V M Litvyak

V Kavokin

A Lemaître

M Vladimirova

S Cronenberger

A Colombier

D Scalbert

K V Kavokin

Simultaneous measurements of nuclear spin heat capacity, temperature and relaxation in GaAs microstructures

Simultaneous measurements of nuclear spin heat capacity, temperature and relaxation in GaAs microstructures

I. INTRODUCTION

In n-doped semiconductors two spin ensembles, donor-bound electron spins and the lattice nuclei spin system (NSS) are coupled via hyperfine interaction [START_REF]Optical Orientation[END_REF]. Their magnetic moments, interaction energies, their coupling to light, to crystal lattice vibrations, as well as to electric and magnetic fields differ dramatically. The resulting physics, that is best studied in n-GaAs, is quite complex. However, if sufficient NSS stability is reached, it may potentially offer novel applications for quantum information technologies [START_REF] Foletti | Universal quantum control of two-electron spin quantum bits using dynamic nuclear polarization[END_REF][START_REF] Bluhm | Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 µs[END_REF][START_REF] Chekhovich | Nuclear spin effects in semiconductor quantum dots[END_REF][START_REF] Urbaszek | Nuclear spin physics in quantum dots: An optical investigation[END_REF][START_REF] Stockill | Quantum dot spin coherence governed by a strained nuclear environment[END_REF][START_REF] Gangloff | Quantum interface of an electron and a nuclear ensemble[END_REF][START_REF] Denning | Collective Quantum Memory Activated by a Driven Central Spin[END_REF][START_REF] Chekhovich | Nuclear spin quantum register in an optically active semiconductor quantum dot[END_REF][START_REF] Gangloff | Witnessing quantum correlations in a nuclear ensemble via an electron spin qubit[END_REF].

In this context, precise knowledge and control of nuclear quadrupole effects in n-GaAs and GaAs-based heterostructures has attracted substantial attention. Indeed, in semiconductors with cubic symmetry containing isotopes with spin I > 1/2 quadrupole interaction induced by strain splits NSS energy states (cf Fig. 1 (a)) and thus strongly affects NSS thermodynamics [START_REF] Abragam | The principles of nuclear magnetism[END_REF][START_REF] Dzhioev | Stabilization of the Electron-Nuclear Spin Orientation in Quantum Dots by the Nuclear Quadrupole Interaction[END_REF][START_REF] Maletinsky | Breakdown of the nuclear-spin-temperature approach in quantumdot demagnetization experiments[END_REF][START_REF] Chekhovich | Structural analysis of strained quantum dots using nuclear magnetic resonance[END_REF]. The effects of strain are particularly important in heterostructures, such as quantum wells [START_REF] Kotur | Ultra-deep optical cooling of coupled nuclear spin-spin and quadrupole reservoirs in a GaAs/(Al,Ga)As quantum well[END_REF] and quantum dots (QDs), and were extensively studied in both single QDs [START_REF] Chekhovich | Nuclear spin effects in semiconductor quantum dots[END_REF][START_REF] Chekhovich | Cross calibration of deformation potentials and gradient-elastic tensors of GaAs using photoluminescence and nuclear magnetic resonance spectroscopy in GaAs/AlGaAs quantum dot structures[END_REF] and QDs ensembles [START_REF] Flisinski | Optically detected magnetic resonance at the quadrupole-split nuclear states in (In,Ga)As/GaAs quantum dots[END_REF]. More recently quadrupole effects in bulk n-GaAs have been evidenced, suggesting the importance of the precise control of both strain and electric field when NSS needs to be efficiently cooled down [START_REF] Litvyak | Warm-up spectroscopy of quadrupolesplit nuclear spins in n-gaas epitaxial layers[END_REF]. Moreover, the parameters of different isotopes determining the relation between the strain tensor elements and the resulting gradients of electric fields remain controversial [START_REF] Chekhovich | Cross calibration of deformation potentials and gradient-elastic tensors of GaAs using photoluminescence and nuclear magnetic resonance spectroscopy in GaAs/AlGaAs quantum dot structures[END_REF][START_REF] Litvyak | Warm-up spectroscopy of quadrupolesplit nuclear spins in n-gaas epitaxial layers[END_REF][START_REF] Sundfors | Experimental gradient-elastic tensors and chemical bonding in III-V semiconductors[END_REF][START_REF] Griffiths | Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance[END_REF]. This is partly due to the lack of non-destructive and non-perturbative experimental techniques, capable of probing nuclear magnetic resonances (NMR) at low and zero magnetic field, where quadrupole splittings between spin levels dominate over Zeeman ones.

Indeed, traditional optically detected NMR experiments addressing QD ensembles are limited to the field range given by the Hanle curve, and suffer from huge broadening due to inhomogeneity of the electric field gradient experienced by the nuclei [START_REF] Flisinski | Optically detected magnetic resonance at the quadrupole-split nuclear states in (In,Ga)As/GaAs quantum dots[END_REF]. On the other hand, since pioneering experiments of Gammon et al resolving PL fine structure in individual QDs [START_REF] Gammon | Fine structure splitting in the optical spectra of single GaAs quantum dots[END_REF], remarkable progress has been achieved in the NSS control in such structures [START_REF] Chekhovich | Nuclear spin effects in semiconductor quantum dots[END_REF]. However, in order to reach measurable splitting a relatively high magnetic field, where Zeeman effect dominates over quadrupole one, should be applied.

Spin noise spectroscopy (SNS) could be promising for this purpose. It has been shown that electron spin noise can non-destructively relay properties of nuclear environment through the statistics of electron spin interaction with the total nuclear polarization [START_REF] Glazov | Spin noise of localized electrons: Interplay of hopping and hyperfine interaction[END_REF][START_REF] Ryzhov | Measurements of nuclear spin dynamics by spinnoise spectroscopy[END_REF][START_REF] Berski | Interplay of Electron and Nuclear Spin Noise in n-Type GaAs[END_REF][START_REF] Kalevich | Dynamic Nuclear Polarization and Nuclear Fields[END_REF][START_REF] Glazov | Electron & Nuclear Spin Dynamics in Semiconductor Nanostructures[END_REF]. SNS is based on the non-perturbative detection of the spin polarization noise via fluctuations of the Kerr or Faraday rotation of the off-resonant probe beam polarization Fig. 1 (b) [START_REF] Alexandrov | Magnetic resonance in the Faraday-rotation noise spectrum[END_REF][START_REF] Römer | Spin noise spectroscopy in semiconductors[END_REF].

It is important to stress that although we aim at probing the NSS, rather than electron spin, what is measured in practice is the electron spin noise spectrum in the megahertz frequency range. This approach differs from the one adopted by Berski et al [START_REF] Berski | Interplay of Electron and Nuclear Spin Noise in n-Type GaAs[END_REF], who measured in n-GaAs directly nuclear spin noise associated with different isotopes in the kilohertz frequency range. Such direct nuclear spin measurements are quite difficult, since NSS does not couple directly with light, but they are particularly useful to address the nuclear spin dynamics at low magnetic fields hardly accessible otherwise.

In electron SNS, the fluctuation spectrum exhibits a pronounced peak at the Larmor frequency corresponding to the total magnetic field experienced by electrons. It is given the sum of the external field B ⊥ and Overhauzer field B N , an effective nuclear field resulting from the hyperfine interaction, see Fig. 1 (d). Therefore any variation of the NSS polarization, either due to relaxation or when external magnetic field is reduced, results in a change of the Larmor frequency and thus in a shift of the electron spin noise peak. This shift is proportional to the Overhauser field, but is not isotope-selective. We have demonstrated that evolution of nuclear spin polarization during optical pumping, relaxation and re-magnetization through zero field can be monitored via this shift with sub-second resolution [START_REF] Ryzhov | Measurements of nuclear spin dynamics by spinnoise spectroscopy[END_REF][START_REF] Kalevich | Dynamic Nuclear Polarization and Nuclear Fields[END_REF][START_REF] Vladimirova | Nuclear spin relaxation in n-GaAs: From insulating to metallic regime[END_REF][START_REF] Vladimirova | Spin temperature concept verified by optical magnetometry of nuclear spins[END_REF].

These experiments performed on in n-GaAs layers revealed that the relaxation rates of optically cooled NSS under magnetic fields below 10 G are dramatically enhanced. We also measured huge heat capacity characterized by so-called local field, also of order of ≈ 10 G. This value is much higher than expected from magnetic dipole-dipole interaction B D ≈ 1.5 G [START_REF] Paget | Low field electron-nuclear spin coupling in gallium arsenide under optical pumping conditions[END_REF]. This result is illustrated in Fig. 1 (b). It shows how the Overhauser field (solid line) and the nuclear spin temperature (dashed line) in a cold NSS vary during remagnetization from zero to high magnetic field. The dependence of the Overhauser field on the magnetic field B is related to the nuclear spin temperature Θ N [START_REF]Optical Orientation[END_REF]: Here γN = i A i γ N i is the average gyromagnetic ratio, A i (γ N i ) is the abundance (gyromagnetic ratio) of i-th isotope (see Table I), h and k B are Planck and Boltzmann constants, respectively, b N is the Overhauser field at saturation of the nuclear magnetization, in GaAs b N = 5.3 T. The magnetic field below which Θ N remains constant reveals the heat capacity of the NSS. In the following it will be referred to as the heat capacity field, B H [32]. The value B H >> B D as shown in Fig. 1 (c) is a fingerprint of the enhanced NSS heat capacity as compared to the traditional spin temperature theory [START_REF]Optical Orientation[END_REF][START_REF] Vladimirova | Spin temperature concept verified by optical magnetometry of nuclear spins[END_REF].

