Z Trabelsi

M Yahyaoui

S Ben Radhia

K Boujdaria

E Zallo

O G Schmidt

P Atkinson

M Chamarro

C Testelin

Neutral, charged excitons and biexcitons in strain-free and asymmetric GaAs quantum dots fabricated by local droplet etching

Keywords: numbers: 73, 21, La, 68, 65, Hb

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Neutral, charged excitons and biexcitons in strain-free and asymmetric GaAs quantum dots fabricated by local droplet etching

Introduction

Strain-free GaAs/AlGaAs quantum dots (QDs) are a relatively new III-V QD family of great interest. One method to grow these dots is droplet epitaxy, and highly symmetric, high optical quality strain-free dots have been fabricated on (111) substrates by this technique [1][2][3]. An alternative QD growth method is self-assembled local droplet etching (LDE) and nanohole in…lling [4][5][6][7]. This technique is particularly attractive since the dot density and emission energy can be controlled independently with the QD size precisely controlled via the hole …lling level [7,8]; both anisotropic and highly symmetric dots can be grown by suitable choice of droplet etching conditions [9,10]; and quantum dot molecules can be realized and implemented in devices [11]. This makes these strain-free GaAs/AlGaAs QDs potentially useful for applications such as QD lasers [12] and solar cells [13]. However, for the realization of QD based devices, the dependence of the QD optical properties, namely, the exciton transition energies and intersublevel energy spacing on the QD shape and size needs to be well understood.

The asymmetry of QDs is of particular interest also for quantum information applications since symmetric dots are attractive due to the low value of exciton …ne-structure splitting (FSS), which makes them promising systems for entangled photon sources, while highly asymmetric dots have the potential to lead to brightening of the long-lived dark exciton state, which is of interest for spin-based quantum processing [14]. Since GaAs/AlGaAs QDs are nominally unstrained and have sharp interfaces with little intermixing, they provide an ideal system to study the role of the geometric anisotropy in determining QD electronic properties, such as the FSS.

Recently, a few studies [5][6][7] have focused on the optoelectronic properties of strain-free GaAs QDs, however theoretical investigations are still required. In our recent work [15], we presented a theoretical study of the optical transitions of di¤erent excitonic complexes in highly symmetric strain-free GaAs quantum dots (QDs) fabricated by epitaxially …lling nanoholes (NHs) in AlGaAs surface. The QD energy levels were calculated using a model of the QD shape and size based on atomic force microscopy (AFM) measurements of the QD, and the calculated levels showed reasonable agreement with photoluminescence measurements demonstrating the suitability of our modelling approach. Nevertheless, the majority of the QDs fabricated by local droplet etching technique are slightly anisotropic [8,[16][17][18].

Hence, taking into account such anisotropy in our modelling approach is necessary. Here, we give a comprehensive outline of our theoretical approach which is supported by a good agreement with optical spectroscopic study, performed on GaAs QDs. Since we are interested in asymmetric QDs, we investigate the exciton FSS, due to the electron-hole (e-h) exchange interaction, as a function of the QD height.

The paper is organized as follows. Section 2 is divided into two parts. In part 2.1., we present micro-photoluminescence ( -PL) measurements of single GaAs QDs obtained by …lling NHs with di¤erent amounts of GaAs. In part 2.2., we outline our theoretical model which includes the QD shape modelling and the calculation of the con…ned states.

The numerical results are presented in section 3 where we perform a comparison between experimental data and theoretical results, and we study the few-particle binding energies and correlation e¤ects. Exchange interaction e¤ects and exciton FSS will also be evaluated and discussed. Our results are summarized in section 4.

Methods

Samples and experimental setup

The samples have been grown by solid source molecular beam epitaxy on GaAs (100) substrate. Details of the gallium droplet etching process used to form the nanohole can be found in Ref. [7]. The nanoholes were then overgrown by a 7 nm Al 0:45 Ga 0:55 As barrier, a GaAs layer with nominal thickness from 0:25 to 3:5 nm, and an upper Al 0:33 Ga 0:67 As barrier.

A 2 minute growth interruption ensured signi…cant GaAs migration towards the bottom of the NH such that the GaAs thickness at the bottom is signi…cantly larger than the nominal deposited thickness.

Figure 1 shows the shape of the NH after growth of the lower AlGaAs barrier (left image, black lines) and after growth of a subsequent 1 nm GaAs layer followed by 2 minute growth interruption (right image, blue lines). It can be seen that the NHs, which have a typical depth of 8 nm before growth of the GaAs layer, are elongated along the [110] direction, with an in-plane anisotropy (ratio of the width along [110] to the width along [110]) given by 2:5. It can be seen that net migration towards the bottom of the hole results in a dot, approximately 4 nm high, con…ned in the NH for a nominally 1 nm deposition. -PL measurements were carried out at 10 K using a 532 nm Nd:YVO4 excitation laser focused down to 1 m 2 and a spectrometer with 40 eV resolution. The dot density was low enough to enable single dot spectroscopy to be carried out without any need for sample processing.

The -PL polarization was analyzed by including a half-wave plate and a linear polarizer before detection. The …ne structure splitting of the exciton can then be observed (see Inset -PL spectra obtained for di¤erent nominal thicknesses are shown on Fig. 2. For each sample, one detects a strong emission peak associated to the neutral exciton of a single QD, labelled X, (more precisely, an electron-hole pair in a neutral QD), determined by polarization measurements. At lower energy, extra peaks are observed; they can be assigned to the biexciton and charged excitons.

To analyze the low energy peaks, a study of the -PL spectrum, versus the excitation intensity, has been performed on a QD with a 1 nm nominal thickness. Figure 2(d) shows the excitation power dependence of the QD emission intensity for the peaks at 1643:5 and 1647:7 meV. When P exc is large, both emission intensities saturate, as expected. Below saturation, from a power law …t (I _ P exc ), it can be seen that the neutral exciton emission has a linear dependence on the excitation power whereas the line at 1643:5 meV has a quadratic dependence which is the signature of the biexciton (XX) complex. The inter-mediate emission, between X and XX, can be attributed to charged excitons (as we will discuss below).

Theoretical model

We …rst model the QD shape and then calculate the single particle energy levels, the single particle wave functions (WFs) and the …nal alignment of the excitonic states by combining a good description of the QD shape and the BenDaniel-Duke Hamiltonian [19]. Since a shape anisotropy induces a splitting of p-like states, we also calculate the s-p x ( s px ) and s-p y ( s py ) energy level spacings.

Shape modelling

The QDs studied here di¤er from the symmetric GaAs/AlGaAs QDs that we have previously modelled [15] in that the dots here are elongated along the (Ox) ([110]) direction (see Fig. 1 Ref. [7]). It is however possible to adapt the QD shape modelling approach used in

Ref. [15] to take into account the QD shape anisotropy as described in Fig. 3. The QDs are modelled by the overlaying of two ellipsoids E 1 and E 2 revolving around the z axis (the growth direction) with di¤erent centers and eccentricities. E 1 (E 2 ) describes the shape of NH before (after) GaAs …lling. These ellipsoids satisfy the following equations:

r 2 ix a 2 0 i + r 2 iy b 2 0 i + (z + d i ) 2 c 2 0 i = 1 (1)
the indices i = 1 and i = 2 are reserved for NH before and after …lling, respectively.

Note that in the symmetric QD case, the cross-section of the two ellipsoids and (xOy)

planes are circles, while they are ellipses of radii R b ( ; z) for the asymmetric QD case :

R b ( ; z) = 1 q cos 2 r 2 ix + sin 2 r 2 iy (2)
where r ix and r iy are the major and minor ellipse axes, respectively. They are de…ned as follows:

r 2 ix (z) = a 2 0 i 1 (z + d i ) 2 c 2 0i r 2 iy (z) = b 2 0 i 1 (z + d i ) 2 c 2 0 i (3)
FIG. 3: Schematic view of the QD shape modeling.

In order to determine the other parameters, namely a 0 i , b 0 i , c 0 i and d i , the same approach as in Ref. [15] is followed. We note that the QD dimensions were based on the AFM data of the AlGaAs NH before and after GaAs …lling [7]. These AFM measurements establish that the hole opening is anisotropic and the pro…les allow us to de…ne an aspect ratio = r 0x r 0y

2:5, where r 0x and r 0y are the radii of the hole opening along the major axis

(x = [110]
) and the minor axis (y = [110]) for the elongated structure, respectively. The NH depth before GaAs …lling is h 1 8 nm and we denote by h 2 the NH depth after …lling. Thus, the QD height is de…ned as h QD = (h 1 h 2 ). h 2 is considered as an adjustable parameter in our calculations.

