S Pizzini

J Vogel

S Rohart

L D Buda-Prejbeanu

E Jué

O Boulle

I M Miron

C K Safeer

S Auffret

G Gaudin

A Thiaville

Chirality-induced asymmetric magnetic nucleation in Pt/Co/AlOx ultrathin microstructures

Keywords: numbers: 75, 70, Ak, 75, 60, Jk

The nucleation of reversed magnetic domains in Pt/Co/AlOx microstructures with perpendicular anisotropy was studied experimentally in the presence of an in-plane magnetic field. For large enough in-plane field, nucleation was observed preferentially at an edge of the sample normal to this field. The position at which nucleation takes place was observed to depend in a chiral way on the initial magnetization and applied field directions. A quantitative explanation of these results is proposed, based on the existence of a sizable Dzyaloshinskii-Moriya interaction in this sample. Another consequence of this interaction is that the energy of domain walls can become negative for in-plane fields smaller than the effective anisotropy field.

Chirality is a fascinating property of nature [1]. It was discovered in 1848 by Pasteur, by correlating the optical activity of molecules in solution to the hemihedral shape of the crystals that they form [2]. More generally, chiral textures often appear in physics as a result of symmetry breaking, either spontaneously for example in macroscopic quantum systems [3], or when stabilized by a chiral interaction as in liquid crystals [4] or magnetism [5]. In the latter case, the transcription of the chirality from the atomic scale to the macroscopic scale of textures may be impeded by the existence of an anisotropy. This is exemplified in liquid crystals, where under a DC magnetic or electric field that induces anisotropy, the cholestericnematic transition takes place [4]. Thus, the detection and quantification of a chiral interaction when it is too weak to give rise to a global chiral texture is difficult.

Magnetism is another prominent field where chiral textures are considered. A chiral magnetic interaction indeed exists, namely an anti-symmetric exchange called Dzyaloshinskii-Moriya interaction (DMI) [5,6], that is allowed when central symmetry is broken. In many compounds having this property, especially the cubic B20 structures with depressed magnetic anisotropy, chiral textures like homochiral spin spirals or 2D skyrmion lattices have been observed, both in reciprocal [7] and in real space [8]. Another class of chiral magnetic systems has recently appeared, namely the few atomic layer thick samples grown on an underlayer with large spinorbit interaction, showing structural inversion asymmetry and perpendicular magnetic anisotropy (PMA) [9][10][11][12]. In these systems, PMA is very strong so that chiral spin spirals are not stable, and as a result DMI has remained unnoticed for about 20 years. However, at magnetic edges like a domain wall (DW) separating two uniformly magnetized domains or at physical edges in a microstructure, the competition of chiral interaction with anisotropy is modified. Indeed, the peculiarities of field and current-induced dynamics of domain walls in such samples [13][14][15] have been found to be consistent with a chiral texture localized on the DW, deriving from the presence of interface-induced DMI [16]. These local chiral magnetization textures, that appear as Néel walls of a fixed chirality, have also been observed recently by lowenergy electron microscopy [17,18], on samples with wide domain walls.

In this Letter we show that chiral interactions can also be detected at the edges of a microstructure: in the presence of an additional in-plane field, nucleation of reversed domains takes place preferentially at one edge of the sample, oriented perpendicular to this field. The side at which nucleation takes place depends on the direction of both the additional field and the initial magnetization. This asymmetry is thus chiral, and we show that DMI can explain this chirality as well as the values of the nucleation field.

The experiments were carried out on Pt(3 nm)/Co(0.6 nm)/AlO x (2 nm) layers patterned by electron beam lithography into two large injection pads connected by micrometric strips. The strips were used for field and current-driven domain wall dynamics (not shown here) while the nucleation experiments were carried out on the pads. The films were deposited on a Si/SiO 2 substrate by magnetron sputtering. Samples were oxidized in situ by oxygen plasma in order to induce PMA [19]. Magnetization reversal was studied using magneto-optical Kerr microscopy. In each experiment, magnetization was first saturated with an out-of-plane magnetic field (H z ). Nucleation of reversed domains was then induced by an opposite H z field pulse, under a DC in-plane field H x . Several samples with similar composition and varying magnetic anisotropy were measured. Figure 1 illustrates an example of the occurrence of chiral nucleation and the symmetry of this effect. When an H z field pulse (amplitude 18 -20 mT and length 50 -100 ms) is applied antiparallel to the initial magnetization direction, magnetization reversal is initiated by the nucleation of a reversed domain at a particular spot of the sample, away from the edges, corresponding a local defect (Fig. 1(a)). When a sufficiently strong inplane field H x is applied at the same time as H z , new nucleation centers appear at one edge of the pad. Starting from positive (↑) saturation (corresponding to dark contrast in the Kerr images), a positive H x field (along the positive x axis) promotes nucleation of reversed ↓ domains at the left edge of the sample (Fig. 1(c-e)). As the amplitude of H x increases, the nucleation probability increases but no nucleation appears at the right edge of the sample, up to µ 0 H x = 260 mT. If either the initial magnetization direction or the H x field direction is reversed, the nucleation takes place at the opposite edge. Figures 1(e-f) show indeed that nucleation takes place at the right edge when a negative H x field is applied starting from the same ↑ saturation. Similarly, nucleation takes place on the right edge when H x is kept positive but the initial magnetization is reversed (↓) (Fig. 1(g-h)). This shows that the observed asymmetry is indeed chiral. The nucleation field H z,n was measured as a function of H x for a sample having an anisotropy field of 700 mT (slightly weaker than that of the sample shown in Fig. 1) both for a defect within the film and at the sample edges. The length of the H z pulse was fixed at 20 ms. For a defect within the film, H z,n was defined as the field for which the domain appears with 100% probability. For the edges, H z,n was defined as the field for which 10-15 domains systematically nucleate [20]. The main result of these measurements, shown in Fig. 2, is that while the nucleation field of a domain within the film is almost H x -field independent, H z,n strongly decreases with the in-plane field for the domains nucleating at the sample edges.

