## 2 Dynamic Nuclear Polarization in Semiconductor Nano-and Microstructures
### Optically Induced DNP in Semiconductors
#### 2.1.3 Fundamentals of the Optically Induced DNP  
Photoexcitation of electrons in a semiconductor under irradiation with _unpolarized_ light (the case of circular polarization is considered in Appendix) produces an equal number of electrons with spin up and down. After that, the Overhauser effect [16] occurs on \({}^{29}\)Si nuclei in the bulk crystalline silicon (c-Si): the dynamic enhancement of the nuclear polarization, obtained by maintaining the electronic polarization off its equilibrium value. The essence of the Overhauser effect consists in the following. A metal sample is placed in a constant magnetic field \(\mathcal{B}_{0}\), which creates the polarization of electron spins. The electron energy in this field \(E~{}=~{}\epsilon_{F}-\mu_{e}\cdot\mathcal{B}_{0}\), is expressed in terms of \(\epsilon_{F}\) is the Fermi energy \(\epsilon_{F}\) and the magnetic moment of the electron spin \(\mathcal{S}:~{}\mu_{e}~{}=~{}-g_{e}\mu_{Be}\mathcal{S}\), where \(\mu_{Be}~{}=~{}\frac{e\hbar}{2m_{e}}\) is the Bohr magneton and \(g_{e}\) is the electron \(g\)-factor ([17], p. 420). It has two eigenvalues  
\[E_{j}~{}=~{}\epsilon_{F}\pm\mu_{Be}\mathcal{B}_{0},j~{}=\uparrow,\downarrow, \tag{1}\]  
where \(g_{e}\) is taken as 2. The electron spins tend to be aligned with the field. The resulting difference in spatial density of electrons with spins down and up \(D~{}\equiv N_{\downarrow}-N_{\uparrow}\) is selected as a measure of the electron spin magnetization \(S_{z}\equiv\mu_{Be}D\). In thermal equilibrium under relatively weak magnetic fields, the value of \(D\) takes on the form ([17], pp. 421 and 435)  
\[D_{0}~{}=~{} \begin{cases}\frac{3N\mu_{Be}\cdot\mathcal{B}_{0}}{2\epsilon_{F}}& \text{at low temperatures;}\\ \frac{N\mu_{Be}\cdot\mathcal{B}_{0}}{kT}&\text{at high temperatures.}\end{cases}  \tag{2}\]  
The metal sample is subjected, furthermore, to a perpendicular alternating magnetic field (in particular, of the microwave radiation) \(\mathcal{B}_{ac}=2\mathcal{B}_{1}\cos(\omega t)\), whose frequency satisfies the _electron spin resonance_ (electron paramagnetic resonance) criterion ([17], p. 488)  
\[\hbar\omega~{}=~{}2\mu_{Be}\mathcal{B}_{0}, \tag{3}\]  
where \(2\mu_{Be}\mathcal{B}_{0}\) is the distance between the eigenenergies given by Equation (1). \(2\mu_{Be}\mathcal{B}_{0}/\hbar\) is the precession frequency for the electron spin magnetic moment \(\mu_{e}\) around the magnetic field \(\mathcal{B}_{0}\). Such a radiation leads to the transitions between electronic states. As a consequence, two kinds of processes take place in the sample--first, relaxation drives the magnetization to its equilibrium value \(S_{z0}\equiv\mu_{Be}D_{0}\) with a typical spin relaxation time \(T_{1}\); second, the alternating field induces more transitions with the photoinduced relaxation time \(\tau\) from the state with a higher population and therefore leads to a decrease of the magnetization:  
\[\frac{dD}{dt}~{}=~{}-\frac{D-D_{0}}{T_{1}}-\frac{D}{\tau}. \tag{4}\]  
In a stationary state, \(\frac{dD}{dt}~{}=~{}1\), the both processes are counterbalancing each other, what gives rise to the steady-state magnetization  
\[\langle S_{z}\rangle~{}=~{}\frac{S_{z0}}{1+T_{1}/\tau}. \tag{5}\]  
At strong alternating magnetic fields, the resonance is saturated, that is \(D=0\). In the general case, a _saturation parameter_  
\[s~{}=~{}1-\frac{\langle S_{z}\rangle}{S_{z0}} \tag{6}\]  
changes from 0 when there is no alternating magnetic field to 1 when it is strong. It is a function of the ratio \(T_{1}/\tau\) only:  
\[s~{}=~{}\frac{T_{1}/\tau}{1+T_{1}/\tau}. \tag{7}\]The amplitude of the microwave field that is necessary to saturate the resonance is inversely proportional to the geometric mean value of the spin relaxation time \(T_{1}\) and the transverse relaxation time \(T_{2}\), which is determined by the integrated width of the resonance line \(w\) through \(w\ =\ \pi\hbar/T_{2}\)[16].