B N = hBb N I(I + 1) γN 3k B Θ N . ( 1
)
Because in GaAs the spin of all isotopes I = 3/2, it was suggested that such an enhanced NSS heat capacity could be explained by the quadrupole splitting of nuclear spin energy levels induced by some small residual strain, see Fig. 1 (b). The corresponding effective field B Q is the one where Zeeman and quadrupole energies become comparable. In re-magnetization experiment illustrated in Fig. 1 (a) one would expect

B H = B 2 D + B 2 Q . ( 2
)
However, in the absence of NMR experiments that could provide direct access to quadrupole energies these ideas remained unconfirmed.

In this work we implement a new technique, NMR detected by electron SNS, that allows us to confirm this hypothesis. NSS absorption spectra in the radiofrequency (RF) range are measured as a function of magnetic field from zero to 100 G. In order to reach the detection sensitivity required for such experiments (sub-second integration times) we use n-GaAs layer embedded in a microcavity [START_REF] Giri | Nondestructive Measurement of Nuclear Magnetization by Off-Resonant Faraday Rotation[END_REF] and adopt the homodyne detection scheme for the for spin noise measurements [START_REF] Cronenberger | Quantum limited heterodyne detection of spin noise[END_REF][START_REF] Sterin | Optical amplification of spin noise spectroscopy via homodyne detection[END_REF][START_REF] Petrov | Increased sensitivity of spin noise spectroscopy using homodyne detection in n-doped gaas[END_REF]. The analysis of the NMR spectra allows us to (i) evaluate quantitatively quadrupole energies and confirm the impact of the quadrupole interaction on the heat capacity of the NSS. The quadrupole energies extracted from the NMR spectra appear to be consistent with heat capacity fields measured in the NSS remagnetization experiments and with the spin temperature theory; (ii) to provide additional insight into relative values of so-called gradient-elastic tensor S ijkl for Ga and As isotopes. This fundamental parameter determines the relation between the strain tensor ϵ kl and the resulting gradients of electric fields at each atomic site V ij = k,l S ijkl ϵ kl . Its values are isotope-dependent and are still actively debated in the literature [START_REF] Chekhovich | Cross calibration of deformation potentials and gradient-elastic tensors of GaAs using photoluminescence and nuclear magnetic resonance spectroscopy in GaAs/AlGaAs quantum dot structures[END_REF][START_REF] Litvyak | Warm-up spectroscopy of quadrupolesplit nuclear spins in n-gaas epitaxial layers[END_REF][START_REF] Sundfors | Experimental gradient-elastic tensors and chemical bonding in III-V semiconductors[END_REF][START_REF] Griffiths | Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance[END_REF].

II. SAMPLE

We study the microcavity sample cut out from the same wafer as in Ref. 30: a Si-doped 3λ/2 GaAs layer (cavity mode at ≈ 830 nm in the studied piece of the sample ) with donor concentration n D ≈ 2 × 10 15 cm -3 sandwiched between two Bragg mirrors, in order to enhance the sensitivity of the SNS (quality factor ≈ 2×10 4 ). The front (back) mirrors are distributed Bragg reflectors composed of 25 [START_REF] Vladimirova | Spin temperature concept verified by optical magnetometry of nuclear spins[END_REF] pairs of AlAs/Al 0.1 Ga 0.9 As layers. The sample is grown on a 400-µm-thick [001]-oriented GaAs substrate and placed in a cold finger cryostat at T=6 K. A 30-spires copper coil is placed on top of the sample to create an oscillating magnetic field (OMF). It is directed at 45 o out of the sample plane [37].

III. NMR AND RE-MAGNETIZATION DETECTED BY THE ELECTRON SPIN NOISE

The principle of NMR and NSS re-magnetization detected by electron spin noise are sketched in Fig. 2. Prior to the measurement, any preexisting polarization in the NSS is erased by OMF B RF = 0.3 G during 20 s, its frequency sweeping between 0 and 50 kHz (stage 1). At second stage the NSS is polarized by optical pumping in the presence of a longitudinal magnetic field B z = 150 G during typically 30s (circularly polarized laser diode emitting at 770 nm). This is followed by 30 s of dark time under the same magnetic field to allow for fast relaxation of the nuclear spins under the donors orbits while keeping bulk nuclei polarized (stage 3).

After that the NSS is adiabatically demagnetized down to zero magnetic field (stage 4). This last stage is important to include in the protocol, despite some additional NSS warm-up that it causes. Indeed it ensures NSS thermalization and a transfer of the optically pumped energy stored in Zeeman reservoir towards internal degrees of freedoms, that is dipole-dipole reservoir. This additional nuclear spin relaxation during stage 4 does not play any role in the experiment, provided that NSS is strongly polarized (which is the case). What matters is to start the detection of the the NSS demagnetisation or absorption each time at the same spin temperature.

The four stages preparation of the NSS is identical for both NMR and re-magnetization experiments,Fig. 2 (a).

The re-magnetization experiments are similar to those reported in Ref. 30. After preparation stages 1 -4, a set of electron spin noise spectra under in-plane magnetic field that increases from B = 0 to B = 100 G is measured every 2 G, see Fig. 2 (b). The measured frequency F L of the electron spin noise spectral peak at a given field B ⊥ is directly related to the Overhauser field B N ,

F L = F B + F N = γ e (B ⊥ + B N ), (3)
where γ e = 0.62 MHz/G is the electron gyromagnetic ratio in GaAs. Representative spectra measured at several values of magnetic field are shown in Fig. 3. The corresponding Larmor frequencies are pointed by arrows. Measurement of each point of the NMR spectrum at a given field B (either B ⊥ or B z ) corresponding to a given RF frequency f consists in three steps illustrated in Fig. 2 (c): detection of electron spin noise and its peak frequency F L in the sample under magnetic field B ⊥ = 100 G in order to evaluate the Overhauser field prior to absorption (stage 5), application during 3 s of the OMF at a given frequency f in the RF range and at a given magnetic field B at which the RF absorption is studied (stage 6), and again detection of electron spin noise and its frequency F L under magnetic field B ⊥ = 100 G (stage 7). This allows us to deduce from Eq. 3 to which extent the Overhauser field has decreased during stage 6.

Typical spin noise spectra measured at stages 5 and 7 are shown in Fig. 4 (b). The corresponding field under study is B = 0. The OMF frequencies applied during stage 6 are f = 1 Hz where NSS absorbs energy very weakly, and f = 18 kHz where NSS absorption is strong. The spectrum measured during stage 7 in the absence of OMF is shown by dashed magenta line for comparison. The shift between the electron spin noise spectral peaks measured during stages 5 and 7 contains two contributions. The first is RF-independent, it is due to some weak, but unavoidable NSS relaxation during measurement. It can be evaluated by performing identical measurements in the absence of the OMF during stage 6, see dashed magenta line in Fig. 4 (b). The second contribution is proportional to the NSS absorption at a given OMF frequency f . It is negligibly small at f = 1 Hz, but when f matches the energy difference between spin states of one of the isotopes (as for f = 18 kHz, where f matches 75 As resonance) the radiation is efficiently absorbed and the spectrum shifts more than in the absence of the OMF field.

The entire absorption spectrum (or NMR spectrum) can be constructed by repeating the protocol (stages 1-7) at different values of f . The details of this procedure are given in Appendix X A. An example of the raw data obtained in a NMR experiment addressing NSS absorption at B = 0 is shown in Fig. 4 (a). Color-encoded set of 40 pairs of electron spin noise spectra measured in the units of the ratio between spin and shot noise are shown as a function of laboratory time (bottom scale). The corresponding OMF frequency f is indicated on the top. For each value of f two spectra are systematically measured, one before application of the OMF (in this case the electron spin noise peak frequency F L does not depend on the OMF frequency f ), and one after. Left scale shows spin noise frequency and right scale the corresponding total magnetic field experienced by the electrons, B ⊥ + B N = F/γ e , where B ⊥ = 100 G. One can see that B N is significantly smaller after application of the OMF at f ≈ 8 kHz and at f ≈ 20 kHz. These frequencies characterize nuclear spin splittings at B = 0, cf Fig. 1 (a). We anticipate, that the higher frequency is related to quadrupole-split states of 75 As which has the highest quadrupole moment, and the lower one to closely lying transitions of 71 Ga and 69 Ga isotopes.

It is important to note that in most of SNS experiments the magnetic field is modulated to eliminate spinindependent signals [START_REF] Übner | The rise of spin noise spectroscopy in semiconductors: From acoustic to GHz frequencies[END_REF]. Such approach is not suitable for studies of the NSS, because nuclear spins get depolarized by non-adiabatically varying magnetic fields. To avoid this effect and also to increase the sensitivity, in this work we use a homodyne detection scheme [START_REF] Cronenberger | Quantum limited heterodyne detection of spin noise[END_REF]. This allows us to use the probe power as low as 150 µW focused on 30 µm-diameter spot. We detect the spectra in two mutually orthogonal linear polarizations, that is either parallel (which contains useful information about electron spin noise) or orthogonal to the local oscillator polarization. The spin noise spectrum is obtained by taking the difference between these spectra, in order to get rid of the background noise.