Con…ned states : outline of the calculation method

To calculate the charge carrier states within the e¤ective-mass theory, we consider the single carrier (electron (e) or hole (h)) Hamiltonian which is written as follows:

H e(h) = T e(h) (r e(h) ) + V conf e(h) (r e(h) ) (4)
where

V conf e(h) (r e(h)
) is the carrier con…nement potential and T (r e(h) ) denotes the carrier kinetic operator. Taking into account the QD shape anisotropy, the cylindrical symmetry is somewhat broken and, in this case, the kinetic energy operator will be decomposed into three components as follows:

T e(h) = T z e(h) + T e(h) + T e(h) (5)
with

8 > > > > < > > > > : T z e(h) = } 2 2 h @ @z 1 m z e(h) @ @z i T e(h) = } 2 2 h 1 @ @ m e(h) @ @ i T e(h) = } 2 2 h 1 2 @ @ 1 m e(h) @ @ i (6)
where ( ; ; z) denotes cylindrical coordinates. m z e(h) and m e(h) are the carriers e¤ective masses along z-and -directions, respectively. The Luttinger parameters, j (j = 1; 2) are calculated via the 40-band k.p theoretical model [20]. The numerical parameters used in our calculations are compiled in Table 1.

GaAs

GaAs/Al 0:45 Ga 0:55 As ; z e(h) functions correspond to the eigenbasis of a large cylinder of height Z C and elliptical base of radius R C ( ) which is given by : R C ( ) = 1=(

1 2 m e (m 0 ) m z h (m 0 ) m h (m 0 ) " V e (eV ) V h (eV )
q cos 2 A 2 + sin 2 B 2 ).
A and B denote the dimensions along x-and y-directions of the elongated cylinder, respectively.

The normalization condition of the carriers WFs imposes the condition (A=B) = = 2:5.

The origin of the z-axis is taken at the midpoint of the cylinder. n `m are given by: 1=2 sin( e(h) )J

8 > > > < > > > :
1 1 RC ( ) e(h) sin( m Z C z e(h) ), for p y -state (7)
n `is the `th root of the n-order Bessel functions J n , and n `are the normalization constants. We should point out that only the matrix elements of the kinetic term T e(h) are a¤ected by the QD shape anisotropy, and these are calculated using Eq. ( 6) and (7). Calculations show that these elements di¤er from one state to another. The other contributions of the kinetic matrix elements, T z e(h) and T e(h) are given in Ref. [21]. Thus, …nding the exact energy eigenvalues of the stationary Schrödinger equation is reduced to the numerical diagonalization of the Hamiltonian matrix. Such a technique has proved itself to be a workable and reliable approach [22,23].

Results

In this section, we check the accuracy of our numerical approach by comparing the theoretical results with experimental data. Firstly, we begin by calculating the carriers con…nement energies as well as the s p x(y) and p x p y spacing energies, supported by a study of the carriers WFs distribution and localization inside the QD. We then calculate the binding energies of the excitonic complexes present in the dot, such as the neutral excitons, charged excitons (X + and X ) and the biexcitons. This is then compared with -PL data on single dots with di¤erent heights. A study of the anisotropic FSS of the ground state (s-s) exciton concludes the section.

Single-particle energy levels

As described previously, the electron and hole energy levels are numerically calculated by diagonalizing H e(h) over the eigenstates basis of a large cylinder. Note that, due to symmetry conditions, the cylinder is laterally elongated with an anisotropy factor equal to that of the QD: = A B = r 0x r 0y = 2:5. Figure 4(a) shows the electron and hole con…nement energies of the s-shell (E e(h) s

), p x -shell (E e(h) px ) and p y -shell (E e(h) py ) states as a function of h QD . As for conventional QDs, the carriers con…nement energies of the studied QDs decrease gradually when increasing the QD size. In contrast to the case of symmetric QDs [15] and as expected a degeneracy lifting of the p-shell states is observed for asymmetric QDs. In Fig. 4(b), we highlight the size e¤ect on the s-p x and s-p y level spacing energies E e;h s px and E e;h s py , respectively. It is known that the energy spacing mainly depends on the lateral size. For the studied QDs, the increase of the QD height leads to an expansion of the QD lateral size due to the shape of the NH and an increase of the WF extensions (see Fig. 6). This explains the decrease of E e;h s px and E e;h s py as h QD increases, as well as the p x -p y spacing energy becoming equal to a few meV. This is summarized in Fig. 4(c) which shows a schematic illustration of single-particle electron and hole energy levels for the GaAs/AlGaAs QDs studied here. We have neglected here the fact that the as-grown QDs had a lower Al 0:45 Ga 0:55 As barrier and an upper Al 0:33 Ga 0:67 As barrier, treating both barriers as Al 0:45 Ga 0:55 As. The s p x energy spacing ranges between 6.7 meV and 11.7 meV for holes, while for electrons it is almost double. The p x p y spacing energy is much lower than s p x one. In order to show more clearly the evolution of the WFs extension with the QD height, we have reproduced the carriers'WFs by Gaussian …ts using the following expressions [START_REF]) and both carriers (electron, hole)[END_REF]: , for p y -state (8) where A n e(h) (n = 0; 1) are the normalization constants. `xe(h) , `ye(h) and `ze(h) are the …t parameters which determine the e(h)-WFs extension in the three directions x, y and z. It can be seen from Fig. 6(a)-(c) that the WF extension always decreases as the QD size decreases. The WF extension along (Oz) is similar for all the states, due to the fact that the main con…nement is along the (Oz) direction. In the xOy plane, the WF is always elongated along (Ox) re ‡ecting the dot asymmetry; however the increase in WF extension with dot height is more rapid along the (Ox) direction than the (Oy) direction for the p x -state so that the p x state become more anisotropic, and more rapid along the (Oy) direction compared to the (Ox) direction for the p y -state such that the p y state becomes more isotropic.

8 > > > > > > > > < > > > > > > > > : e(h) 0 2 = A 0 e(h

Exciton transition energy: comparison with experiment

In this section, we compare the ground state transition energy, E 0 (X) experimentally observed, with our numerical results. The exciton energy can be calculated as follows:

E 0 (X) = E g + E e 0 + E h 0 V 0 eh (9)
where E g is the GaAs QD band gap energy. The electron and hole con…nement energies, E e 0 and E h 0 , are de…ned relative to the bottom of the con…nement potential. V 0 eh is the electron-hole Coulomb energy treated in the perturbative approach and it is given by [START_REF] Grundmann | [END_REF]:

V 0 eh = e 2 4 " 0 " Z Z j e 0 ( ! r e )j 2 h 0 ( ! r h ) 2 j ! r e ! r h j d 3 ! r e d 3 ! r h ( 10
)
where " is the average dielectric constant of GaAs [26].

In Fig. 7(a), we show the dependence of the QD emission energy as a function of the dot height h QD ( h QD = (8 nm h 2 ), where 8 nm is the average NH depth, and h 2 is an input to the calculation). The calculated QD emission energy can then be compared to the measured ensemble QD emission energy for a given nominal thickness, which allows the QD height for a given nominal thickness to be estimated. This is shown in Fig. 7(b), where the experimental data points of ensemble emission vs. nominal GaAs thickness is shown, together with the estimated QD height vs. nominal thickness taken from the calculated emission energies. It can be seen that the QD height is much greater than the nominal thickness due to the fact that there is net migration towards the bottom of the nanohole. (black points). The curves are guides to eye.

Excitonic complexes binding energies

We present a systematic calculation of the few-particle binding energies in order to assign excitonic complexes to the peaks observed in single QD -PL spectra. We …rst calculate the excitonic complexes binding energies in terms of direct Coulomb interactions and then the e¤ect of correlation energies will be included. The electron-hole exchange interaction e¤ects on the excitonic binding energy will be discussed thereafter.

The s-shell direct Coulomb interactions between an electron and a hole (V 0 eh ), and between two electrons (V 0 ee ) and two holes (V 0 hh ) are calculated using Eq. ( 10), and plotted as a function of h QD in Fig. 8(a).

