Element-sensitive measurement of the hole–nuclear spin interaction in quantum dots

Abstract

It has been proposed that valence-band holes can form robust spin qubits^1,2,3,4 owing to their weaker hyperfine coupling compared with electrons^5,6. However, it was demonstrated recently^7,8,9,10,11 that the hole hyperfine interaction is not negligible, although a consistent picture of the mechanism controlling its magnitude is still lacking. Here we address this problem by measuring the hole hyperfine constant independently for each chemical element in InGaAs/GaAs, InP/GaInP and GaAs/AlGaAs quantum dots. Contrary to existing models^10,11 we find that the hole hyperfine constant has opposite signs for cations and anions and ranges from −15% to +15% relative to that for electrons. We attribute such changes to the competing positive contributions of p-symmetry atomic orbitals and the negative contributions of d-orbitals. These findings yield information on the orbital composition of the valence band¹² and enable a fundamentally new approach for verification of computed Bloch wavefunctions in semiconductor nanostructures¹³. Furthermore, we show that the contribution of cationic d-orbitals leads to a new mechanism of hole spin decoherence.

Main

Owing to the s-type character of the Bloch wavefunction, the hyperfine interaction of the conduction band electrons is isotropic (the Fermi contact interaction) and is described by a single hyperfine constant A, positive (A>0) for most III–V semiconductors and proportional to the electron density at the nucleus. In contrast, for valence-band holes the contact interaction vanishes owing to the symmetry properties of the wavefunction, and the non-local dipole–dipole interaction dominates^{10,11,13,14,15}. As a result, the sign, magnitude and anisotropy of the hyperfine interaction depend on the actual form of the valence-band Bloch wavefunction, which is usually not available with sufficient precision. Thus, predicting the properties of the hole hyperfine coupling using first-principle calculations remains a difficult task.

In this work we perform direct measurements of the hyperfine constants that describe the hole hyperfine interaction with nuclear spins polarized along the growth axis of the structure (that is, the diagonal elements of the hole hyperfine Hamiltonian). This is achieved by simultaneous and independent detection of the electron and hole Overhauser shifts using high-resolution photoluminescence spectroscopy of neutral quantum dots. In contrast to previous work⁹, we now also apply excitation with a radiofrequency oscillating magnetic field, which allows isotope-selective probing of the valence-band hole hyperfine interaction¹⁶. Using this technique we find that in all studied materials, cations (gallium, indium) have a negative hole hyperfine constant, whereas it is positive for anions (phosphorus, arsenic), a result attributed to the previously disregarded contribution of the cationic d-shells into the valence-band Bloch wavefunctions.

Using the experimentally measured diagonal components of the hyperfine Hamiltonian (hole hyperfine constants) we calculate its non-diagonal part. We show that the admixture of the d-shells has a major effect on the symmetry of the hyperfine Hamiltonian in quantum dots: unlike pure heavy holes constructed only of p-symmetry shells for which the hyperfine interaction has an Ising form¹¹, the d-shell contribution results in non-zero non-diagonal elements of the hyperfine Hamiltonian. We predict this to be a major source of heavy-hole spin decoherence.

Our experiments rely on detection of photoluminescence of both bright and dark neutral excitons^9,17,18,19 formed by electrons ↑(↓) with spin ±1/2 and heavy holes with momentum ±3/2 parallel (antiparallel) to the growth axis O z (Fig. 1a). As the quantum dots contain of the order of 10⁵ nuclei, non-zero average nuclear spin polarization of the kth isotope 〈I_z^k〉 along the O z axis can be treated as an additional magnetic field acting on the electron and hole spins. The coupling strength of the electron to the nuclear spins of isotope k is described by the hyperfine constant A^k. The additional energy of the exciton state with electron spin ↑(↓) is equal to +(1/2)ΔE_e^k (−(1/2)ΔE_e^k), where the electron hyperfine shift induced by the kth isotope is defined as

with ρ^k describing the relative concentration of the kth isotope. For the heavy-hole states the hyperfine interaction is described using a constant C^k expressed in terms of the normalized heavy-hole hyperfine constant γ^k as C^k = γ^kA^k. The variation of the energy of the exciton with hole spin is +(1/2)ΔE_h^k (−(1/2)ΔE_h^k), where the hole hyperfine shift is

