Christopher A Broderick
email: c.broderick@umail.ucc.ie

Simone Mazzucato
email: mazzucat@insa-toulouse.fr

Hélène Carrère

Thierry Amand

Hajer Makhloufi

Alexandre Arnoult

Chantal Fontaine

Omer Donmez

Ayşe Erol

Muhammad Usman

Hejer Makhloufi

Eoin P O'reilly

Xavier Marie

Anisotropic electron g factor as a probe of the electronic structure of GaBi x As 1 -x / GaAs epilayers

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

The incorporation of small concentrations of bismuth (Bi) into GaAs to form the highly-mismatched dilute bismide alloy GaBi x As 1-x results in a rapid reduction of the band gap (E g ) with increasing Bi composition x, of 70 -80 meV per % Bi replacing As in the alloy at low x. 1,2 This decrease in E g is accompanied by a strong increase of the spin-orbit-splitting energy (∆ SO ), 3 which leads to a band structure in which ∆ SO > E g for x 10%. 4,5 This band structure condition makes dilute bismide alloys an attractive candidate material system for applications in GaAs-based semiconductor lasers operating at mid-infrared wavelengths due to the possibility to suppress the dominant non-radiative recombination pathway. 6 Moreover, the increased spin-orbit coupling in dilute bismide alloys could lead to the development of spintronic devices based on the Rashba spin-orbit interaction requiring shorter channel lengths or reduced modulation electric field strengths compared to existing GaAs-based devices. 7 As a result of this and other potential applications, research interest in dilute bismide alloys has been steadily increasing in recent years.

As a result of the large differences in size and chemical properties between Bi atoms and the As atoms they replace, Bi acts as an isovalent impurity in GaAs which strongly perturbs the valence band (VB) structure. An explanation of the observed strong, compositiondependent bowing of E g and ∆ SO in GaBi x As 1-x has been proposed 2,4,8 in the form of a band anti-crossing interaction between the extended VB edge states of the GaAs host matrix, and highly localized Bi-related impurity states which are resonant with the GaAs VB. The situation in disordered GaBi x As 1-x alloys is more complicated, where the reduction in symmetry brought about through alloy disorder has significant effects on the electronic properties. 4,9 The Landé g factor of conduction electrons (g * ) is a key parameter in semiconductor materials, describing the response of the electron spins to externally applied magnetic fields, as well as providing useful information regarding the symmetry of the electron and hole states at the Γ point in the Brillouin zone. Therefore, in addition to providing information pertinent to evaluating the potential of specific materials for spintronic applications, analysis of the electron spin properties and g factor gives insight into the material band structure, providing valuable data for material characterization and modeling. 2,8,9 In particular, g * is extremely sensitive to the separation in energy between the conduction band (CB) minimum and the light-hole (LH), heavy-hole (HH) and spin-splitoff (SO) bands, as well as hybridization between these bands caused by, e.g., a reduction of crystal symmetry due to pseudomorphic strain. 10 As a result, calculation of g * provides a stringent test of theoretical models of the material band structure. 11 The Larmor precession of electron spins in the presence of a magnetic field leads to quantum beating of the circular polarization of the photoluminescence (PL). Using time-resolved PL spectroscopy the frequency of these quantum beats can be determined, allowing the relevant component of g * to be determined directly for a given orientation of the externally applied field. 12 As a result of the long coherence times associated with the spin quantum beats observed in the PL polarization, this technique enables highly accurate determination of g * . 11,13 Recently, in Ref. 14, we experimentally studied the temperature-dependent spin dynamics of a single GaBi x As 1-x /GaAs epilayer with x = 2.2% and presented the first measurement of the transverse component of the electron g factor (g * ⊥ ) in this emerging material system. Our measurements showed that the magnitude of g * ⊥ increases strongly with increasing Bi composition x in GaBi x As 1-x , primarily due to the increase in the strength of the spin-orbit coupling with increasing x. In this article we significantly extend upon the analysis of Ref. 14 by studying a number of samples with Bi compositions ranging up to x = 3.8%. This allows us to examine the evolution of g * with x, which enables us to identify general trends in the spin properties of GaBi x As 1-x alloys. By performing time-resolved PL measurements with different orientations of the applied magnetic field, we further demonstrate that g * is strongly anisotropic in GaBi x As 1-x epilayers.

This experimental analysis is supported by the development of a detailed theoretical model for the calculation of g * in GaBi x As 1-x and related alloys. The theoretical model is based upon a well-established, atomistic sp 3 s * tight-binding Hamiltonian which we have previously applied to the study of the electronic structure of ordered 8 and disordered 4,9 GaBi x As 1-x alloys, as well as GaBi x As 1-x -based quantum well heterostructures. 15 Using this model we identify and quantify the causes of the observed anisotropy and strong variation of g * with x in GaBi x As 1-x . Our theoretical calculations explain the measured variation of g * with Bi composition x in terms of the effects of epitaxial strain and Bi-induced mixing of VB states due to alloy disorder. By employing large supercell calculations containing 4096 atoms we account for the effects of alloy disorder in a realistic manner and hence obtain a theoretical description of g * which is in very good agreement with experiment.

The remainder of this paper is organized as follows. In Section II we describe the sample growth and characterization, as well as the details of the experimental measurements. This is followed in Section III by an outline of our theoretical model. We then present the results of our combined experimental and theoretical study in Section IV, beginning in Sections IV A and IV B with time-integrated and time-resolved experimental results, before presenting our theoretical results in Section IV C. Finally, in Section V we conclude.

II. EXPERIMENTAL DETAILS

We have investigated five undoped GaBi x As 1-x epilayers with Bi compositions up to x = 3.8%, which were grown on (001)-oriented GaAs substrates by solid-source molecular beam epitaxy. Full details of the growth parameters can be found in Refs. 16 and 17. The Bi composition in each epilayer was determined using x-ray diffraction (XRD). Additionally, the XRD measurements indicated that the GaBi x As 1-x epilayers were elastically strained. The structural details of each of the epilayer samples are listed in Table I.

A combination of time-integrated and time-resolved PL spectroscopy was used to investigate the optical response and spin properties of each of the samples. Optical excitation was provided via 1.5 ps long pulses generated by a mode-locked Ti:Al 2 O 3 laser with 80 MHz repetition frequency. The excitation wavelength was fixed at 795 nm and the beam was focused on a 50 µm diameter spot on the sample surface. The polarization of the exciting beam (linear or circular) was set by a combination of a linear polarizer and a quarter-wave plate, while the degree of circular polarization was obtained by passing the PL signal through another quarter-wave plate followed by a linear polarizer. The PL signals were recorded using a S1 photocathode streak camera with an overall temporal resolution of 8 ps.

For the measurement of the anisotropic electron g factor continuous external magnetic fields up to 0.73 T were applied parallel to (Voigt configuration), and at a 45 • angle from (tilted configuration), the plane of the GaBi x As 1-x epilayer. 12 We note that, in this range of magnetic field intensities, the binding energy of excitons is not strengthened by Landau quantization effects. 18 The incident laser beam was circularly polarized. In the Voigt configuration, the light propagation was along the sample growth (z-) direction while in the tilted configuration, due to the high refractive index of GaBi x As 1-x , the light propagated at an angle of approximately ∼ 12 • from the growth direction. In this case, the circular polarization of the exciting light was still close to 100%, and the mean electron spin orientation was almost entirely along the z-axis.

The samples were mounted in a closed cycle He-cooled cryostat, and the temperature varied from 100 to 300 K. Based on an analysis of the low temperature PL 17 the temperatures and excitation intensities for the experiments were chosen in order to minimize both exci-tonic effects and the effects of radiative recombination via localized states. 19,20 Therefore, all measurements of g * were carried out at temperatures of 100 K and above with excitation powers exceeding 10 mW, corresponding to incident photon fluxes at the sample surface of at least 2.5 × 10 12 cm -2 .

Since the value of g * is strongly dependent upon the material band gap and spin-orbit-splitting energies, 10,21 photo-modulated reflectance (PR) measurements were carried out on each of the samples, which enabled both E g and ∆ SO to be measured accurately. 22,23 The PR measurements were carried out in the bright configuration, in which the samples were illuminated by quasiwhite light from a 50 W halogen lamp. A 10 mW beam from a 514 nm Ar-ion laser chopped at a frequency of 170 Hz was used as the modulation source. The reflected light was dispersed through a 0.5 m monochromator and detected by a thermoelectrically cooled InGaAs photodiode. The AC and DC components of the reflectance (R) and the relative change in the reflectance ( ∆R R ) were simultaneously acquired using a lock-in amplifier. 24 The PR spectra measured in this manner were then analyzed using low-field PR line-shape functions based on the Aspnes third-derivative functional form. 25 Based on this analysis, the dependence of E g and ∆ SO on the Bi composition x was determined directly from the measured PR spectra.

III. THEORETICAL MODEL

In this section we outline the theory of the effects of Bi incorporation on the electron g factor in GaBi x As 1-x . The calculation of g * is based on an extension of the method outlined in Ref. 10, which considered the effects of pseudomorphic strain on the transverse and longitudinal components of g * via the strain-induced hybridization of the LH and SO band edge states at Γ.

We extend this approach here to consider the hybridization of the extended states of the GaAs VB edge with Bi-related localized states, which we describe using large supercell calculations. A disordered GaBi x As 1-x alloy contains a large range of these localized states which are associated not only with isolated Bi atoms, but also with Bi-Bi pairs and larger clusters of Bi atoms. We have previously shown 4,9 that these alloy disorder and hybridization effects play a crucial role in determining the details of the VB structure in GaBi x As 1-x .

