Abstract
It has been proposed that valenceband 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 psymmetry atomic orbitals and the negative contributions of dorbitals. These findings yield information on the orbital composition of the valence band^{12} and enable a fundamentally new approach for verification of computed Bloch wavefunctions in semiconductor nanostructures^{13}. Furthermore, we show that the contribution of cationic dorbitals leads to a new mechanism of hole spin decoherence.
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Owing to the stype 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 valenceband holes the contact interaction vanishes owing to the symmetry properties of the wavefunction, and the nonlocal 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 valenceband Bloch wavefunction, which is usually not available with sufficient precision. Thus, predicting the properties of the hole hyperfine coupling using firstprinciple 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 highresolution photoluminescence spectroscopy of neutral quantum dots. In contrast to previous work^{9}, we now also apply excitation with a radiofrequency oscillating magnetic field, which allows isotopeselective probing of the valenceband hole hyperfine interaction^{16}. 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 dshells into the valenceband Bloch wavefunctions.
Using the experimentally measured diagonal components of the hyperfine Hamiltonian (hole hyperfine constants) we calculate its nondiagonal part. We show that the admixture of the dshells has a major effect on the symmetry of the hyperfine Hamiltonian in quantum dots: unlike pure heavy holes constructed only of psymmetry shells for which the hyperfine interaction has an Ising form^{11}, the dshell contribution results in nonzero nondiagonal elements of the hyperfine Hamiltonian. We predict this to be a major source of heavyhole 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 10^{5} nuclei, nonzero 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 heavyhole states the hyperfine interaction is described using a constant C^{k} expressed in terms of the normalized heavyhole hyperfine constant γ^{k} as C^{k} = γ^{k}A^{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^{k}ΔE_{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).
We start by presenting results for strainfree 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}.
We have also performed isotopesensitive 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 ^{71}Ga (circles), ^{75}As (squares) and for the total hyperfine shifts of ^{69}Ga and ^{115}In(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 previously^{9} we find γ^{In} = −12.5±3.0%, consistent with our previous results obtained without isotope selection^{9}. 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 valenceband 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 dottodot variation of γ^{In} observed in InP quantum dots (see Table 1). Such variation may also arise from the dottodotdependent spatial separation of electron and hole wavefunctions^{8}.
From the measurements without radiofrequency pulses (similar to earlier isotopenonselective 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 indiumrich InGaAs dots emitting at E_{PL}∼1.30 eV the value of γ≈−9% has been reported^{8}. 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 protocols^{4}.
We now turn to analysis of the experimental results presented in Table 1. Firstprinciple calculation of the valenceband 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 valenceband states constructed from atomic porbitals (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 wavefunction^{20}: in particular we consider the contribution of the dshell states (l = 2). Both p and dorbitals are schematically depicted in Fig. 3a. We assume that the heavyhole states can be taken as linear combinations of pshells with weight α_{p} and dshells 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).
It follows from equation (3) that unlike the pshell, the dshell 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 valenceband states with sorbitals due to quantumdot symmetry reduction would lead to a positive contribution to γ^{k} and thus cannot account for the negative hyperfine constants^{11}. 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 3dshells, respectively, taken with effective orbital radii^{11,21}. The resulting calculated dependence of γ^{k} on dshell 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 ptype, whereas for the cation gallium a significant contribution of the dshell (∼20%) is required to account for the negative hole hyperfine constant measured experimentally.
The nonzero contribution of the dsymmetry 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 heavyhole states and . This is in contrast to the case of pure heavy holes constructed only of psymmetry states for which the hyperfine interaction has an Ising form^{11}: in that case the symmetry of the system is artificially raised to spherical, resulting in hyperfine interaction conserving angular momenta. The inclusion of the dshells 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 nonzero nondiagonal elements coupling heavy holes with the opposite spins.
