Cross calibration of deformation potentials and gradient-elastic tensors of GaAs using photoluminescence and nuclear magnetic resonance spectroscopy in GaAs/AlGaAs quantum dot structures

Lattice matched GaAs/AlGaAs epitaxial structures with quantum dots are studied at $T=4.2$ K under static uniaxial stress applied either along the [001] or [110] crystal directions. We conduct simultaneous measurements of the spectral shifts in the photoluminescence of the bulk GaAs substrate, which relate to strain via deformation potentials $a$ and $b$, and the quadrupolar shifts in the optically detected nuclear magnetic resonance spectra of the quantum dots, which relate to the same strain via the gradient-elastic tensor $S_{ijkl}$. Measurements in two uniaxial stress configurations are used to derive the ratio $b/a=0.242\pm 0.008$ in good agreement with previous studies on GaAs. Based on the previously estimated value of $a\approx -8.8$ eV we derive the product of the nuclear quadrupolar moment $Q$ and the $S$-tensor diagonal component in GaAs to be $Q{S}_{11}\approx +0.758\times {10}^{-6}$ V for ${}^{75}\mathrm{As}$ and $Q{S}_{11}\approx -0.377\times {10}^{-6}$ V for ${}^{69}\mathrm{Ga}$ nuclei. In our experiments the signs of $S_{11}$ are directly measurable, which was not possible in the earlier nuclear acoustic resonance studies. Our $Q{S}_{11}$ values are a factor of $\sim 1.4$ smaller than those derived from the nuclear acoustic resonance experiments [Phys. Rev. B 10, 4244 (1974)]. The gradient-elastic tensor values measured in this work can be applied in structural analysis of strained III-V semiconductor nanostructures via accurate modeling of their magnetic resonance spectra.

Figure 1

(a) Sample structure showing the sequence of GaAs and ${\mathrm{Al}}_{0.5}{\mathrm{Ga}}_{0.5}\mathrm{As}$ epitaxial layers. (b) Schematic of the experiment geometry showing orientation of a sample, direction of the static magnetic field $B_z$, radio frequency field $B_{\text{rf}}$, and the direction of the photoluminescence excitation and collection. External stress is applied either along the [001] direction or the [110] direction. (c) AFM image of a sample grown under similar conditions as the studied structure but without overgrowing the ${\mathrm{Al}}_{0.5}{\mathrm{Ga}}_{0.5}\mathrm{As}$ layer with nanoholes. (d) Profiles along the lines in image (c), which are oriented along the [001] and $\left[1\overline{1}0\right]$ directions. (e) Typical photoluminescence spectrum at $B_z=0$ showing emission from the quantum well (QW), a single quantum dot (QD), as well as emission from the GaAs substrate which includes free-exciton emission and impurity-induced recombination. The excitation power is high enough to observe PL of both the ground and excited QD states. Square-root vertical scale is used to reveal weak spectral features.

Figure 2

Effect of strain on bulk GaAs photoluminescence (PL) and quantum dot NMR spectra. (a) Free-exciton PL from a GaAs substrate measured at $B=0$ T under excitation with a photon energy $\sim 1.54$ eV and intensity $\sim 5\times {10}^6$ ${\mathrm{W}/\mathrm{m}}^2$ in an unstressed sample (top), sample stressed along [001] (middle), and sample stressed along [110] (bottom). In an unstressed sample emission has negligible splitting between peaks detected in two orthogonal linear polarizations (${\pi }_1$, ${\pi }_2$). Stress along [001] splits the luminescence spectrum into two nonpolarized peaks corresponding to emission of light (LH) and heavy (HH) hole excitons. Stress along [110] splits luminescence spectrum into a stronger peak with partial linear polarization corresponding to emission from a predominantly LH exciton, and a weak linearly polarized peak from a predominantly HH exciton (parts of the spectra with $\times 10$ vertical magnification are shown to reveal the HH peak). (b) Nuclear magnetic resonance spectra of ${}^{75}\mathrm{As}$ nuclei measured with ${\sigma }^{+}$ polarized optical excitation at $B_z=8$ T (${\nu }_0\approx 58.46$ MHz) on GaAs/AlGaAs quantum dots in an unstressed (top), [001] stressed (middle), and [110] stressed (bottom) samples. Each NMR spectrum in (b) is measured from the same spot as the corresponding GaAs PL spectrum in (a). Spectra are offset vertically for clarity. Well-resolved NMR triplets arising from quadrupolar effects are observed. In an unstressed sample small quadrupolar shift ${\nu }_Q$ is observed due to the residual strain of the GaAs/AlGaAs heterostructure. Under [001] ([110]) stress, the resulting strain shifts the $-3/2\leftrightarrow -1/2$ satellite transition to lower (higher) frequency corresponding to negative (positive) ${\nu }_Q$. (c) Energies of HH (solid symbols) and LH (open symbols) bulk GaAs PL peaks plotted against quadrupolar shift ${\nu }_Q$ measured in multiple quantum dots in an unstressed (circles), [001]-stressed (triangles), and [110]-stressed (squares) samples. GaAs PL energies and NMR frequencies are derived from the spectra using Lorentzian and Gaussian peak fitting, respectively. Variation of ${\nu }_Q$ and PL energies in each sample is due to the inhomogeneity of strain across the sample surface—the exception is the four points at ${\nu }_Q\approx +200$ kHz that were measured at an increased stress along [110].