Our electronic structure calculations are based upon an atomistic sp 3 s * tight-binding Hamiltonian for GaBi x As 1-x , which includes the effects of spin-orbit coupling. In this model the on-site energies are taken to depend on the overall neighbor environment and the interatomic interaction matrix elements are taken to vary with the relaxed nearest-neighbor bond length d as d0 d η , where d 0 is the equilibrium bond length in the equivalent binary compound and η is a dimensionless scaling parameter (the value of which depends on the type and symmetry of the interaction). The bond angle dependence of the interaction matrix elements is represented using the two-center integrals of Slater and Koster. 26 The supercell calculations were implemented using the NanoElectronic MOdeling package NEMO-3D. 27,28 Full details of the tight-binding model can be found in Ref. 4.

The Bi-induced mixing of VB states can be quantified by the fractional GaAs Γ character of the zone center alloy states. For a given GaBi x As 1-x zone center state |ψ j,1 having energy E j,1 , the fractional GaAs Γ character is given by 4,8

G Γ (E j,1 ) = g(Ei,0) i=1 | ψ j,1 |ψ i,0 | 2 , (1)
where |ψ i,0 is the unperturbed GaAs zone center host matrix state having energy E i,0 and g(E i,0 ) is the degeneracy of E i,0 in the unstrained bulk material. G Γ (E j,1 ) is therefore a direct measure of the fraction of the host matrix GaAs states at energy E i,0 which have mixed into the alloy state |ψ j,1 .

The GaBi x As 1-x epilayers under investigation were grown in a state of biaxial compressive stress, so that an appropriate macroscopic strain must be included in the supercell calculations in order to correctly describe the epilayer electronic properties. For a pseudomorphic epilayer under biaxial stress, the longitudinal and transverse components of the strain tensor are related through Poisson's ratio σ by 29

⊥ = -2σ 1 -σ , (2)
with the longitudinal strain in the plane of the epilayer given by

= a 0 -a(x) a(x) , (3)
where a 0 and a(x) are, respectively, the lattice constants of the GaAs substrate and the GaBi x As 1-x epilayer.

The GaBi x As 1-x lattice constants along the x, y and z directions in the strained epilayer are then given by a x = a y = a 0 and a z = (1 + ⊥ ) a(x).

We interpolate the GaBi x As 1-x lattice and elastic constants linearly between those of the end point binary compounds [START_REF] Vurgaftman | [END_REF]31 in order to calculate and ⊥ for a given Bi composition x, and use the lattice constants a x , a y and a z to set the initial atomic positions in the supercell. Beginning with a GaAs supercell in which the atomic positions are defined in this manner, a disordered GaBi x As 1-x supercell is generated by replacing As atoms by Bi atoms at randomly chosen sites on the anion sublattice. Then, in order to impose a macroscopic pseudomorphic strain on the supercell, we fix the positions of the atoms on the exterior boundaries and perform a relaxation of the positions of the atoms in the interior of the supercell using a valence force field model based on the Keating potential. 32,33 In a pseudomorphically strained epilayer of a conventional III-V alloy, the transverse (g * ⊥ ) and longitudinal (g * ) components of g * are given in the limit of small strain by 10

g * ⊥ g 0 = 1 - 2m 0 |P | 2 3 2 1 E B - 1 E C , (4)
g * g 0 = 1 - m 0 |P | 2 2 1 E A - 1 3E B - 2 3E C , (5)
where m 0 and g 0 are, respectively, the free electron mass and g factor. In Eqs. ( 4) and ( 5) the indices A, B and C denote, respectively, the HH, LH and SO band edge states at Γ, while E A , E B and E C denote the differences in energy between each of these states and the Γ 6c states at the CB minimum. The inter-band momentum matrix element P between the s-like Γ 6c states at the CB minimum and the p-like Γ 8v states at the VB maximum is defined by

P = - i m 0 s| p m |m (6)
where m = x, y, z and p m denotes the components of the momentum operator. In Ref. 10 the strain-induced hybridization between the zone center LH and SO states was calculated explicitly, leading to additional terms in Eqs. ( 4) and ( 5) which are proportional to the degree of mixing between them. In our analysis of pseudomorphically strained GaBi x As 1-x alloys here, we include such state mixing in our calculations by projecting each alloy state onto the unstrained GaAs HH, LH and SO states, as described by Eq. ( 1). Using unstrained GaAs states in this way allows us to explicitly include the alloy-and strain-induced modifications to g * in our theoretical calculations.

In a GaBi x As 1-x alloy the extended VB edge states of the GaAs host matrix hybridize with a large number of alloy VB states, 8 with the details of the VB mixing depending strongly on alloy disorder. 4,9 As discussed above, this hybridization can be quantified by the fractional GaAs Γ character. Furthermore, the zone center alloy states |ψ j,1 form a complete basis set in which the extended host matrix VB states |ψ i,0 can be expanded. Therefore, in order to explicitly account for the effects of Bi-and strain-induced mixing of the HH-like VB states in a disordered GaBi x As 1-x supercell, we make the replacement

1 E A → j G HH Γ (E j,1 ) E CB,1 -E j,1 , (7)
where E CB,1 is the GaBi x As 1-x CB edge energy, the sum runs over the zone center supercell states and G HH Γ (E j,1 ) is the fractional GaAs HH character of the supercell VB state |ψ j,1 , calculated by taking |ψ i,0 = |ψ HH,0 in Eq. (1).

Making the equivalent replacements for E -1 B and E -1

C

in Eqs. ( 4) and ( 5) in terms of G LH Γ (E j,1 ) and G SO Γ (E j,1 ), calculated using the unstrained GaAs LH and SO states |ψ LH,0 and |ψ SO,0 in Eq. ( 1), we arrive at the following expressions for g * ⊥ and g * in a disordered, pseudomorphically strained GaBi x As 1-x alloy

g * ⊥ g 0 = 1 - 2m 0 |P | 2 3 2 j G LH Γ (E j,1 ) -G SO Γ (E j,1 ) E CB,1 -E j,1 , (8)
g * g 0 = 1 - 2m 0 |P | 2 3 2 j 3 2 G HH Γ (E j,1 ) -1 2 G LH Γ (E j,1 ) -G SO Γ (E j,1 ) E CB,1 -E j,1 , (9)
which reduce respectively to Eqs. ( 4) and ( 5) for x = 0.

Since we are concerned with describing the variation of g * with Bi composition x, and because Bi incorporation primarily affects the VB, we have neglected the effects of the Γ 7c and Γ 8c CB states in our analysis of g * . The effects of these bands on g * are significantly less important than the effects of the Γ 7v and Γ 8v VB states that we have included explicitly in our model, due to their weaker relative coupling to the Γ 6c states at the CB minimum. 21 In addition, the analysis presented here can be straightforwardly modified to include contributions from these and other remote bands.

We have previously shown 4,34 that in order to accurately describe the effects of Bi-induced mixing of VB states in ordered (In)GaBi x As 1-x alloys, large supercells containing 2000 atoms must be considered. The use of supercells containing several thousand atoms further enables an accurate description of the effects of alloy disorder to be obtained, due to the large scope for the formation of different local configurations of group V atoms at fixed alloy composition. We therefore perform our calculations on supercells containing 4096 atoms. For I. The measured spectra for samples 1 -5, which have all been normalized, are shown using red, black, green, blue and light blue lines, respectively. The sharp features visible at 1.42 eV correspond to emission from the GaAs substrate.

a given Bi composition x we consider four separate supercells, each of which contains a different statistically random spatial distribution of Bi atoms. We then obtain our theoretical values for the longitudinal and transverse components of g * at each Bi composition x by averaging over the results obtained for each distinct supercell at that fixed composition.

IV. RESULTS

A. Time-integrated photoluminescence and photo-modulated reflectance

Steady-state PR and PL measurements have been used to investigate the optical properties of each of the GaBi x As 1-x epilayers. Measurements of the lowtemperature PL at low excitation intensities displayed strong localization effects, characterized by an "s-shape" variation of the PL peak energy with temperature up to approximately 100 K. 17,19,35,36 As stated in Section II, in order to investigate the g factor associated with conduction electrons (as opposed to that associated with localized excitons) the PL measurements were performed at temperatures ≥ 100 K and with sufficiently high excitation intensities to ensure at most a marginal contribution to the PL signal from localized states. 17,20 Fig. 1 shows the measured PL spectra at T = 300 K for all five of the epilayers listed in Table I. The measured red-shift of the PL peak over the investigated composition range is approximately 75 meV per % Bi, in agreement with previous experimental 2, 3 and theoretical 4,8 results. Comparing each of the GaBi x As 1-x PL features to the observed sharp features at 1.42 eV (associated with I.

The measured values were obtained using photo-modulated reflectance spectroscopy, while the calculated values were obtained using the theoretical model outlined in Section III.

emission from the GaAs substrate) we see that the spectral width of the band edge PL increases with increasing Bi composition. This increase in the PL linewidth is primarily a result of the effects of alloy disorder, which causes strong inhomogeneous broadening of the VB edge states in GaBi x As 1-x . 9 The variation of the band gap and spin-orbit-splitting energy with Bi composition are crucial in determining the magnitude of g * . In order to determine ∆ SO precisely steady-state PR measurements were performed on all of the samples. The results of the PR measurements are summarized in Fig. 2, which shows the measured variation of E g and ∆ SO with x. Also shown in Fig. 2 are the results of large supercell tight-binding calculations of E g and ∆ SO , which were performed using the theoretical model outlined in Section III. We note that the theoretical model accurately captures the strong, composition dependent bowing of E g and ∆ SO , and is in good agreement with the experimental data across the investigated composition range.

