin with eigenvalues $\pm \hbar/2$. Wolfgang Pauli derived what are now known as the Pauli matrices in 1927 \cite{pauli1927quantum} to describe the quantum mechanical behaviour of spin-$\frac{1}{2}$ particles. These three $2\times2$ Hermitian matrices represent the spin angular momentum operators along the x-, y-, and  z-axes, allowing the spin angular momentum components to be modelled mathematically and satisfy the correct commutation relations for angular momentum. They also fit into the framework of quantum mechanics, obeying the necessary algebra for operators acting on a two-level quantum system. The dimensionless Pauli matrices are:

% operator -  (which for the Heisenberg formalism is a matrix)

\[
\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad
\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad
\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\]

\bigskip


The actual spin angular momentum operators in quantum mechanics for a spin-$\frac{1}{2}$ in the $i$-direction (where $i$ can be $x$, $y$ or $z$) are given by:
\[
S_i=\frac{\hbar}{2}\sigma_i
\]


The eigenstates of the Pauli matrices represent the quantum states of a particle with spin aligned along a particular axis. For $\sigma_z$, which is the most commonly used Pauli matrix, the eigenstates are:
\[
\text{Spin-up along \(z\)-axis:} \quad \begin{pmatrix} 1 \\ 0 \end{pmatrix}
\]
\[
\text{Spin-down along \(z\)-axis:} \quad \begin{pmatrix} 0 \\ 1 \end{pmatrix}
\]

corresponding to spin being aligned or anti-aligned with the \(z\)-axis, and their eigenvalues are $\pm 1$, which correspond to $\pm\frac{\hbar}{2}$ when considering the physical spin angular momentum. Similarly, the eigenstates of \(\sigma_x\) and \(\sigma_y\) correspond to spin being aligned along the \(x\)- and \(y\)-axes, respectively, with the same eigenvalue structure. The spin of a particle affects how its wavefunction changes under rotation. When the particle is rotated by an angle $\theta$ around the axis of its spin, its quantum state picks up a phase factor of $e^{is\theta}$, where $s$ is the spin of the particle.



In reality, the quantum state $\psi$ of a spin-\(\frac{1}{2}\) particle is not necessarily purely spin-up or spin-down, but a two-component ``spinor", which is essentially a linear combination of the spin-up and spin-down states along the \(z\)-axis: 


\begin{equation} \label{eq:spinor}
\psi= \begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} = \psi_1 \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \psi_2 \begin{pmatrix} 0 \\ 1 \end{pmatrix}
\end{equation}



% pauli and dirac eqn
where $\psi_1$ and $\psi_2$ are complex numbers which satisfy  \(|\psi_1|^2 + |\psi_2|^2 = 1\), representing the ``spin-up" and ``spin-down" components of the spinor wavefunction respectively. We can connect this back to our discussion on quantum computing in Chapter \ref{chap:intro}, whereby quantum computers process qubits which are coherent superpositions of two binary classical bits. In this context, spin polarisation of an electron is the ideal physical implementation of a qubit.

Pauli replaced the original one-component wavefunction in the Schroedinger equation (and thus transformed the Hamiltonian into a $2\times2$ matrix) with this two-component spinor, giving us what's known as the Pauli equation. However, the Pauli equation is completely non-relativistic. By combining the work of Pauli and Klein and Gordon \cite{klein1926quantentheorie,gordon1926comptoneffekt}, Paul Dirac produced the Dirac equation \cite{dirac1928quantum}, which included electron spin as a crucial component of his derivation of relativistic quantum mechanics. Dirac’s equation defined electrons (and all fermions) as ``spin-$\frac{1}{2}$'' particles, which obey the Pauli Exclusion Principle \cite{pauli1924pauli}, which states that no two fermions can occupy the same quantum state within an atom. This work provided a formal foundation for electron spin.