It was demonstrated previously that heavy–light hole mixing can result in a nonIsing form of the hyperfine interaction^{10,11}. However, our estimates show that for gallium in GaAs the contribution of the dshells to the nondiagonal matrix elements of the hyperfine Hamiltonian dominates over the effect of the heavy–light hole mixing even if the valenceband states have a lighthole 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 dshells is observed (resulting in γ^{k}<0). Thus, the dorbital contribution will be a source of heavyhole 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 core^{13}. To estimate this volume we limit the integration in equation (3) to a sphere of a radius r_{0}, which makes M_{l} (and hence γ^{k}) a function of r_{0}. The dependence of M_{l}(r_{0}/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 4pshells 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 dshells in describing the valenceband states is well recognized^{20,22}, and it has been shown that the dsymmetry contribution originates mainly from cations. However, the previous reports^{12,23,24} predicted a much larger dsymmetry 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 tightbinding or pseudopotential methods^{25} 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 semiconductors^{25}. However, as true firstprinciple calculation of the manybody 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, Xray photoemission spectra). The experimental data on the valenceband 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 isotopeselective, 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 groupIV bulk semiconductors where dark excitons are observed^{26} and hyperfine shifts can be induced and detected^{27}.
A rigorous modelling of the hyperfine parameters^{13} has not been carried out so far for the valencebands 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 valenceband hyperfine coupling. In particular, the potential effect of large inhomogeneous elastic strain (present in selfassembled 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) quantumdot 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 (chargecoupled device) camera.
Detection of the hyperfine shifts, required to measure hole hyperfine constants is achieved using pump–probe techniques^{9} (see timing diagram in Fig. 1b). Nuclear spin polarization is prepared with a long (∼6 s) highpower optical pump pulse. Following this, a radiofrequency oscillating magnetic field is switched on to achieve isotopeselective 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 lowpower 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 isotopeselective measurement of the hole hyperfine constant.
The concept of the valenceband 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}〉. Nonzero 〈I_{z}^{k}〉 is induced by optical nuclear spin pumping: circularly polarized light of the pump laser generates spinpolarized 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 polarization^{9}. According to equations (1) and (2), hole and electron hyperfine shifts depend linearly on each other (ΔE_{h}^{k} = γ^{k}ΔE_{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 isotopeselective depolarization of nuclear spins depend on quantumdot material. For example, in the case of GaAs/AlGaAs quantum dots, we take into account that both ^{69}Ga and ^{71}Ga 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 ^{69}Ga and ^{71}Ga polarization (which improves the measurement accuracy). In contrast, in InGaAs/GaAs quantum dots, ^{115}In and ^{69}Ga NMR spectra overlap owing to straininduced quadrupole effects^{16,30}, and γ^{In} is extracted by calculating the hyperfine shifts of ^{69}Ga from the measured hyperfine shifts of ^{71}Ga. Further details of the isotopeselective experimental techniques can be found in Supplementary Section S3.
Theoretical model.
Firstprinciple calculation of the valenceband 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 shortrange 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 valenceband maximum (corresponding to heavyhole 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 F_{2} representation of the T_{d} point group relevant to bulk zincblende 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_{p}x/r, Y _{p}y/r (corresponding to ptype 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 dshell states (l = 2). To calculate the hyperfine constants (equation (3)) we assume that the heavyhole 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 F_{2} representation (dshell states have the form X_{d}y z/r^{2}, Y _{d}x z/r^{2}). Further details of the theoretical model can be found in Supplementary Sections S4 and S5.
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Acknowledgements
The authors are grateful to M. Nestoklon, A. J. Ramsay, D. N. Krizhanovskii and M. Potemski for fruitful discussion and to D. Martrou for help with the GaAs sample growth. This work has been supported by EPSRC Programme Grants Nos EP/G001642/1 and EP/J007544/1, the Royal Society, ITN SpinOptronics and ITN S^{3}NANO. M.M.G. was supported by the RFBR, Russian Federation President Grant NSh5442.2012.2 and EU project SPANGL4Q.
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A.B.K., M.H., P.S. and A.L. developed and grew the samples. E.A.C. and A.I.T. conceived the experiments. E.A.C. developed the techniques and carried out the experiments. E.A.C., M.M.G. and A.I.T. analysed the data. E.A.C., M.M.G., A.I.T. and M.S.S. wrote the manuscript with input from all authors.
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Chekhovich, E., Glazov, M., Krysa, A. et al. Elementsensitive measurement of the hole–nuclear spin interaction in quantum dots. Nature Phys 9, 74–78 (2013). https://doi.org/10.1038/nphys2514
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DOI: https://doi.org/10.1038/nphys2514
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