High-resolution electrical detection of free induction decay and Hahn echoes in phosphorus-doped silicon

Paramagnetic centers in a solid-state environment usually give rise to inhomogeneously broadened electron paramagnetic resonance (EPR) lines, making conventionally detected free induction decay (FID) signals disappear within the spectrometer dead time. Here, high-resolution experimental results of an electrically detected FID of phosphorus donors in silicon epilayers with natural isotope composition are presented, showing Ramsey fringes within the first $150$ ns. An analytical model is developed to account for the data obtained as well as for the results of analogous two-pulse echo experiments. The results of a numerical calculation are further presented to assess the capability of the method to study the exchange interaction between the donor spins and paramagnetic interface defect states. In the current samples, an upper limit of $5$ MHz is obtained for the average exchange interaction.

Figure 1

First derivative spectrum of the relative change $\Delta I/I$ of the photocurrent in a cwEDMR experiment on ${}^{\mathrm{nat}}$Si:P with a magnetic field modulation amplitude of $0.2$ mT. The red curve represents a fit using Gaussian line shapes and equal amplitudes of both ${}^{31}\mathrm{P}$ resonance lines. The constituent lines of the fit are shown below for better visibility.

Figure 2

Electrically detected free induction decay or Ramsey experiment on the high-field hf(${}^{31}\mathrm{P}$) resonance. (a) Contour plot of $\Delta Q$ as a function of the external magnetic field $B_0$ and the free evolution time $\tau$. White dashed curves mark the positions of local extrema of $\Delta Q$ described by Eq. (2). The black dashed line indicates the position where the cross-sectional chart shown in panel (b) is taken. (b) Cross section of $\Delta Q$ along the evolution time axis at $B_0=349.25$ mT. Red dashed curves illustrate the Gaussian-shape damping of the oscillation amplitude.

Figure 3

Oscillations in the evolution time domain. (a) Cross sections of $\Delta Q$ along the evolution time axis taken at different values of $B_0$. The damping of the oscillations indicates that the spin ensemble dephases. Red curves are fits based on the model given by Eq. (4). (b) Frequency ${\nu }_{\mathrm{FID}}$ of the damped oscillations as a function of the external magnetic field $B_0$ shows a linear behavior as expected from Eq. (5).

Figure 4

Electrically detected two-pulse Hahn echo on the high-field hf(${}^{31}\mathrm{P}$) resonance with ${\tau }_1=300$ ns. (a) Contour plot of $\Delta Q$ as a function of the external magnetic field $B_0$ and the free evolution time ${\tau }_2$. Black dashed lines mark the hyperbola pattern according to Eq. (6). Red and blue dashed lines indicate positions where cross-sectional diagrams shown in panel (b) are taken. (b) Cross section of $\Delta Q$ along the evolution time axis at resonance condition (red line, $B_0=350.0$ mT) and for the off-resonant case (blue line, $B_0=350.5$ mT). Please refer to the text for details.

Figure 5

Oscillations in the evolution time domain. (a) Cross sections of $\Delta Q$ along the evolution time axis taken at different values of $B_0$. Red curves are fits based on the model given by Eq. (6). (b) The frequency ${\nu }_{\mathrm{echo}}$ of the damped oscillations as a function of the external magnetic field $B_0$ shows a linear behavior as expected from Eq. (7).

Figure 6

(a) Experimental results of the EDFID measurement showing all spectral features. (b) Simulation for $J=0$ MHz. In both simulation and experimental results, the characteristic Ramsey pattern can be clearly seen at the position of the high-field hf(${}^{31}\mathrm{P}$) resonance, whereas the patterns at the low-field hf(${}^{31}\mathrm{P}$) and the ${\mathrm{P}}_{\mathrm{b}0}$ resonances are more complicated due to mutual interference.

Figure 7

(a) and (b) Fourier transformation of the EDFID tomography experiment and simulation shown in Fig. 6. The linear dependence of the oscillation frequency on $B_0-B_{\mathrm{res}}$ described by Eq. (5) is clearly seen in the frequency domain. The two ${}^{31}\mathrm{P}$ resonances and the broader ${\mathrm{P}}_{\mathrm{b}0}$ resonance are marked by black and white dashed lines, respectively. (c) and (d) For better visibility, details of the FFT spectrum near the high-field hf(${}^{31}\mathrm{P}$) resonance are shown after subtracting a background of the form of Eq. (4) from the experimental and simulated data. Again, the low-field hf(${}^{31}\mathrm{P}$) resonance and the ${\mathrm{P}}_{\mathrm{b}0}$ resonance are marked by black and white dashed lines. An additional resonance, indicated by the solid white line, can be seen in the experimental data. Its spectral position is in accordance with the small central line observed in cwEDMR (see Fig. 1).

Figure 8

Simulation of the EDFID experiment at the high-field hf(${}^{31}\mathrm{P}$) resonance for different exchange coupling parameter. An oscillation of the signal in the time domain with the coupling frequency is expected. This oscillation is masked by the exponential decay of the signal due to dephasing and therefore can only be resolved for $J>5$ MHz.

Figure 9

(a) Cross section along the $\tau$ axis at the high-field resonance field of ${}^{31}\mathrm{P}$ of the experimental data (open circles) and the simulated data (solid lines) for different exchange coupling parameter between ${}^{31}\mathrm{P}$ and ${\mathrm{P}}_{\mathrm{b}0}$. (b)–(d) Simulated data as in (a) for isotopically purified ${}^{28}\mathrm{Si}$ samples with different concentrations of ${}^{29}\mathrm{Si}$. Here, more oscillations caused by the weak coupling can be seen due to the longer dephasing time.