B. Measurement of g * via time-resolved photoluminescence spectroscopy

As a result of the T d point group symmetry present in unstrained bulk III-V compounds such as GaAs, g * is isotropic and independent of the direction of the applied magnetic field. 37 However, in the presence of a reduction in symmetry, caused for example by quantumconfinement effects 12,38,39 or in the presence of pseudomorphic strain, 10 g * becomes anisotropic. Since the GaBi x As 1-x epilayers under investigation were grown on GaAs substrates, they are in a state of biaxial compressive stress. As a result of the increase in the GaBi x As 1-x lattice constant with increasing x, the difference between the transverse and longitudinal components of g * is expected to increase with increasing Bi composition.

Polarization-resolved PL spectroscopy can be used both to determine the magnitude of g * , as well as to investigate its anisotropy. According to the optical selection rules in bulk material, when the pump beam propagates with circular polarization along the growth (z-) direction, the electron spins also align along z. Assuming rapid spin dephasing of holes, [40][41][START_REF]Optical Orientation (Modern Problems in Condensed Matter Science[END_REF] the degree of electron spin polarization is obtained from the circular polarization P c of the luminescence

P c = I + -I - I + + I -, (10)
where I + and I -are, respectively, the intensities of the co-and counter-polarized (with respect to the polarization of the exciting beam) PL components. When an external magnetic field is applied in the Voigt configuration, Larmor precession of spins around B occurs, giving rise to a quantum beating of the electron spins and thus of P c . This behaviour is clearly visible in Fig. 3, which shows the measured temporal evolution of the co-and counter-polarized PL components of sample 1 (x = 1.16%) at T = 100 K, in the presence of a transverse magnetic field of 0.73 T. The measured temporal evolution of the degree of PL circular polarization P c is also shown in Fig. 3. The latter is well described by assuming an exponential spin dephasing multiplied by a cosinusoidal oscillation describing the Larmor precession of electron spins about B at angular frequency ω P c (t) = P 0 e -t/τ cos (ωt + φ) ,

where τ is the electron spin dephasing time, P 0 is the initial degree of PL circular polarization, and φ is a constant phase shift. The Larmor angular frequency of the electron spin precession is linked to the g factor tensor and applied field by

ω = µ B B g * , (12)
where µ B is the Bohr magneton. Experimentally, ω is obtained via analysis of the timeresolved PL polarization, which allows the magnitude of the relevant component of g * to be determined using Eq. ( 12) for a given relative orientation of the epilayer and applied magnetic field. In Ref. 14 we considered only the transverse component g * ⊥ of the electron g factor, obtained by measuring ω in the Voigt configuration. Since there is no Larmor precession of electron spins when the applied magnetic field is parallel to the growth direction, determination of the longitudinal g factor g * must be undertaken in a tilted configuration in which the applied magnetic field has two components, one in the layer plane as in the Voigt configuration, as well as an additional component along the growth (z-) direction. By orienting the epilayer at 45 • with respect to the orientation of B, the magnitude of g * can be determined via where g * 45 • is the g factor determined in the 45 • (tilted) configuration. 12 Full details of the analysis leading to Eq. ( 13) are given in the Appendix.

g * = 2g * 2 45 • -g * 2 ⊥ , (13)
A comparison of the measured polarization dynamics of sample 1 at T = 100 K in the Voigt and tilted configurations is shown in Fig. 4, for an applied magnetic field with B = 0.73 T. We note that the Larmor frequency of the observed quantum beats is different in the two cases, with ω 45 • > ω ⊥ , which indicates that g * is anisotropic and also that |g * | > |g * ⊥ |. The inset of Fig. 4 shows the measured variation of the quantum beating (Larmor) frequency as a function of the flux density of the applied field in the tilted configuration, confirming that ω increases linearly with B. By performing a linear fit to the variation of the measured Larmor angular frequency as a function of the magnetic flux density, g * 45 • can be determined from the slope of ω 45 • (B) as per Eq. ( 12).

Both g * ⊥ and g * increase strongly in magnitude with increasing Bi composition x, departing from the isotropic GaAs Landé g factor g * = -0.44. [START_REF] Oestreich | [END_REF] The measured values of g * ⊥ and g * for sample 1 (x = 1.16%) and sample 4 (x = 3.04%) are plotted as a function of temperature in Fig. 5. A reduction in the magnitude of both g * ⊥ and g * is observed with increasing temperature, which is consistent with previously observed trends in GaAs. [START_REF] Oestreich | [END_REF][44][45] We observe a larger difference between g * ⊥ and g * in sample 4 than in sample 1 due to the fact that the pseudomorphic strain in the epilayer increases with increasing Bi composition x, leading to stronger anisotropy in g * . Detailed explanation of this behaviour, as well as the dependence of g * on x, is provided by the results of our theoretical calculations in Section IV C. Our justification for the choice of a negative sign for the transverse and longitudinal components of g * in GaBi x As 1-x is based upon our previous measurement of g * ⊥ in Ref. 14, as well as the fact that g * is known to be negative in GaAs. 21 We note that the uncertainty associated with the measurement of g * increases with increasing temperature. The decrease in the magnitudes of g * ⊥ and g * with increasing temperature results in an increase in the period of the Larmor precession, which becomes comparable to the electron spin dephasing time at higher temperatures. Rapid decrease of the spin dephasing time has been observed not only with increasing temperature, but also with increasing Bi composition x in GaBi x As 1-x , as expected based on the theoretical analysis of Ref. 46. The latter is a result of the strong enhancement of the spinorbit coupling in GaBi x As 1-x with increasing x. Spin dephasing therefore reduces the accuracy with which the quantum beating frequency, and hence g * , can be measured at high temperature. We further note that, at all temperatures and Bi compositions, the uncertainty associated with the measurement of g * is greater than that related to g * ⊥ due to the increased noise present in measurements of the time-resolved PL in the tilted configuration compared to the Voigt configuration. This increased noise in the PL measured in the tilted configuration, combined with dephasing of the electron spins on a timescale comparable to the Larmor period at high temperature, meant that it was not possible to measure g * at 300 K. Similarly, the increased uncertainty associated with the measured values of g * ⊥ and g * at higher x is due to the Bi-induced reduction of the spin dephasing time.

C. Theoretical results and comparison with experimental measurements of g *

In order to investigate the effects of alloy disorder and pseudomorphic strain on the anisotropy of the electron g factor, we have used the theoretical model described in Section III to calculate the variation of the transverse and longitudinal components of g * as a function of x. The theoretical calculations were performed on a series of disordered 4096-atom supercells, which are either freestanding (unstrained) or under compressive pseudomorphic strain. In the free-standing supercells all atomic positions are allowed to relax freely, with no constraints applied to the atoms at the supercell boundaries. In order to directly quantify the effects of pseudomorphic strain on the electron g factor, the same spatial distributions of Bi atoms were used at each composition x in the unstrained and strained supercell calculations.

Due to the fact that the tight-binding model which we use underestimates the magnitude of the inter-band momentum matrix element P by approximately 25%, 8 and since we neglect the effects of the Γ 7c and Γ 8c CB states in our calculations of g * , we obtain a suitable value for P by fitting to the experimental value of g * in GaAs, where the values of E g and ∆ SO are taken directly from a tight-binding calculation with x = 0. Following this procedure we obtain a value of P = 10.13 eV Å for GaAs at T = 300 K, close to the commonly accepted value of supercell. The supercell was placed under a compressive pseudomorphic strain, corresponding to epitaxial growth on a GaAs substrate. Red lines show the calculated distribution of Γ character in a Bi-free Ga2048As2048 supercell. The zero of energy is taken at the GaAs valence band maximum. The black vertical arrows in each panel denote the weighted average energies of the HH-, LH-and SO-like alloy valence states, calculated using Eq. ( 7) for each of EA, EB and EC .

10.48 eV Å. [START_REF] Vurgaftman | [END_REF] This value of P is then kept constant in all calculations of g * ⊥ and g * as the Bi composition x is varied. Based on the analysis of Ref. 44, the variation of the GaAs host matrix band gap with temperature is taken to depend solely on the effects of lattice dilatation. Fig. 6 shows the calculated distribution of the Γ character associated with the extended GaAs host matrix HH, LH and SO band edge states over the full spectrum of zone center alloy states, for a disordered 4096-atom Ga 2048 Bi 82 As 1966 (x = 4%) supercell under compressive pseudomorphic strain. The top, middle and bottom panels of Fig. 6 show, respectively, the G HH Γ (E j,1 ), G LH Γ (E j,1 ) and G SO Γ (E j,1 ) spectra calculated using Eq. ( 1). For comparative purposes, the corresponding Γ character of the unperturbed GaAs host matrix is also shown in Fig. 6. As is expected for an epilayer under compressive strain, we see that the topmost alloy VB state is predominantly HH-like, having > 50% GaAs HH Γ character. We additionally note that both the LH and HH band edges in the alloy are strongly inhomogeneously broadened, which is evidenced by the large number of alloy VB states over which the GaAs VB edge Γ character is distributed.

We have previously demonstrated that the strong inhomogeneous broadening of the GaBi x As 1-x VB edge states is a result of (i) the strong broadening of local-ized states associated with isolated Bi atoms due to the large density of GaAs VB states with which they are resonant, 8,47 and (ii) the formation of a range of localized states associated with Bi-Bi pairs and larger clusters of Bi atoms, which lie above the GaAs VB edge in energy and hybridize strongly with the alloy VB edge states. 9 In contrast to the character of the alloy VB edge, we note that the feature associated with the Γ character of the GaAs SO band edge states, visible in the bottom panel of Fig. 6, remains relatively sharp in GaBi x As 1-x . The calculated distribution of GaAs SO character shows that the strong inhomogeneous broadening of the LH-like alloy VB states minimizes the expected strain-induced mixing between the LH-and SO-like states. Further calculations that we have undertaken show strain-induced mixing between the LH and SO states in ordered supercells; the strong inhibition of such mixing observed in Fig. 6 is therefore due to the disorder-induced broadening of the GaAs LH states in the disordered alloy supercells.