By taking the same values of ρ^k in equations (1) and (2) we assume for simplicity a uniform distribution of the average nuclear spin polarization and concentration of chemical elements within the volume of the quantum dot. According to equations (1) and (2), hole and electron hyperfine shifts depend linearly on each other (ΔE_h^k = γ^kΔE_e^k). In the experiment, hyperfine shifts are measured from photoluminescence spectra of the quantum dot (see Fig. 1a) and the slope of the resulting dependence of ΔE_h^k on ΔE_e^k is used to determine γ^k (see further details in Methods).

Figure 1: Optical techniques for isotope-selective measurement of the hole hyperfine constants.

a, Photoluminescence spectra of a single neutral InGaAs/GaAs quantum dot in a magnetic field B_z≈8.0 T. For low-power optical excitation, four photoluminescence lines are observed in each spectrum corresponding to all possible combinations of the electron spin states (↑,↓) and hole states () forming two bright excitons (, ) and two dark excitons (, ) that have a small (<0.01) admixture of bright states making dark states visible in photoluminescence spectra^9,17. To demonstrate independent detection of electron and hole hyperfine shifts two spectra are shown corresponding to negative (open symbols) and positive (solid symbols) nuclear spin polarization 〈I_z〉induced on the dot by pumping with σ⁺ and σ⁻ polarized light, respectively. b, Timing diagram of the pump–probe experiment used in the measurements of the hole hyperfine constants: nuclear spins are polarized by a high-power optical pump pulse. Following this, a radiofrequency oscillating magnetic field is switched on to achieve isotope-selective depolarization of nuclear spins. Finally, the sample is excited with a low-power probe laser pulse, during which the photoluminescence spectrum of both bright and dark excitons (similar to that in a) is measured. See further details of experimental techniques in Methods.

We start by presenting results for strain-free GaAs/AlGaAs quantum dots. The dependence of ΔE_h^k on ΔE_e^k for k = As (squares) and k = Ga (circles) is shown in Fig. 2a for GaAs quantum dot A1. It can be seen that the dependences for both Ga and As are linear as predicted by equations (1) and (2). Fitting gives the following values for the hole hyperfine constants γ^Ga = −7.0±4.0% and γ^As = +15.0±4.5%. Similar measurements were performed on three other GaAs quantum dots. The resulting values are given in Table 1. As the variation between different dots is within experimental error, we take average values for all dots yielding γ^Ga = −7.5±3.0% and γ^As = +16.0±3.5%. We thus conclude that different chemical elements have opposite signs of the hole hyperfine constants: they are positive for arsenic and negative for gallium. This is an unexpected result in comparison with previous theoretical studies^10,11 and experiments insensitive to individual chemical elements where negative values of γ have been found in InP and InGaAs quantum dots^8,9.

Figure 2: Experimental results of the isotope-selective hole hyperfine measurements.

a,b, Dependence of the hole hyperfine shift ΔE_h^kon the electron hyperfine shift ΔE_e^k for different isotopes in GaAs quantum dot A1 (a) and InGaAs quantum dot B1 (b). Solid lines show fitting: the slopes correspond to the relative hole–nuclear hyperfine constants γ^k (see details in Methods). We find γ^Ga≈−7.0%, γ^As≈+15.0% for GaAs quantum dot A1 and γ^Ga≈−6.5%, γ^As≈+10.5% for InGaAs quantum dot B1. As the NMR resonances of ⁶⁹Ga and ¹¹⁵ In in InGaAs cannot be resolved¹⁶, we measure the total hyperfine shifts $\Delta E_{\text{e}}^{{\text{In+}}^{\text{69}}\text{Ga}}$ and $\Delta E_{\text{h}}^{{\text{In+}}^{\text{69}}\text{Ga}}$ produced by these isotopes. Further analysis gives γ^In≈−16.0% for quantum dot B1 (see Supplementary Section S3B). The dashed line in b is a guide to the eye.

