Nuclear magnetic resonance inverse spectra of InGaAs quantum dots: Atomistic level structural information

A wealth of atomistic information is contained within a self-assembled quantum dot (QD), associated with its chemical composition and the growth history. In the presence of quadrupolar nuclei, as in InGaAs QDs, much of this is inherited to nuclear spins via the coupling between the strain within the polar lattice and the electric quadrupole moments of the nuclei. Here, we present a computational study of the recently introduced inverse spectra nuclear magnetic resonance technique to assess its suitability for extracting such structural information. We observe marked spectral differences between the compound InAs and alloy InGaAs QDs. These are linked to the local biaxial and shear strains, and the local bonding configurations. The cation alloying plays a crucial role especially for the arsenic nuclei. The isotopic line profiles also largely differ among nuclear species: While the central transition of the gallium isotopes have a narrow linewidth, those of arsenic and indium are much broader and oppositely skewed with respect to each other. The statistical distributions of electric field gradient (EFG) parameters of the nuclei within the QD are analyzed. The consequences of various EFG axial orientation characteristics are discussed. Finally, the possibility of suppressing the first-order quadrupolar shifts is demonstrated by simply tilting the sample with respect to the static magnetic field.

Figure 1

Inverse NMR spectra of binary InAs, and alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QDs, under the conditions $B_0=5$ T, $f_{\mathrm{gap}}$ = 200 kHz, $T_{\mathrm{nuc}}$ = 3 mK, ${\sigma }^{+}$ optical pumping. Inset shows corresponding QD atoms over the (100) cross section.

Figure 2

(Upper panel) The contribution of individual isotopic species under the same conditions as Fig. 1. (Lower panel) The effect of optical pumping helicity $({\sigma }^{+}/{\sigma }^{-})$ on the inverse spectra, under the same conditions as Fig. 1 other than $f_{\mathrm{gap}}=800$ kHz.

Figure 3

The evolution of alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD inverse spectra with respect to $f_{\mathrm{gap}}$ for ${\sigma }^{+}$ optical pumping, $B_0=5$ T, $T_{\mathrm{nuc}}=3$ mK.

Figure 4

The dependence of CT line profiles on the external magnetic field for all the isotopes. Full width at half maximum $\left(\Gamma \right)$ values are included in the legend boxes. To superimpose their peaks, curves for different magnetic fields are displaced in frequency. Alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD is considered with $f_{\mathrm{gap}}$ = 1 kHz.

Figure 5

Standard deviation $\left(\sigma \right)$, skewness, and kurtosis of the CT for all isotopes contained in the alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD. The inverse spectra are computed with $f_{\mathrm{gap}}=1$ kHz.

Figure 6

Contributions to inverse spectra from the nuclei as a function of their major EFG orientations with respect to static magnetic field, denoted by the angle $\theta$. (Left panels) ${}^{75}\mathrm{As}$ nuclei; (right panels) ${}^{115}\mathrm{In}$ nuclei. (a) Alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD, (b) InAs QD. For all cases $B_0=5$ T, $f_{\mathrm{gap}}$ = 200 kHz, and $T_{\mathrm{nuc}}$ = 3 mK under ${\sigma }^{+}$ optical pumping. For the panels on the right, to improve visibility the color scale maxima are set to a quarter of their ordinary values.

Figure 7

Two-dimensional EFG histograms. (Top two rows) EFG axial tilting $\theta$ versus the major EFG, $V_{ZZ}$ (in units of $3\times {10}^{21}$ $\mathrm{V}/\mathrm{m}{}^2$). (Bottom two rows) EFG axial tilting $\theta$ versus the EFG biaxiality $\eta$. In each group, (upper rows) ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD; (lower rows) binary InAs QD. Color code represents the number of nuclei in the logarithmic scale.

Figure 8

Same as Fig. 6, but with $B_0=8$ T, $f_{\mathrm{gap}}=1$ kHz to focus on CT. The color scale maxima are set to a quarter of their ordinary values to improve visibility. For information purposes, full isotopic spectra filled in red are placed on top; the vertical thin white line indicates the position of the pure Zeeman frequency for each case.