IV. NMR SPECTRA AND THEIR INTERPRETATION

A set of color-encoded NMR spectra measured under in-plane static magnetic field B ⊥ ∥ [110] and at B z ∥ [001], is shown in Fig. 5 (a-b). In order to understand these spectra in terms of the OMF-induced transitions between spin states of the three GaAs isotopes, we must calculate the energy spectrum of the NSS. It is determined by both quadrupole interaction and Zeeman effects: Ĥi = Ĥi Q + Ĥi Z , were Ĥi Q ( Ĥi Z ) describes quadrupole (Zeeman) interaction [START_REF]The dipole-dipole interaction which is known to broaden the NMR transitions, but does[END_REF]. The energy levels of each isotope are given by the eigenvalues of the Hamiltonian Ĥi , and the NMR transitions frequencies by their differences f i kl = (E i k -E i l )/h. The OMF induces spin transitions between a pair of states k and l of i-th isotope, if the OMF frequency matches the energy difference between their energy levels E k and E l . The probability of corresponding transition is given by P i kl ∝ M 2 kl,i , where M kl = ⟨Ψ k |H OMF | Ψ l ⟩ is the matrix element of the Hamiltonian describing Zeeman interaction of the nuclear spins with the OMF.

The quadrupole Hamiltonian of the i-th isotope in the presence of the in-plane biaxial strain can be written as [START_REF] Abragam | The principles of nuclear magnetism[END_REF]:

Ĥi Q = E i QZ 2 Î2 z - I(I + 1) 3 + E i QR 2 √ 3 Î2 x -Î2 y ) + E i QI 2 √ 3 Îx Îy + Îy Îx , ( 4
)
where

E i QR = E i Q⊥ cos2ζ i and E i QI = E i Q⊥ sin2ζ i .
It is determined by three parameters: E i QZ -the quadrupole energy along growth axis, E i Q⊥ -the in-plane quadrupole energy, and ζ i -the angle between the principal axis of the electric field gradient tensor V ij and the [100] crystal axis. The ensemble of the absorption spectra shown in Fig. 5 (a-b) can be understood by fitting this simple model to the observed transition frequencies. This yields 3 parameters per isotope. They are summarized in Table I together with other relevant parameters (abundances, quadrupole moments and gyromagnetic ratios) taken from the literature. Assuming that all transitions have Lorenzian shape broadened by dipoledipole interaction (∆f D = 1.5 kHz), we calculate the resulting absorption spectra at different values and two orientations of the magnetic field as shown in Fig. 5 (cd).

One can see that most of salient features of the measured spectra are faithfully reproduced. Indeed, at B = 0, see also measured and calculated spectra presented separately in Fig. 6 (a), the 75 As absorption (higher frequency line) dominates the spectrum. At small fields parallel to the growth axis B z the 75 In the presence of the longitudinal field B z the splitting between central line and its satellites is the smallest for 71 Ga and the highest for 75 As. By contrast, under an inplane field B ⊥ this splitting vanishes for 75 As. This later fact results from the sign difference in the in-( E QI ) and out-of-plane (E QZ ) quadrupole parameters for Ga and As. We will show below that this implies a particular sign relation between different elements of the gradientelastic tensor elements S 11 , S 44 for Ga and As.

Two more observations could be drawn from the com-parison between the model and measured spectra. First, the generic Lorenzian broadening ∆f D = 1.5 kHz common for all transition fits the data satisfactorily. The fact that this linewidth is identical for all transitions and comparable with the one expected from dipole-dipole broadening (see also Fig. 1 (a)) suggests that NSS is efficiently thermalized at B ≈ B D . Second, the relative intensities of Ga and As absorption are not faithfully reproduced by the model: at low magnetic field it underestimates Ga absorption as compared to As, and at high field the situation is inverted. We have no solid physical explanation for this discrepancy, which was also reported in Ref. [START_REF] Berski | Interplay of Electron and Nuclear Spin Noise in n-Type GaAs[END_REF], but not in Ref. [START_REF] Litvyak | Warm-up spectroscopy of quadrupolesplit nuclear spins in n-gaas epitaxial layers[END_REF]. Nevertheless it could simply result from ill-defined direction of the OMF field.

V. RE-MAGNETIZATION OF THE NSS IN THE PRESENCE OF THE QUADRUPOLE EFFECTS

The above measurements show that quadrupole interaction is much stronger in this sample than dipoledipole interaction. In this section we compare quantitatively the quadrupole splittings and effective field B Q that they should induce with the actual heat capacity field B H characterizing NSS re-magnetization.

The spin temperature theory states that in a NSS isolated from the lattice an equilibrium state characterized by the temperature Θ N will be established within a characteristic time T 2 ≈ h/γ N B D [START_REF] Abragam | Spin temperature[END_REF]. On time scales larger than T 2 and shorter than T 1 , and provided that T 2 ≪ T 1 , where T 1 is spin-lattice relaxation time, the NSS can be considered as isolated from the lattice. If NSS is prepared at temperature Θ N i under magnetic field B i and subjected to a slowly varying magnetic field, such that dB/dt < B D /T 2 , then Θ N changes obeying universal expression [START_REF] Vladimirova | Spin temperature concept verified by optical magnetometry of nuclear spins[END_REF][START_REF] Abragam | Spin temperature[END_REF][START_REF] Kalevich | Onset of a nuclear polarization front due to optical spin orientation in a semiconductor[END_REF][START_REF] Dyakonov | Spin Physics in Semiconductors[END_REF]:

Θ N (B 2 + B 2 H ) 1/2 = Θ N i B i . ( 5
)
where Θ N is related to B N via Eq. 1.

It was assumed for many years that the heat storage in bulk semiconductor NSS is dominated by dipoledipole interaction. Our recent measurements of the Overhauser field in the presence of the slowly varying external magnetic field overturned this assumption and pointed out possible contribution of strain-induced quadrupole interaction, see Eq. ( 2) [START_REF] Vladimirova | Spin temperature concept verified by optical magnetometry of nuclear spins[END_REF]. The effective field due to quadrupole interaction can be calculated as [START_REF] Kalevich | Dynamic Nuclear Polarization and Nuclear Fields[END_REF]:

B Q = 4 5 i A i (E i 2 Q⊥ + E i 2 QZ ) i A i γ 2 N i . ( 6
)
This yields B N MR Q = 16 G using the quadrupole energies extracted from the NMR spectra. This value needs to be compared with B H from Eqs. (2, 5) that fits the remagnetization experiments.

Figure 7 shows two sets of the spin noise spectra measured in a re-magnetization experiment starting from B i = 0 for two different initial nuclear spin temperatures obtained by optical pumping during 4 minutes (a) and 30 s (b). The peak in the electron spin noise spectrum shifts from zero at B = 0 to ≈ 230 MHz (≈ 100 MHz) at B ⊥ = 100 G in the NSS cooled during 120 (30) seconds. Longer optical cooling yields lower NSS temperatures and higher Overhauser field reached in the former case, while the contribution from the external field is identical in these two experiments. It is shown by the yellow line. One can see that the rapid growth of the frequency at small fields is followed by much slower growth at higher fields. The latter is essentially due to the external field. In the framework of the spin temperature theory, the characteristic magnetic field where the slope changes is of the order of heat capacity field B H . Red lines calculated using Eqs. ((1), 2), ( 3) and ( 5) assuming B H = 14 ± 2 G and Θ N i as a fitting parameter are shown in Fig. 7 (10 21 V/m 2 ) 32 [START_REF] Sundfors | Experimental gradient-elastic tensors and chemical bonding in III-V semiconductors[END_REF], 9 [START_REF] Griffiths | Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance[END_REF] 32 [START_REF] Sundfors | Experimental gradient-elastic tensors and chemical bonding in III-V semiconductors[END_REF], 9 [START_REF] Griffiths | Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance[END_REF] 68 [START_REF] Sundfors | Experimental gradient-elastic tensors and chemical bonding in III-V semiconductors[END_REF], 48 [START_REF] Griffiths | Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance[END_REF] 2.1 [START_REF] Sundfors | Experimental gradient-elastic tensors and chemical bonding in III-V semiconductors[END_REF], 5.3 [START_REF] Griffiths | Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance[END_REF],1.9 [ = 16 G. Importantly, NMR and re-magnetization measurements were done at very close points on the sample surface. Indeed, it has been shown that the strain varies in the plane of GaAs substrates, epilayers, quantum wells and microstructures [START_REF] Litvyak | Warm-up spectroscopy of quadrupolesplit nuclear spins in n-gaas epitaxial layers[END_REF][START_REF] Yamada | Quantitative photoelastic measurement of residual strains in undoped semi-insulating gallium arsenide[END_REF][START_REF] Eickhoff | Mapping of strain and electric fields in GaAs/Al x Ga 1-x As quantum-well samples by laserassisted NMR[END_REF][START_REF] Brunetti | Linear dichroism in a GaAs microcavity[END_REF][START_REF] Abbarchi | Discretized disorder in planar semi-conductor microcavities: Mosaicity effect on resonant Rayleigh scattering and optical parametric oscillation[END_REF][START_REF] Lastras-Martínez | Microscopic optical anisotropy of exciton-polaritons in a GaAs-based semiconductor microcavity[END_REF]. A different set of NMR experiments performed at another point yielded a smaller value of B N MR Q = 12 G, correlated with smaller B Q = 9 ± 2 G extracted from the corresponding re-magnetization experiments.