In the absence of pseudomorphic strain g * is expected to be isotropic, with g * ⊥ = g * . 10 Examining the results of our free-standing supercell calculations in Fig. 7 we see that this isotropy holds in GaBi x As 1-x alloys, even in the presence of strong alloy disorder. In addition to a rapid, monotonic increase in the magnitude of g * with increasing x, we also calculate that the reduced local symmetry present in the disordered alloy supercells has a negligible effect on the isotropy of g * . In all of the supercells considered the calculated values of g * ⊥ and g * are approximately equal, even in the presence of significant alloy disorder at x = 4%. For example, the maximum calculated difference between the magnitudes of g * ⊥ and g * in an individual disordered supercell at x = 4% was 0.034, representing a disorder-induced variation of approximately 2% in the magnitude of g * , so that the g factor can be considered effectively isotropic even in strongly disordered GaBi x As 1-x alloys.

Further examining Fig. 7 we see that this situation changes drastically in the presence of pseudomorphic strain, in which a strong anisotropy of the g factor appears so that g * ⊥ = g * . As in the unstrained case we see that the magnitudes of both g * ⊥ and g * increase strongly and monotonically with increasing x. We also calculate that (i) the magnitude of g * exceeds that of g * ⊥ for nonzero x, in agreement with experiment, and (ii) the magnitude of g * is close to the isotropic value of g * calculated in the absence of strain, both of which are expected for epitaxial layers under compressive strain. 10 Based on these comparative calculations we conclude that while the symmetry of g * is essentially unaltered by the effects of alloy disorder, the effects of pseudomorphic strain are crucial for determining the experimentally observed anisotropy of g * in GaBi x As 1-x epilayers, as is generally expected for III-V alloys. 10 Fig. 8 shows the measured and calculated variation of g * ⊥ and g * with x at T = 200 K, where the theoretical calculations include the appropriate composition depen- Red open symbols denote free-standing calculations in which no macroscopic strain has been applied to the supercells. Black closed symbols denote calculations in which a compressive pseudomorphic strain corresponding to epitaxial growth on a GaAs substrate has been applied to the supercells.

dent pseudomorphic strain. In addition to the experimental data for samples 1 -5, the measured value of g * ⊥ at x = 2.2% has been included from Ref. 14 for completeness. We see that the theoretical model reproduces the observed enhancement of g * ⊥ with increasing x to a high degree of accuracy, but that the description of g * is comparatively less accurate. One possible source of the observed discrepancy between the measured and calculated values of the longitudinal component of the g factor is the difficulty related to obtaining a suitably strong PL signal for the determination of the Larmor frequency in the 45 • tilted configuration. This is reflected in large experimental errors associated with the measured values of g * , as discussed in Section IV B. Additionally, reciprocal space maps measured around the asymmetric (224)+ direction did not show any evidence of plastic relaxation in the epilayers. However, transmission electron diffraction measurements performed on sample 5 (x = 3.83%) showed the presence of a low density of interfacial dislocations, which indicates that plastic relaxation has started to occur. The onset of plastic relaxation in samples with higher Bi compositions may also partially explain the observed discrepancy between the measured and calculated values of g * .

In Ref. 14 it was shown that using Eq. ( 4) to calculate the variation of g * ⊥ with x, by setting E B = E g (x) and E C = E g (x) + ∆ SO (x), strongly overestimates the experimentally observed increase in the magnitude of g * ⊥ with increasing x. The failure of this simple approach can be understood in terms of the results presented in Fig. 6, which show that (i) the separation in energy between the CB and LH band edges is not equal to E g (x) due to the strain-and disorder-induced splitting of the alloy VB edge, and (ii) the weighted average of the LH contributions to g * lie at lower energies in the VB of the disordered alloy, due to the strong, Bi-induced hybridization of the alloy VB states.

In the theoretical model of Section III, the effects of alloy disorder on the calculated variation of g * ⊥ and g * with x can be determined by considering the distribution of host matrix Γ character shown in Fig. 6. In the disordered alloy the VB edge GaAs Γ character is distributed over a large number of alloy VB states, with the details of the distribution determined by the precise disorder present in the alloy. The details of this distribution of the GaAs Γ character onto VB states lying below the alloy VB edge therefore plays a crucial role in determining the calculated variation of g * ⊥ and g * with x since, by Eqs. ( 8) and ( 9), the contribution from a given alloy VB state to a specific component of g * is weighted by its associated host matrix Γ character. Specifically, mixing of this Γ character into VB states with greater separation in energy from the alloy CB minimum results in contributions from terms such as those in Eq. ( 7) with larger denominators, the net result of which is to act against the stronger increase in g * ⊥ and g * that would be expected based solely upon the measured variation of E g and ∆ SO with x.

To further illustrate this point within the context of our theoretical model we have used the values of g * ⊥ and g * calculated using Eqs. ( 8) and ( 9) at T = 200 K for the supercell of Fig. 6, as well as the weighted average energy of the sharp SO-related feature calculated using the GaAs SO character spectrum (denoted by a vertical arrow in the bottom panel of Fig. 6), to solve for the values of E A and E B in Eqs. ( 4) and ( 5) that would be required to reproduce these same values of g * ⊥ and g * . The vertical arrows in the top and middle panels of Fig. 6 show the HH-and LH-like band edge energies obtained from the values of E A and E B calculated in this manner (equivalent to calculating E A from Eq. ( 7), with a similar expression for E B in terms of G LH Γ (E j,1 )). Examining the calculated distribution of GaAs HH and LH character in the disordered supercell, we see that the values of E A and E B required to reproduce g * ⊥ and g * as obtained from Eqs. ( 8) and ( 9) lie at significantly lower energy than the alloy VB states which account for the majority of the GaAs VB edge character. As a result, simple calculations based on Eqs. ( 4) and ( 5) that do not take into account the strong, Bi-induced mixing of VB states in a disordered GaBi x As 1-x alloy necessarily break down unless unrealistic values of E A and E B are used to represent the energies of the alloy VB edge states. We conclude that alloy disorder therefore plays a key role in determining the variation of the magnitudes of g * ⊥ and g * with x in GaBi x As 1-x . Based on this analysis we conclude that pseudomorphic strain plays a key role in determining the magnitudes of g * ⊥ and g * at a given Bi composition in the disordered alloy since, when compared the equivalent unstrained material, compressive strain acts to preferentially redistribute the GaAs HH character onto higher energy alloy VB states.

We note that the theoretical calculations of g * presented here, as well as further calculations that we have undertaken, indicate a monotonic decrease of the electron effective mass at the CB minimum (m * c ) with increasing x. While this is in agreement with the experimental measurements presented in Ref. 48, it contradicts those presented in Ref. 49 which identified an unexpected increase of m * c with increasing x. Therefore, while conflicting sets of experimental data exist on the evolution of m * c with x in GaBi x As 1-x alloys and there appears as yet to be no consensus in the literature, our theoretical analysis supports those experimental data that indicate a decrease in m * c with increasing x. However, we note that the Γ 7c and Γ 8c CB states (the effects of which are neglected in our current analysis) play a significantly more important role in determining m * c than they do in determining g * . 21 As a result, a detailed discussion of the effects of Bi incorporation on m * c is beyond the scope of the current work.

Overall, the theoretical model we developed in Section III is capable of describing the anisotropy of g * in pseudomorphically strained GaBi x As 1-x epilayers, predicting correctly that |g * | > |g * ⊥ | in addition to describing the observed strong enhancement of the magnitudes of g * ⊥ and g * with increasing x. We conclude that the theoretical calculations are in very good agreement with the observed experimental trends, which reinforces the validity of the detailed picture of the electronic structure of disordered GaBi x As 1-x alloys provided by the tightbinding model. 4,9 V. CONCLUSIONS

We have presented a detailed experimental and theoretical study of the composition dependence of the electron Landé g factor g * in GaBi x As 1-x /GaAs epilayers, for Bi compositions up to x = 3.8%. The optical properties of these epilayers were characterized using a combination of time-integrated photoluminescence and photo-modulated reflectance measurements. The observed rapid reduction of the GaBi x As 1-x band gap, and increase in the spin-orbit-splitting energy, with increasing x is in line with that noted previously by several authors.

The transverse and longitudinal components of g * were measured via time-resolved photoluminescence spectroscopy, which was used to determine the frequency of the quantum beating of the luminescence circular polarization associated with the Larmor precession of electron spins about an externally applied magnetic field. By varying the orientation of the epilayer samples with respect to the direction of the applied magnetic field we demonstrated that (i) g * becomes highly anisotropic in pseudomorphically strained GaBi x As 1-x alloys, and (ii) the magnitudes of the transverse (g * ⊥ ) and longitudinal (g * ) components of g * increase strongly with increasing Bi composition x.

In order to explain this behaviour we have developed a theoretical model for the description of g * in GaBi x As 1-x alloys. The theoretical model is based upon atomistic tight-binding calculations of the electronic structure of large GaBi x As 1-x alloy supercells and explicitly takes account of the effects of alloy disorder and pseudomorphic strain, which are known to play key roles in determining the details of the valence band structure in dilute bismide alloys. Calculations undertaken using this model have revealed that the rapid increase in the magnitude of both g * ⊥ and g * with increasing x in GaBi x As 1-x is explained by a combination of the strong decrease (increase) in the band gap (spin-orbit-splitting energy) with increasing x, as well as hybridization of the extended states of the GaAs host matrix valence band edge with the large range of localized Bi-related states present in a disordered GaBi x As 1-x alloy.