All these quantities decrease when h QD increases which is attributed to the reduced overlap of the carriers WFs and the increase of the WFs extension already noted in Fig. 6. It can also be seen that the hole-hole Coulomb interaction is always smaller then the electron-electron interaction. This can be explained by the larger extension of the hole WF in the (Ox) and (Oy) directions compared to that of the electron (see Fig. 6(a)). Our results satisfy the following relationship [27]: V 0 ee > jV 0 eh j > V 0 hh over the entire dot height range studied here. This …nding is in contrast to the symmetric QDs case where a change in order of the Coulomb contributions was observed [15]. The absence of such a reversal in the asymmetric QDs studied here is mainly due to the larger lateral dimensions of the dot.

Both electron and hole carriers are con…ned within even the thinnest QDs, and their WFs evolve in a monotonic way which explains the monotonic behavior of V 0 ee , jV 0 eh j and V 0 hh with respect to h QD . We denote by (X `) (`= ) the binding energy of trions and by (XX) that of biexcitons. These quantities have been estimated using the Hartree-Fock approximation in terms of direct Coulomb energies (see Eq. ( 6) of Ref. [15]). Figure 8(b) shows the evolution of (X `) and (XX) as a function of h QD . The important points in this …gure are summarized as follows: (i) (X ) is always negative, thus X forms an unbound state.

(X ) becomes more negative as h QD increases. (ii) X + shows a trend opposite to X .

Its binding energy increases with QD height and it is positive, meaning that X + forms a bound state. (iii) XX shows a binding energy close to zero, regardless of h QD . Since the binding energy of a few-particle state is approximated by the sum of attractive and repulsive interactions between all pairs of particles, the same magnitude of the (X ) and (X + ) but with opposite trends is expected. This result agrees well with the V 0 ee , jV 0 eh j and V 0 hh ordering seen in Fig. 8(a). It is important to note however that the few-particle binding energies cannot be fully evaluated using only the direct Coulomb interaction. Consideration of the correlation e¤ects is paramount to provide a good estimation of the excitonic complexes binding energies. This has been demonstrated in several previous works either in strained QDs [28] or unstrained ones [6,15].

In the following, we calculate the few-particle binding energies by evaluating the Coulombic interaction expressions to the second order of the perturbation theory. The procedure is detailed in Ref. [15] (Eqs ( 7)-( 9)). We have summarized our theoretical results of the correlation energies ( c (X), c (X `) (`= ), and c (XX)) as well as the binding energies c (X `)(`= ); c (XX) of the excitonic complexes in Figs. 9(a) and 9(b), respectively. It can be seen that the few-particle correlation energies and binding energies are mutually correlated. A sensitive dependence of the correlation energies on the h QD is plainly shown in Fig. 9(a). Compared to the symmetric QDs case where the set of the correlation energies ( c ) vary monotonically with the QD height, in asymmetric QDs the correlation energies …rst increase and then subsequently decrease with increasing height because c depends both on the s p splitting energies and the Coulomb interaction energies V n i m j 00 ij , (see Eq. ( 7) of Ref. [15]). The Coulomb coupling V n i m j 00 ij depends strongly on the WF overlap and extension. From our study (see Fig. 6), the WF extensions are continuously increasing and so V n i m j 00 ij is continuously decreasing, with increasing h QD . For the s-p splittings, E s p x(y) , we observe two regimes (see Fig. 4(b)): below 5 nm, the splitting is almost constant, while at higher QD thickness, this splitting decreases rapidly. The correlation term being proportional to V n i m j 00 ij 2 and inversely proportional to E s p x(y) , one can conclude that at low thickness ( E s p x(y) almost constant), the correlation term decreases following the Coulomb contribution, while at high thickness, the rapid decrease of E s p x(y) leads to an increase of j c j.

We present in Fig. 9(b) the calculated binding energies when including correlation e¤ects.

It is clear that all the calculated few-particles binding energies are now positive, consistent with experimental observations. The addition of the correlation e¤ects leads to a change from unbound to bound for X and XX states as was previously shown for symmetric GaAs QDs [15]. The sequence of excitonic peaks, arranged in order of increasing emission energy is XX, X + , X , X and is similar to that obtained for bow-tie shaped GaAs QDs, where the X + was also found to have a binding energy of around 3 meV [17]. In contrast, X + is typically unbound in strained InAs QDs [27]. This di¤erence can be related to the larger lateral size of the GaAs QDs, which increases the contribution of the correlation e¤ect on the exciton binding energy resulting in a bound positive trion for these dots [15,31].

-PL measurements of QDs show a strong emission associated with the neutral exciton and two weaker lines at lower energy (see Fig. 2). The main emission line is attributed to the exciton X. The lowest energy line is attributed to the biexciton XX, since its emission intensity shows a quadratic dependence on the excitation power. This observation is in agreement with our calculations, which predicts that XX has the highest binding energy.

The theoretical binding energies, for all the excitonic complexes are plotted on Fig. 9(b) together with the experimental values for XX. Our calculations reproduce quite well the XX binding energy which is in the 3 5 meV range. While a good agreement is obtained for low QD thickness, a discrepancy is observed at high QD thickness (h QD > 5:5 nm). This deviation is related to the smaller s and p-states splitting when h QD increases and the fact that the contribution of the d-states has not been included in these calculations. Finally, for the intermediate emission line, the binding energy varies from 1:3 to 3:2 meV, with a mean value equal to 2:6 meV. This allows this emission line to be assigned to the positively charged exciton X + , according to our calculation.

Exchange interaction e¤ects: …ne structure splitting

Here, we study the role of the e-h exchange interaction taking into account the shape anisotropy, on the excitonic properties. In particular, we investigate the contribution of the long-range exchange interaction which can be separated into two di¤erent terms acting on bright exciton states: (i) a diagonal term (" exch d ) which induces an additional energy correction, the same for both bright excitons; (ii) the o¤-diagonal term (" exch od ) which splits the bright excitons and will give the exciton …ne-structure splitting: F SS = 2 " exch od . " exch d and exch od are given by

" exch d = ~2 4m 0 E P E 2 g Z dq(q 2 x + q 2 y )V q Z d 3 r ne (r) n h (r)e iq:r 2 (11)
" exch od = ~2 4m 0 E P E 2 g Z dq(q x + iq y ) 2 V q Z d 3 r ne (r) n h (r)e iq:r 2 (12)
where E P = 23:81 eV [20] is the related energy to the Kane matrix element P = m0 hSj p x jiXi = m0 hSj p y jiY i = m0 hSj p z jiZi. V q = 1 (2 ) 3 e 2 " 0 "r 1 q 2 is the Fourier transform of the Coulomb potential. We then evaluate numerically, " exch d and " exch od by using the Gaussian functions. It should be noted that previous studies on III-V QDs have shown that both the short range exchange interaction and the contribution of the valence band mixing to the long range exchange interaction have only a very weak e¤ect on the magnitude of the bright exciton splitting, and so these e¤ects have not been included here [30].

Our calculations show that the correction to the exciton binding energy due to e-h exchange interaction, " exch d , ranges between 14 eV and 30 eV ( inset of Fig. 10). The magnitude of this term is quite negligible, in comparison with the other contributions (direct Coulomb interaction and correlation energy) and consequently, it does not impact on the excitonic binding energy. We show in Fig. 10 calculated and measured FSS values as a function of the X emission energy. The FSS has been measured on 5 QD families, with emission energy ranging from 1.58 to 1.72 eV. The FSS varies from 17 to 155 eV. In contrast, GaAs/AlGaAs QDs grown in quasi-symmetric LDE nanoholes [10] showed a FSS varying from 4 to 12 eV, for the same exciton energy range. For the same QD size, one then observes a FSS one order of magnitude larger, signature of the asymmetric nanohole shape. Our calculations predict a FSS increase from 16 to 36 eV for the same energy range, which qualitatively agrees with the experimentally increase of the FSS observed here, and on other quasi-isotropic [10] or anisotropic [31] GaAs/AlGaAs QDs. A quantitative agreement is missing for the thinnest QDs. For small nominal thicknesses, the QD shape will strongly depend on the NH bottom, whose symmetry is probably reduced under the e¤ect of inhomogeneous etching at the start of the droplet etching process. Moreover, as shown on AFM images (Fig. 1), the pro…le along (Oy) is not symmetric meaning that the xOz plane is no more a mirror plane: the C 2v symmetry, assumed in our model, is broken to a C s symmetry. This extra anisotropy may favor an increase of the FSS in small QDs, needing more complex calculations. ) and o¤-diagonal (" exch od ) terms of the long-range exchange interaction as a function of h QD .