Figure 9

The effect of selected nuclei on the CT asymmetry based on their EFG axial tilting away from the static magnetic field, for the As nuclei at 5 T (top) and In nuclei at 8 T. Painted curves show the original contribution with all the nuclei. For each case maxima are normalized to unity to assist the asymmetry comparison. Alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD is considered.

Figure 10

A schematic illustration for the typical EFG components and orientations for the indium (black solid arrows) and arsenic (red dashed arrows) nuclei of alloy ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ vs compound InAs QDs.

Figure 11

The effect of sample tilting on the inverse spectra. (a) Binary InAs, (b) alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD, both for $B_0=5$ T, $f_{\mathrm{gap}}=200$ kHz, $T_{\mathrm{nuc}}=3$ mK. Faraday geometry is preserved, i.e., optical pumping and magnetic field are parallel, whereas the sample (growth axis) is tilted away from these, as shown in the inset. Upper row displays the two-dimensional spectra of the tilting angle versus RF frequency; lower row shows the one-dimensional spectra at zero and $52{}^{\circ }$ tilting angles.

Figure 12

The isotope-resolved and total nuclear spin polarizations along the static magnetic field for the alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD. (a) Using the original gyromagnetic ratio for arsenic, (b) with the ${}^{75}\mathrm{As}$ gyromagnetic ratio artificially increased to the average value of ${}^{69}\mathrm{Ga}$ and ${}^{71}\mathrm{Ga}$, i.e., ${\gamma }_{\mathrm{As}}\to {\gamma }_{\mathrm{Ga}}$. Linestyle sets are the same for each panel.

Figure 13

Zeeman interaction having a unidirectional axis (left) versus QI with a bidirectional character (center). In actual QD samples the local major quadrupole axis is somewhat tilted because of shear strain with respect to magnetic field in Faraday geometry (right), which can accordingly reduce their competition.

Figure 14

For the alloy ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ and binary InAs QD histograms of the three EFG parameters. (Top row) The angular tilting of the major EFG axis away from the magnetic field $\theta$ (inset). (Middle row) Major EFG $V_{ZZ}$ in units of $3\times {10}^{21}$ $\mathrm{V}/\mathrm{m}{}^2$. (Bottom row) EFG biaxiality $\eta$. The results here refer to the case after the QD strain relaxation.

Figure 15

The switching of the major EFG axis of the center gallium nuclei (indicated by black double arrows) from parallel (top) to perpendicular (bottom) orientation with respect to the growth axis (also the direction of the static magnetic field) by a change in a second-nearest-neighbor atom. Color coding is as follows: indium in gray, gallium in purple, arsenic in yellow.

Figure 16

Dependence of arsenic resonance of CT asymmetry on the nuclear quadrupole parameters and alloy mole fraction. In all panels dotted (red) lines refer to original ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ QD with $f_{\mathrm{gap}}=1$ kHz, $B_0=5$ T, $T_{\mathrm{nuc}}=3$ mK, ${\sigma }^{+}$ optical pumping. Solid (blue) lines demonstrate the cases after a modification in material parameters, $Q,S_{44},S_{11}$, as well as the case for binary InAs QD. Each peak is set unity to compare the lineshapes.

Figure 17

Atomistic shear ${\epsilon }_S$ (left) and biaxial ${\epsilon }_B$ (right) strain distributions for the ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ (top) and InAs (bottom) QDs, cut through both (100) and (010) planes. The lens-shaped QD boundaries can easily be identified on the InAs QD from the enclosing bright shear strain regions corresponding to the interfaces with the host matrix as well as the wetting layer.

Figure 18

Demonstration on a single ${}^{115}\mathrm{In}$ nucleus of the effects of various EFG parameter combinations on the CT (left) and ST (right) frequencies. For convenience Zeeman frequency is subtracted to highlight QI. The major EFG $V_{ZZ}$ is in units of $3\times {10}^{21}$ $\mathrm{V}/\mathrm{m}{}^2$. (a)–(c) are for $\eta =0.5$; (d) shows both $\eta =0$ and $\eta =1$ cases with the latter in dash-dotted lines; (d) and (e) are for $V_{ZZ}=-1$.