\bigskip



\bigskip
\section{Spin-Orbit Coupling}

While Pauli matrices and spinors provide a compact and elegant framework for describing the quantum mechanics of spin-$\frac{1}{2}$ particles, they do so in the context of idealised, isolated systems. In real materials, however, electron spin does not evolve in isolation. One of the most significant couplings between spin and the surrounding environment arises through spin-orbit coupling (SOC). SOC (or spin-orbit interaction) is a relativistic interaction of a particle's spin with its motion inside a potential, such as the electric field of the host lattice described by the periodic potential $V$. An electron orbiting a nucleus feels the electric field because of the nucleus' positive charge. Consequently, a magnetic field with flux density $\vec{B}$ will appear in the rest frame of the electron through a Lorentz transformation \cite{bandyopadhyay2008introduction}. This magnetic field, in turn, interacts with the intrinsic magnetic moment $\vec{\mu}_e$ associated with the particle's spin. The result is a coupling between the particle's spin and its orbital angular momentum. The energy of this interaction is given by:
\begin{equation} \label{eq:Esoc}
E=-\vec{\mu}_e\cdot\vec{B}
\end{equation}

where:
\begin{equation}
    \vec{B}=\frac{\vec{\varepsilon} \times \vec{v}}{2c^2\sqrt{1-v^2/c^2}}
\end{equation}

where $\vec{\varepsilon}$ is the electric field seen by the electron and $ \vec{v}$ is its orbital velocity. It's called the spin-orbit interaction since $\vec{\mu}_e$ arises from the spin and $\vec{B}$ arises from the orbital motion. Given that for a free electron, $\vec{\mu}_e=-\mu_B\vec{\sigma}$, where $\mu_B=e\hbar/2m$ is the Bohr magneton, we can use $\vec{v}=\vec{p}/m$ to rewrite Eq. \ref{eq:Esoc} to describe the SOC Hamiltonian:
\begin{align} \label{eq:hsoc}
    H_{SOC} &=  \frac{e\hbar}{4m^2c^2\sqrt{1-v^2/c^2}} \left( \vec{\varepsilon} \times \vec{p} \right) \cdot \vec{\sigma} \nonumber\\
    &=-\frac{e\hbar}{4m^2c^2} ( \nabla V \times \vec{p} )  \cdot \vec{\sigma}
\end{align}

where the electric field is related to the electric potential $V$ by $\vec{\varepsilon}=-\vec{\nabla} V$, and we assume $v<<c$. SOC leads to the splitting of atomic energy levels that would otherwise be degenerate, a phenomenon known as spin-orbit splitting. The magnitude of SOC depends on the charge of the nucleus and the angular momentum of the electron's orbit. Therefore, heavier elements with larger atomic numbers generally exhibit stronger SOC effects.



Spin-orbit coupling can arise from either the Rashba interaction \cite{rashba1960spin,bychkov1984oscillatory} or the Dresselhaus interaction \cite{dresselhaus1955spin}, which vary depending on the nature of the electric field felt by the electrons. The Rashba interaction occurs if there is a strong electric field due to an externally applied electric field or an internal potential gradient caused by band discontinuity in a heterostructure \cite{bandyopadhyay2008introduction}. To examine this effect, let's rewrite Eq. \ref{eq:hsoc} by introducing the g-factor, which had been previously assumed to be $2$ (for a free electron), whilst also applying the vector identity $\left(\vec{A}\times\vec{B}\right)\cdot\vec{C}=-\vec{A}\cdot\left(\vec{C}\times\vec{B}\right)$:
\begin{equation}
    H_{R}=-\frac{g e\hbar}{8(m^*(\vec{r}))^2c^2}  \varepsilon(\vec{r}) \cdot (\vec{\sigma}\times\vec{p} )  
\end{equation}

where $H_R$ is the Rashba Hamiltonian, and we have replaced the electron mass with the effective mass $m^*$ since we are in a crystal. In the presence of a magnetic field, the momentum operator $\vec{p}$  must be replaced by $\vec{p}+e\vec{A}$ for vector potential $\vec{A}$. This gives a general expression for the Hamiltonian describing the Rashba interaction:
\begin{equation}
    H_{R}=\eta(\vec{r}) \cdot (\vec{\sigma}\times(\vec{p}+e\vec{A}) ) 
\end{equation}

where we define:
\begin{equation} \label{eq:eta}
    \eta(\vec{r})=-\frac{g e\hbar}{8(m^*(\vec{r}))^2c^2}  \varepsilon(\vec{r})
\end{equation}

Eq. \ref{eq:eta} is simplistic insofar as it neglects band structure effects in a solid \cite{bandyopadhyay2008introduction}. Accounting for such effects, Eq. \ref{eq:eta} reads:
\begin{equation} \label{eq:eta1}
    \eta(\vec{r})=-\frac{e\hbar}{m^*(\vec{r})} \frac{\pi\triangle_s(2E_g+\triangle_s)}{E_g(E_g+\triangle_s)(3E_g+2\triangle_s)}   \varepsilon(\vec{r})
\end{equation}


where $E_g$ is the bandgap of the semiconductor and $\triangle_s$ is the spin-orbit splitting in the valence band.