Conclusions

We present here an experimental and theoretical study of the electronic states of asymmetric strain-free GaAs/AlGaAs QDs grown by the droplet etching and in…lling method.

Both the lateral and vertical size of these QDs can be adjusted by the thickness of the …lling layer. AFM data allowed us to give a realistic description of the anisotropic QD shape.

Using the e¤ective mass approximation and the BenDaniel-Duke Hamiltonian, we have pro-vided a detailed calculation of the charge carriers energy levels which includes new terms, compared to the symmetric case, arising from the shape anisotropy. The calculated exciton transition energies correspond to realistic dot heights in the nanoholes, demonstrating the suitability of our theoretical approach. For a series of QD height ranging between 3:3 nm < h QD < 7:5 nm, the emission energy of the QD s-state decreases from 1:77 eV to 1:57 eV.

The energy di¤erence between s and p x (p y ) band to band transition decreases from 32:6 meV (XX) down to 19:7 meV (YY) with increasing h QD .

-PL measurements of QDs of di¤erent nominal thicknesses exhibit high optical quality with sharp excitonic peaks. In order to get a meaningful identi…cation of these spectra, we have calculated the few-particle binding energies. We have shown that the direct Coulomb interactions are insu¢ cient to reproduce the experimental …ndings and that the inclusion of correlation e¤ects allows the positive binding energy of the negatively charged exciton and of the biexciton to be explained. Binding states of the di¤erent excitonic complexes have been shown to be in reasonable agreement with experimental observations. For the thickest QDs, an improved agreement is expected by including d and higher excited states in the evaluation of the correlation e¤ects. Our model of the anisotropic QD shape leads to a calculated FSS ranging from 16 to 36 eV, as the dot height decreases from 7:5 nm to 3:3 nm. The trend of increasing FSS with decreasing QD height is in qualitative agreement with experimental observations; however the calculated FSS underestimates the experimental values. The discrepancy between calculation and FSS increases as the QD height decreases, which is likely due to the fact that our model considers a C 2v symmetry which is lost in the smallest QDs due to higher degree of asymmetry at the bottom of the nanohole. Further studies, out of the scope the present work, should extend the model to C s symmetry QDs.

Fig. 1 :

 1 Fig. 1: AFM images of an in-situ etched nanohole where the growth was stopped after: overgrowth of 7 nm Al 0:45 Ga 0:55 As bottom barrier layer (left image), overgrowth of the bottom AlGaAs barrier layer and 1 nm GaAs layer followed by a 2 minute growth interruption (right image). Corresponding surface pro…les along the [110] and [110] directions, which are the average from ten measured nanoholes, are shown for the bottom barrier layer (black line) and top of the GaAs layer (blue line). The gray shaded area indicates the location and form of the QD.

Fig. 2 (

 2 Fig. 2(b)).

FIG. 2 :

 2 FIG. 2: (a)-(c) -PL spectra of single QDs with di¤erent sizes. The nominal thickness is given for each QD. The highest energy line for each dot is the neutral exciton emission, characterized by polarisation measurements not shown here; inset (b) shows orthogonally polarised exciton emission. (d) Intensity of the exciton (black dot) and biexciton (red dot) emission lines, respectively at 1647.7 and 1643.5 meV ; a power law …t, below saturation, leads to the exponents = 1.0 and 2.1, respectively.

  ) ; e(h) ; z e(h) = 0 `J0 0 RC ( ) e(h) sin( m Z C z e(h) ), for s-state 1 `m e(h) ; e(h) ; z e(h) = 1 ` A+B A 1=2 cos( e(h) )J 1 1 RC ( ) e(h) sin( m Z C z e(h) ), for p x -state 1 `m e(h) ; e(h) ; z e(h) = 1 ` A+B B

FIG. 4 :

 4 FIG. 4: (a) Calculated single particle, electron and hole, con…nement energies of the s, p x and p y states with respect to h QD . (b) Calculated s-p x and s-p y splitting energies of electrons and holes as a function of h QD . (c) Band alignment of the GaAs/Al 0:45 Ga 0:55 As QD. The s-p x and p x -p y splitting energies are shown for both electron and hole

3. 2 .FIG. 5 :

 25 FIG. 5 : Electron and hole WFs of GaAs/Al 0:45 Ga 0:55 As QDs with r 0x =75 nm and r 0y =30 nm. (a) Evolution of

FIG. 6 :

 6 FIG. 6: (a)-(c) The evolution of the s, p x and p y carriers WFs extension with respect to h QD .

FIG. 7 :

 7 FIG. 7: (a) Calculated (black curve) and measured (red points) excitonic transition energy of the ground state. (b) Measured excitonic transition energy as a function of the nominal GaAs thickness deposited (red points), and corresponding calculated QD height (h QD )

FIG. 8 :

 8 FIG. 8: (a) Direct Coulomb interactions versus h QD . (b) Calculated binding energies of exciton complexes as a function of h QD without including correlation e¤ects.

Fig. 9 :

 9 Fig. 9: (a) The variation of the correlation energies of the exciton complexes as well as the neutral exciton, c , with respect to h QD . (b) Calculated binding energies of exciton complexes as a function of h QD while including correlation e¤ects. The wine-coloured stars indicate the -PL experimental data.

Fig. 10 :

 10 Fig. 10: Calculated (stars) and measured (circles) values of the FSS as a function of the X emission energy for di¤erent QDs. The inset shows the calculated values of the diagonal ( " exch d

TABLE I :

 I Numerical values of the input parameters used in this work. Masses are given in free

	7.03 2.33 0.067	0.422	0.107 12.47 0.390	0.206
	electron mass unit
	The carriers energy levels are calculated by diagonalizing the Hamiltonian H e(h)
	using the matrix method over a known and …nite eigenstate basis.	The eigen-
	functions of the Hamiltonian are expanded in a Fourier-Bessel …nite series as
	`max;mmax X	C n `m	n `m
	`>0;m>0

n (r e(h) ) = e(h) ; e(h) ; z e(h) , with C n `m are the basis coe¢ cients. The n `m e(h) ; e(h)

  ) e

			x 2 e(h)	y 2 e(h)	(z e(h) zm e(h)	) 2
			`2 x e(h) e	`2 y e(h) e		`2 z e(h)	, for s-state
	e(h) 1	2	= A 1 e(h) x 2 e(h) e	x 2 e(h) `2 x e(h) e	y 2 e(h) `2 y e(h) e	(z e(h) zm e(h) `2 z e(h)	) 2	, for p x -state
	e(h) 1	2	= A 1 e(h) y 2 e(h) e	x 2 e(h) `2 x e(h) e	y 2 e(h) `2 y e(h) e		(z e(h) zm e(h) `2 z e(h)	) 2


Ricardo J Andrade

Sandro R Freitas

François Hug

Michel W Coppieters

Eva Sierra-Silvestre

Antoine Nordez
email: antoine.nordez@univ-nantes.fr

Spatial variation in mechanical properties along the sciatic and tibial nerves: An ultrasound shear wave elastography study

Keywords:

Ultrasound shear wave elastography has become a promising method in peripheral neuropathy evaluation. Shear wave velocity, a surrogate measure of stiffness, tends to increase in peripheral neuropathies regardless of etiology. However, little is known about the spatial variation in shear wave velocity of healthy peripheral nerves and how tensile loading is distributed along their course. Sixty healthy young adults were scanned using ultrasound shear wave elastography. Five regions of the sciatic (Sciatic PROXIMAL , Sciatic DISTAL ) and tibial nerve (Tibial PROXIMAL , Tibial INTERMEDIATE , and Tibial DISTAL ) were assessed in two hip positions that alter nerve tension: 1) neutral in supine position; and 2) flexed at 90 • . Knee and ankle remained in full-extension and neutral position. We observed spatial variations in shear wave velocity along the sciatic and tibial nerve (P < 0.0001). Shear wave velocities were significantly different between all nerve locations with the exception of Sciatic DISTAL vs. Tibial INTERMEDIATE (P = 0.999) and Tibial PROXIMAL vs. Tibial INTERMEDIATE (P = 0.708), and tended to increase in the proximal-distal direction at both upper and lower leg segments. Shear wave velocity increased with hip flexion (+54.3%; P < 0.0001), but the increase was not different among nerve locations (P = 0.233). This suggests that the increase in tensile loading with hip flexion is uniformally distributed along the nerve tract. These results highlight the importance of considering both limb position and transducer location for biomechanical and clinical assessments of peripheral nerve stiffness. These findings provide evidence about how tension is distributed along the course of sciatic and tibial nerves.