Table 1 Experimentally measured photoluminescence energies E_PL and hole hyperfine constants γ^k for different chemical elements k in several GaAs and InGaAs quantum dots.

We have also performed isotope-sensitive measurements of the hole nuclear interaction in InGaAs/GaAs quantum dots. The dependence of ΔE_h^k on ΔE_e^k for InGaAs quantum dot B1 is shown in Fig. 2b for ⁷¹Ga (circles), ⁷⁵As (squares) and for the total hyperfine shifts of ⁶⁹Ga and ¹¹⁵In(triangles). The values of γ^k obtained from the fitting (see details in Methods) are summarized in Table 1. Similar to GaAs, we find that arsenic has a positive hole hyperfine constant whereas for gallium and indium it is negative.

Applying the isotope selection techniques to InP/GaInP quantum dots studied previously⁹ we find γ^In = −12.5±3.0%, consistent with our previous results obtained without isotope selection⁹. Similar to GaAs and InGaAs quantum dots, we find a large positive constant for anions (phosphorus) γ^P = +18.0±8.0%.

The values of γ presented in Table 1 describe the hyperfine interaction of the valence-band states that are in general mixed states of heavy and light holes. However, as we show in detail in Supplementary Section S5, such mixing cannot account for the opposite signs of γ^k observed for the cations and anions, but might be the reason for the dot-to-dot variation of γ^In observed in InP quantum dots (see Table 1). Such variation may also arise from the dot-to-dot-dependent spatial separation of electron and hole wavefunctions⁸.

From the measurements without radiofrequency pulses (similar to earlier isotope-non-selective experiments of refs 8, 9), we find that in GaAs quantum dots the total hole hyperfine shift (induced by all isotopes) is positive and amounts to γ≈+5% relative to the total electron hyperfine shift. For the studied InGaAs quantum dots where indium and gallium concentrations are estimated to be ρ^In≈20% and ρ^Ga≈80% (see ref. 16), we find negative γ≈−4%, whereas for more indium-rich InGaAs dots emitting at E_PL∼1.30 eV the value of γ≈−9% has been reported⁸. This suggests that for quantum dots with a particular indium concentration (ρ^In∼10%) one can expect close to zero (γ≈0) total hole hyperfine shift induced by nuclear spin polarization along the O z direction. Hole spin qubits in such structures will be insensitive to static nuclear fields that are induced by the optical control pulses and cause angle errors in spin rotations. Such spin qubits will benefit from a simplified implementation of the coherent control protocols⁴.

We now turn to analysis of the experimental results presented in Table 1. First-principle calculation of the valence-band hyperfine coupling requires integration of the hyperfine Hamiltonian using explicit expressions for the Bloch wavefunctions. However, it has been shown that reasonable estimates can be obtained using a simplified approach^10,11,15: Bloch wavefunctions can be approximated by linear combinations of hydrogenic wavefunctions (further details may be found in Methods). Previous calculations^10,11,15 based on this approach considered valence-band states constructed from atomic p-orbitals (with orbital momentum l = 1), which yields positive hole hyperfine constant γ^k>0 for all chemical elements, in contradiction with our experimental findings.

This disagreement can be overcome by taking into account the contribution of shells with higher orbital momenta l, resulting in more accurate approximation of the hole wavefunction²⁰: in particular we consider the contribution of the d-shell states (l = 2). Both p- and d-orbitals are schematically depicted in Fig. 3a. We assume that the heavy-hole states can be taken as linear combinations of p-shells with weight α_p and d-shells with weight α_d (|α_p|²+|α_d|² = 1). Calculation of the relative hole hyperfine constant yields:

where positive integrals M_l (l = p,d) depend on the hydrogenic radial wavefunctions R_l(r) corresponding to the shell with orbital momentum l and normalized by the density (4π)⁻¹|S(0)|² of the conduction band electron wavefunction at the nuclear site (see further information in Methods and Supplementary Section S4).