The Rashba interaction is vital to spintronics since it can be tuned by and external electric field $\vec{\varepsilon}$. It is the basis of many spintronics devices, such as spin-FETS and spin-based quantum gates \cite{bandyopadhyay2000self}. While the Rashba effect is a splitting of otherwise degenerate bands due to SOC and asymmetry of the crystal potential, the Dresselhaus effect examines SOC by means of an electric field which arises from crystallographic inversion asymmetry. Since we are interested in GeSn, which has inversion symmetry due to its crystal structure, we will not discuss the Dresselhaus interaction in detail, and instead recommend Ref. \cite{dresselhaus1955spin} for a complete derivation of the Dresselhaus Hamiltonian.



\section{Spin Relaxation}

In spintronics, it is essential for a spin-polarised electron population to retain its spin orientation for as long as possible. At the same time, for practical control and manipulation, it should also be possible to change the spin orientation efficiently when needed. Achieving  both stability and tunability requires a detailed understanding of the mechanisms through which electron spins lose their initial alignment or coherence over time. In any realistic crystal, electrons are continually scattered by lattice imperfections, phonons, other electrons, and magnetic impurities, which not only hinders the electron’s motion but also perturbs its spin state. A key parameter in this context is the \textit{spin relaxation time} (also known as spin-lattice relaxation), which quantifies the rate at which a non-equilibrium spin-polarised population returns to equilibrium. 

In general, the relaxation of the $z$-component of the spin can be modelled by the Bloch equations, which govern the time evolution of the expectation value of the spin operator:
\begin{equation}
    \frac{d\langle S_z\rangle}{dt} = -\frac{\langle S_z\rangle-S_{z,eq}}{T_1}
\end{equation}


where $T_1$ is the longitudinal spin-relaxation time, describing the time for which a spin decays equilibrium, or to put it another way, the time taken for the spins to align back to the magnetic field's direction (along the $z$-axis). $S_{z,eq}$ is the equilibrium spin value in the presence of the magnetic field. This equation describes the exponential return of the spin to its equilibrium value, governed by interactions with the surrounding lattice. In this work, we do not concern ourselves with the transverse relaxation time $T_2$, which characterises the decay of spin coherence in the $x$ and $y$ directions, is typically much smaller than $T_1$ and plays a minor role in the relaxation mechanisms we implement in this work (see Section \ref{subsec:ey}).

In this section, we provide an overview of key relaxation mechanisms, with particular emphasis on the Elliott–Yafet mechanism.





\subsection{Spin Relaxation Mechanisms - Elliott-Yafet} \label{subsec:ey}

There are four major spin relaxation mechanisms of electrons in semiconductor conduction bands: the D'yakonov-Perel (DP) mechanism \cite{dyakonov1971current}, the Elliott-Yafet (EY) mechanism \cite{elliott1954theory,yafet1963g}, the Bir-Aronov-Pikus (BAP) mechanism and hyperfine interactions with nuclear spins. The BAP interaction specifically involves electron-hole interactions, and often occurs in p-type semiconductors. Because of the weak electron-hole exchange coupling in group-IV materials, the BAP interaction is not of concern in this work. The hyperfine interaction involves the interaction between an electron’s magnetic moment and the magnetic fields generated by nuclear spins. The vast majority of isotopes of  group-IV semiconductors have no nuclear spin, therefore we neglect this interaction also. While both the DP and EY mechanisms arise from SOC, the DP process depends on both the Dresselhaus and Rashba interactions. As a result, the DP mechanism is of particular importance for semiconductor materials lacking inversion symmetry, such as zinc-blende structures. We therefore do not concern ourselves with the DP process, a clear explanation of which can be found in \cite{patibandla2010competing}.








\subsubsection{The Elliott and Yafet Processes}

The EY mechanism is valid for metals and semiconductors (metals in the following) with inversion symmetry and weak SOC. It is split into two parts: the Elliott process and Yafet process. Elliott \cite{elliott1954theory} in 1954 was the first to consider the effect of SOC on general band theory. In EY theory, spin-up and spin-down states maintain degeneracy due to time-reversal invariance, unless disrupted by a magnetic field. This is outlined by Kramer's theorem \cite{kramers1930theorie}, which basically states that if a physical process is allowed under certain conditions, then the time-reversed process is also allowed, meaning that if the system is time-reversal invariant, then for every energy eigenstate $\psi$  characterised by quantum numbers $\textbf{k},\mathbf{\sigma}$, there exists another state with the same energy, but with opposite spin, i.e. $\psi_{\textbf{k},\mathbf{\sigma}} = \psi_{-\textbf{k},-\mathbf{\sigma}}$. 