Introduction

Various animal models of peripheral neuropathies, such as those associated with metabolic syndrome [START_REF] Layton | Differences between collagen morphologies, properties and distribution in diabetic and normal biobreeding and Sprague-Dawley rat sciatic nerves[END_REF], trauma [START_REF] Zhu | Evaluation of the crushed sciatic nerve and denervated muscle with multimodality ultrasound techniques: an animal study[END_REF] or anti-neoplastic drugs [START_REF] Bober | Paclitaxel alters sensory nerve biomechanical properties[END_REF], have shown that peripheral nerve stiffness tends to be higher regardless of etiology. Using ultrasound shear wave elastography, preliminary human studies have demonstrated that shear wave velocity of nerves, a surrogate measure of nerve stiffness [START_REF] Schrier | Shear wave elastography of the median nerve: a mechanical study[END_REF], increases in both compression [START_REF] Kantarci | Median nerve stiffness measurement by shear wave elastography: a potential sonographic method in the diagnosis of carpal tunnel syndrome[END_REF] and diabetic [START_REF] Dikici | Evaluation of the Tibial Nerve with Shear-Wave Elastography: A Potential Sonographic Method for the Diagnosis of Diabetic Peripheral Neuropathy[END_REF] neuropathies. However, these clinical investigations focused on a single nerve region and did not account for potential heterogeneity of mechanical changes along the course of peripheral nerves. Although spatial heterogeneity in shear wave velocity has recently been reported in healthy upper limb nerves [START_REF] Greening | Posture-induced changes in peripheral nerve stiffness measured by ultrasound shear-wave elastography[END_REF][START_REF] Rugel | Influence of limb position on assessment of nerve mechanical properties by using shear wave ultrasound elastography[END_REF], a comprehensive elasticity mapping of sciatic and tibial nerves remains to be investigated.

Length-dependent neuropathies, such as diabetic polyneuropathy, are typically characterized by clinical manifestations primarily located distally, but with a proximal progression [START_REF] Watson | Peripheral Neuropathy: A Practical Approach to Diagnosis and Symptom Management[END_REF]. Furthermore, similar to remote neuro-immune inflammatory findings observed in focal mononeuropathies [START_REF] Schmid | Local and remote immune-mediated inflammation after mild peripheral nerve compression in rats[END_REF], increased sciatic nerve shear wave velocity has also been reported far away from the lesion site in lumbar radiculopathies [START_REF] Çelebi | Sonoelastographic evaluation of the sciatic nerve in patients with unilateral lumbar disc herniation[END_REF][START_REF] Neto | Noninvasive Measurement of Sciatic Nerve Stiffness in Patients With Chronic Low Back Related Leg Pain Using Shear Wave Elastography[END_REF][START_REF] Wang | The relationship of the shear wave elastography findings of patients with unilateral lumbar disc herniation and clinical characteristics[END_REF]. Collectively, these observations highlight a need for improved understanding of the spatial variations in ultrasound shear wave velocity along the course of peripheral nerves which could improve the ability to distinguish normal regional differences in nerve mechanical properties from those arising from damage and, potentially, regeneration.

It has previously been suggested that healthy peripheral nerves exhibit regional variations in their ultrastructure, including number of fascicles [START_REF] Sunderland | The intraneural topography of the radial, median and ulnar nerves[END_REF][START_REF] Sunderland | The intraneural topography of the sciatic nerve and its popliteal divisions in man[END_REF], proportion of epineural tissue [START_REF] Sunderland | The anatomy and physiology of nerve injury[END_REF] and collagen fibre diameter [START_REF] Mason | An ultrastructural and biochemical analysis of collagen in rat peripheral nerves: the relationship between fibril diameter and mechanical properties[END_REF], which likely to ditacte the shear wave velocity. Peripheral nerves undergo considerable mechanical loads during limb movements and body postures [START_REF] Boyd | Altered tibial nerve biomechanics in patients with diabetes mellitus[END_REF][START_REF] Ellis | Is there a relationship between impaired median nerve excursion and carpal tunnel syndrome? A systematic review[END_REF]. Both animal [START_REF] Phillips | Peripheral nerves in the rat exhibit localized heterogeneity of tensile properties during limb movement[END_REF] and cadaver [START_REF] Mahan | Altered ulnar nerve kinematic behavior in a cadaver model of entrapment[END_REF] studies reported that healthy peripheral nerves exhibit heterogeneous mechanical responses to (physiological) tensile stresses, being this response influenced by a nerve's proximity to joint regions. Shear wave velocity, as measured by using ultrasound shear wave elastography, is also a good indicator of tensile loading of soft tissues [START_REF] Hug | Elastography for Muscle Biomechanics: Toward the Estimation of Individual Muscle Force[END_REF] such as peripheral nerves [START_REF] Andrade | Non-invasive assessment of sciatic nerve stiffness during human ankle motion using ultrasound shear wave elastography[END_REF]. For example, recent studies have demonstrated that nerve-directed movements or limb positions can modulate the shear wave velocity of healthy human nerves [START_REF] Andrade | Chronic effects of muscle and nerve-directed stretching on tissue mechanics[END_REF][START_REF] Andrade | Non-invasive assessment of sciatic nerve stiffness during human ankle motion using ultrasound shear wave elastography[END_REF][START_REF] Rugel | Influence of limb position on assessment of nerve mechanical properties by using shear wave ultrasound elastography[END_REF]. However, it remains unclear how shear velocity varies along the length of resting healthy nerves in-vivo, and how it behaves under tensile loads (i. e., stretching).

Despite utrasound shear wave elastography has become a promising non-invasive method for characterising changes in peripheral nerve mechanics that are associated with altered tissue structure and composition [START_REF] Wee | Ultrasound elastography for the evaluation of peripheral nerves: a systematic review[END_REF], the spatial variations in mechanical properties of healthy nerves are not well understood in-vivo. This study aimed to: 1) investigate the spatial variations in shear wave velocity along the sciatic and tibial nerve in healthy individuals; and 2) examine the influence of tensile loading on the spatial variations in shear wave velocity. Based on structural [START_REF] Mason | An ultrastructural and biochemical analysis of collagen in rat peripheral nerves: the relationship between fibril diameter and mechanical properties[END_REF] and mechanical [START_REF] Mahan | Altered ulnar nerve kinematic behavior in a cadaver model of entrapment[END_REF][START_REF] Phillips | Peripheral nerves in the rat exhibit localized heterogeneity of tensile properties during limb movement[END_REF] findings from both animal and cadaver studies, we hypothesized that sciatic and tibial nerve would exhibit regional differences in shear wave velocity along their length and that tensile loading would not be uniformly distributed.

Methods

We analyzed data collected at baseline of a published randomized controlled study [START_REF] Andrade | Chronic effects of muscle and nerve-directed stretching on tissue mechanics[END_REF]. Considering the large number of outcome measures and main study purposes, the analysis of the spatial variations in shear wave velocity were not examined in the previous publication.

Participants

Sixty healthy young adults (29 females, 31 males; age: 20.5 ± 2.0 years, height: 172.2 ± 9.1 cm, weight: 63.8 ± 9.0 kg) with no selfreported history of lower limb neuropathy, or musculoskeletal disorders, participated in this study. Prior to testing, written informed consent was obtained from all participants. The study was approved by the Institutional Ethics Review Board.