Note that the NSS relaxation enhancement at small fields has also been attributed to the presence of quadrupole effects in the similar samples [START_REF] Vladimirova | Nuclear spin relaxation in n-GaAs: From insulating to metallic regime[END_REF][START_REF] Kotur | Nuclear spin warm up in bulk n-GaAs[END_REF]. This so-called quadrupole relaxation is induced by fluctuating donor charges, and the resulting enhancement of the relaxation rate as compared to the relaxation rate at strong field 1/T ∞ is given by the same heat capacity field B H , see Eq. ( 2):

1 T 1 = 1 T ∞ + 1 T Q ; 1 T Q ∝ 1/(B 2 + B 2 H ). ( 7
)
Fitting the experimental values of the NSS relaxation rate (Fig. 7, inset) to the Eq. ( 7) returns the value B r Q = 9 G which is slightly smaller, but still close to the the values returned by NMR and re-magnetization experiments. These results strongly support the ideas put forward in Refs. 29, 30, and 50. Namely, the electric field gradients in the NSS containing isotopes with the spin I > 1/2 lead to quadrupole effects which are detrimental to both nuclear spin memory and adiabatic demagnetization efficiency.

VI. ESTIMATION OF GRADIENT-ELASTIC TENSOR ELEMENTS

In this section we further analyze the quadrupole parameters determined from NMR experiments and extract relevant information on the stress intensity and orientation, as well as on the elements of the gradientelastic tensor S ijkl -an important material parameter relating electric field gradients on the atomic sites with the strain experienced by the crystal. In the crystals with cubic symmetry like GaAs, it can be reduced to two components, S 11 and S 44 . The interest in this tensor has emerged recently, both in view of non-destructive characterization of the strain, and in the context of exploration of coherent electron-nuclei spin dynamics in GaAs quantum dots.

The three isotope-dependent parameters that we extract from the NMR spectra are related to the three isotope-independent parameters which characterize the in-plane biaxial stress experienced by the crystal: values of the pressure applied in two orthogonal directions (p 1 , p 2 ) and the angle ς between the stress principal axis and [100] crystallographic axis. Their derivation is given in Section X B [START_REF]The same formula wher used[END_REF]: -1)

E i QZ = - 3eQ i S i 11 4I (2I
p 1 + p 2 C 11 -C 12 , ( 8
)
E i Q⊥ cos2ζ i = 3 √ 3eQ i S i 11 4I (2I -1) p 1 -p 2 C 11 -C 12 cos (2ς) , (9)
E i Q⊥ sin2ζ i = √ 3eQ i S i 44 2I (2I -1) p 1 -p 2 C 44 sin (2ς) . ( 10
)
Here e is the absolute value of the electron charge, C . Since both Ga isotopes experience the same electrostatic environment, these ratios are expected to be identical for 71 Ga. This appears to be the case within our experimental precision. In the following we denote both isotopes as Ga, to simplify the notations.

The obtained values of the gradient elastic tensor elements are summarized in Table I the discrepancy between the existing values is very substantial. Our value 2.5 ± 0.4 is rather close to that of Refs. 18 and 19, and much smaller that the result of Ref. [START_REF] Griffiths | Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance[END_REF]. At this stage we don't have any explanation for these discrepancies, but such a strong dispersion of S 44 measured in different experiments suggests that further studies are mandatory to resolve this issue.

VII. EVALUATION OF THE BUILT-IN STRESS AND DISORDER

In order to estimate the stress parameters p 1 , p 2 and ς Eqs. ( 8)- [START_REF] Gangloff | Witnessing quantum correlations in a nuclear ensemble via an electron spin qubit[END_REF] are not sufficient and we need to use the values of S 11 measured elsewhere. We rely on recent values of Chekhovich et al [START_REF] Chekhovich | Cross calibration of deformation potentials and gradient-elastic tensors of GaAs using photoluminescence and nuclear magnetic resonance spectroscopy in GaAs/AlGaAs quantum dot structures[END_REF], [52]. For the point where the complete set of NMR spectra has been measured (see Fig. 5) this yields p 1 + p 2 = 29.5 MPa, p 1p 2 = 14 MPa, ς = 55 o degrees, bearing in mind that relative sign of p 1 + p 2 , p 1p 2 and ς is not uniquely defined. Measurements performed at other points of the sample revealed variation of the quadrupole energies up to 30%. This is consistent with the disorder effects reported in GaAs microcavities [START_REF] Brunetti | Linear dichroism in a GaAs microcavity[END_REF][START_REF] Abbarchi | Discretized disorder in planar semi-conductor microcavities: Mosaicity effect on resonant Rayleigh scattering and optical parametric oscillation[END_REF][START_REF] Lastras-Martínez | Microscopic optical anisotropy of exciton-polaritons in a GaAs-based semiconductor microcavity[END_REF], as well as in bulk GaAs samples [START_REF] Litvyak | Warm-up spectroscopy of quadrupolesplit nuclear spins in n-gaas epitaxial layers[END_REF][START_REF] Yamada | Quantitative photoelastic measurement of residual strains in undoped semi-insulating gallium arsenide[END_REF].

More specifically, in our microcavity sample due to the difference of lattice constants between GaAs and AlAs plastic relaxation is likely to occur in the Bragg mirrors having ≈ 4 µm thickness. This relaxation is eventually accompanied by the formation of dislocations. Therefore, the lattice constant of the bottom Bragg mirror would be different from that of GaAs cavity that we probe optically. It is not possible to give a quantitative estimation, but this difference should result in a strain in the cavity layer.

VIII. CONCLUSIONS

In this work we develop and implement a new technique that allows us to probe NMR by electron spin noise spectroscopy. These experiments aim to quantify quadrupole effects in the NSS of n-GaAs. The NSS absorption spectra in a n-GaAs epilayer embedded in a microcavity are measured under magnetic field in the range from zero to 100 G, either parallel or perpendicular to the growth axis. The ensemble of the spectra determines unambiguously the parameters of the quadrupole Hamiltonian, and thus relevant splittings between nuclear spin states for each of three GaAs isotopes. These measurements are particularly relevant at zero and weak magnetic fields, because the spin states splittings are dominated by quadrupole interaction rather than by Zeeman effect, and only few experimental methods give access to zero-and low-field NMR [START_REF] Litvyak | Warm-up spectroscopy of quadrupolesplit nuclear spins in n-gaas epitaxial layers[END_REF][START_REF] Blanchard | Lower than low: Perspectives on zero-to ultralow-field nuclear magnetic resonance[END_REF].

Our results establish a connection between the residual strain giving rise to quadrupole splittings, the increased heat capacity and nuclear spin relaxation rates at low and zero field. Indeed, from the quadrupole parameters measured by NMR we estimate that heat capacity field limiting NSS cooling by adiabatic demagnetization and increasing NSS spin relaxation rate is of order of B N MR Q = 16 ± 2 G. This value is close to B Q = 14 ± 2 G measured at very close point on the sample surface in a set of separate adiabatic demagnetization experiments using SNS as a detection tool. More precise measurements are complicated due to variation up to 30% of the quadrupole energies and thus nuclear warm-up rates induced by inhomogeneity of the local strain on the scale of several millimeters across the sample surface. Nevertheless our results strongly support the model of quadrupole-limited NSS cooling [START_REF] Vladimirova | Spin temperature concept verified by optical magnetometry of nuclear spins[END_REF] and quadrupole-driven NSS warm-up [START_REF] Vladimirova | Nuclear spin relaxation in n-GaAs: From insulating to metallic regime[END_REF][START_REF] Kotur | Nuclear spin warm up in bulk n-GaAs[END_REF].

From the quadrupole hamiltonian we also obtain an estimation of the gradient-elastic tensor that relates gradients of the electrostatic potential on each nuclear site with the strain tensor. Since we don't have an independent measurement of the stress present in the sample, we are limited to the determination of only relative values of the two relevant gradient-elastic tensor elements for different isotopes, S An OMF B 1 cos(2πf t) along z-axis creates an energy flux towards NSS, Q(f , t), corresponding to the energy change per one spin given by

Q(f , t) = ∂E N SS ∂t = dB 1 dt i A i hγ N i I zi , (11)
where index i runs over all isotopes and I zi is an average projection of the i-th isotope spin on the OMF direction

I zi = B 1 hγ N i χ ′ f cos(2πf t) -χ ′′ f sin(2πf t) . (12)
Here χ ′ f = χ ′ -f and χ ′′ f = -χ ′′ -f are real and imaginary parts of the NSS susceptibility, respectively. Therefore, the averaged over a period energy flux towards NSS Q(f ) is related to its absorption

A(f ) = f i A i χ ′′ f i as Q(f ) = πB 2 1 A(f ). (13)
On the other hand the warm-up rate of the NSS in the presence of the OMF reads [START_REF]Optical Orientation[END_REF]:

1 T w (f ) = k B Θ N C N Q(f ) = πB 2 1 k B Θ N C N A(f ). ( 14
)
Here C N is the heat capacity of the NSS

C N = I(I + 1) 3 B 2 + B 2 H h 2 i A i hγ 2 N i , (15)
B is an arbitrary external static field and B H is the heat capacity field introduced in Eq. ( 2). It can be rigorously determined as

B 2 H = 3 I(I + 1) Sp Ĥ2 SS i A i hγ 2 N i , ( 16
)
where ĤSS is the total Hamiltonian of all nuclear interactions excluding Zeeman [START_REF] Kalevich | Dynamic Nuclear Polarization and Nuclear Fields[END_REF]. Thus, NSS absorption at a given spin temperature is proportional to the rate of NSS warm-up induced by an OMF. T w (f ) can be extracted from the experiments described in Section III in a following way. Following Ref. [START_REF]Optical Orientation[END_REF], Ch. 5 we can write a detailed balance equation for the inverse nuclear spin temperature β = 1/k B Θ N in the presence of the OMF:

β = - 1 T t (f ) (β -β t ) , (17)
where

1 T t (f ) = 1 T w (f ) + 1 T 1 ( 18
)
is the total relaxation rate including OMF-independent relaxation at rate T 1 , β t = β 0 (1 + T w (f )/T 1 ) -1 is the steady-state spin temperature and β 0 is the steady-state spin temperature for B 1 = 0. Nuclear spin temperature is related to the Overhauser field-induced spin noise frequency component F N via Eqs. ( 1) and (3). Therefore, from Eqs. ( 17) we obtain

1 T t (f ) = 1 ∆t 6 log F N (t 7 ) F N (t 5 ) , ( 19
)
where ∆t 6 is the duration of the 6-th stage of the NMR experiment, and F N (t 5 ) and F N (t 7 ) are frequencies measured at 5-th and 7-th stages of the NMR experiment, respectively. Similarly, if B 1 = 0 Eq. ( 17) becomes

β = - 1 T 1 (β -β 0 ) , ( 20
)
and

1 T 1 = 1 ∆t 6 log F 0 N (t 7 ) F 0 N (t 5 ) , ( 21
)
where F 0 N (t 5 ) and F 0 N (t 7 ) are frequencies measured at 5th and 7-th stages of the identical experiment but without OMF (B 1 = 0). 1/T w (f ) is recalculated from Eqs. ( 18), ( 19) and ( 21) for each OMF frequency. Since in high-temperature approximation A(f ) ∝ β, 1/T w (f ) does not depend on temperature and thus pertinently characterizes NSS absorption. Therefore, 1/πB 2 1 T w (f ) constitutes an NMR spectrum, or, equivalently, NSS absorption spectrum. Such spectra, normalized to unity and encoded in colors, are shown in Fig. 5 (a,b) for various values of static magnetic field.

B. Relation between the in-plane stress and energetic parameters of the quadrupole Hamiltonian.

The second rank tensors describing stress σ ij and strain ε kl are related by the forth rank elasticity tensor C ijkl :

σ ij = k,l C ijkl ε kl . ( 22
)
In cubic crystals there are only three nonzero components of the elasticity tensor: By expressing components of the stress tensor in Eq. ( 24) via strain components with Eqs. ( 22) and ( 23), one obtains a system of linear equations for strain tensor components that yields the following solutions (see also problem 3 to 10th section of Chapter 1 in [54]):

ε
In the general case, the stress can be applied along two orthogonal axes in the plane. If the angle ς defines the direction of one of the two principal axes of the stress tensor in the plane, principal values of this tensor being p 1 and p 2 , then due to linearity of Eqs. [START_REF] Kalevich | Dynamic Nuclear Polarization and Nuclear Fields[END_REF]

ε zx = ε xz = ε zy = ε yz = 0 ε xy = ε yx = (p 1 -p 2 ) • cos ς sin ς C 44 . ( 26
)
Thus, with the knowledge of the strain components one can find the energetic parameters of the quadrupole Hamiltonian [START_REF] Abragam | The principles of nuclear magnetism[END_REF]:

E QZ =

Figure 1 .

 1 Figure 1. (a) Schematic representation of quadrupole-split nuclear spin levels. Arrows indicate NMR transitions, (b) Sketch of the donor-bound electron spin interacting with the underlying nuclear spins. (c) Overhauser field B N and the corresponding nuclear spin temperature Θ N as functions of the magnetic field according to spin temperature theory. B H is a field that characterizes NSS heat capacity. (d) Fluctuations of the electron spin detected via Kerr rotation in time domain within 0.5 GHz frequency band and the corresponding Fourier power spectrum.

Figure 2 .

 2 Figure 2. Time-sequence used for NSS preparation (a) and re-magnetization (b) and NMR (c) experiments.

Figure 3 .

 3 Figure 3. SN spectra measured at different values of the magnetic field during adiabatic re-magnetization. Arrows point the corresponding Larmor frequencies F L .

Figure 4 .

 4 Figure 4. (a) 40 pairs of color-encoded SN spectra required for NSS absorption measurement at B = 0. Each pair is recorded within steps 5 and 7 of the NMR experiment that includes both preparation and detection. The measurements are realised in a single run one after another for different OMF frequencies f . Bottom scale indicates the laboratory time, and the top scale the corresponding OMF frequency. Such set of measurements allows us to reconstruct the NMR spectrum at B = 0. Intensities are given in the units of spin to shot noise ratio. (b) Pairs N = 1 (f = 1 Hz) and N = 18 (f = 18 kHz) of the spin noise spectra shown in (a). At f = 1 Hz NSS absorbs energy very weakly, so that the corresponding stage 7 spectrum is identical to the one measured in the absence of the OMF (dashed magenta line) and its peak position is essentially governed by NSS relaxation. The peak of the spectrum corresponding to N = 18 is shifted to much lower frequency, this shift is governed by NSS absorption.

Figure 5 .

 5 Figure 5. Color-encoded NSS absorption spectra measured (a, b) and calculated (c, d) at different values of magnetic field parallel to either [ 110] (b) or [001] (a) crystallographic axis. All the spectra are normalized to unity.

Figure 6 .

 6 Figure 6. NSS absorption spectra measured (symbols) and calculated (lines) at B = 0 (a), at B z = 100 G, that is parallel to [001] crystal axis (b) and at B ⊥ = 100 G, parallel to [ 110]. All the spectra are normalized to unity.

  As absorption line splits into two lines, that shift linearly with B z . By contrast, under in-plane field B ⊥ the 75 As absorption line shifts quadratically. At strong field B = 100 G we identify for each isotope a triplet of lines, including a central line and a pair of satellites. Measured and calculated spectra at B = 100 G oriented either along [ 110] or [001] crystallographic axis are shown in Fig. 6 (b) and (c), respectively. The strongest absorption is observed for 71 Ga.

Figure 7 .

 7 Figure 7. Color-encoded SN spectra (in signal to shot noise ratio units) during the adiabatic demagnetization procedure preceded by preparation of nuclei including either 4 minutes (a) or 30 s optical pumping (b). Red lines are fit to spin temperature model assuming B H = 14 ± 2 G and taking into account spin relaxation. Yellow line shows the external field contribution to the spin noise frequency, F N . Inset shows the inverse nuclear spin relaxation time as a function of the magnetic field (symbols) measured under the pumping conditions of (b). Solid line is a lorentzian fit to the data used to calculate red lines.

C 11 =

 11 C xxxx = C yyyy = C zzzz ; C 12 = C xxyy = C yyxx = C xxzz = C zzxx = C yyzz = C zzyy ; C 44 = C xyxy = C yzyz = C xzxz . (23)In a crystalline plate with the normal (z-axis) parallel to the [001] axis, application of the pressure p in the structure plane at the angle ς to the [100] axis results in appearance of the following components of the stress tensor:σ xx =pcos 2 ς σ yy = psin 2 ς σ xy = σ yx = p cos ς sin ς σ zz = σ zx = σ xz = σ zy = σ yz = 0 (24)

  xx = p • (C 11 + 2C 12 ) cos 2 ς -C 12 (C 11 -C 12 ) (C 11 + 2C 12 ) ε yy = p • (C 11 + 2C 12 ) sin 2 ς -C 12 (C 11 -C 12 ) (C 11 + 2C 12 )ε zz = -p • C 12 (C 11 -C 12 ) (C 11 + 2C 12 ) ε zx = ε xz = ε zy = ε yz = 0 ε xy = ε yx = p • cos ς sin ς C 44 .

  on top of the color-encoded spin-

		69 Ga	71 Ga	75 As	75 As/Ga
	Q (10 -30 m 2 ) [43]	10.7	17.1	31.4
	Abundance	0.3	0.2	0.5
	γ N (kHz/G) [43] C 11 (10 -10 N/m 2 ) [44] C 12 (10 -10 N/m 2 ) [44] C 44 (10 -10 N/m 2 ) [44]	0.82 12 5.4 6.2	0.64 12 5.4 6.2	0.45 12 5.4 6.2
	E QZ /h (kHz)	-10±0.5	-6±0.5	20±0.5
	E ⊥ /h (kHz)	-3.5±0.5	-2.2±0.5	12.5±0.5
	ζ (degrees with [100] axis)	-68±5	-68±5	-33±5
	S 11 (10 21 V/m 2 ) S 11 ( 75 As) / S 11 ( Ga) [this work]	-31.7[19], -22 [16] -31.7[19], -22 [16] 34[19], 24.2 [16] -1.07 [19], -1.1 [16, 18] -1.1±0.1
	S 44

Table I .