The presence of a nearby band results in a mixing of the spin-up and spin-down states within the conduction band by SOC, preserving energy degeneracy. In other words, the spin-up state contains the spin-down state and vice versa (as we saw in Eq. \ref{eq:spinor}). Examining Eq. \ref{eq:spinor} in more detail, the degenerate Bloch states corresponding to the lattice momentum $\textbf{k}$ in the presence of the spin-orbit coupling can be expressed as:
\begin{align} 
    \Psi_{\textbf{k},n,\uparrow}(\textbf{r}) &= (\alpha_{\textbf{k},n}(\textbf{r})\ket{\uparrow} + \beta_{\textbf{k},n}(\textbf{r})\ket{\downarrow}) \cdot e^{i\textbf{k}\textbf{r}} \nonumber \\
    \Psi_{\textbf{k},n,\downarrow}(\textbf{r}) &= (\alpha^*_{-\textbf{k},n}(\textbf{r})\ket{\downarrow} - \beta^*_{-\textbf{k},n}(\textbf{r})\ket{\uparrow}) \cdot e^{i\textbf{k}\textbf{r}} \label{eq:SOC}
\end{align}


where $n$ signifies the band index, $\uparrow$ and $\downarrow$ signify the effective spins, $\ket{\uparrow}$ and $\ket{\downarrow}$ are the unperturbed spin states with component of angular momentum $\pm 1/2$, while $\alpha_{\textbf{k},n}$, $\alpha^*_{\textbf{-k},n}$, $\beta_{\textbf{k},n}$ and $\beta^*_{\textbf{-k},n}$ contain complex coefficients that determine the probability amplitudes for finding the electron in the spin-up and spin-down states, respectively  \cite{fabian2007semiconductor}. Since spin mixing is often small, $\lvert \beta_{\textbf{k},n} \rvert \ll \lvert \alpha_{\textbf{k},n} \rvert$ and $\lvert \beta^*_{\textbf{k},n} \rvert \ll \lvert \alpha^*_{\textbf{k},n} \rvert$. These coefficients are determined by the $L$-matrix element of the SOC for the conduction band, which refers to the matrix elements of the orbital angular momentum operator $\vec{L}=\vec{r}\times\vec{p}$, and the corresponding energy separation $\Delta$: $\lvert \beta_{\textbf{k},n}/\alpha_{\textbf{k},n} \rvert \approx L/\Delta$ \cite{kiss2016elliott}. This is essentially the \textit{Elliott process}; this admixture of the Pauli up and down spin states in the Bloch state. 


Overhauser \cite{overhauser1953paramagnetic} was the first to pinpoint the modulation of the spin–orbit interaction by phonons as the mechanism for spin relaxation due to incoherent electron–phonon (e-ph) scattering processes. His work, combined with that of Elliott, was furthered by Yafet \cite{yafet1963g}, who generalised Elliott's theory for a range of relaxation mechanisms and at lower temperatures. The \textit{Yafet process} arises from a spin-orbit interaction where the SOC of the conduction electrons to the lattice potential can be influenced by lattice vibrations. This results in an interaction where the electron spin becomes coupled to the quantum of the phonon, causing a scattering process in which the electron spin becomes entangled with the vibrational modes of the lattice. Yafet calculated the spin–flip matrix element due to the e-ph interaction (as a function of electronic momentum transfer \textbf{q}) which included both the Overhauser and Elliott contributions, revealing that the first few orders in \textbf{q} vanish due to a cancellation of Overhauser and Elliott contributions. 


This combination of Overhauser and Elliott contributions is called the \textit{EY mechanism}. For materials such as group-IV semiconductors, where inversion symmetry is present, and in low mobility materials, where momentum-relaxation scattering events are frequent, the EY mechanism is the dominant spin-relaxation mechanism, especially at high temperatures ($>100K$). 