Shear wave elastography acquisition

Ultrasound shear wave elastography provides real-time and high spatial resolution elasticity mapping by measuring the propagation velocity of shear waves generated remotely using ultrasonic focused beams [START_REF] Gennisson | Ultrasound elastography: principles and techniques[END_REF]. Nerve shear wave velocities are typically used as a surrogate measure of musculoskeletal tissue stiffness [START_REF] Hug | Elastography for Muscle Biomechanics: Toward the Estimation of Individual Muscle Force[END_REF]. Considering an elastic, homogeneous and isotropic medium, the relationship between the shear wave velocity (V 2 ) and the shear modulus (μ) arise from the following equation: μ = ρV 2 , where ρ is the density (assuming ρ = 1000 kg m -3 ) [START_REF] Bercoff | Supersonic shear imaging: a new technique for soft tissue elasticity mapping[END_REF]. The Young's modulus (E) can then be derived from the following equation: E = 3μ. Preliminary shear wave elastography studies on peripheral nerves have reported outcomes in either shear modulus [START_REF] Dikici | Evaluation of the Tibial Nerve with Shear-Wave Elastography: A Potential Sonographic Method for the Diagnosis of Diabetic Peripheral Neuropathy[END_REF][START_REF] Paluch | Shear-wave elastography: a new potential method to diagnose ulnar neuropathy at the elbow[END_REF] or shear wave velocity [START_REF] Andrade | Chronic effects of muscle and nerve-directed stretching on tissue mechanics[END_REF][START_REF] Neto | Noninvasive Measurement of Sciatic Nerve Stiffness in Patients With Chronic Low Back Related Leg Pain Using Shear Wave Elastography[END_REF] as commercial systems can display all these metrics. However, the Young's modulus calculation from the shear wave velocity do not hold for anisotropic musculoskeletal tissues, such as the peripheral nerves [START_REF] Gennisson | Viscoelastic and anisotropic mechanical properties of in vivo muscle tissue assessed by supersonic shear imaging[END_REF]. In addition, for a thin and stiff layer, the propagation of the shear waves can be guided (i.e., when they experience successive reflections on the nerve boundaries) and this phenomena has the potential to affect the relationship between the shear modulus and the shear wave velocity [START_REF] Brum | In vivo evaluation of the elastic anisotropy of the human Achilles tendon using shear wave dispersion analysis[END_REF]. For this reason, this study reports the values in terms of shear wave velocity, and we further examined the influence of nerve thickness on nerve shear wave velocity.

A 38-mm linear array transducer (L10-2 MHz; Super Linear, Aix-en-Provence, France) coupled to an Aixplorer ultrasound scanner (version 6.1; Supersonic Imagine, Aix-en-Provence, France) was used for all measurements. Shear wave elastography scans were performed in MSK preset with penetration mode, 100% opacity, no temporal smoothing (persistence = OFF), and intermediate spatial smoothing -5/9. All ultrasound data were collected by the same experienced examiner (RJA) using minimal transducer pressure. The maps of the shear wave velocity were obtained at 1 sample/s with a spatial resolution of 1 × 1 mm.

All ultrasound scans were performed in a single experimental session on the participant's right lower limb. Each participant was imaged in two different positions in a random order: (1) hip neutral and (2) hip flexed at 90 • . For the assessments in hip neutral, participants lay prone on a dynamometer system with both hips positioned in neutral rotation (Biodex 3 Medical, NY, USA). For the assessments in hip flexion, participants sat in long sitting on the dynamometer chair. For both hip testing conditions, participant's right foot was attached to a footplate with the knee fully extended. The lateral malleolus was aligned with the axis of the dynamometer system and considered as an estimate of the center of rotation of the ankle. The ankle joint was then positioned in neutral position, defined as the angle of 90 • between the footplate and the shank. For both testing positions, the contralateral limb was supported in approximately 45 • knee flexion. Based on previous cadaver and shear wave elastography investigations [START_REF] Coppieters | Strain and excursion of the sciatic, tibial, and plantar nerves during a modified straight leg raising test[END_REF][START_REF] Greening | Posture-induced changes in peripheral nerve stiffness measured by ultrasound shear-wave elastography[END_REF], the hip neutral position was expected to induce minimal axial tension on the sciatic and tibial nerve whereas hip flexion would increase substantially the nerve tension.

Spatial variations in shear wave velocity were examined along the course of the sciatic and tibial nerve. Measurements were performed at both thigh and lower leg segments. Skin markings were used to divide the sciatic and tibial nerve into five different regions for ultrasound examination purposes (Fig. 1): Sciatic PROXIMAL , Sciatic DISTAL , Tibial - PROXIMAL , Tibial INTERMEDIATE , and Tibial DISTAL . Four anatomical landmarks were used to locate these regions: (i) the midpoint between the ischial tuberosity and the greater trochanter; (ii) the point where the sciatic nerve bifurcates into tibial and common fibular nerve; (iii) the lateral femoral condyle; and (iv) the medial malleolus (Fig. 1). B-mode ultrasound was used to locate the anatomical references. The sciatic nerve was then divided into two equidistant regions between points (i) and (ii): Sciatic PROXIMAL and Sciatic DISTAL . The midpoint of each region was identified and used as a reference for transducer position. The Tibial PROXIMAL was imaged at the midpoint between (ii) and (iii). The distance from point (iv) and (iii) was measured. Tibial INTERMEDIATE and Tibial DISTAL were scanned at 50% and 10% of this distance, respectively.

Prior to the ultrasound assessments, five passive ankle rotations (from 40 • plantar flexion to 0 • ) were performed at 5 • /s to pre-condition the neuromusculoskeletal structures acting on ankle joint, including the sciatic and tibial nerve. This conditioning procedure was completed for both testing positions. The ankle was then passively positioned in a resting angle (neutral position) and the examination of shear wave velocity was performed in a randomized order for all the five nerve regions. Briefly, the nerve was first identified in the transverse plane. The transducer was subsequently rotated 90 • into the longitudinal plane and aligned with the direction of the nerve fibers. The center of the transducer was aligned with the midpoint of each nerve region. The shear wave velocity map was placed over the nerve and its location remained unchanged during the duration of each clip. The size of the elastography map was defined as the largest size allowed by the scanner. A 10 s shear wave elastography clip was obtained for each region. After recording one clip, the transducer was then immediately moved to the next nerve region, and the procedure for data acquisition was repeated. Real-time shear wave elastography and B-mode windows (Fig. 1) were acquired simultaneously to assess both the nerve shear wave velocity (elastography window) and nerve geometry (B-mode window). The Bmode window was used to: 1) identify region of interest (ROI) for shear wave velocity calculations; and 2) measure the nerve thickness. Participants were instructed to remain completely relaxed during the ultrasound scans, and hence with their lower limb muscles in the passive state.

Data analysis

All data were analyzed offline using customized Matlab (Version 18; The MathWorks Inc, Natick, MA, USA) scripts. Data remained coded so that the data analyses could be undertaken in a blinded manner for testing position and nerve region. Shear wave elastography clips were exported in "mp4" format exported from the system. Clips were then converted (1 Hz) into JPEG images. ROIs were then manually traced (image-by-image) on the adjacent B-mode window using a custom script (ElastoGUI, Laboratory MIP, University of Nantes, France). ROIs were defined as the largest nerve area within the nerve hyperechoic boundaries (epineurium) of the sciatic and tibal nerves, a 2D approximation of the nerve structure based on its thickness. Next, each pixel of the color map (shear wave velocity) was converted into a shear wave velocity value based on the scanner's scale (0-18.3 m/s). Shear wave velocity values within the ROI were averaged to obtain a representative shear wave velocity value (m/s) per frame. Mean shear wave velocity extracted from each frame was then averaged over consecutive frames (total of 10 s), and this value was used for statistical analysis. The area of the ROIs was also calculated and averaged over consecutive images. The percentage of signal void (pixels within shear wave elastography map without shear wave velocity value) and saturation (pixels with maximum shear wave velocity value -18.3 m/s) were quantified for each nerve ROI. If present, void areas were excluded from the analysis. Nerve thickness was calculated from the ROI coordinates and calibrated using a known distance on the image. Thickness was measured as the mean of six different straight lines between superficial and deep nerve boundaries (epineurium) along the longitudinal view of each nerve Bimage.

Statistical analysis

All statistical analyses were conducted in GraphPad Prism (version 8; GraphPad Software, Inc., CA, US). Data are presented as mean ± standard deviation. Differences in nerve shear wave velocity and thickness were assessed using two-way repeated-measures ANOVA with nerve region (Sciatic PROXIMAL , Sciatic DISTAL , Tibial PROXIMAL , Tibial INTERMEDIATE , and Tibial DISTAL ) and hip position (hip neutral, hip flexed) as fixed factors. Post-hoc analyses were performed using Bonferroni tests for multiple comparisons. To investigate the potential influence of nerve thickness on shear wave velocity, univariate linear regression analysis was used to find the best fit line of shear wave velocity to nerve thickness. Pearson correlation coefficients were used to evaluate the relationships between nerve shear wave velocity and body heigh, weight and BMI. This analysis was performed for both Hip neutral and Hip flexed positions and included all nerve locations. The level of significance was set at P < 0.05.