 I Quadrupole parameters obtained in this work and those taken from the literature, as well as other GaAs parameters used in the calculations: stiffness tensor elements, isotope abundancies, quadrupole moments and gyromagnetic ratios. noise spectra. The magnetic field dependent NSS relaxation on the time scale of the measurement (50 s) is taken into account in this calculation. The corresponding values of 1/T 1 measured in a separate set of experiments are shown in the inset. One can see that within the experimental accuracy, which is mainly given by the unavoidable warm-up of the NSS during measurements, B

		18]
	S 44 ( 75 As) / S 44 ( Ga) [this work]	2.5±0.4

H ≈ B Q = 14 ± 2 G matches quite well the value B N MR Q

  11 , C 12 and C 44 are known stiffness tensor components, E i Qz , E i Q⊥ and ζ i are the parameters of the Hamiltonian (4) determined from the fit to the NMR spectra. Their values are given in Table I. Eqs. (8)-[START_REF] Gangloff | Witnessing quantum correlations in a nuclear ensemble via an electron spin qubit[END_REF] are not sufficient to fully characterize five independent parameters (p 1 , p 2 , ς, S i 11 , S i 44 ) for each isotope. However, they allow us to ascertain the ratio between the values of gradient elastic tensor components S

	69 Ga 11 /S 11 75 As	and S 44 /S 69 Ga 44 75 As

  , and compared with previous results. The component S 11 of the gradientelastic tensor contributes to both in and out-of-plane quadrupole energies, see Eqs 8-9. In particular, it is proportional to E QZ , that dominates over in-plane quadrupole splitting E Q⊥ in our sample. One can see

	that S Ga 11 /S 11 75 As clusions of Refs. 16, 18, and 19, corroborating their re-= -1.1 ± 0.1 matches quite well the con-
	sults. However for S Ga 44 /S 44 75 As

  ± 0.4, rather close to that of Refs. 18 and 19, and much smaller than the result of Ref.[START_REF] Griffiths | Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance[END_REF]. Understanding of discrepancies in the S 44 values involved in the presence of an in-plane shear strain calls for further studies.

	11 and S 44 . It appears = -1.1 ± 0.1 matches quite well the con-clusions of Refs. 16, 18, and 19, while for S Ga that S Ga 11 /S 75 As 11 44 /S 75 As 44 , where a big discrepancy exist between the values in the literature, we get S Ga 44 /S 75 As 44 = 2.5 IX. ACKNOWLEDGEMENTS The authors are grateful to R. Cherbunin and V. K. Kalevich for enlightening discussions, and ac-knowledge financial support from Russian Founda-tion for Basic Research grant N 22-42-09020, French national research agency grant ANR-21-CE30-0049, French Embassy in Moscow (Ostrogradski fellowship for young researchers 2020), and French RENATECH network. X. APPENDIX A. Determination of the NSS absorption from the electron spin noise peak frequencies.

  we get: ε xx = (C 11 + 2C 12 ) p 1 cos 2 ς + p 2 sin 2 ς -C 12 (p 1 + p 2 ) (C 11 -C 12 ) (C 11 + 2C 12 ) ε yy = (C 11 + 2C 12 ) p 1 sin 2 ς + p 2 cos 2 ς -C 12 (p 1 + p 2 ) (C 11 -C 12 ) (C 11 + 2C 12 ) ε zz = -C 12 (p 1 + p 2 ) (C 11 -C 12 ) (C 11 + 2C 12 )


Herve Rivano
email: herve.rivano@sophia.inria.fr

Fabrice Theoleyre
email: fabrice.theoleyre@insa-lyon.fr

Fabrice Valois
email: fabrice.valois@insa-lyon.fr

Capacity Evaluation Framework and Validation of Self-Organized Routing Schemes

Assuming a given network topology and a routing protocol, this work is focused on the capacity evaluation of routing protocols based on either a self-organization scheme or a flat approach. To reach this goal, we propose to use linearprogramming formulation to model radio resource sharing as linear constraints. Four models are detailed to evaluate the capacity of any routing scheme in wireless multihops networks. First, two models of fairness are proposed: either each node has a fair access to the channel, or the fairness is among the radio links. Besides, a pessimistic and an optimistic scenarios of spatial re-utilization of the medium are proposed, yielding a lower bound and an upper bound on the network capacity for each fairness case. Finally, using this model, we provide a comparative analysis of some flat and self-organized routing protocols.

I. INTRODUCTION

Wireless multihops networks are promised to a large spectrum of usages (sensor, ad-hoc or mesh networks) [START_REF] Mario | From battlefields to urban grids: New research challenges in ad hoc wireless networks[END_REF]. A fundamental challenge is to provide a tight evaluation of the capacity of a network to conceive adequate applications. Indeed, the radio interferences yield bandwidth sharing mechanisms that are less local and more sophisticated than in wired or cellular networks. Moreover, multihops routes induce strong bandwidth reduction: each forwarded packet creates interferences, burdening other transmissions. [START_REF] Gupta | The capacity of wireless networks[END_REF] was the first work investigating this problem, proposing an asymptotic study. Only few propositions study the trade-off between capacity and network topology.

Routing protocols have been widely studied in wireless multihops networks leading to mainly flat approaches (OLSR, DSR. . . ) Recently, new propositions suggest to apply routing protocols on a self-organization structure [START_REF] Theoleyre | Virtual structure routing in ad hoc networks[END_REF]. A selforganization creates a hierarchy in the network, based for example on clusters or backbones. However, this approach could reduce the capacity, i.e. the maximum achievable throughput: some radio links are never used and some nodes, acting as leaders, route most of the control traffic. Thus, a fine capacity evaluation is required to quantify this possible drawback. The contribution of this paper is to propose a flexible framework to evaluate the capacity, the network topology and the routes being given.

Our first contribution is to develop four linear programming models of radio resource sharing in 802.11-like networks. Two medium access fairness hypothesis are considered (nodes fairness and links fairness) and we present a pessimistic and an optimistic resource sharing in each case. The models remain valid for any wireless multihops network (e.g. sensor or mesh networks). The second contribution is to compare the capacity of flat and self-organized approaches. This comparative study evaluates the maximum and the max-min capacities achievable by OLSR flat routing scheme [START_REF] Clausen | Optimized link state routing protocol (OLSR)[END_REF], VSR [START_REF] Theoleyre | Virtual structure routing in ad hoc networks[END_REF], and Wu & Li [START_REF] Wu | A dominating-set-based routing scheme in ad hoc wireless networks[END_REF] self-organized approaches.

This paper is organized as follows. Section II summarizes related works about capacity of ad hoc networks. Sections III and IV are dedicated to our linear models of radio resource sharing based on radio interferences and nodes interactions. These models provide lower and upper bounds of the capacity of the network under two hypothesis of medium access fairness. Section V presents the results. We then conclude the article and expose some perspectives.

II. RELATED WORK

[2] presents a pioneering work in this area. The authors study the capacity of ad hoc networks, i.e. the maximal achievable throughput. Let n be the number of nodes. The authors demonstrate that in optimal conditions, the global capacity is O 1 √ n . Even in an optimal case, the capacity per node decreases when the number of nodes increases. The authors presented an asymptotic study on the capacity of ad hoc networks independent of any routing protocol, or distributed scheduling technology. This constitutes its strength and its weakness: the capacity evaluation does not model neither a MAC layer behavior, nor the correlation between routing and the capacity. Indeed, the MAC layer regulates the bandwidth sharing. In the same way, the capacity is inherently dependent from the routing protocols: long routes can increase interferences but can also spread the load in the network. In the opposite way, shortest routes will not optimize the capacity [START_REF] De Couto | Performance of multihop wireless networks: shortest path is not enough[END_REF]. Several articles [START_REF] Kozat | Throughput capacity of random ad hoc networks with infrastructure support[END_REF], [START_REF] Liu | On the capacity of hybrid wireless networks[END_REF], [START_REF] Zemlianov | Capacity of ad hoc wireless networks with infrastructure support[END_REF] propose to extend the work [START_REF] Gupta | The capacity of wireless networks[END_REF] to hybrid networks, using the same interference model. Hybrid networks allow to deploy Access Points (AP) in the ad hoc network, and a part of the traffic is sent to Internet through the AP. However, the capacity reaches an asymptotic In [START_REF] Jain | Impact of interference on multi-hop wireless network performance[END_REF], the capacity of ad hoc networks is modeled using linear programs (LP). Their work focuses on the capacity estimation for a given network topology and workload. The capacity is modeled as a multicommodity flow problem. However, the complexity of such a formulation resides in the capacity estimation of each link. The authors estimate the maximum capacity in searching sets of links which can convey traffic simultaneously (i.e. they are not interfering). However, their work models an unfair MAC layer.

[11] proposes a scheduling scheme maximizing the throughput from a group of sources to a group of destinations. The authors propose a scheduling of radio links such that no two simultaneously activated links are interfering. A greedy algorithm allocates slots to the different links according to the decreasing euclidean length of the links, this order constituting a sufficient condition of feasibility. However, such an approach could under-estimate the capacity and does not introduce any fairness model.

III. HYPOTHESIS AND NOTATION

In this section we introduce the generic form of our linear models for radio resource sharing in wireless IEEE 802.11like networks. The capacity is evaluated for a given network topology and routing protocol. We will discuss later of the fairness assumption.

More precisely, a discrete-event simulation process provides the input data of our models: the network topology, the routes defined by the routing protocol, and the rates of control traffic which is locally broadcasted by each node to its neighborhood. We consider the end to end data transmissions as network flows that induce load on the links they cross, depending directly on the routes created by the routing protocol. The union of the link constraints yields to global network constraints. Besides, the resource sharing is translated in local set of constraints for each node, depending directly on the radio topology. An optimistic and pessimistic models are introduced. Finally, an objective function maximizing the throughput, i.e. estimating the capacity, is used.

In order to develop a linear model of the radio resource sharing, we do some standard assumptions. The radio channel delivers a constant bandwidth and does not corrupt data transmissions. The MAC layer avoids all collisions and the bandwidth required for collision avoidance mechanisms is neglected. A communication is bidirectional: each radio transmission is acknowledged. When control traffic is broadcasted by a node to its neighborhood, all neighbors receive it, which forces 2-neighbors of the source to be silent.