\subsubsection{Relaxation Rates/Times}



Equation \ref{eq:fgr} below describes Fermi's Golden Rule \cite{fermi1950nuclear}, which is used calculate the transition rate of a quantum system from one state to another due to a weak perturbation, such as a phonon or impurity, causing the momentum of the electron to change from $\textbf{k}$ to $\textbf{k'}$:
\begin{equation} \label{eq:fgr}
    \Gamma(\textbf{k'},\textbf{k})= \frac{2\pi}{\hbar} \left| M_{\textbf{k}\textbf{k'}}\right|^2 \delta (E_f-E_i)  
\end{equation}



where $M_{\mathbf{k}, \mathbf{k'}} = \langle \psi_\mathbf{k'} | H' | \psi_\mathbf{k} \rangle$ represents the strength of the interaction between the states $\psi_\mathbf{k}$ and $\psi_\mathbf{k'}$ caused by a particular scattering mechanism determined by the interaction Hamiltonian $H'$, while the Dirac delta function ensures energy conservation between the initial ($i$) and final ($f$) states. Yafet aimed to calculate the spin–flip transition rate for two Kramer's degenerate bands including SOC. Using Eq. \ref{eq:SOC} and Eq. \ref{eq:fgr}, the scattering probability of preserving/flipping a spin can be described as:
\begin{align}
    P_{\textbf{k},\ket{\uparrow} \rightarrow \textbf{k'},\ket{\uparrow}} &= \frac{2\pi}{\hbar} \delta (E_\textbf{k}-E_\textbf{k'}) (V_{\textbf{k}\textbf{k'}})^2 \lvert \alpha^*_{\textbf{k'},n} \alpha_{\textbf{k},n}\rvert ^2 \nonumber \\
   P_{\textbf{k},\ket{\uparrow} \rightarrow \textbf{k'},\ket{\downarrow}} &= \frac{2\pi}{\hbar} \delta (E_\textbf{k}-E_\textbf{k'}) (V_{\textbf{k}\textbf{k'}})^2 \lvert \beta^*_{\textbf{k'},n} \alpha_{\textbf{k},n}\rvert ^2 \label{eq:Pflip}
\end{align}

%noindent{If we write the Bloch states in Eqs. \ref{eq:SOC} as: $\ket{\Psi_{\textbf{k},n,\sigma}}=\ket{u_{\textbf{k},n,\sigma}}e^{i\textbf{k}\cdot r}$, then Eqs. \ref{eq:Pflip} can be expressed as: 
%}
%\begin{align*}
 %   P_{\textbf{k},n,\sigma \rightarrow \textbf{k'},n,\sigma'} &= \frac{2\pi}{\hbar} \delta (E_\textbf{k}-E_\textbf{k'}) (V_{\textbf{k}\textbf{k'}})^2 \ \lvert \bra{u_{\textbf{k},n,\sigma}} u_{\textbf{k'},n,\sigma'} \rangle \rvert ^2 \\
 %  P_{\textbf{k},n,\sigma \rightarrow \textbf{k'},n,\sigma} &= \frac{2\pi}{\hbar} \delta (E_\textbf{k}-E_\textbf{k'}) (V_{\textbf{k}\textbf{k'}})^2 \ \lvert \bra{u_{\textbf{k},n,\sigma}} u_{\textbf{k'},n,\sigma} \rangle \rvert ^2 
%\end{align*}

where for an impurity scattering potential $V$ which varies slowly across the unit cell, the scattering matrix element $M_{\textbf{k}\textbf{k'}}$ can be approximated as:
\begin{align}
    \bra{\Psi_{\textbf{k},\uparrow}} V \ket{\Psi_{\textbf{k'},\uparrow}} &\approx V_{\textbf{k}\textbf{k'}} \ (\alpha^*_{\textbf{k'},n} \alpha_{\textbf{k},n}) \nonumber \\
    \bra{\Psi_{\textbf{k},\uparrow}} V \ket{\Psi_{\textbf{k'},\downarrow}} &\approx V_{\textbf{k}\textbf{k'}} \ (\beta^*_{\textbf{k'},n} \alpha_{\textbf{k},n})
\end{align}

with $V_{\textbf{k}\textbf{k'}}=\int e^{i(\textbf{k}-\textbf{k'})\cdot \textbf{r}} \ V(\textbf{r}) \ d^3r$, which represents the spin-flip and non-spin-flip transition elements.



%The scattering rates $R$ and $R_s$ can be written as:
%\begin{align*}
%     R &= \int d^3k d^3k' \frac{\delta(\textbf{k}-\textbf{k}_F)}{4\pi k_F^2} \frac{\delta(\textbf{k'}-\textbf{k}_F)}{4\pi k_F^2} P_{k,n,\sigma \rightarrow k',n,\sigma} \\
%     R_s &= \int d^3k d^3k' \frac{\delta(\textbf{k}-\textbf{k}_F)}{4\pi k_F^2} \frac{\delta(\textbf{k'}-\textbf{k}_F)}{4\pi k_F^2} P_{k,n,\sigma \rightarrow k',n,\sigma'}
% \end{align*}

 % I don't use this, I use a thermal average and get the inverse for the relaxation time!