Results

Shear wave velocity

We observed a low occurrence of saturated values (maximum saturation in hip neutral = 0%; maximum saturation in hip flexed = 23.1% and interquartile range = 0.4%) and very small void areas (maximum saturation in hip neutral = 0.8% and interquartile range = 0%; maximum saturation in hip neutral = 2.1% and interquartile range = 0%) on the shear wave velocity measurements (Table 1). The segmented shear wave elastography ROI area per nerve region was 1.2 ± 0.2 cm 2 for Sciatic PROXIMAL , 1.4 ± 0.3 cm 2 for Sciatic DISTAL ,1.4 ± 0.3 cm 2 for Tibial PROXIMAL , 1 ± 0.3 cm 2 for Tibial INTERMEDIATE , and 0.9 ± 0.2 cm 2 for Tibial DISTAL . There were no significant correlations between shear wave velocity and BMI or body weight. A significant weak positive correlation was found between shear wave velocity and height (data comprised all nerve regions and hip positions: r = 0.091, p = 0.027). When subgrouping by nerve region and hip position, this correlation was only positive for Sciatic DISTAL (grouped hip neutral and hip flexed: r = 0.23, P = 0.013; hip flexion only: r = 0.3, P = 0.018; hip neutral only: r = 0.33; P = 0.009).

There was no significant interaction between hip position and nerve region (F(4,236) = 1.4; P = 0.233) on shear wave velocity. Significant main effects of nerve region (F(4,236) = 47.8; P < 0.0001) and hip position (F(1,59) = 310; P < 0.0001) were observed. Specifically, shear wave velocity varied systematically with spatial location regardless of the hip position. Post hoc analysis revelad that shear wave velocities were significantly different between all nerve locations, with the exception of Sciatic DISTAL vs. Tibial INTERMEDIATE (P = 0.999) and Tibial PROXIMAL , vs. Tibial INTERMEDIATE (P = 0.708). Overall, shear wave velocities tended to increase in the proximal-distal direction at both thigh and lower leg segments and were highest in nerve regions near the knee (Tibal PROXIMAL ) and ankle (Tibial DISTAL ) joints (Fig. 2A).

In addition, the main effect of hip position showed a large increase in shear wave velocity (+54.3 ± 45.7%, on average) when the hip was flexed compared to the neutral position (Fig. 3). The absence of significant interaction suggest that the spatial variations in shear wave velocity along the anatomic course of the nerve were not altered by the addition of tensile stress to the nerve (i.e., hip flexion). Notably, hip flexion induced a similar mean (SD) increase in shear wave velocity across all nerve regions: +2.4 ± 1.6 m/s for Sciatic PROXIMAL ; +2.8 ± 1.9 m/s for Sciatic DISTAL ; +2.9 ± 2.1 m/s for Tibial PROXIMAL ; +2.5 ± 1.9 m/ s for Tibial INTERMEDIATE , and + 2.9 ± 1.8 m/s Tibial DISTAL .

Nerve thickness

The analysis of variance revealed no main effect of hip position (F (1,59) = 1.9; P = 0.171) neither an interaction between region and hip position (F(4,236) = 1.1; P = 0.369). A significant main effect of region on nerve thickness (F (4, 236) = 87.3; P < 0.0001) was observed. Overall, it suggests that nerve was thicker along the thigh than at lower leg (Fig. 3B), and that these regional differences were not significantly affected by nerve tensioning. Shear wave velocity was not significantly correlated to the nerve thickness (Hip neutral: r = 0.028, p = 0.623; Hip flexed: r = -0.074, p = 0.199).

Discussion

The results of this study show that shear wave velocity varies spatially along the course of the sciatic and tibial nerve. Shear wave velocities increased in a proximal to distal pattern at both upper and lower leg semgents. In contrast, the regional variations were not affected by the overall tensioning of the nerve. This suggests that axial tension is uniformly distributed along the course of healthy sciatic and tibial nerve when subjected to tensile loads (i.e., stretching).

Averaged shear wave velocities ranged between 4.6 and 6.5 m/s for hip neutral and 7.1-9.4 m/s and hip flexed (Table 1). These values are comparable to those described in previous studies [START_REF] Andrade | Non-invasive assessment of sciatic nerve stiffness during human ankle motion using ultrasound shear wave elastography[END_REF][START_REF] Greening | Posture-induced changes in peripheral nerve stiffness measured by ultrasound shear-wave elastography[END_REF][START_REF] Neto | Noninvasive Measurement of Sciatic Nerve Stiffness in Patients With Chronic Low Back Related Leg Pain Using Shear Wave Elastography[END_REF] for healthy sciatic and tibial nerve. As expected, shear wave velocity increased with the hip flexion across all nerve locations which suggests an increase in sciatic and tibial nerve tensioning [START_REF] Boyd | Impact of movement sequencing on sciatic and tibial nerve strain and excursion during the straight leg raise test in embalmed cadavers[END_REF][START_REF] Coppieters | Strain and excursion of the sciatic, tibial, and plantar nerves during a modified straight leg raising test[END_REF] and demonstrates the elastic nature of peripheral nerves [START_REF] Topp | Structure and biomechanics of peripheral nerves: nerve responses to physical stresses and implications for physical therapist practice[END_REF]. Furthermore, it highlights the importance of considering the hip position on either clinical or biomechanical assessments of lower limb nerves regardless the location where the ultrasound scans are performed.

Because of their stiff nature and finite thickness with respect to the shear wavelength, the propagation of the shear waves has been proven to be guided in tendons [START_REF] Brum | In vivo evaluation of the elastic anisotropy of the human Achilles tendon using shear wave dispersion analysis[END_REF]. This phenomenon cannot be excluded when assessing relatively thin peripheral nerves, and it could bias the nerve shear wave velocities. For example, a thicker nerve may be associated to a higher shear wave velocity for a given shear modulus and vice versa. We speculate that the phenomenon of guided waves may be minor in peripheral nerves for two reasons: 1) the fact that shear waves propagate much faster in tendons than in peripheral nerves; and 2) the smaller difference in shear wave velocity between nerves and the surrounding tissues, as depicted shear wave elastography maps in Fig. 1. This can be partly confirmed by the absence of correlations between the shear wave velocity and nerve thickness. In addition, our data showed that thickness was not affected by the hip angle and the nerve region with the second highest shear wave velocity (i.e., tibial DISTAL ) was also the thinnest, as opposed to the guided wave propagation behavior. We are therefore confident that the spatial variations in shear wave velocity were not biased by the guided wave behavior. However, the regional variations in nerve thickness adds a layer of complexity when interpreting the shear wave velocity mapping. As shown in Table 2, four of ten comparisons between nerve locations may have been influenced by concomitant variations in nerve thickness and nerve shear wave velocity. Future studies should investigate the influence of the guided wave behavior when using ultrasound shear wave elastography to assess the mechanical properties of peripheral nerves. Notably, six of the comparisons between locations showed significant differences in shear wave velocity that were unlikely to have been affected by nerve thickness. Based on these observations, we believe that the sciatic and tibial nerve displays large regional changes in mechanical properties.

Our results showed that shear wave velocity increased in a proximal to distal pattern at both upper and lower leg segments, being greatest in locations near knee and ankle joints (Fig. 2). This finding suggests that sciatic nerve tract exhibits heterogeneous mechanical properties along its course. These results could be due to regional variations in their ultrastructure such as thinner and more numerous nerve fascicles wrapped by a thicker perineurium near joints [START_REF] Hadzic | Hadzic's Textbook of Regional Anesthesia and Acute Pain Management[END_REF][START_REF] Sunderland | The connective tissues of peripheral nerves[END_REF], and an increased epineurial tissue content at joints regions [START_REF] Sunderland | The connective tissues of peripheral nerves[END_REF]. Proximity to joint regions has also been associated to distinct nerve material properties [START_REF] Phillips | Peripheral nerves in the rat exhibit localized heterogeneity of tensile properties during limb movement[END_REF]. For example, Tibial - PROXIMAL was assessed on a highly mobile (knee) joint region, subjected to higher localized strains that can lead to distinct shear wave velocities. Similarly, we found spatial variations in sciatic and tibial nerve thickness (Fig. 2B). This finding strongly suggests that sciatic and tibial nerves exhibit a heterogeneous geometry as they course distally. Our nerve thickness outcomes, which are equivalent to dorsal to ventral measurement in a cross-sectional view, are in line with cadaveric [START_REF] Gustafson | Human distal sciatic nerve fascicular anatomy: implications for ankle control using nerve-cuff electrodes[END_REF] and ultrasound [START_REF] Bruhn | Visualization of the course of the sciatic nerve in adult volunteers by ultrasonography[END_REF] findings. In particular, demonstrating that sciatic nerve is elliptical proximally and becomes circular as it courses distally.

