Pulsed low-field electrically detected magnetic resonance

We present pulsed electrically detected magnetic resonance (EDMR) measurements at low magnetic fields using phosphorus-doped silicon with natural isotope composition as a model system. Our measurements show that pulsed EDMR experiments, well established at $X-\text{band}$ frequencies (10 GHz), such as coherent spin rotations, Hahn echoes, and measurements of parallel and antiparallel spin pair life times are also feasible at frequencies in the megahertz (MHz) regime. We find that the Rabi frequency of the coupled ${}^{31}\mathrm{P}$ electron-nuclear spin system scales with the magnetic field as predicted by the spin Hamiltonian, while the measured spin coherence and recombination times do not strongly depend on the magnetic field in the region investigated. The usefulness of pulsed low-field EDMR for measurements of small hyperfine interactions is demonstrated by electron spin echo envelope modulation measurements of the ${\mathrm{P}}_{\mathrm{b}0}$ dangling-bond state at the $\mathrm{Si}/{\mathrm{SiO}}_2$ interface. A pronounced modulation with a frequency at the free Larmor frequency of hydrogen nuclei was observed for radio frequencies between 38 and 400 MHz, attributed to the nuclear magnetic resonance of hydrogen in an adsorbed layer of water. This demonstrates the high sensitivity of low-field EDMR also for spins not directly participating in the spin-dependent transport investigated.

Figure 1

(a) Breit-Rabi diagram of the ${}^{31}\mathrm{P}$ donor. Due to the level mixing, electron- and nuclear-spin-like transitions become observable in a pulsed low-field EDMR experiment as shown in (b). The spin system was excited with a 300 ns-long square rf pulse with a frequency of 48.6 MHz. The resonance line of the nuclear-spin-like transition is broader than the one of the electron-spin-like transition because the magnetic field dependence of the transition frequency is weak as shown in (a). (d) Rabi oscillations measured on the electron-spin-like, the nuclear-spin-like, and the dangling-bond transition (exemplarily shown for an rf of 48.6 MHz). The solid black lines represent fits with exponentially damped cosines and a linear background accounting for the reduced spectral width of the longer pulses [28]; the data are vertically offset for better visibility. Further, Rabi oscillations were measured as a function of the rf power level, resulting in a linear dependence of the Rabi frequencies $({\omega }_{\mathrm{Rabi}}/2\pi )$ on the rf magnetic field $B_1$ as shown in the inset; the ${\mathrm{P}}_{\mathrm{b}0}$ transition was used for the calibration of the $B_1$ field and the experimental values for ${\gamma }_{\mathrm{x},\mathrm{eff}}/\gamma$ were extracted from the slopes of the linear $B_1$-Rabi-frequency relationship of the electron- and nuclear-spin-like transitions. (c) The effective gyromagnetic ratio ${\gamma }_{\mathrm{x},\mathrm{eff}}/\gamma$, which determines the coupling of a spin transition to the rf driving field, is given by $cos(\eta /2)$ and $sin(\eta /2)$ for electron- and nuclear-spin-like transitions, respectively, and plotted against the magnetic field (solid lines). The blue and green data points shown were extracted from the $B_1$ dependence of the Rabi frequencies of the electron- and nuclear-spin-like transitions, cf. the inset in (d).

Figure 2

(Top) Pulse sequence employed for low-field spin echo measurements. (a) Spin echo decays of the ${\mathrm{P}}_{\mathrm{b}0}$, ${\mathrm{P}}_{\mathrm{e}}$, and ${\mathrm{P}}_{\mathrm{n}}$ transitions at an rf of 50.5 MHz with stretched exponential fits. (b) Effective coherence times of the ${\mathrm{P}}_{\mathrm{e}}$ and ${\mathrm{P}}_{\mathrm{n}}$ transitions as a function of ${\gamma }_{z,\mathrm{eff}}/\gamma$. (c) Effective coherence time of the ${\mathrm{P}}_{\mathrm{b}0}$ transition as a function of the magnetic field.

Figure 3

(Top) Inversion recovery pulse sequence used for the measurements of ${\tau }_{\mathrm{ap}}$ in the low-field region. The $\pi$ pulse used for the inversion of the spin system has a larger excitation bandwidth than the $\pi$ pulse in the echo for reasons discussed in the text. (a) Measurements with the pulse sequence sketched on top performed on the ${\mathrm{P}}_{\mathrm{b}0}$, ${\mathrm{P}}_{\mathrm{e}}$, and ${\mathrm{P}}_{\mathrm{n}}$ transitions at an rf of 50.5 MHz with stretched exponential fits. (b) Effective recombination times $T_{\mathrm{IR},\mathrm{eff}}$ of the ${\mathrm{P}}_{\mathrm{e}}$ and ${\mathrm{P}}_{\mathrm{n}}$ transitions as a function of ${\gamma }_{\mathrm{x},\mathrm{eff}}/\gamma$. The dashed line shows the theoretically expected ${(\gamma /{\gamma }_{\mathrm{x},\mathrm{eff}})}^2$ dependence of the recombination time; the discrepancy between theory and experiment is discussed in the text. (c) Effective recombination time of the ${\mathrm{P}}_{\mathrm{b}0}$ transition as a function of the magnetic field.

Figure 4

(Top) Pulse sequence used to determine the lifetime of parallel spin pairs. (a) Measurement of the lifetime of parallel spin pairs ${\tau }_{\mathrm{p}}$ for the ${\mathrm{P}}_{\mathrm{b}0}$, ${\mathrm{P}}_{\mathrm{e}}$, and ${\mathrm{P}}_{\mathrm{n}}$ transition at an rf of 50.5 MHz with stretched exponential fits. The data sets are offset for better visibility. (b) The parallel spin pair recombination time as a function of the magnetic field; the exponent was fixed to $n=0.9$.

Figure 5

(Top) Pulse sequence used in the electrically detected stimulated echo decay measurements. (a) Stimulated echo decays of the ${\mathrm{P}}_{\mathrm{b}0}$ transition measured at different radio frequencies, revealing a pronounced modulation of the echo envelope with a period $T_{\mathrm{ESEEM}}$. (b) ESEEM frequency ${\nu }_{\mathrm{ESEEM}}$ as obtained from fits of the data in (a) plotted against the Zeeman splitting, i.e., the employed radio frequency $f_{\mathrm{rf}}$, revealing a linear dependence; the solid black line represents a linear fit to the data through the origin. The free nuclear Larmor frequencies of ${}^{31}\mathrm{P}$ (dotted black line), ${}^{29}\mathrm{Si}$ (dashed black line), and ${}^1\mathrm{H}$ (dashed orange line) are shown for comparison. (c) and (d) The ESEEM signal is simulated based on a point-dipole approximation assuming randomly located H atoms in a 1 nm-thick slab. The plots show the resulting echo signal in normalised units at the same resonance frequencies used in the experiment (c) and for different densities of H atoms (d). To avoid overlapping, the curves in (c) and (d) are vertically offset and the data points at $T=0\mu \mathrm{s}$ are not shown.