%after a double integral of the transition probabilities with wave-vector $\textbf{k}_F$ on the Fermi surface. }



The relaxation time $\tau$ of a scattering event is essentially the inverse of the scattering rate. The general form of the scattering rate due to an e-ph interaction ($(\tau_{n\mathbf{k}}^{e-ph})^{-1}$) can be expressed as:
\begin{equation}
\frac{1}{\tau_{n\mathbf{k}}^{e-ph}} = \frac{2\pi}{\hbar} \sum_{n'\mathbf{q}, \nu} \left|M_{n n' \nu}(\mathbf{k},\mathbf{q})\right|^2 \ (N_{\mathbf{q}, \nu} + \frac{1}{2} \pm \frac{1}{2}) \ \delta(\epsilon_{n\mathbf{k}} - \epsilon_{n'\mathbf{k}+ \mathbf{q}} \pm \hbar \omega_{\mathbf{q}, \nu}) 
\end{equation}

where $M_{n n' \nu}(\mathbf{k},\mathbf{q})$ is the e-ph matrix element, $q$ is the phonon wave vector such that $\mathbf{k'}=\mathbf{k}+\mathbf{q}$, $\nu$ is the phonon branch, $\omega_{\mathbf{q}, \nu}$ is the phonon frequency and $N_{\mathbf{q}, \nu}$ is the phonon occupation number. Yafet showed the spin-flip e-ph relaxation times $\tau_{n\mathbf{k}}^{flip}$ could be calculated using a similar expression \cite{yafet1963g,park2020spin}:
\begin{equation}
\frac{1}{\tau_{n\mathbf{k}}^{flip}} = \frac{4\pi}{\hbar} \sum_{n'\mathbf{q}, \nu} \left|M^{flip}_{n n' \nu}(\mathbf{k},\mathbf{q})\right|^2 \ (N_{\mathbf{q}, \nu} + \frac{1}{2} \pm \frac{1}{2}) \ \delta(\epsilon_{n\mathbf{k}} - \epsilon_{n'\mathbf{k}+ \mathbf{q}} \pm \hbar \omega_{\mathbf{q}, \nu}) 
\end{equation}



The temperature-dependent spin-relaxation time $\tau_s(T)$ is easily obtained as an ensemble average (average over \textbf{k}) of the spin-flip relaxation times \cite{yafet1963g,park2020spin} by integration over the Fermi-surface:
\begin{equation} \label{eq:tauav}
    \tau_s(T)=  \left< \frac{1}{\tau_{n,k}^{flip}} \right>^{-1}_T = \left( \frac{\sum_{n\mathbf{k}} \frac{1}{\tau_{n,k}^{flip}} \left( -\frac{\partial f_{n\mathbf{k}}}{\partial E}  \right) 
 d\mathbf{k}}{\sum_{n\mathbf{k}} \left( -\frac{\partial f_{n\mathbf{k}}}{\partial E}  \right)  d\mathbf{k}}  \right)^{-1}
\end{equation}


where $f$ represents the electron distribution function. When the EY mechanism is the dominant scattering mechanism, Elliott showed a direct proportionality correlation between $\tau_{e-ph}$  and $\tau_s$ using first-order time-perturbation treatment:
 \begin{equation} \label{spinrate}
     \tau_{e-ph}\approx\alpha \left( \frac{L}{\Delta_s} \right)^2 \tau_s
 \end{equation}

where $\alpha$ is a band-dependent constant near unity. Eq. \ref{spinrate} is suitable for evaluation from ab initio input. Yafet showed this expression to be temperature-independent, given that $E_g$ and $\Delta_s$ have weak temperature dependences \cite{bandyopadhyay2008introduction} (in other words, $\tau_s$ and $\tau_{e-ph}$ have the same temperature dependence). The direct proportionality between $\tau_{s}$  and $\tau_{e-ph}$ is to be expected, since the EY mechanism is caused by momentum relaxation. Therefore, with an increase in temperature, the spin-relaxation rate via the EY mechanism also increases due to enhanced e-ph scattering. 