Figure 3: Model calculations of the hole hyperfine constants.

a, Schematic representations of p- and d-orbitals that transform according to the F₂ representation of the T_d point group of the crystal symmetry. b, Calculated dependence of the relative hole hyperfine constant γ^k of Ga and As as a function of d-shell contribution |α_d|² (lines). Horizontal bands show experimentally measured confidence intervals for γ^k for GaAs quantum dots (see Table 1). c, Dependence of integrals M_l(r₀) (see equation (3)) on the upper integration limit r₀ for 3d- and 4p-shells for both Ga and As. Values of M_l(r₀) are normalized by their values at , and r₀ is normalized by the distance between Ga and As nuclei r_Ga–As≈0.245 nmin GaAs. The rapid saturation of the M_l(r₀) variations shows that the major contribution to the hole hyperfine interaction (>95%) arises from a small volume with a radius of r₀≲0.15×r_Ga–As around the nucleus.

It follows from equation (3) that unlike the p-shell, the d-shell gives rise to a negative contribution to γ^k: importantly the sign of the hyperfine interaction is totally determined by the angular symmetry of the wavefunction, whereas the radial part R_l(r) affects only the magnitude of the contribution. We note that any hybridization of the valence-band states with s-orbitals due to quantum-dot symmetry reduction would lead to a positive contribution to γ^k and thus cannot account for the negative hyperfine constants¹¹. To obtain numerical estimates we consider GaAs material and approximate S(r), R_p(r), R_d(r) with radial hydrogenic wavefunctions corresponding to 4s-, 4p- and 3d-shells, respectively, taken with effective orbital radii^11,21. The resulting calculated dependence of γ^k on d-shell admixture |α_d|² is shown in Fig. 3b for k = Ga and k = As nuclei. On comparing this with the experimental results of Table 1 (shown by the horizontal bands in Fig. 3b), we conclude that the symmetry of the wavefunction at the anions (arsenic) is close to pure p-type, whereas for the cation gallium a significant contribution of the d-shell (∼20%) is required to account for the negative hole hyperfine constant measured experimentally.

The non-zero contribution of the d-symmetry orbitals has a further unexpected effect on the hole hyperfine interaction: we find (see Supplementary Section S5) that the hyperfine interaction induces spin flips between the heavy-hole states and . This is in contrast to the case of pure heavy holes constructed only of p-symmetry states for which the hyperfine interaction has an Ising form¹¹: in that case the symmetry of the system is artificially raised to spherical, resulting in hyperfine interaction conserving angular momenta. The inclusion of the d-shells reduces the symmetry of the system down to that of the real crystal (described by the T_d point group). Under these conditions the hyperfine interaction does not conserve angular momentum and has non-zero non-diagonal elements coupling heavy holes with the opposite spins.

It was demonstrated previously that heavy–light hole mixing can result in a non-Ising form of the hyperfine interaction^10,11. However, our estimates show that for gallium in GaAs the contribution of the d-shells to the non-diagonal matrix elements of the hyperfine Hamiltonian dominates over the effect of the heavy–light hole mixing even if the valence-band states have a light-hole contribution as large as ∼30% (see Supplementary Section S5). A similar effect is expected for the other materials studied, because for all of them significant contribution of the cation d-shells is observed (resulting in γ^k<0). Thus, the d-orbital contribution will be a source of heavy-hole spin dephasing even in the absence of mixing with light holes and should be taken into account when analysing experimentally measured hole spin coherence times.