Our theoretical analysis has additionally revealed that the reduced symmetry in strongly disordered GaBi x As 1-x alloys has a negligible effect on the isotropy of g * . The observed strong anisotropy of g * in the bulklike epitaxial layers studied here is then attributed to the effects of compressive strain, as is the case for other III-V material systems.

Overall, our measured and calculated results are in very good agreement and our combined experimental and theoretical analysis has elucidated important factors concerning the strongly perturbative effects of Bi incorporation on the electronic structure GaBi x As 1-x alloys.

  FIG.1: Time-integrated photoluminescence spectra measured at T = 300 K, for the five GaBixAs1-x samples listed in TableI. The measured spectra for samples 1 -5, which have all been normalized, are shown using red, black, green, blue and light blue lines, respectively. The sharp features visible at 1.42 eV correspond to emission from the GaAs substrate.

FIG. 2 :

 2 FIG.2: Measured (red closed circles) and calculated (black closed circles) variation of the band gap (Eg) and spin-orbitsplitting energy (∆SO) at T = 300 K as a function of Bi composition x for the GaBixAs1-x/GaAs epilayers listed in TableI. The measured values were obtained using photo-modulated reflectance spectroscopy, while the calculated values were obtained using the theoretical model outlined in Section III.

FIG. 3 :

 3 FIG. 3: Green and blue lines, left y-axis: Temporal evolution of the co-and counter-polarized photoluminescence components of sample 1, measured in the Voigt configuration at T = 100 K with B = 0.73 T. Open red circles and black line, right y-axis: Temporal evolution of the luminescence circular polarization Pc and an exponentially damped cosinusoidal fit to Pc based on Eq. (11), respectively.

FIG. 4 :

 4 FIG. 4: Comparison of the temporal evolution of the photoluminescence polarization of sample 1, measured in the transverse (Voigt; red open circles) and 45 • (tilted; black closed circles) configurations at T = 100 K with B = 0.73 T. Inset: Measured variation of the Larmor frequency ν (= ω 2π ) with the flux density B of the externally applied magnetic field.

FIG. 5 :

 5 FIG. 5: Measured variation of the transverse (black closed squares) and longitudinal (red open circles) components of the electron g factor with temperature in sample 1 (x = 1.16%, top panel) and sample 4 (x = 3.04%, bottom panel) .

FIG. 6 :

 6 FIG.6: Black lines: Calculated distribution of the Γ character associated with the GaAs heavy-hole (top panel), light-hole (middle panel) and spin-split-off-hole (bottom panel) band edge states over the full spectrum of alloy zone center valence states in a disordered, 4096-atom Ga2048Bi82As1966 (x = 4%) supercell. The supercell was placed under a compressive pseudomorphic strain, corresponding to epitaxial growth on a GaAs substrate. Red lines show the calculated distribution of Γ character in a Bi-free Ga2048As2048 supercell. The zero of energy is taken at the GaAs valence band maximum. The black vertical arrows in each panel denote the weighted average energies of the HH-, LH-and SO-like alloy valence states, calculated using Eq. (7) for each of EA, EB and EC .

FIG. 7 :

 7 FIG.7: Calculated variation of the transverse and longitudinal components of the electron g factor (denoted respectively using circles and squares) at T = 300 K, as a function of Bi composition x. Red open symbols denote free-standing calculations in which no macroscopic strain has been applied to the supercells. Black closed symbols denote calculations in which a compressive pseudomorphic strain corresponding to epitaxial growth on a GaAs substrate has been applied to the supercells.

FIG. 8 :

 8 FIG.8: Comparison between the measured and calculated variation of the transverse and longitudinal components of the electron g factor at T = 200 K, as a function of Bi composition x in a series of pseudomorphically strained GaBixAs1-x/GaAs epilayers. Red closed circles (squares) and black closed circles (squares) denote the measured and calculated values of g * ⊥ (g * ), respectively. The measured value of g * ⊥ at x = 2.2% from Ref. 14 is shown using a blue closed circle.

TABLE I :

 I Structural parameters for the GaBixAs1-x/GaAs epilayers investigated, including layer thicknesses and compositions, as well as the expected growth uncertainties in these quantities provided by the Bruker analysis software LEPTOS.

	Sample	Thickness (nm)	Bi composition, x (%)
	1	233 ± 1	1.16 ± 0.01
	2	230 ± 10	1.80 ± 0.05
	3	258 ± 1	2.34 ± 0.01
	4	147 ± 1	3.04 ± 0.02
	5	265 ± 1	3.83 ± 0.02

Acknowledgements CB, MU and EOR acknowledge support from the Irish Research Council (EMBARK Postgraduate Scholarship no. RS/2010/2766 to CB), the European Union Seventh Framework Programme (BIANCHO; project no. FP7-257974) and Science Foundation Ireland (project no. 10/IN.1/I299). OD and AE acknowledge support from the Scientific Research Projects Coordination Unit of Istanbul University (project nos. 27643, 31160 and 44649) and the Ministry of Development of the Turkish Republic (project no. 2010 K121050). LPCNO and LAAS members acknowledge support from Toulouse Tech InterLabs 2013, as well as the support of the LAAS technology platform (RENATECH) for facilitating the MBE growth.

Appendix

In order to directly determine the components of the electron g factor tensor g * , non-resonant time-resolved PL measurements were carried out in the presence of an externally applied magnetic field. We describe here the theory underlying the procedure used to obtain the transverse and longitudinal components g * ⊥ and g * of the anisotropic g factor from the observed quantum beating of the circular polarization P c of PL from the GaBi x As 1-x epilayers.

The temporal evolution of the components of the electron spin S = (S x , S y , S z ), in the presence of a tilted magnetic field B = (B x , 0, B z ), is given by 12

where i = x, y, z and the anisotropic spin dephasing times τ i are given along the transverse and longitudinal directions by τ x = τ y = τ ⊥ and τ z = τ respectively. The vector ω is the Larmor angular frequency of the gyrating electron spins, the components of which are given by

The first term on the right hand side of Eq. ( 14) describes the coherent evolution resulting from the coupling of the electron spin S to the magnetic field B, while the second term describes the spin dephasing dynamics. Under experimental conditions in which 50 ω 1 2

if we take an initial spin vector S 0 = (0, 0, S 0 ) and ignore the effects of spin dephasing, then we calculate the zcomponent of the electron spin by solving Eq. ( 14) to obtain

where ω = ω 2 x + ω 2 z . In Voigt configuration, when B has only one component, along the x-axis, S z (t) is given by

Experimentally, the Larmor angular frequency ω is extracted from the quantum beats of the circular polarization of the PL, P c (t) = -2S z (t), as a function of the intensity of the applied magnetic field.

In the Voigt and 45 • titled configurations -in which the magnetic fields are directed at 90 • and 45 • to the growth (z-) direction respectively, with B = (B, 0, 0) and

Therefore, once the values of g * ⊥ and g * 45 • are determined by measuring the frequency of the quantum beats in the PL circular polarization for each of the Voigt and 45 • tilted configurations, g * is determined straightforwardly from Eq. ( 22) as

which is Eq. ( 13) in the main text.


Bertrand Busson

Martti Kauranen

Colin Nuckolls

Thomas J Katz

André Persoons

Quasi phase matching in chiral materials

Keywords: 42.65.Ky, 42.70.Nq, 78.66.Qn

The second-order nonlinear optical coefficients associated with chirality differ in sign for the two mirror-image forms (enantiomers) of a chiral material. Structures comprised of alternating stacks of the enantiomers can therefore be used for quasi-phase-matched frequency conversion, as we demonstrate here by second-harmonic generation from Langmuir-Blodgett films of a helicenebisquinone. Such structures could lead to new types of frequency converters in which both the second-order nonlinear response and quasi phase matching arise from the chirality of a material rather than its polar order.

Frequency conversion of lasers is a significant application of second-order nonlinear optics. To do this efficiently, a material must fulfill two basic requirements. First, it must be noncentrosymmetric. Otherwise it will not have a second-order nonlinear response [START_REF] Boyd | Nonlinear Optics[END_REF].

Second, it must allow the nonlinear source polarization and the generated field to propagate through it in phase. The interaction must be phase-matched [START_REF] Boyd | Nonlinear Optics[END_REF].

In organic materials, molecular noncentrosymmetry is relatively easy to achieve. Macroscopic centrosymmetry, however, must usually be broken artificially. This can be done by various techniques, such as by aligning molecular dipoles in an external electric field (poling) or by depositing Langmuir-Blodgett or self-assembled films [START_REF] Prasad | Introduction to Nonlinear Optical Effects in Molecules and Polymers[END_REF][START_REF] Ch | Organic Nonlinear Optical Materials[END_REF][START_REF] Marks | [END_REF]. All these techniques result in polar order, the net alignment of molecular dipoles along an axis, the polar axis of the material. Such materials are often unstable, and their nonlinear response therefore decreases with time.

Phase matching is also hard to realize because the refractive indices of materials change with wavelength. In second-harmonic generation, for example, the wave vectors of the fundamental and second-harmonic beams  k and 

        / ) n n ( 4 k k 2 k 2 2    
, where   is the fundamental wavelength.

Consequently, after a distance

k L c   
, known as the coherence length, the source polarization and generated field are out of phase.

Perfect phase matching can be achieved in birefringent materials, because their refractive indices depend on the polarizations of the interacting fields [START_REF] Boyd | Nonlinear Optics[END_REF]. However, in such materials the energy flux and wave propagation vectors usually are not parallel, and their usable thickness is thus limited. In addition, only rarely can their largest nonlinear coefficients be applied for phase matching.

Because of these limitations, an alternative to perfect phase matching, known as quasi phase matching, is frequently used [5,6]. In this procedure, the signs of the nonlinear optical coefficients are reversed after every coherence length (so-called lowest-order quasi phase matching), or after any odd multiple of it. This restores the phase relation between the source polarization and the generated field, allowing the nonlinear signal to grow quasi-continuously.

