## 2 Dynamic Nuclear Polarization in Semiconductor Nano-and Microstructures
### Optically Induced DNP in Semiconductors
#### 2.1.3 Fundamentals of the Optically Induced DNP
Using NMR of \({}^{29}\)Si, the spin-lattice relaxation time was determined as a function of mobile carrier concentration (sample resistivity) [18]. \(T_{1}\) was established to be inversely proportional to the carrier concentration for the high-conductivity samples, approaching an asymptotic value in the purer samples (Figure 4). Analysis of relaxation times in semiconductors [19] led to the conclusion that the relaxation of \({}^{29}\)Si in the exhaustion and intrinsic ranges should be accomplished by the hyperfine interaction between the nuclear spins and the spins of the mobile electrons or holes. However, there was a significant uncertainty in the density of mobile carriers. No differences were observed because of the crystal growth conditions (some crystals were pulled from quartz crucibles, some zone-refined without crucibles). The asymptotic values of \(p\)- and \(n\)-type Si were substantially different. Figure 4: Plot of \({}^{29}\)Si spin-lattice relaxation times versus mobile carrier concentration. (Reprinted figure with permission from Reference [18] © (1956) by the American Physical Society). The low-field relaxation of nuclear hyperpolarization was studied in undoped and highly doped Si MPs at room temperature following removal from high field [20]. For nominally undoped particles (Figure 5a), two relaxation time scales were identified for ambient fields above 0.2 mT. The slower, \(T_{1,s}\), was roughly independent of ambient field; the faster, \(T_{1,t}\), decreased with increasing ambient field. A model assumed the nuclear spin relaxation to occur at the Si MPs surface via a 2-electron mechanism in a good agreement with the experimental data in what concerns the magnetic-field independence of the slow relaxation time. For boron-doped particles (Figure 5b), a single relaxation time scale was observed. This suggested that for doped particles, mobile carriers and bulk ionized acceptor sites, rather than paramagnetic surface states, were the dominant relaxation mechanisms. Relaxation times for the undoped particles were not affected by tumbling in a liquid solution [20]. Nuclear polarization is most efficiently performed via the _hyperfine interaction_ with electrons localized in a donor impurity [21]. The key issue is the hyperfine dipolar interaction between the magnetic moments of electron \(\mu_{e}\) and nucleus \(\mu_{n}~{}=~{}\mu_{Bn}\delta_{n}\mathbf{I}\) (\(\mu_{Bn}~{}=~{}\frac{e\hbar}{2m_{n}c}\)) separated in space by a vector \(\mathbf{r}\)  
\[H~{}=~{}-\frac{8\pi}{3}\mu_{e}\cdot\mu_{n}\delta(\mathbf{r}). \tag{8}\]  
Figure 5: **(a)** Model showing the spin reservoirs and relaxation pathways in undoped silicon particles. Nuclei in the core region of the particle relax by transferring their magnetization to the nuclei in the shell region, nearer to the surface, by spin diffusion. The nuclei near the surface can relax quickly due to strong dipolar coupling to paramagnetic defects at the Si/SiO\({}_{2}\) interface. **(b)** Rate model showing dominant nuclear relaxation pathways of direct relaxation on mobile holes and short diffusion to non-ionized B (i.e., bound holes) throughout the bulk of the boron-doped particle. Long-distance spin diffusion to surface impurities is relatively slow and direct coupling of the nuclei to the lattice is weak. (Reprinted figures with permission from Reference [20] © (2011) by the American Physical Society). Here, \(I\) and \(\mu_{Bn}\) (for nuclei with \(I=\frac{1}{2}\)) are the nuclear spin angular moment and nuclear Bohr magneton, respectively. As shown in Figure 6, the action of the hyperfine interaction is twofold. (i) Every nucleus is subjected, next to the applied magnetic fields, to a magnetic field of polarized electron spins fluctuating due to the aforementioned processes of spin relaxation. The magnitude of this field is proportional to the average (over the fluctuations) electron spin.