To model interferences, we use the transmitter-receiver model [START_REF] Kumar | Algorithmic aspects of capacity in wireless networks[END_REF]: two links (A, B) and (C, D) can be activated simultaneously if no edge (X, Y ) X={A,B},Y ={C,D} exists. The conflict graph could be deducted from the interference model: a node of the conflict graph corresponds to each link in the radio graph. Then, two nodes of the conflict graph have a common edge iif the corresponding radio links are interfering. We will see further how to adapt the following proposition to cope with other interference models.

The following notation is used:

• BW is the available radio bandwidth. This gives the maximum amount of data that can be sent by one terminal. • f (p) is the throughput of the data sent on the path p.

• Let u be a node. T (u) is the total amount of traffic sent by u:

T (u) = v∈Γ(u) T (u, v) + T c (u) with -Γ k (u) is the k-neighborhood of u, i.e. the set of the nodes at most k hops far from u. Γ 1 (u) is written Γ(u) for short. Note that ∀k, u ∈ Γ k (u) -∆ k (u) is the size of the k-neighborhood of u: ∆ k (u) = |Γ k (u)|. -T (u, v) is the unicast traffic on the radio link (u, v) (i.e. p (u,v) f p ). -T c (u)
is the broadcast traffic for control and topology maintenance sent by u to its neighborhood.

IV. MODELS OF RADIO RESOURCE SHARING

The MAC layer of 802.11-like networks provides a fair access to the channel: when two nodes competes to access the channel, they have the same probability to communicate first. This fairness is quite more complicated to express when the whole network is considered.

In the following, two models of the channel fairness are developed. The first model is node-oriented: nodes access the channel following a uniform distribution. The second model is link-oriented: the point-to-point communication between neighbors have the same probability to access the medium. For each model of fairness, we propose a pessimistic radio resource sharing computing a lower bound on the capacity, and an optimistic radio resource sharing LP yielding an upper bound.

A. Node-oriented fairness 1) A pessimistic radio resource sharing scenario: A lower bounding linear program is obtained when local radio resource sharing constraints model a pessimistic behavior of the MAC layer. The worst case for contention is when a node broadcasts control traffic. Indeed these packets have to be received by all its neighbors, which requires that all 2-neighbors of the source keep silent. This silence includes refusing any incoming communication request, since it requires to send an acknowledgment for the received packet.

Thus, a pessimistic resource sharing does not distinguish broadcast and unicast communications. This guarantees that data communications are successful, but overestimates interferences.

The node-oriented fairness is modeled by an equal share of the bandwidth among 2-neighboring nodes. Therefore, the available bandwidth to a node is constrained by the cardinality of each 2-neighborhood set it belongs to.

This pessimistic MAC local behavior can be described by linear constraints as follows. Suppose we consider a node c, called the center of its 2-neighborhood.

• The radio bandwidth is uniformly distributed between all nodes in potential contention: the center c and its whole 2-neighborhood, Γ 2 (c):

∀c, ∀u ∈ Γ 2 (c), T (u) ≤ BW ∆ 2 (c) (1)
Note, that for each center c, a set of ∆ 2 (c) 2 equations is given. In consequence, the capacity T (u) allocated to a node u is constrained on average by ∆ 2 (u) equations (one centered in each 2-neighbor). • In the capacity allocated to one node, all the control traffic and the unicast transmissions must be scheduled. Additionally, a node allocates an equal capacity to each of the communication links with its neighbors:

∀u, ∀v ∈ Γ(u) -{u}, T (u, v) ≤ T (u) -T c (u) ∆(u) -1 (2)
The equation ( 2) models that each node manages its available bandwidth, while (1) models that the radio medium capacity is shared among the nodes. Finally, two nodes can send data simultaneously only if they are sufficiently distant, at least 3 hops, and a low loaded node is given the same bandwidth than a fully loaded one. This set of local constraints yields LP 1, whose solutions lower bound the total amount of traffic supported by the network.

Linear Program 1 (Pessimistic model): Maximize Objective function on P Subject to Equation set [START_REF] Mario | From battlefields to urban grids: New research challenges in ad hoc wireless networks[END_REF] node c, the center Equation set [START_REF] Gupta | The capacity of wireless networks[END_REF] ∀ node u Traffic management for p ∀ path p

2) An optimistic radio resource sharing scenario: The pessimistic radio resource sharing model leads to over-estimate the interferences. Some simultaneous communications could be possible in a more realistic protocol, but are forbidden by our previous model. Hopefully, many protocols, like IEEE 802.11, achieve a better repartition. For example, in fig. 1, the transmissions (A→E) and (B→G) can be active simultaneously, because the packets are not to be understood by C.

Indeed, a set of communications that can occur simultaneously is an independent set of the conflict graph. An optimistic resource sharing model would consider that a maximal independent set of the conflict graph is active. That would upperbound the capacity of the network since the existence of a distributed process realizing such an optimal call scheduling is assumed. Furthermore, describing all the maximal independent sets of a graph is an NP-hard problem. Our optimistic model avoids this problem by considering the constraints on simultaneous communications at a local view point. In other words, 2 One equation for each 2-neighborhood sets it belongs to the model assumes that the local maximal independent sets can be combined into a global one. This obviously weakens the constraints, yielding an upper bound on the global capacity of the network.

Suppose that the channel is free. When one neighbor of the center, u, sends a packet to v, all neighbors of v has to keep silent. When u finishes the transmission, v sends a MAC acknowledgment, blocking all neighbors of u. Suppose that another node u wants to send a packet. If u is neither neighbor of u nor of v, u can initiate a communication. Nevertheless, for the communication to succeed, the destination has to be outside the neighborhood of u or v. This behavior is strongly linked to the combinatorial concept of independent set.

Indeed, such a contention-free communication set is an independent set, maximal for inclusion, of the graph L 1,2 (L (G c )), defined as follows:

• G c is the graph of the 2-neighborhood of c.

• LG = L(G c ) is the linegraph of G c
, that is the graph with one vertex per arc of G c , and a link between any two vertices whose corresponding arcs are adjacent.

• L 1,2 (
LG) is the graph with the same vertices as LG, and a link between any two 1-neighboring or 2-neighboring vertices. L 1,2 (LG) is the conflict graph with the transmitterreceiver interference model. Independent vertices (i.e. pairwise non adjacent vertices) of L 1,2 (L (G c )) correspond to contention-free communications. An inclusion-wise maximal independent set is therefore an inclusion-wise maximal set of communications that can be activated simultaneously.

Eventually, the MAC layer achieves a fair sharing of the bandwidth among the maximal independent sets. Fairness of bandwidth sharing among the nodes must be respected. Let BW (I) be the bandwidth given to the independent set I ∈ I, the set of all maximal independent sets of L 1,2 (L (G c )). BW (I) is proportional to P (I), the probability of I to be selected. P (I) depends on the links it is composed of and from their order of activation. The bandwidth in the neighborhood of c is shared as follows:

BW (I) = P (I) •   BW -T (c) - u∈Γ(c)-{c} T c (u)   (3) ⇒ BW ≥ T (c) + I∈I BW (I) + u∈Γ(c)-{c} T c (u) (4)
The total bandwidth allocated to a communication link (u, v) is the sum of the bandwidth allocated to each independent set including (u, v):

T (u, v) ≤ I (u,v) BW (I) (5)
T (u, v) ≤   BW -T (c) - x∈Γ(c)-{c} T c (x)   • I (u,v) P (I) (6)
Moreover, I (u,v) P (I) is exactly the probability for the communication link (u, v) to be activated by the channel. This quantity is hence denoted P (u, v) in the following.

∀(u, v) ∈ Γ 2 (c) -{c}, T (u, v) ≤   BW -T (c) - x∈Γ(c)-{c} T c (x)   • P (u, v) (7)
Unfortunately, on arbitrary network topologies, P (I) and P (u, v) can not be computed unless the whole set I is known, and I has an exponential size. We therefore build a stochastic estimation of P (u, v), denoted f req(u, v) in the following.

These frequencies f req(u, v) must absolutely take into account the fairness among the nodes. Hence, we propose the following algorithm to construct an independent set:

• While at least one not blocked node exists, take randomly one, say u • choose randomly one neighbor v of u which is not blocked -If v exists, activate the communication (u, v) and mark all the neighbors of u and v as blocked else, mark u as blocked If this algorithm is repeated n times, f req(u, v) is equal to the proportion of the cases where the link (u, v) was selected. Note that each link is directed: the link (u, v) will not receive the same amount of traffic as (v, u).

In order to complete the model, control and topology maintenance traffic generated by the routing algorithms is to be completely taken into account. The equation set [START_REF] Kozat | Throughput capacity of random ad hoc networks with infrastructure support[END_REF] models that when a neighbor of the center c emits control traffic, c has to stop any radio activity. On the other hand, the 2-neighbors of c will include their control traffic into their allocated bandwidth.

The bandwidth is distributed per link: a node is not allowed to distribute locally the traffic to each of its link. Indeed, the capacity allocation takes into account the specificities of each link, and particularly the fact that several links can send information simultaneously. If the node chooses itself to redistribute the capacity of an unloaded link to another of its links, the interference constraints could be violated. Such a behavior must be avoided.

The last optimistic aspect of this model is that the combination of the local constraints might not yield a feasible share of the global capacity. As a matter of fact, the union of the local independent sets might not be a global independent set. In other words, the global constraints are stronger that the union of the local ones. The linear program 2 neglects this fact, yielding an upper bound on the global capacity of the network.