Although the efficiency of quasi phase matching is lower than that of perfect phase matching, this is compensated by the ability to use the material's largest nonlinear coefficients. In polar materials, quasi phase matching is achieved by periodically inverting the direction of the polar axis, for example by poling the material in a periodic electric field [6].

However, polar order is not required for a material to be noncentrosymmetric [7,8].

Chiral molecules occur as two mirror-image forms (enantiomers) [START_REF] Barron | Molecular Light Scattering and Optical Activity[END_REF], and materials comprised of enantiomerically pure chiral molecules are always noncentrosymmetric. Accordingly, they give rise to second-order nonlinear optical effects even when there is no polar order [7,8], and the nonlinear coefficients associated with chirality can be large [8,[START_REF] Verbiest | [END_REF]. However, nonlinear processes relying on chirality have suffered from low overall efficiency because they have been observed only in experiments that are not phase-matched. But this limitation is not fundamental, because for enantiomers the nonlinear coefficients associated with chirality are opposite in sign [11]. This means that quasi-phase-matched frequency conversion should be possible in periodically alternating stacks of enantiomers.

In this Letter, we prepare such alternating stacks of enantiomers and demonstrate that they can be used for quasi-phase-matched second-harmonic generation. A series of samples was investigated in which equally thick stacks of the two enantiomers alternate. Quasi phase matching was verified by selecting the second-harmonic signal associated with chirality and measuring its intensity as a function of the thickness of the stacks. It was also shown that when the thickness of the stacks equals the coherence length, the second-harmonic signal grows continuously with the number of stacks.

To analyze the experiments, a model for the structure was considered that consists of p stacks, each of thickness L, in which the sign of the second-order nonlinearity alternates (Fig. 1a). The equations of Fejer et al. [6] were used, but they were generalized to account for linear absorption at the second-harmonic wavelength. When the conversion efficiency is sufficiently low, the amplitude of the fundamental field can be assumed to be constant. The growth of the second-harmonic amplitude 2 E along the z direction is then governed by the

equation kz i 2 2 e ) z ( d E dz dE       , ( 1
)
where  is a constant numerical factor proportional to the squared amplitude of the fundamental beam,  is the linear absorption coefficient at the second-harmonic wavelength, and k  is the wave vector mismatch. The nonlinear coefficient

) z ( d
reverses sign between successive stacks and is therefore given by

0 1 j d ) 1 ( ) z ( d   
where 0 d is a constant and j, where

p j 1  
, is the label of the stack.

Integrating Eq. ( 1) between

0 z  and pL z  gives   kpL i p pL L ) k i ( L ) k i ( 0 2 e ) 1 ( e e 1 e 1 k i d ) pL ( E                     . ( 2
)
In the particular case

k / L L c    
, the intensity becomes

2 c pL 2 c L c L 2 2 2 0 2 c 2 c 2 ) 1 e ( e 1 e 1 ) k ( d ) pL ( E ) pL ( I                        . ( 3
)
According to Eq. ( 2) the output intensities from a series of films with fixed number of stacks p, but with increasing thickness L, are expected to grow until c L L  and then to decrease.

When the thickness of each stack is equal to the coherence length, the intensity is expected to grow with the number of stacks, p, [Eq. ( 3)]. The growth should be quadratic when the material does not absorb the second-harmonic light (  0).

For our experiments, we used Langmuir-Blodgett films [START_REF] Roberts | Langmuir-Blodgett films[END_REF] of the helicenebisquinone shown in Fig. 1b [START_REF] Nuckolls | [END_REF]. Its pure enantiomers are known to aggregate in appropriate solvents and in Langmuir-Blodgett films into supramolecular helical arrays [START_REF] Nuckolls | [END_REF][14][15]. In Langmuir-Blodgett films, the nonlinear coefficients associated with chirality are large, as high as 50 pm/V, and approximately ten times larger than the coefficients not associated with chirality (that is, those that would not vanish even if the sample were achiral) [START_REF] Verbiest | [END_REF].

The Langmuir-Blodgett films of the helicenebisquinone are known to be anisotropic in the plane of the substrate [START_REF] Verbiest | [END_REF] and, when prepared by horizontal dipping, have 2 C symmetry, all molecular layers having the same orientation [START_REF] Verbiest | [END_REF]14,16]. In this symmetry, both nonlinear coefficients that are associated with chirality and those that are not can be nonvanishing.

However, the former account for most of the nonlinear response when the fundamental beam is p-polarized and the second-harmonic light detected is s-polarized. To further reduce the possible influence of the coefficients that are not associated with chirality, the samples were prepared by vertical dipping. This results in Y-type deposition [START_REF] Roberts | Langmuir-Blodgett films[END_REF], that is, deposition in which each molecular layer is oriented opposite to the previous one. Such films are expected to have 2 D symmetry, and therefore only coefficients associated with chirality nonvanishing [START_REF] Boyd | Nonlinear Optics[END_REF]. Vertical dipping has the added benefit that it allows multilayers to be prepared more rapidly.

Films were prepared by spreading molar chloroform solutions of the enantiomers of the helicenebisquinone (the samples were more than 98% enantiomerically pure) onto a milli-Q water subphase (resistivity 18 M cm) in a KSV trough. After the solvent had evaporated at room temperature, the monolayers were slowly compressed to a surface pressure of 20 mN m -1 . Films were deposited at low speed (10 mm/min) onto one side of hydrophobic glass substrates. The mean value of the transfer ratio (> 0.9, the same for the two enantiomers) indicates that the material deposits very well. Multilayers were prepared by depositing a stack with a given number of molecular layers of the (P)-(+)-enantiomer, following it by a stack with the same number of layers of the (M)-(-)-enantiomer, and repeating the sequence as necessary (Fig. 1a). Below, we refer to the samples by their sequence of stacks (e.g., P/M/P/M).

The Langmuir-Blodgett procedure allows films to be prepared with a thickness that can be controlled to a precision of one molecular layer. A drawback is that very large number of layers must be deposited to attain the thickness of even one coherence length, usually 1-10 m for visible and near-infrared wavelengths. However, the coherence length is very much shorter if the second-harmonic light is detected in the reflected instead of the more common transmitted direction [17]. In this case, the wave vectors of the fundamental and second-harmonic fields are antiparallel, and the phase mismatch becomes

        / n 8 k 4 k k 2 k 2   . The coherence length is then ca. nm 100 ñ 8 / k / L c      
. Thus the need to deposit very large numbers of molecular layers was avoided by conducting the experiments in the reflection geometry.

The experimental setup is shown in Fig. 2. A large (ca. 3 mm diameter) fundamental beam, provided by a Q-switched Nd:Yag laser (   = 1064 nm, 50 Hz repetition rate, 8 ns pulse length), irradiated the film sides of the samples. For reasons explained above, the fundamental beam was p-polarized with respect to the sample, and only s-polarized secondharmonic light (532 nm) was detected in the reflected direction. The experimental geometry is such that the signal grows along the surface normal [18], which is therefore the z axis in Eq.

( ). This means that to analyze the results, the experimental parameters must be projected along this direction. The linear absorption coefficient of the samples at normal incidence was measured by a UV-visible spectrophotometer to be / 10 1.21

3     molecular layer.
The first step was to determine the coherence length, c L . This was done by measuring the second-harmonic response from a series of structures in which the thickness of the stacks was varied. In films comprised of only one enantiomer, such measurements could be hindered by interference between the part of the second-harmonic light emitted directly in the reflected direction and the part first emitted in the transmitted direction but then reflected by the film/substrate-and substrate/air interfaces. The latter part is absent in P/M samples because the coherence length in transmission is much larger than the thickness of the samples, and thus the opposite nonlinear responses of the P and M stacks cancel. In addition, the presence of equal amounts of the two enantiomers cancels any appreciable effects that could arise from the linear optical activity [START_REF] Barron | Molecular Light Scattering and Optical Activity[END_REF] of the material. Therefore, P/M samples were used for this measurement.

The experiments were carried out at two angles of incidence, 9.9° and 44.3°, and the data were fitted to Eq. ( 2) for p = 2 (Fig. 3). The agreement between the experimental data and the fitted curves is very good. The data thus show the expected characteristics of quasi phase matching. The coherence lengths obtained from the fitted curves for the two angles of incidence are 43.65 and 48.45 molecular layers, respectively. Assuming that the refractive indices at the fundamental and second-harmonic wavelengths are equal, these results imply that the refractive index of the helicenebisquinone films is 1.57 and that their molecular layers are 1.95 nm thick [19].

The experiments were repeated at the two angles of incidence previously used, but with a series of films comprised of four stacks (P/M/P/M). Since the absorption coefficient, refractive index, and coherence lengths for the two angles of incidence were now known, the data were fitted to Eq. ( 2) for p = 4 by using the amplitudes of the phase matching curves as the only adjustable parameter. The experimental and fitted results, also displayed in Fig. 3, are again in very good agreement.

Finally, when the thickness of each stack was one coherence length, the goal of quasiphase-matched frequency conversion was reached: the second-harmonic intensity grew continuously with the number of stacks. Samples with 48 molecular layers per stack were used for the experiment. If the refractive index is taken to be 1.57, this thickness equals one coherence length when the angle of incidence is 42.6°. A new series of samples comprised of 1, 2, 3, 4 and 6 stacks was investigated at this angle of incidence. As displayed in Fig. 4, the experimental results follow Eq. ( 3), providing further evidence of quasi phase matching. The growth of the second-harmonic intensity with the number of stacks is somewhat less than quadratic only because the samples absorbed light at the second-harmonic wavelength.