The initial derivations of Elliott and Yafet were based on expressions of wave functions for alkali metals (which included Li, Na and K). They also examined the wavefunctions of Si, predicting that the spin relaxation time is of the order of $10^{-9}s$ at $300K$. Appelbaum et al \cite{appelbaum2007electronic,huang2007coherent} experimentally determined $\tau_s$ for Si in 2007 and their results were in excellent agreement with the theoretical value. Li, Song and Dery \cite{li2012intrinsic} calculated the spin relaxation time of Ge to also be in the nanosecond range (see Hamaya et al for a detailed list of previously determined spin-relaxation times of electrons and holes in Ge \cite{hamaya2018spin}). In metals, Monod and Beunue \cite{monod1979conduction} provided experimental evidence that confirmed that $\tau_s$ and $\tau_{e-ph}$ show the same temperature dependence. Often, $\tau_{e-ph}$ is of the order of picoseconds, while usually $\tau_s$ is several nanoseconds long. In Table \ref{tab:T1}, we give a list of spin relaxation times previously determined theoretically or by experiment.
%A more detailed expression is given by Pitkus and Titkov \cite{pikus1984spin}, who calculated an approximate expression for $\tau_s$ using the EY mechanism in bulk semiconductors using Eq. \ref{eq:tauav}, assuming the carriers are in a non-degenerate population:

%\begin{equation}
%    \frac{1}{\tau_s}=A \left( \frac{k_B T}{E_g} \right)^2 \Lambda^2 \left( \frac{1-\Lambda/2}{1-\Lambda/3} \right)^2 \frac{1}{\tau_{e-ph}}
%\end{equation}

%where $\Lambda=\nabla_s/(E_g+\nabla_s)$.  


Perhaps surprisingly, both long spin lifetimes and short spin lifetimes can be advantageous, depending on the application. Generally speaking, materials with long spin lifetimes are sought after for applications such as spin-based transistors, quantum computing and spin interconnects. In contrast, other spintronics applications such as ultra-fast optical switches require short spin lifetimes \cite{marchionni2021inverse,vzutic2004spintronics}.


\bigskip

\begin{table}[H] \label{tab:T1}
\centering
\begin{tabular}{lcc}
\toprule
\textbf{Material} & \textbf{Spin Relaxation Time (\(\tau_s\))} & \textbf{Reference} \\
\midrule
Si  & $\approx$ 1 ns & \cite{appelbaum2007electronic,elliott1954theory,cheng2010theory} \\
Ge         & $\approx$ 1 ns    &    \cite{tang2012electron,hamaya2018spin,li2012intrinsic}   \\
Ge$_{0.95}$Sn$_{0.05}$   & $\approx$ 0.1 ns  &  \cite{marchionni2021inverse} \\
GaAs & 50 ps - 2 ns     & \cite{seymour1980time,ohno1999spin,oertel2008high} \\
GaSb  &  5-10 ps        & \cite{song2002spin,hautmann2009temperature} \\
InAs  &   $\approx$ 20 ps         & \cite{song2002spin,boggess2000room} \\
InSb  &  10 - 20 ps & \cite{song2002spin,bandyopadhyay2014coherent,murzyn2003suppression} \\
Bilayer Graphene   &  100 ps - 2 ns  & \cite{yang2011observation,idzuchi2015revisiting} \\
\bottomrule
\end{tabular}
\caption{Spin relaxation times (\(\tau_s\)) for different materials at room temperature ($300K$). }
\label{tab:spin_relaxation}
\end{table}






\subsubsection{Developments of the EY Mechanism}

Although the EY mechanism has been successful in describing spin relaxation, particularly in light semiconductors and metals with inversion symmetry, it has had several limitations and challenges that have been addressed in the decades since its initial development; most of which have been since the early 2010s. In 2012, Restrepo and Windl \cite{restrepo2012full} produced a new parameter-free first-principles method to obtain spin-relaxation times for phonon and impurity scattering. They demonstrated the possibility of calculating spin-relaxation times without adjustable parameters and without restrictions concerning the nature of the band gap, making it applicable to any type of impurity or defect on spin transport. They applied their method to group-IV materials silicon, diamond, and graphite, producing results that matched well with experiment. This generalised the EY mechanism, ensuring that it was no longer materials-specific.