Several studies have investigated the shear wave velocity of the sciatic nerve including the response to passive joint motion and various limb postures [START_REF] Andrade | Non-invasive assessment of sciatic nerve stiffness during human ankle motion using ultrasound shear wave elastography[END_REF][START_REF] Greening | Posture-induced changes in peripheral nerve stiffness measured by ultrasound shear-wave elastography[END_REF], adaptations to acute and chronic tensile loads in either healthy [START_REF] Andrade | The potential role of sciatic nerve stiffness in the limitation of maximal ankle range of motion[END_REF][START_REF] Andrade | Chronic effects of muscle and nerve-directed stretching on tissue mechanics[END_REF] or impaired [START_REF] Neto | Shear Wave Elastographic Investigation of the Immediate Effects of Slump Neurodynamics in People With Sciatica[END_REF] nerves. These studies have typically examined the shear wave velocity at one single location along the course of sciatic and tibial nerves. This does not account for a potential heterogenous nature of shear waves along the length of lower limb nerves, as shown it has been shown in the upper limbs [START_REF] Rugel | Influence of limb position on assessment of nerve mechanical properties by using shear wave ultrasound elastography[END_REF]. The present study constitutes the first comprehensive mapping of the human sciatic and tibial nerve shear wave velocity and demonstrates that transducer location should be standardized for accurate comparison between individuals or conditions. The latter is of major clinical importance when assessing impaired peripheral nerves, especially in length dependent neuropathies such as diabetic neuropathy or chemotherapy-induced peripheral neuropathy.

Our observations also showed that hip flexion induced a similar magnitude increase in shear have velocity along the sciatic and tibial nerve. In contrast with our hypothesis, this finding suggests that forces are homogeneously distributed along healthy nerves and well beyond the location where the stress is applied. Future studies are required to determine whether such tensile properties are altered in damaged nerves or in the presence of interfacing injured muscles, as well as their clinical relevance for non-pharmacological rehabilitation strategies [START_REF] Ellis | Neurodynamics: is tension contentious?[END_REF].

We would like to acknowledge the following methodological considerations. First, the epineurium was excluded from the ROI used to quantify the shear wave velocity. Nevertheless, a recent nerve model suggests a tightly coupled meso-epi-perineurium interacting with a loosely coupled peri-endoneurium [START_REF] Sung | Decoupled epineurial and axonal deformation in mouse median and ulnar nerves[END_REF]. Therefore, the meso-epineurium connections have the potential to modulate the shear wave velocity of the perineurium, and consequently contribute to the spatial variations observed along the course of the nerve tract. Second, skin marks were used to identify the nerve regions where ultrasound scans were performed. This procedure was performed with the hip in neutral position. However, sciatic and tibial nerve may stretch and slide proximally due to the hip flexion [START_REF] Boyd | Impact of movement sequencing on sciatic and tibial nerve strain and excursion during the straight leg raise test in embalmed cadavers[END_REF][START_REF] Robinson | How Much Sciatic Nerve Does Hip Flexion Require? Can[END_REF]. It is therefore possible that the scanned nerve locations differed slightly between hip neutral and hip flexion due to nerve sliding. Third, the hip neutral position might not have been sufficient to fully unload the sciatic nerve tract due to the knee (extension) and ankle (neutral) angles. This limitation arises from the methodology adopted in the published randomized controlled study [START_REF] Andrade | Chronic effects of muscle and nerve-directed stretching on tissue mechanics[END_REF], and could explain some of the variability in shear wave velocity observed in hip neutral. In addition, it is possible that the spatial variations could have been influenced by the hip postures adopted in this study. We acknowledge that we have not compared fully unloaded versus loaded nerve configurations. Furthermore, peripheral nerves exhibit longitudinal undulations in both the nerve core and its fascicles that unravel during stretching [START_REF] Shah | Tissue Biomechanics: Whales Have Some Nerve[END_REF] such as that imposed upon the sciatic nerve tract by the hip flexion. This load-bearing mechanism is likely to modify the overall tissue anisotropy and, thus, to have affected the shear wave velocities. Further studies are required to: 1) compare fully unloaded and loaded sciatic and tibial nerves by manipulating ankle, knee, and hip joints and evaluate their respective usefulness to clinical diagnostic performance; and 2) investigate a larger range of ages.

Conclusion

The present study showed spatial-dependent differences in shear wave velocity in the sciatic and tibial nerve. Although shear wave velocity largely increases with the hip flexion along the course of the sciatc and tibial nerve, the spatial variations in nerve shear wave velocity were not altered by nerve tensioning. This further suggest that axial loads are distributed unfirmly along the course of healthy human nerves when subjected to stretching. Overall, our findings contribute to a better understanding of peripheral nerve biomechanics in-vivo. In addition, they demonstrate the importance of considering both transducer location and limb positioning when using ultrasound shear wave elastography for

Table 2

Inter-regional comparisons for shear wave velocity and thickness and their contribution to the interpretation of the (Young's) modulus in light to the guided wave propagation behavior. Shear wave velocity comparisons highlighted in yellow may have been influenced by nerve thickness and, thus, conclusions about changes in mechanical properties between locations can't be established. Shear wave velocity comparisons highlighted in green indicate that thickness did not influence the shear wave velocity and, consequently, interpretations about spatial variations in mechanical properties can be drawn with confidence.

SWV: shear wave velocity; Modulus: Young's modulus; ⇧: increase; ⇩: decrease; ⇔: no change; ?: unknown.

Fig. 1 .

 1 Fig. 1. Typical example of shear wave elastography measurements per nerve location and hip posture. Shear wave elastography mode allows to simultaneously acquire B-mode and shear wave elastography windows. Numbers represent the location where ultrasound scans were performed. 1: Sciatic PROXIMAL ; 2: Sciatic DISTAL ; 3: Tibial PROXIMAL ; 4: Tibial INTERMEDIATE ; 5: Tibial DISTAL ; (i) midpoint between the ischial tuberosity and the greater trochanter; (ii) the point where sciatic nerve bifurcates into tibial and common peroneal nerves; (iii) the lateral femoral condyle; and the (iv) medial malleolus.

Fig. 2 .

 2 Fig. 2. Panel A. Spatial variations in shear wave velocities (m/s) across the different regions of the sciatic nerve tract: Sciatic PROXIMAL , Sciatic DISTAL , Tibial PROXIMAL , Tibial INTERMEDIATE , and Tibial DISTAL . *P < 0.0001. Panel B. Spatial variations in thickness (mm) across the sciatic nerve tract. *P < 0.0001. For both panels, data corresponds to the ANOVA's effect of region and combines both hip neutral and hip flexed positions. Violin plots show.

Fig. 3 .

 3 Fig. 3. Effects of hip angle position on shear wave velocity across five different regions of the sciatic nerve tract: Sciatic PROXIMAL , Sciatic DISTAL , Tibial PROXIMAL , Tibial INTERMEDIATE , and Tibial DISTAL . Shear wave velocity was significantly greater in hip flexion compared to hip neutral position for all nerve regions. Horizontal lines correspond to the mean ± SD and individual data is shown. *P < 0.0001.

Table 1

 1 Shear wave velocity (m/s), percentage of ROI saturation, and percentage of ROI void areas used for shear wave elastography assessments. Data is displayed as mean ± SD for each nerve region and hip posture.

	Nerve region	Hip position
		Hip	Hip
		neutral	flexed

ROI: region of interest.

Acknowledgements

RJA was supported by a Griffith University Postdoctoral Fellowship Scheme award. This study was financially supported by the Région des Pays de la Loire (QUETE project, no. 2015-09035) and the University of Nantes (Interdisciplinary program). We thank Guillaume Le Sant, Lilian Lacourpaille, Jean-Baptiste Quillard, Sébastien Guillard, Thomas Icre, and Kaoutar Kabbaj for their support in data collection.

biomechanical and clinical examinations of the sciatic and tibial nerve.

Availability of data and materials

The datasets used are available from the corresponding author on reasonable request.

CRediT authorship contribution statement

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.