In summary, the experiments show that quasi-phase-matched frequency conversion can be achieved in structures in which stacks of the two enantiomers of a chiral molecule alternate.

In such structures, no polar order need be established, for both noncentrosymmetry and quasi phase matching are consequences of the chirality of the material. In the present work, quasi phase matching was demonstrated for second-harmonic generation in reflection. However, quasi phase matching by chirality should be limited neither to this particular process nor to the reflection geometry. Nor should it be limited to Langmuir-Blodgett films. It should apply to all frequency conversion processes, also in transmission, and to other structures in which Experimental setup. The fundamental beam (1064 nm) is p-polarized with respect to the sample and applied at an angle of incidence . The nonlinear interaction with the film produces second-harmonic light (532 nm). The s-polarized component of the second-harmonic signal is detected in the reflected direction using a photomultiplier. Experimental results for the second-harmonic intensities as functions of the total number of molecular layers of the quasi-phase-matched structures. The thickness of each stack is 48 molecular layers, calculated to be equal to one coherence length at the 42.6° angle of incidence used in this measurement. Circles represent experimental points, the solid line fits them to the theoretical function given by Eq. ( 3).

Figure captions Figure 1 .

 1 Figure captions

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.Experimental results for the second-harmonic intensities as functions of the total number of molecular layers of the P/M (circles) and P/M/P/M (squares) structures for (a) 9.9° and (b) 44.3° angles of incidence. Solid lines fit the experimental results to the theoretical function given by Eq. (2). The intensity scales of the four curves, arbitrary and independent of each other, are normalized to unity.

Figure 4 .

 4 Figure 4.

Figure 4

 4 Figure 1

enantiomers alternate. It should thereby make possible the design of completely new types of frequency converters.

We acknowledge the support of this work by the Belgian Government (IUAP P4/11), the Fund for Scientific Research -Flanders (FWOV G.0338.98, 9.0407.98), and the U.S.

National Science Foundation (CHE9802316). We acknowledge contributions of S. Van Elshocht and T. Verbiest to early work that led to the experiments described. The AFM experiments were performed by G. de Schaetzen, the ellipsometric measurements by H. Bender. B. B. acknowledges the support of the European Union and M. K. that of the Academy of Finland.

* email: Martti.Kauranen@fys.kuleuven.ac.be, fax: +32 16 32 79 82


Thomas Belhadj

Claire-Marie Simon

Thierry Amand

Pierre Renucci

Olivier Krebs

Aristide Lemaître

Paul Voisin

Xavier Marie

Bernhard Urbaszek

Controlling the polarization eigenstate of a quantum dot exciton with light

Keywords: numbers: 72.25.Fe, 73.21.La, 78.55.Cr, 78.67.Hc Keywords: Quantum dots, hyperfine interaction

We demonstrate optical control of the polarization eigenstates of a neutral quantum dot exciton without any external fields. By varying the excitation power of a circularly polarized laser in microphotoluminescence experiments on individual InGaAs quantum dots we control the magnitude and direction of an effective internal magnetic field created via optical pumping of nuclear spins. The adjustable nuclear magnetic field allows us to tune the linear and circular polarization degree of the neutral exciton emission. The quantum dot can thus act as a tunable light polarization converter.

Semiconductor quantum dots (QDs) are nanometer sized objects that contain typically several thousand atoms resulting in a confinement of electrons in all three spatial directions. The absence of translational motion prolongs the carrier spin lifetimes as compared to bulk (3D) and quantum well (2D) structures [START_REF]Spin Physics in Semiconductors[END_REF]2,3,4,5]. As a result a large number of schemes for QD spin based qbit manipulations have been proposed [START_REF]Semiconductor Quantum Bits[END_REF]. After optical excitation, a conduction electron and a valence hole form a neutral exciton X 0 in the dot. For the model system of self assembled InGaAs QDs in GaAs, the anisotropic electron hole Coulomb exchange interaction for the QD symmetry C 2v gives rise to a bright X 0 doublet of eigenstates |X and |Y polarized along the [ 1 10] and [110] crystallographic directions, respectively [START_REF] Bayer | [END_REF]8]. To characterize the strength of the anisotropic Coulomb exchange an effective magnetic field B AEI in the QD plane acting on the exciton pseudo spin can be introduced [9,10,11].

In analogy to the electron and the proton in a hydrogen atom, the electron in a QD is also interacting with the magnetic moment of the nuclear spins of the atoms that form the dot [START_REF] Abragam | Principles of Nuclear Magnetism[END_REF]. The electron-hole Coulomb exchange interaction cancels out in the ground state of a singly charged exciton as for example the X + (2 holes + 1 electron) as the holes form a spin singlet [START_REF] Braun | [END_REF]. Under suitable excitation conditions, the electron polarization created through optical pumping of the X + exciton can be transferred to the nuclear spins in the dot via the hyperfine interaction even at zero applied magnetic field, giving rise to an effective magnetic field B N that can in turn stabilize the electron spin [START_REF]Spin Physics in Semiconductors[END_REF]14,15,16].

In this Letter we demonstrate optical control of the polarization eigenstate of a neutral quantum dot exciton X 0 in the absence of any external magnetic or electric field. We show novel effects resulting from the combined effect of the effective nuclear magnetic field B N and the * Corresponding author : urbaszek@insa-toulouse.fr Coulomb exchange interaction (i.e. B AEI ) on the electron spin in an InGaAs QD: the control over B N via non-resonant optical pumping allows us to orientate the pseudo spin of a neutral exciton and therefore achieve substantial optical orientation, previously only reported for charged excitons. As compared to charged excitons, we show that the robust electron spin injection for X 0 has the advantage that in the presence of B AEI the quantum dot can act as a tunable light polarization converter. The degree of circular to linear polarization conversion can be adjusted through a slight variation in excitation laser power, which could provide a new approach to switching the polarization of QD based single photon emitters [17]. We show that the build-up of B N is possible due to the presence of charged excitons X + appearing under non-resonant pumping conditions.

The sample consists of: GaAs substrate, 20 nm of GaAlAs, 98 nm GaAs, delta doping Si 10 9 cm -2 , 2 nm GaAs, InGaAs wetting layer (WL) and QDs, 100 nm GaAs, 20 nm of GaAlAs, 5nm GaAs. Although the samples are intentionally n-doped, detailed spectroscopic analysis shows that (residual) p-type doping prevails, leading to the observation of neutral and singly positively charged excitons. The photoluminescence (PL) and PL excitation (PLE) measurements at 4K on individual QDs were carried out with a confocal microscope build around attocube nano-postioners connected to a spectrometer and a charge coupled device (CCD) camera. The signal to noise ratio of 10 4 obtained by placing a solid immersion lens on the sample allows to obtain a spectral precision of +/-1 µeV for the transition energy by fitting the spectra with Lorentzian lineshapes. The excitation energy E laser of a continuous wave Ti-Sapphire laser is varied between 1.38 and 1.48 eV, covering the heavy hole and light hole to electron transitions in the WL [18,19] .The circular polarization degree of the QD PL is defined as P c = (I + -I -)/(I + + I -), where I +(-) is the σ +(-) polarized PL intensity integrated over the spectral domain covering the X 0 doublet (X + singlet) emission. The linear polarization degree is defined as P l = (I X -I Y )/(I X + I Y ). Figure 1.A shows a typical PL spectrum for E laser =1.44 eV exciting the heavy hole to electron transition in the WL about 90meV above the QD neutral exciton X 0 emission. Three transitions are observed: X 0 , the neutral biexciton 2X 0 (2 electrons + 2 holes) which is blue shifted [18], and the positively charged exciton X + . The X + shows as anticipated no fine structure and a P c in the order of 50% under strong pumping with circularly polarized light (not shown) [20]. The attribution of the three transitions is in agreement with the power dependence of the PL intensity shown in figure 1B.

Under linearly polarized excitation, the two bright

X 0 states |X = |⇑,↓ +|⇓,↑ √ 2 and |Y = |⇑,↓ -|⇓,↑ i √ 2
are separated in energy by δ 1 ≡ E X -E Y ≃ -9µeV for dot I due to B AEI [START_REF] Bayer | [END_REF]. Here ⇑ (⇓) stands for the heavy hole pseudo spin up (down) and ↑ (↓) for the electron spin up (down) projections onto the z -axis, which is also the light propagation axis and the sample growth axis [21]. The same splitting |δ 1 | is found for the 2X 0 , but as expected with order of the peaks reversed. It is important to note A surprising power dependence of P c and P l for X 0 following circularly polarized excitation in the WL is shown in figure 1C andD. We observe an increase from P c ≃ 0 up to 22% when exciting with P exc above 4µW (figure 1C), so substantial optical orientation has been achieved. Even more intriguing, we observe under circularly polarized excitation that the linear polarization increases abruptly with Laser power from P l ≃ 0 to 17% before gradually decreasing for P exc > 1µW, (figure 1D). These effects do not purely depend on laser power but also polarization, as can be seen in figures 1C and D: exciting with linearly polarized light results in P c ≃ P l ≃ 0 for X 0 with no dependence on P exc [START_REF]PL experiments show that a linearly polarized excitation does result in linearly polarized PL[END_REF].

With only the exchange interaction B AEI present the X 0 eigenstates |X and |Y are linearly polarized. Excitation with circularly polarized light should result in beats in the time domain between | ⇑, ↓ and | ⇓, ↑ as those are not the X 0 polarization eigenstates [START_REF] Flissikowski | [END_REF]24]. So assuming (i) an exponential radiative decay for the X 0 with a characteristic time τ r = 700ps [3] and (ii) an exciton spin lifetime τ s ≫ τ r the measured P c in cw PL would be P c = P 0 c (1 + ω 2 τ 2 r ) -1 with ω = δ 1 and P 0 c is the P c created in the dot at time t=0 for X 0 [10]. For the measured δ 1 ≃ -9µeV one would only expect P MAX c ≃ 1%, and not 22% as found in the experiment. Concerning P l , circularly polarized excitation should result in PL with I X = I Y and hence P l = 0 which is in contradiction to the 17% measured.