The chiral behavior of magnetization reversal cannot be explained by simply invoking a local reduction of anisotropy along the sample edges, which would keep the symmetry between opposite edges. In order to explain the observed chiral nucleation, a phenomenon which breaks the symmetry of the system when an H x field is applied has to be invoked. A possible origin of this phenomenon is the presence in non-centrosymmetric Pt/Co/AlO x stacks of a non-vanishing DM interaction, which has already been invoked to explain the stabilization of chiral Néel walls, called Dzyaloshinskii domain walls (DDW) [16]. We thus quantitatively investigate this hypothesis using two models.

Zero temperature model. In real samples (i.e. including defects), magnetization reversal is controlled by few defects acting as nucleation centers [27]. Chiral nucleation requires defects with a chiral micromagnetic structure around them. DMI provides such a state at sample edges, inducing locally a tilt of the magnetization [28]. When an in-plane field normal to an edge is applied, the tilt angle depends on its orientation (parallel or anti-parallel to the field) and preferential nucleation at one edge can be expected. Using the same 1D model in the x direction normal to the edge as in [28], the edge tilt angle θ is given by

m x = sin θ = ±∆ D 2A + H x H K0 , ( 1
)
where D is the DMI constant, A the exchange constant, H K0 = 2K 0 /(µ 0 M s ) the anisotropy field (K 0 is the effective anisotropy constant), ∆ = A/K 0 the nominal domain wall width and the ± sign refers to the two edges of the sample along x. For D = 0 an easy and a hard branch develop, corresponding to the reversal at the two edges of the microstructure. Inset: complete astroids.

In the absence of thermal fluctuations, the solution for the onset of magnetization instability at the edge can be mapped [20] to a solution of the Stoner-Wohlfarth model [29]. Figure 3

(b) shows the reversal field H z versus H x (normalized to H K0 ), calculated for different D/D c0 (with D c0 = 4
√ AK 0 /π ≡ σ 00 /π giving the onset of the spontaneous formation of magnetization cycloids ). For D = 0, the standard Stoner-Wohlfarth astroid is obtained and no difference occurs between the sample edge and the center. For a finite D, the edge reversal field with H x = 0 decreases by a factor [1 -(2/π)D/D c0 ]. As H x is applied, the astroid splits into two branches revealing the difference between the two sample sides: on the side where m x is initially parallel (resp. anti-parallel) to H x the tilt is larger (resp. smaller) and the reversal field decreases (resp. increases) with H x . This model is in qualitative agreement with the experimental results: i) it explains why in the presence of DMI magnetic nucleation is observed only at one sample edge, ii) it explains qualitatively the decrease of nucleation field as the H x field amplitude is increased. However, the calculated values of the nucleation field are about one order of magnitude larger than the experimental ones. Finite temperature model. In a macroscopic sample, magnetization reversal occurs via the creation of reversed domains followed by the propagation of domain walls. This is described by the so-called 'droplet model' [30,31] well-known for first order phase transitions. Let us first consider the creation of a cylindrical domain of radius R inside the film. The free energy of this droplet is:

E/t = 2πRσ 0 -2µ 0 M s H z πR 2 , ( 2
)
where t is the film thickness, σ 0 = σ 00 (1 -D/D c0 ) the domain wall energy density in the presence of DMI [16], and H z the applied magnetic field. The critical droplet radius is R c = σ 0 /(2µ 0 M s H z ). Below R c the droplet collapses, whereas above R c the domain increases its size by DW propagation. This gives rise to an energy barrier for the nucleation of the droplet:

