Spin noise of conduction electrons in $n$-type bulk GaAs

We report a comprehensive study of stochastic electron spin fluctuations—spin noise—in lightly doped ($n$-type) bulk GaAs, which are measured using sensitive optical magnetometry based on off-resonant Faraday rotation. Frequency spectra of electron spin noise are studied as a function of electron density, magnetic field, temperature, probe-laser wavelength and intensity, and interaction volume. Electron spin lifetimes ${\tau }_s$ are inferred from the width of the spin noise spectra and are compared to direct measurements of ${\tau }_s$ using conventional Hanle-effect methods. Both methods reveal a strong and similar dependence of ${\tau }_s$ on the wavelength and intensity of the probe laser, highlighting the undesired influence of sub-bandgap absorption effects on the nominally “nonperturbative” spin noise measurements. As a function of temperature, the spin noise power increases approximately linearly from 1.5 to 30 K, as expected for degenerate electrons obeying Fermi-Dirac statistics, but with an additional zero-temperature offset. Finally, as the cross-sectional area of the probe laser shrinks and fewer electrons are probed, the measured Faraday rotation fluctuations due to electron spin noise are shown to increase, as expected.

Figure 1

(Color online) (a) The spin noise experiment, showing linear polarizer (LP), polarization beam splitter (PBS), and half-wave plate $\left(\lambda /2\right)$. (b) Raw spin noise data from an $n:\text{GaAs}$ wafer (sample B) for transmitted probe-laser intensities from 0–4 mW. Data are not offset; the increasing noise power density arises from increasing photon shot noise. At each laser intensity, two spectra are acquired: one in the target transverse magnetic field (here, $B_x=175\text{G}$) and one in a background field $\left(B_x>1000\text{G}\right)$. (c) Their difference reveals the extra noise power density due to fluctuating, precessing electron spins, shown in units of ${\text{V}}^2/\text{Hz}$. The 4 mW spectrum is also expressed as a Faraday rotation power density (${\text{rad}}^2/\text{Hz}$; right axis). Inset: The integrated spin noise in units of volts (or square-root of the integrated power) scales linearly with laser intensity. All these spectra have the same total integrated Faraday rotation noise: $\sim 22.5\mu \text{rad}$.

Figure 2

(Color online) (a) Electron spin noise power spectra at 10 K from $n:\text{GaAs}$ wafers A and C at $B_x=0,50,100,150,200,250,300\text{G}$ $\left(\text{black}\to \text{red}\right)$. (b) The center frequency of these spin noise peaks versus $B_x$ for all three $n:\text{GaAs}$ wafers. The slope reveals the electron $g$ factor, $\left|g_e\right|$.

Figure 3

(Color online) (a) The electron spin noise power in sample B at 10 K using transmitted probe-laser intensities of 0.5, 1, 2, and 4 mW ($\lambda =830.1\text{nm}$; spectra offset for clarity). (b) The spin lifetime ${\tau }_s$ inferred from the width $\Gamma$ of the noise spectra decreases with increasing probe-laser intensity. (c) Measuring ${\tau }_s$ in sample B via optical spin orientation and “conventional” Hanle-effect methods using probe intensities from 0.125 to 4.0 mW. $\left(\lambda =830.1\text{nm}\right)$ (d) The inferred ${\tau }_s$ from these Hanle data also decreases with increasing probe-laser intensity.

Figure 4

(Color online) (a) The normalized optical transmission spectrum, $T/T_0$, through $n:\text{GaAs}$ sample B at 10 K using different probe-laser intensities (the 10 K band gap of GaAs is ~1.515 eV). (b) The same data plotted as an optical absorption (and assuming 18% reflection by the sample). The Urbach absorption tail persists well below the nominal GaAs band gap. Absorption at these subgap energies can reduce the electron spin lifetime that is measured by spin noise spectroscopy.

Figure 5

(Color online) (a) A direct comparison of the electron spin lifetime ${\tau }_s$ measured by spin noise methods (black points) and by conventional Hanle-effect methods (red points) in $n:\text{GaAs}$ sample B at 10 K. ${\tau }_s$ is shown on a log-log scale as a function of the probe-laser intensity that is absorbed by the sample. When plotted in this way, data from experiments using various probe-laser wavelengths and intensities collapse onto common curves (lines are guides to the eye). For both methods, ${\tau }_s$ decreases significantly with absorbed probe-laser intensity, likely due to electron-hole creation by the probe-laser itself. (b) A similar comparison from $n:\text{GaAs}$ sample C.

Figure 6

(Color online) (a) Electron spin noise power spectra from $n:\text{GaAs}$ sample B at 10 K, using sub-band-gap probe-laser wavelengths from 830.6 to 850.4 nm. For all spectra, the transmitted probe-laser intensity was 3 mW and $B_x=57\text{G}$. (b) The integrated electron spin noise ($\sqrt{⟨{\theta }_F^2⟩}$, in microradians) versus probe-laser photon energy $h\nu$, for three probe-laser intensities. For comparison, the continuous red trace (right axis) shows ${\theta }_F\left(\nu \right)$ induced by intentional optical orientation of electron spins in sample B using an additional 1.58 eV circularly polarized pump laser. ${\theta }_F\left(\nu \right)$ decays as $\sim {\Delta }^{-1}$, assuming a GaAs band edge at 1.515 eV. (c) A similar comparison from sample C.

Figure 7

(Color online) (a) The temperature dependence of the integrated electron spin noise power ($⟨{\theta }_F^2⟩$, in units of $\mu {\text{radians}}^2$) from $n:\text{GaAs}$ sample C. Temperature sweeps corresponding to red and black points used different probe-laser wavelengths, intensities, and spot sizes, but are scaled so as to overlap. (b) and (c) Similar temperature dependencies from $n:\text{GaAs}$ samples B and A. Dotted lines are linear guides to the eyes.

Figure 8

(Color online) Log-log plot of the integrated Faraday rotation fluctuations due to electron spin noise ($\sqrt{⟨{\theta }_F^2⟩}$, in units of microradians) in $n:\text{GaAs}$ sample A as a function of the cross-sectional area of the probe-laser spot on the sample. The line shows a $1/\sqrt{\text{area}}$ dependence, as expected for spin noise [see Eq. (4)]. Inset: The laser spot size was varied by translating the sample near the probe-laser focus and assuming Gaussian optics.