Temporal and spatial evolution of nuclear polarization in optically pumped InP

The electron-nuclear interaction in optically pumped NMR of semiconductors manifests itself through changes in spectral features (resonance shifts, linewidths, signal amplitudes) and through the magnitude of the nuclear-spin polarization. We show that these spectral features can provide a measure of the parameters that govern the optical pumping process: electron-nuclear cross-relaxation rate, Bohr radius and fractional occupancy of the optically relevant defect (ORD), and electron polarization at the ORD. Applying a model of the spatial and temporal evolution of the nuclear spins under optical pumping to ${}^{31}\mathrm{P}$ in semi-insulating InP we find an ORD Bohr radius of 6 nm, independent of the electron polarization used to fit the data, confirming the ORD is a shallow donor. For an electron polarization of $-0.15$, the ORD fractional occupancy is 0.02, leading to an electron-nuclear cross-relaxation time of 0.20 s and a hyperfine frequency shift of 8.1 kHz for super-bandgap irradiation. Allowing the electron polarization to vary in the model constrained to the hyperfine shift data, we find the fractional occupancy and electron-nuclear cross-relaxation rate to be approximately inversely proportional to the electron polarization. From the long-time evolution of the nuclear polarization we calculate an ORD density of $5\times {10}^{15}{\text{cm}}^{-3}$.

Figure 1

Plot of normalized ${}^{31}\mathrm{P}$ NMR signal amplitude (blue circles) and nuclear polarization (red squares), defined as $\langle I_z\rangle /I$, vs temperature for 800 s irradiation with ${\sigma }_{+}^B$ light at 1.428 eV.

Figure 2

Plot of normalized ${}^{31}\mathrm{P}$ NMR signal amplitude as a function of laser power normalized to the saturation laser power. The signals (open circles, 1.428 eV; filled circles, 1.408 eV) were obtained by illuminating the sample for 200 s with ${\sigma }_{+}^B$ light. The normalization parameters were obtained by fitting the data to the integral over $d^{†}$ of Eq. (16). The saturation parameters ${\mathrm{\Phi }}_S$ are 0.3 and 1.6 ${\mathrm{W}/\mathrm{cm}}^2$ at 1.428 and 1.408 eV, respectively. Inset: Polarization as a function of normalized depth, from Eq. (16), for $\mathrm{\Phi }/{\mathrm{\Phi }}_S$ of 2 (dashed red line) and 11 (solid blue line) corresponding to the typical experimental situation 3.4 ${\mathrm{W}/\mathrm{cm}}^2$ pumping power for 1.408 and 1.428 eV, respectively.

Figure 3

The top two graphs show complex ${}^{31}\mathrm{P}$ NMR echo data starting with the midpoint of the echo and the graphs below give the corresponding spectra; for clarity only the real spectral data are shown. For long pumping time data, as shown in the rightmost graphs, the signals are very similar for data acquired with the laser light off (blue circles) and on (orange stars). In contrast, for short pumping time data, as shown in the leftmost graphs, the light-on signal decays more quickly than the light-off signal and is shifted in frequency, revealing the effects of the polarized electron. While the data shown here are for sub-bandgap irradiation, super-bandgap irradiation shows similar trends. The solid lines in the upper graphs correspond to fits to the data, as described by Eqs. (22) and (23).

Figure 4

Plots of ${}^{31}\mathrm{P}$ NMR spectral data as a function of pumping time for sub-bandgap data (orange filled circles) and super-bandgap data (black open squares). The lines, from top plot to bottom plot, represent scaled nuclear polarization, scaled square root of the second moment of the line shape ($÷2$ super-bandgap, $\times 2$ sub-bandgap), and absolute first moment of the line shape ($\times 2$ sub-bandgap), respectively, obtained from the model. As shown in the top figure the amplitude of the echo signal is linear with pumping time. In contrast, for pumping times on the order of 20 s or less, the spectral behavior changes quite dramatically as a function of pumping time. The narrowing of the spectral line with pumping time is demonstrated in the middle graph, while the bottom graph shows how $\mathrm{\Delta }f$ decreases for increasing ${\tau }_L$. While sub-bandgap data and super-bandgap data show similar behavior, the broadening, defined as $\mathrm{\Gamma }/\left(2\pi \right)$, for sub-bandgap behavior is more pronounced.

Figure 5

The ${}^{31}\mathrm{P}$ NMR signal grows with irradiation time; the nuclear polarization, defined as $\langle I_z\rangle /I$, grows at an even faster rate. Sub-bandgap (blue filled circles) irradiation leads to slower signal buildup compared to super-bandgap (blue open circles), for the same power, 3.4 ${\mathrm{W}/\mathrm{cm}}^2$. Higher power irradiation, 6 ${\mathrm{W}/\mathrm{cm}}^2$ (green triangles), results in faster buildup and lower power irradiation, 1 ${\mathrm{W}/\mathrm{cm}}^2$ (red squares), results in slower buildup of signal. Signal-weighted polarization buildup for super-bandgap irradiation at 3.4 ${\mathrm{W}/\mathrm{cm}}^2$ is plotted as shaded blue stars. The signal (normalized) and polarization share the same axes. Inset: Plot of the resonance frequency difference between signals from irradiation with ${\sigma }_{+}^B$ and ${\sigma }_{-}^B$ light vs total pulse angle for 3200-s irradiation time at 3.4 ${\mathrm{W}/\mathrm{cm}}^2$. The slope of the line gives an initial signal-weighted polarization of 12%.

Figure 6

Plot of least-squares residuals between the signal buildup model [Eq. (4)] and measurements of ${}^{31}\mathrm{P}$ NMR signal amplitude for 1.428 eV irradiation at 3.4 ${\mathrm{W}/\mathrm{cm}}^2$ as a function of $r_{\max }^{*}$ and $T_{1L}\left(0\right)$ for $J_z=-0.15$. The area between the red contours is the area of minimum residuals. The shaded area is the region where the model predicts nuclear polarization between 10% and 14%, for 3200 s of irradiation at 1.428 eV, consistent with the measured value (Fig. 5). Inset: The shaded regions are where the model predicts nuclear polarization between 10% and 14%, for 3200 s of irradiation at 1.428 eV for several values of $J_z$. The value of $r_{\max }^{*}$ remains between 5 and 6 for all values of $J_z$.

Figure 7

The left set of graphs shows the frequency shift in terms of $f_0$ expected at a given radius $r/a_0$, and $cos\theta$, for contact dipolar interaction (top) and noncontact interaction (bottom). The right graph shows a comparison of the two shift contributions for $\theta =0$.