The observed optical orientation and polarization con-version can not be explained without invoking new X 0 eigenstates. We will argue in the following that nonresonant optical pumping has created a dynamic nuclear spin polarization (DNP) that acts on the electron spin like an effective internal magnetic field of several hundred mT along the z -axis (see figure 2A andB). The coupling of the nuclear spins to the electron spin via the Fermi-contact interaction (neglected here for hole spins [START_REF] Abragam | Principles of Nuclear Magnetism[END_REF]) can be expressed as

HHF = N X k A k " I k z Sz + I k -S+ + I k + S- 2 « ( 1
)
and H HF = A I S ≡ g e µ B B N S, where I k and S are the spin operator for nucleus k (out of N ≃ 10 4 -10 5 ) and for the electron spin, respectively. g e is the longitudinal electron g factor and I MAX z for In, Ga and As is 9/2, 3/2 and 3/2, respectively. The combined effect of an external longitudinal magnetic field and B AEI on the bright exciton doublet are detailed in [3,[START_REF] Bayer | [END_REF]9,11,25]. Here we simply replace the Zeeman Hamiltonian by H HF resulting in a Zeeman splitting (called Overhauser shift) purely due to

B N = (0, 0, B N ) of δ n = g e µ B B N .
The presence of a magnetic field component along the z -axis will result in (i) a splitting δ 2 1 + δ 2 n of the bright X 0 doublet and (ii) new eigenstates |+ = α|X + iβ|Y and

|-= β|X -iα|Y where α 2 = 1 (α 2 → 1/2) for B N =0 (B N → ∞) and α = α(δ 1 , δ n ) and β = β(δ 1 , δ n ). Assum- ing that τ r ≫ Ω -1
where Ω = δ 2 1 + δ 2 n we find:

Pc(δN ) = 4α 2 β 2 P 0 c = δ 2 n P 0 c /(δ 2 n + δ 2 1 ) (2) P l (δN ) = 2αβ(α 2 -β 2 )P 0 c = -δnδ1P 0 c /(δ 2 n + δ 2 1 ) (3)
With our fitting procedure we can extract δ n for X 0 as a function of Laser power for σ + excitation (see figure 1E) [START_REF]doublet for which the energy splitting is much smaller than its linewidth[END_REF]. An identical splitting with opposite sign is found for σ -excitation (not shown) and for linearly polarized excitation we measure δ n ≃ 0, as B N = 0. As the dependence of δ n on P exc is non-linear, it is more instructive to plot P c (figure 2.A) and P l (figure 2.B) achieved for X 0 as a function of the created field B N ∝ δ n (assuming an electron g-factor of |g e | = 0.48 [START_REF] Braun | [END_REF]). The experimental curves in figure 2 are very well reproduced using |P 0 c |=33% as the only fitting parameter in equations 2 and 3. The measured P c of ± 22% for σ± excitation represents 65% of the maximum achievable P 0 c =±33% for δ n → ∞, although only a low nuclear polarization of roughly 5% is achieved as δ n ≃ 10µeV. We demonstrate a wide range of tunability for the circular to linear conversion as we go from P l ≃ 0 to the theoretical limit of maximum conversion

P l = P 0 c /2 for |δ n | = |δ 1 |. For |δ n | > |δ 1 | P l
decreases in both theory and experiment. For P l not all experimental points are on the theoretical curve and we notice a slight asymmetry between σ + and σ -excitation. Our simple model does not take into account strain induced heavy hole -light hole coupling which results in X 0 eigenstates which are already at B N = 0 different from |X and |Y [11,[START_REF] Koudinov | [END_REF]. This could be at the origin of the observed discrepancy.

In the following we discuss the origin of the DNP that builds up through a simultaneous spin flip of an electron spin with a nuclear spin through the fluctuating term (I k -S + +I k + S -) in equation 1 [START_REF]Spin Physics in Semiconductors[END_REF][START_REF] Braun | [END_REF]. This flip-flop process is repeated in time and a steady state nuclear polarization is reached. This process is very costly in energy for an electron in a neutral exciton X 0 [5], as the bright and dark states (for example | ⇑, ↓ and | ⇑, ↑ ) are separated in energy by the isotropic exchange energy of up to δ 0 ≃ 500µeV at zero magnetic field in InGaAs dots [28]. As a result the probability for electron-nuclear spin flip-flops is very low, so we argue that (i) the electron of the X 0 does not transfer its spin to the nuclei (see below) (ii) X 0 is robust against decoherence by fluctuating nuclear spins which acts through the same flip-flop term. In the literature nuclear spin effects on carriers forming the X 0 have been studied in applied magnetic fields much larger than B AEI where the Zeeman effect dominates [5,25]. As shown below, the X 0 is not at the origin of the DNP, but merely experiences the existing field B N in the dot.

The PL in figure 1A shows that the dot is occupied by X 0 some of the time, by X + for the rest of the time. The different origins of the charged excitons are discussed in the literature [29]. In a simple picture, assuming that the dot contains a doping hole, the capture process for electrons (which are less likely to be trapped by potential fluctuations of the WL) could be faster than for holes and an X 0 is formed. If no hole arrives within t ≤ τ r , the X 0 will recombine, if a hole is trapped for t ≤ τ r , the X + exciton is formed. Alternatively, a hole could tunnel into or out of the dot during τ r to a nearby acceptor. During the radiative lifetime of the X + electron-nuclear spin flip-flop processes are far more likely as compared to the X 0 case, because in the absence of Coulomb exchange the energy difference between the X + states | ⇑⇓, ↓ and | ⇑⇓, ↑ is only ≃ δ n . As B N is essentially constant over at least ms [30], the electron spin of the X 0 experiences the same B N as the electron in the X + .

To demonstrate this point, we compare PLE measurements on the dot investigated already in figures 1 and 2 (dot I, see figure 3 upper part) with an additional dot II with a considerably larger splitting δ 1 = 34µeV (see figure 3 lower part). High values of P c in the order of 50% are created for the X + in both dots when exciting with E laser =1.425 to 1.465 eV. When approaching the low energy tail of the density of states of the WL at E laser ≃1.41 eV, the carrier absorption rate is too low to create nuclear polarization [START_REF] Braun | [END_REF]. At E laser ≥ 1.48 eV the P c drops in absolute value and even changes sign as the light hole transitions in the WL are excited [19,31]. δ n changes sign accordingly, which demonstrates that nuclear spin orientation in a QD can also be controlled via the optical selection rules in the WL. Comparing figures 3G and 3H for dot II shows clearly that the P c created for the X + is transferred to the nuclear spins. In stark contrast, the P c for the X 0 is on average zero in figure 3E. The neutral exciton X 0 in both dots is subject to a nuclear field of several hundred mT (figure 3B and3F), created by the charged exciton state X + , but for dot II B AEI ≫ B N , so the projection of the total effective magnetic field onto the z -axis is too small to induce optical orientation. Although the values of δ 1 differ by a factor of 3.5 due to different QD size, shape, composition and strain, the values of δ 0 are less sensitive to the exact dot symmetry as they originate mainly from the short range Coulomb exchange [START_REF] Bayer | [END_REF]32]. Similar values of δ 0 ≫ δ 1 , δ n of typically a few hundred µeV can be assumed for both dots I and II. This means that the P c shown in figure 3A for dot I is due to the B N present in the dot, and not vice versa [START_REF]In the case of non-resonant excitation build-up of DNP in the wetting layer can not be excluded[END_REF].

In summary, optical orientation of neutral excitons X 0 in single QDs in the absence of any applied fields is achieved as an effective nuclear magnetic field B N is constructed through non-resonant optical pumping. Varying B N in the presence of a constant B AEI due to Coulomb exchange allows efficient and tunable conversion of circularly to linearly polarized light mediated by a single QD. Considering the slow evolution of B N [30] and the robustness of the electron spin during energy relaxation, our all optical approach could evolve in future experiments to orientate both the nuclear and the electron spins electrically in QD based Spin-Light Emitting Diodes [34,35,36].

We thank ANR, IUF and DGA for financial support.

FIG. 1 :

 1 FIG.1:(Color online) QD I. E laser = 1.44 eV. (A) PL spectrum at 4K for P=2.5µW and linear π X excitation and detection. (B) Integrated PL intensity as a function of laser power for the X 0 (triangles), X + (stars) and 2X 0 (squares). The circular polarization degree Pc (C), the linear polarization degree P l (D) and the Overhauser shift δn (E) are shown as a function of laser power for X 0 . Solid circles (hollow squares) represent σ + (π X ) laser polarization.

FIG. 2 :

 2 FIG. 2: (Color online) QD I. E laser = 1.44 eV. δ1 = -9µeV Circular polarization degree Pc (B) and Linear polarization degree P l (C) for the X 0 states as a function of the effective nuclear field BN for σ + (σ -) laser polarization shown as solid circles (hollow triangles) assuming |ge| = 0.48. Solid lines are calculated with equations 2 and 3. Dashed vertical line: measured value of |δ1|.

FIG. 3 :

 3 FIG. 3: (Color online) PL detection energy dot I (II) ≃ 1.358 eV (≃ 1.342 eV). Circular polarization degree Pc and Overhauser shift δn as a function of laser energy for σ + (σ -) laser polarization shown as solid circles (hollow triangles) for (A,B) X 0 of dot I, (C,D) X + of dot I, (E,F) X 0 of dot II and (G,H) X + of dot II.