Figure 3

Data of Fig. 2 plotted in terms of the arithmetic average PL energy of the LH and HH excitons (solid symbols, left scales) and the difference of the LH and HH exciton PL energies (open symbols, right scales) as a function of ${}^{75}\mathrm{As}$ quadrupolar shift ${\nu }_Q$. (a) Results for the unstressed (circles) and [001]-stressed (triangles) samples. LH-HH splitting is well described by a linear function with a slope $k_{\left[001\right]}^{-}=46.5\pm 0.8\mu \mathrm{eV}$/kHz (dashed line). (b) Results for the unstressed (circles) and [110]-stressed (squares) samples. The average LH-HH PL energy is well described by a linear function with a slope $k_{\left[110\right]}^{+}=54.8\pm 1.5\mu \mathrm{eV}$/kHz (solid line).

Figure 4

Derivation of the sign of the gradient elastic tensor. (a) Photoluminescence spectra of a QD neutral exciton measured at $B_z=8$ T under ${\sigma }^{+}$ (solid line) and ${\sigma }^{-}$ (dashed line) polarized excitation at $\sim 1.65$ eV. The ${\sigma }^{+}$ excitation predominantly populates the exciton state with hole momentum $j_z=+3/2$ ($\Uparrow$) and electron spin $s_z=-1/2$ ($\downarrow$), while ${\sigma }^{-}$ excitation predominantly populates the $\Downarrow \uparrow$ exciton. Electron spin $\mathbf{s}$ can be transferred to a nuclear spin $\mathbf{I}$ via hyperfine interaction (inset) resulting in dynamic nuclear polarization, which in turn leads to hyperfine shifts of the exciton transitions. (b) Schematic of spin effects under ${\sigma }^{+}$ excitation which increases population of the $s_z=-1/2$ excitons (thicker horizontal line) and reduces population of the $s_z=+1/2$ exciton (thinner line). Dynamic nuclear polarization enhances the populations of the $I_z=-3/2$ and $I_z=-1/2$ nuclear spin states. This is observed as a hyperfine shift of the $s_z=-1/2$ excitons to higher energy and enhanced amplitude of the $-3/2\leftrightarrow -1/2$ NMR transition with frequency $\frac{\gamma B_z}{2\pi }+\frac{eQ}{2h}{S}_{11}{\epsilon }_b$. (c) Excitation with ${\sigma }^{-}$ light leads to the opposite sign of electron and nuclear spin polarizations, enhancing the $+1/2\leftrightarrow +3/2$ NMR transition at frequency $\frac{\gamma B_z}{2\pi }-\frac{eQ}{2h}{S}_{11}{\epsilon }_b$. By matching the signs of the exciton spectral shifts in (a) and the sign of the NMR spectral shifts [cf. Fig. 2] it is possible to deduce the sign of the gradient elastic tensor component $S_{11}$ (see Sec. 4b).

Figure 5

(a) NMR spectra of a single quantum dot measured at high magnetic field $B_z\approx 5.5$ T on ${}^{75}\mathrm{As}$ nuclei (${\nu }_0\approx 40.27$ MHz, dashed line) and ${}^{69}\mathrm{Ga}$ nuclei (${\nu }_0\approx 56.46$ MHz, solid line). Both isotopes are spin-3/2 and exhibit well resolved quadrupolar triplets with different splittings ${\nu }_Q$. Spectra are offset vertically for clarity. (b) Dependence of the ${}^{69}\mathrm{Ga}$ quadrupolar splitting ${\nu }_Q^{{}^{69}\mathrm{Ga}}$ on the ${}^{75}\mathrm{As}$ splitting ${\nu }_Q^{{}^{75}\mathrm{As}}$ measured on different individual quantum dots in an unstressed sample as well as in samples stressed along [001] or [110] crystal axes (symbols). Linear fitting is shown by the solid line and its slope $k_{{}^{69}\mathrm{Ga}/{}^{75}\mathrm{As}}=-0.497\pm 0.009$ gives an estimate of the ratio $\left(Q^{{}^{69}\mathrm{Ga}}{S}_{11}^{{}^{69}\mathrm{Ga}}\right)/\left(Q^{{}^{75}\mathrm{As}}{S}_{11}^{{}^{75}\mathrm{As}}\right)$ between the products of the quadrupolar moments $Q$ and gradient elastic tensor components $S_{11}$ of ${}^{69}\mathrm{Ga}$ and ${}^{75}\mathrm{As}$ in GaAs.