% Kiss - heavy metals, large SOC generalisation
The EY mechanism assumes weak spin-orbit coupling in the system (compared to say the DP mechanism). In heavy materials with a strong spin-orbit interaction (such as lead or bismuth), this assumption breaks down. Strong SOC can lead to different spin relaxation mechanisms not captured by the EY mechanism. This brought into question the first-order perturbation treatment used by Elliott.
In 2015, Kiss et al \cite{kiss2016elliott} generalised the EY mechanism for materials with small SOC compared to energy scales such as band separations and kinetic energy. They implemented a $4$-band model which introduced band-specific contributions to the spin relaxation time, meaning that each of the four bands contributed differently to the overall relaxation process. With their model, they recovered the usual EY theory for weak SOC, but more importantly showed using experimental data on gold (provided by Monod and Janossy \cite{monod1977conduction}) that the spin-relaxation rate approached the momentum-relaxation rate in the limit of strong SOC.



% Baral - yafet not considering spin mixing
The density matrix $\rho$, shown in Eq. \ref{eq:denmat}, provides a framework for analysing a large ensemble of spinors with an initial statistical distribution at time $t=0$ and tracking their evolution over time by examining the evolution of $\rho$. The solution of this evolution is governed by the Bloch equations of motion. $T_1$ dictates how the diagonal elements (population differences) relax to equilibrium, and $T_2$ describes the decay of off-diagonal elements (coherences), representing dephasing of superpositions between spin-up and spin-down states. We can express $\rho$ as:
\begin{equation} \label{eq:denmat}
\rho = \ket{\psi}\bra{\psi}= \begin{pmatrix} \alpha \\ \beta\end{pmatrix} \begin{pmatrix}  \alpha^* & \beta^*   \end{pmatrix} =\begin{pmatrix} |\alpha|^2 & \alpha \beta^* \\ \alpha^* \beta & |\beta|^2 \end{pmatrix} 
\end{equation}

where we recall from Eq. \ref{eq:SOC} that $|\alpha|^2+|\beta|^2=1$. In Yafet's work, the coherences (off-diagonal elements) were overlooked. Consequently, spin mixing wasn't accounted for, so that pseudo-spin polarisation and actual spin polarisation were treated as equivalent, and thus a $k-$dependent expectation value of $\hat{S}_z$ is not achieved. He therefore could not correctly account for the amount of spin-flips in each scattering transition. This is especially problematic in materials in which the Brillouin zone contains regions of very high spin mixing \cite{steiauf2010extension}, known as spin hotspots. In 2016, Baral et al \cite{baral2016re} developed a torque operator due to the e-ph interaction that determines the spin dynamics; deriving a modified expression for the EY spin relaxation time using a reduced density matrix which included the off-diagonal elements of $\rho$. This approach achieved a more accurate description of the dynamics of the spin vector.




% Park and Berardi - 1st principles - no direct prop assumption
The EY mechanism relies on empirical models and symmetry analysis, which are laborious approaches even for simple systems \cite{park2020spin}. Even the more modern methods used by Kiss et al depend on empirical models. In addition, methods such as those used by Restrepo and Windl assume direct proportionality between spin-flip and momentum-scattering interactions \cite{park2020spin}, while other methods \cite{kurpas2016spin} depend on the direct proportionality between the relaxation times. In 2019, Park, Zhou and Bernardi \cite{park2020spin} presented a first-principles approach for computing the spin-flip matrix elements and the spin-phonon relaxation times without any assumptions on the relationship between the matrix elements or the spin-flip and momentum scattering. They treated the spinor wavefunctions and SOC through fully-relativistic density functional theory, applying their method to Si and diamond; producing results which compared well with experiment. This opened up the possibility to predict spin-phonon relaxation times in a wide range of materials.


% We know that spin relaxation via EY increases with temperature. However, experiments in some semiconductors showed that spin relaxation could exhibit different temperature dependencies, necessitating the inclusion of additional mechanisms (e.g., D'yakonov-Perel).




In conclusion, it was the discovery and subsequent understanding of electron spin that not only deepened our knowledge of quantum mechanics, thus laying the foundation for the development of spintronics. It wasn't until the late $20^{th}$ century that it was realised that spin could be used to process digital information encoded with binary bits. As discussed in Chapter \ref{chap:intro}, spintronics devices harnesses the spin of electrons, using electron spins as information carriers. Since we have proposed GeSn as a potential semiconductor in spintronics devices, we will examining its electronic properties on a fundamental level in a later chapter.


% With spin-based devices functionality depending on the deliberate generation and retrieval of a non-equilibrium net spin population, one of the primary goals of spintronics is to develop devices that can efficiently exploit the spin degree of freedom for information processing while overcoming challenges such as maintaining spin coherence over longer distances. 