The hyperfine interaction is particularly strong in the small volume around the atomic core¹³. To estimate this volume we limit the integration in equation (3) to a sphere of a radius r₀, which makes M_l (and hence γ^k) a function of r₀. The dependence of M_l(r₀/r_Ga–As) on the radius of the integration sphere, normalized by the distance between nearest Ga and As neighbours r_Ga–As≈0.245 nm, is shown in Fig. 3c for 3d- and 4p-shells for both Ga and As (because converges it is normalized by its limiting value ). It can be seen that the main contribution to the integral (>95%) comes from the small volume within a sphere with a radius of ∼0.15×r_Ga–As, whereas the outer volume gives only a minor contribution owing to the rapid decrease of the dipole–dipole interaction strength with increasing distance. Thus, hyperfine coupling can be used to probe the structure of the wavefunction in the atomic core.

Our results on Bloch wavefunction orbital composition are in general agreement with existing theoretical models: the importance of d-shells in describing the valence-band states is well recognized^20,22, and it has been shown that the d-symmetry contribution originates mainly from cations. However, the previous reports^12,23,24 predicted a much larger d-symmetry contribution (|α_d|² exceeding 50%) than estimated in our work (|α_d|²∼20%). Such deviation might be due to the simplified character of our calculations and/or due to the intrinsic limitations of the wavefunction modelling techniques such as tight-binding or pseudo-potential methods²⁵ that fail to reproduce the wavefunction structure in the vicinity of the nucleus.

Theoretical modelling of the microscopic wavefunctions allows band structures to be calculated and thus is of importance both for fundamental studies and technological applications of semiconductors²⁵. However, as true first-principle calculation of the many-body wavefunction is highly challenging, empirical approaches are normally used. They ultimately rely on fitting model parameters to describe the set of experimental data (for example, energy gaps, effective masses, X-ray photoemission spectra). The experimental data on the valence-band hyperfine parameters obtained in this work provide a means for probing the hole Bloch wavefunction: they allow direct analysis of the wavefunction orbital composition in the close vicinity of the nuclei, where theoretical modelling is the most difficult. Furthermore, our experimental method is unique in being isotope-selective, thus allowing independent study of cation and anion wavefunctions. The techniques developed in this work for quantum dots have the potential to be extended to other semiconductor systems, for example, bound excitons in III–V and group-IV bulk semiconductors where dark excitons are observed²⁶ and hyperfine shifts can be induced and detected²⁷.

A rigorous modelling of the hyperfine parameters¹³ has not been carried out so far for the valence-bands states of III–V semiconductor nanostructures. Progress in this direction will provide a better understanding of the mechanisms controlling the sign and magnitude of the valence-band hyperfine coupling. In particular, the potential effect of large inhomogeneous elastic strain (present in self-assembled quantum dots) on the microscopic Bloch hole wavefunction needs to be examined. This may be a possible route to engineering of holes with reduced hyperfine coupling.

Methods

Samples and experimental techniques.

Our experiments were performed on undoped GaAs/AlGaAs (ref. 28), InP/GaInP (ref. 9) and InGaAs/GaAs (ref. 16) quantum-dot samples without electric gates (further details can be found in Supplementary Section S1). The photoluminescence spectra of neutral quantum dots placed at T = 4.2 K, in an external magnetic field B_z normal to the sample surface, were measured using a double spectrometer and a CCD (charge-coupled device) camera.

Detection of the hyperfine shifts, required to measure hole hyperfine constants is achieved using pump–probe techniques⁹ (see timing diagram in Fig. 1b). Nuclear spin polarization is prepared with a long (∼6 s) high-power optical pump pulse. Following this, a radiofrequency oscillating magnetic field is switched on to achieve isotope-selective depolarization of nuclear spins (radiofrequency pulse duration varies between 0.15 and 35 s depending on the material). Finally, the sample is excited with a low-power short (∼0.3 s) probe laser pulse, during which the photoluminescence spectrum (see Fig. 1a) of both bright and dark excitons is measured. In all experiments the durations of the radiofrequency and probe pulses are much smaller than the natural decay time of the nuclear polarization. See further details of experimental techniques in Supplementary Sections S2 and S3.

Techniques for isotope-selective measurement of the hole hyperfine constant.