Linear Program 2 (Optimistic model):

Maximize

Objective function on P Subject to Equation set [START_REF] Kozat | Throughput capacity of random ad hoc networks with infrastructure support[END_REF] ∀ link (u, v) ∈ E Traffic management for p ∀ path p

B. Link-oriented fairness

The following section is dedicated to presenting an evolution of the previous linear programs that models a fair allocation of bandwidth among radio links. Consequently, the nodes which have more active links will receive more bandwidth and bottlenecks could be potentially avoided.

In the following, we use the following notations.

• LG: the linegraph of G

• γ k : the k-neighborhood in LG of a link e in G. Each link is directed. • δ k :|γ k | 1)
A pessimistic resource sharing scenario: This model operates a pessimistic allocation of the bandwidth among the links. More precisely, when focusing on a link e, the bandwidth is distributed among the set of links which are neighbors of e in the conflict graph. In our case, the transmitter-receiver interference model induces that a link conflicts with each link in γ 2 (e). The capacity is then uniformly distributed to the links as follows.

∀e ∈ E, ∀f ∈ γ 2 (e), T (f ) ≤ BW -(u,x)∈γ2(e) T c (u) δ 2 (e) (8)
Finally, we obtain the following linear program LP 3. Linear Program 3 (Pessimistic model):

Maximize

Objective function on P Subject to Equation set [START_REF] Liu | On the capacity of hybrid wireless networks[END_REF] link e Traffic management for p ∀ path p Note the lower bounds with link-oriented and node-oriented fairness are not comparable. A different behavior of the MAC layer is modeled. Thus the capacity of the network depends on the protocol chosen to allow concurrent access to the medium.

2) An optimistic resource sharing scenario: The upper bound with link-oriented fairness knows less modifications. The only change relies in the probability for a link to be activated, and therefore the capacity that is allocated to each link. Indeed, the algorithm computing the frequencies, f req(u, v), capacity allocates to each link in the 2-neighborhood of one node must be changed into the following one:

• While at least one not blocked link exists, take randomly one, say (u, v) • Mark as blocked any link (x, y) such that (a, b) and/or (b, a) a∈{x,y},b∈{u,v} exists (i.e. (x, y) and (u, v) are interfering).

If this algorithm is repeated n times, f req(u, v) equals the proportion of the cases where the link (u, v) has been selected. Remind that the links are directed: the link (u, v) may not receive the same amount of bandwidth as (v, u). However, fairness among links is respected. The remaining description of the upper bound remains unchanged. The linear program LP 2 keeps on holding, with the new values of f req(u, v).

C. Flexibility of the models

All previous models where detailed with the transmitterreceiver model to represent interferences. However, our approach is generic and applies to any arbitrary interference model. One would only need to change the definition of Γ k (e) and γ k (e) sets into "sets of nodes (resp. links) interfering with node (resp. link) e". One can therefore use these models as a framework for benchmarking interference models.

In the following, our models assume the transmitter-receiver interference model and allows to compare the capacity of flat and self-organized routing protocols.

D. Objective functions

The previously defined linear programs describe how the radio resource is shared among nodes and links. The capacity of a network is then the maximum quantity of information that can be transported by the network. In the following, we define two notions of network capacity, and therefore two objective functions. The first one is the standard sum of transported traffics while the second one provides a user-wise throughput guaranty.

1) Max-Sum objective: The capacity is often described as the maximal sum of throughput achievable in the network, as follows.

M ax

p∈P f (p) (9)
This objective is global and does not address the distribution of the throughput among the flows. Consequently, the less interfering flows are privileged and some flows will be null because they consume too many resources. In practice, the positive flows are almost only single hop flows. There are almost no application where this notion of capacity is relevant. However, this formulation gives an upper bound of the global achievable capacity.

2) Max-Min objective: A telecommunication operator deploying such a network could be interested in providing a guarantee on the minimum bandwidth available to each user. In such a case, the objective is to maximize this flow-wise capacity as follows.

M ax (M in p∈P f (p)) (10)
Of course, this strong guaranty decreases the global achievable throughput. Nevertheless, such a notion of capacity is relevant to numerous applications in ad hoc and hybrid networks.

V. RESULTS

Our objective is to estimate the capacity of different routing protocols based on either a self-organized scheme or a flat approach. To reach such a goal, the behavior of some routing protocols were simulated with OPNET Modeler. More precisely, topologies from 20 to 60 nodes with an average degree of 10 nodes were generated. Then, OLSR (relevant to represent flat routing protocols), Wu & Li (a backbone is built, and all the routing is done through this backbone) and VSR (routing protocol where the backbone is only used for the control traffic) were implemented and simulated on these topologies. The computed routes and the overhead were directly extracted from the simulations to be used in our LP formulation. We chose to model an ad-hoc traffic pattern: a node has a route toward each other node in the network (n(n-1) routes at all).

First, the general evolution of the capacity in MANET is evaluated with the link-oriented fairness and the max-min objective (fig. 2). With a flat routing protocol, the capacity per flow decreases when the number of nodes increases: the number of flows grows (there exist n(n -1) routes), creating potentially more contention. Consequently, the bandwidth allocated to each flow will surely decrease, corroborating the results of [START_REF] Gupta | The capacity of wireless networks[END_REF]. Oppositely, the total throughput sent across the network remains constant: many flows will with high probability pass through the center of the network since shortest paths are used, creating a bottleneck. Finally, we can note that the optimistic and the pessimistic models present a very close capacity.

The capacity of ad-hoc networks according to different routing protocols is evaluated with the max-min objective and with a link-oriented fairness bandwidth sharing (fig. 3). The flat routing protocol presents the highest capacity: shortest routes in the initial network limit the route length, and consequently present less interferences. VSR presents the same capacity as OLSR: the backbone is only used for control traffic, and although not shortest routes are used, the impact on the capacity is restricted. Oppositely, Wu & Li presents the lowest capacity: it computes shortest routes through the backbone, lengthening the average route.

In a second time, the capacity is evaluated with the nodeoriented fairness bandwidth sharing (fig. 4). The capacity of OLSR and VSR is very slightly impacted by a different fairness model. On the other hand, the capacity presented by Wu & Li decreases: all the nodes receive the same bandwidth although backbone nodes carry most traffic. Consequently, a bottleneck in the backbone appears.

Finally, we maximize the global network throughput using the max-sum objective (fig. 5). Some flows are authorized to receive more traffic than others, discriminating the flows which create many interferences(i.e. multihops flows). Consequently, less flows are active simultaneously, and the center of the network constitutes no more a bottleneck: the capacity is stable or increases with the optimistic bandwidth sharing. Oppositely, the pessimistic resource sharing tends to overestimate the interferences, limiting the spatial re-utilization in small networks. Besides, we can remark that OLSR and VSR provide a very close capacity, whatever the objective function is. On the other hand, Wu & Li routing performs much lower

VI. CONCLUSION AND FUTURE WORK

The goal of this work is to study the capacity of routing protocols based on virtual structure or flat approaches. Thus, we propose a generic capacity evaluation framework used to benchmark routing protocols and to validate the selforganization approach in routing protocols.

Our capacity evaluation framework is made of four linear programs. A set of linear constraints model the local radio resource sharing principles. These local constraints are combined so as to define a global behavior of the network. We provide linear models for two MAC layer fairness hypotheses. One assumes that the probability to access the radio channel is uniformly distributed among the nodes, while the other one assumes a uniform distribution of medium access on the radio links. For each of these settings, we propose a pessimistic and an optimistic scenarios of spatial-reutilization of the medium, hence providing an upper and a lower bound.

Using these models, we compare the capacity provided by a flat routing protocol (OLSR) and two routing protocols based on virtual backbones. The results highlight that the capacity provided by self-organization schemes keeps very close to the performances of standard flat protocols, if the virtual structure More benchmarking have to be done, including other flat routing protocols (AODV, DSR, DSDV. . . ), even though we conjecture that there performances should theoretically be similar to those of OLSR since they all compute almost shortest paths. Multi-path routing protocols should also be studied. Besides, this work is a first step toward the design of routing protocols maximizing the capacity.

Fig. 1 .

 1 Fig. 1. The 2-Neighborhood of the node C maximum even when the number of AP becomes arbitrary high.In[START_REF] Jain | Impact of interference on multi-hop wireless network performance[END_REF], the capacity of ad hoc networks is modeled using linear programs (LP). Their work focuses on the capacity estimation for a given network topology and workload. The capacity is modeled as a multicommodity flow problem. However, the complexity of such a formulation resides in the capacity estimation of each link. The authors estimate the maximum capacity in searching sets of links which can convey traffic simultaneously (i.e. they are not interfering). However, their work models an unfair MAC layer.[11] proposes a scheduling scheme maximizing the throughput from a group of sources to a group of destinations. The authors propose a scheduling of radio links such that no two simultaneously activated links are interfering. A greedy algorithm allocates slots to the different links according to the decreasing euclidean length of the links, this order constituting a sufficient condition of feasibility. However, such an approach could under-estimate the capacity and does not introduce any fairness model.

Fig. 2 .

 2 Fig. 2. Comparison of the objective functions in an ad-hoc flat network (link-oriented fairness)

Fig. 3 .

 3 Fig. 3. Capacity with the max-min objective (link-oriented fairness)

Fig. 4 .Fig. 5 .

 45 Fig. 4. Capacity with the max-min objective (node-oriented fairness)

This work is partly supported by the European Commission, project IST-15964. The views given herein represent those of the authors and may not necessarily be representative of the views of the project consortium as a whole.