The concept of the valence-band hyperfine constant measurement is based on detecting hole hyperfine shift ΔE_h^k (equation (2)) as a function of electron hyperfine shift ΔE_e^k (equation (1)) by varying the nuclear spin polarization 〈I_z^k〉. Non-zero 〈I_z^k〉 is induced by optical nuclear spin pumping: circularly polarized light of the pump laser generates spin-polarized electrons that transfer their polarization to nuclei^16,17,29 through the hyperfine interaction. The magnitude of 〈I_z^k〉 is controlled by changing the degree of circular polarization⁹. According to equations (1) and (2), hole and electron hyperfine shifts depend linearly on each other (ΔE_h^k = γ^kΔE_e^k) with a slope equal to the normalized hole hyperfine constant γ^k. The electron (hole) hyperfine shift of a chosen (kth) isotope is deduced from a differential measurement: the spectral splitting between excitons with opposite electron (hole) spins (Fig. 1a) is measured with a radiofrequency pulse that depolarizes only the kth isotope and without any radiofrequency pulse. The difference between these two splittings is equal to ΔE_e^k (ΔE_h^k). As an example, Fig. 2 shows measurements where electron hyperfine shift ΔE_e^k for isotope k is found as the difference of the energy splitting of the and excitons measured without radiofrequency excitation and the splitting of the same excitons but measured after erasure of the nuclear polarization corresponding to the kth isotope by the radiofrequency pulse. In the same way, the hole hyperfine shift is measured as .

The actual techniques for isotope-selective depolarization of nuclear spins depend on quantum-dot material. For example, in the case of GaAs/AlGaAs quantum dots, we take into account that both ⁶⁹Ga and ⁷¹Ga isotopes have nearly equal chemical properties resulting in equal values of the relative hole hyperfine interaction constants ${\gamma }^{{69}_{Ga}}={\gamma }^{{71}_{Ga}}={\gamma }^{Ga}$. Thus, measurement of γ^Ga can be accomplished by erasing both ⁶⁹Ga and ⁷¹Ga polarization (which improves the measurement accuracy). In contrast, in InGaAs/GaAs quantum dots, ¹¹⁵In and ⁶⁹Ga NMR spectra overlap owing to strain-induced quadrupole effects^16,30, and γ^In is extracted by calculating the hyperfine shifts of ⁶⁹Ga from the measured hyperfine shifts of ⁷¹Ga. Further details of the isotope-selective experimental techniques can be found in Supplementary Section S3.

Theoretical model.

First-principle calculation of the valence-band hyperfine coupling requires integration of the hyperfine Hamiltonian using explicit expressions for the Bloch wavefunctions. Each nucleus is coupled to a hole that spreads over many unit cells. However, it has been shown that the main effect arises from the short-range part of the dipole–dipole interaction^11,15 (that is, coupling of the nuclear spin with the wavefunction within the same unit cell). This allows a simplified approach to be used: the Bloch functions of the valence-band maximum (corresponding to heavy-hole states) can be taken in the form and , where |↑〉, |↓〉 are spinors with corresponding spin projections on the O z axis and , are orbitals that transform according to the F₂ representation of the T_d point group relevant to bulk zinc-blende crystals (such as GaAs). Here, the and orbitals are decomposed into a real radial part R(r) and angular parts X(θ,φ), Y (θ,φ). As a first approximation, the angular parts of the orbitals can be taken in the form X_px/r, Y _py/r (corresponding to p-type states with orbital momentum l = 1), and R_p(r) can be approximated by hydrogenic radial functions^10,11,15.

To explain the opposite signs of the hole hyperfine constants observed experimentally we also need to take into account the contribution of the d-shell states (l = 2). To calculate the hyperfine constants (equation (3)) we assume that the heavy-hole orbitals can be taken as normalized linear combinations of the form and , where α_l are weighting coefficients (|α_p|²+|α_d|² = 1) and all orbitals X_l corresponding to orbital momentum l transform according to the same F₂ representation (d-shell states have the form X_dy z/r², Y _dx z/r²). Further details of the theoretical model can be found in Supplementary Sections S4 and S5.