E B = πσ 0 2 t 2µ 0 M s H z . (3)
In an Arrhenius model with attempt time τ 0 , the nucleation field for a waiting time τ = τ 0 e p reads then:

H n,f ilm = πσ 0 2 t 2µ 0 M s pk B T (4)
Under the assumption that the magnetic droplet structure is completely rigid (i.e no magnetization rotation in the domains nor in the domain wall, no droplet shape optimization), the application of an in-plane field H x does not modify the energy of the droplet: the Zeeman energy gained within the half-droplet having a DW magnetization component parallel to H x is compensated by the loss of energy within the half-droplet with opposite magnetization. This agrees with the results of Fig. 2 which show that the nucleation field for the reversed domain away from the edges does not vary with H x . On the other hand, the energy of a half-droplet nucleating at one edge of the sample is modified by the in-plane field. By again assuming a rigid droplet structure, its energy reads:

E/t = πR(σ 0 ∓ 2∆µ 0 M s H x ) -µ 0 M s H z πR 2 (5)
where the Zeeman energy associated to the in-plane field H x within the DW volume has been included in the DW energy. The ∓ sign refers to the gain/loss of Zeeman energy for a domain wall having its magnetization parallel/anti-parallel to the applied in-plane field, respectively, i.e. to the two sample edges. In analogy with the nucleation within the film, the nucleation field at the edges is then:

H n,edge = π(σ 0 ∓ 2∆µ 0 M s H x ) 2 t 4µ 0 M s pk B T (6)
This shows that the presence of DMI gives rise to different nucleation fields for DW having magnetization parallel or anti-parallel to H x . In a sample with DMI, the DDW created at the two edges starting from saturation have opposite magnetization: this explains the nucleation of reversed domains only at one side of the sample and the symmetry of the effect when in-plane field direction and magnetization saturation are reversed. Note that experimentally, nucleation at the hard side of the sample is never observed at larger H z fields, as the magnetization is always reversed by propagation of the domain walls formed at the easy side for smaller nucleation fields. Figure 4 (lines) shows the variation of the nucleation field as a function of H x , calculated for a droplet within the film and a half-droplet having its magnetization parallel to the applied field. The used magnetic parameters for Pt/Co/AlO x are A = 16 pJ/m, M s =1.1 MA/m, µ 0 H K0 = 700 mT, and

D = 2.2 mJ/m 2 (D/D c0 = 0.7).
It can be seen that the droplet model including the presence of large DMI provides an excellent understanding of the measurements: not only the different variation of H z,n vs. H x within the film and at the easy edge is accounted for, but also the order of magnitude of the reversal fields at the edge is in quantitative agreement with the experiments. In the film, in order to account for the local reduction of the anisotropy field at the defect and therefore reproduce the experimental values of the nucleation field, the energy of the domain wall was reduced by a factor ≈ 0.4 as done previously [32]. Note that the difference between the theoretical and the experimental curve while approaching the x -axis can be explained by the non-perfect compensation of the tilt of the H x field with respect to the sample surface.

Lifting the first two restrictions of the rigid model can be performed semi-analytically, using the 'small circle' model (the wall magnetization distribution is assumed to belong to a plane, that cuts the order parameter sphere along a small circle). This provides accurate DW energies, as was shown long ago [START_REF] Hubert | Theorie der Domänenwände in geordneten Medien[END_REF] and checked again here. Once the orientation-dependent DW energy is known, the optimal droplet shape is obtained using the Wulff construction [START_REF] Desjonquères | Concepts in Surface Physics[END_REF]. In the case of the half droplet, in full analogy with the calculation of the contact angle of a liquid droplet on a surface, the difference of edge energies for up and down domain magnetization (that can be analytically calculated with the same model as used in the Zero temperature section) was introduced in the Wulff construction. For each value of H x , the droplet shapes were first computed (see insets in Fig. 4). Inside the film, an asymmetric elongation along the in-plane field is seen. At the edge, a significant elongation perpendicular to the field takes place, as the DW oriented perpendicular to the field has a much reduced energy and can expand its length. With the shape fixed, the determination of the critical droplet size was then performed [20]. The numerical results of this semi-analytical model, shown in Fig. 4 (symbols) for the case D/D c0 = 0.7, are very close to experiments. The new feature is the decrease of the nucleation field at the defect in the film, as the in-plane field increases. Comparing calculations and experimental data showed that, within this model, D ≥ 0.7D c0 is required to get similar evolutions with in-plane field magnitude [20].

The decrease to zero of the nucleation field, seen both experimentally and in the model, is due to the decrease to zero of the DDW energy under a sufficiently large inplane field. In the case of an in-plane field normal to the DDW, the DW profile can be analytically calculated for an arbitrary value of the in-plane field. For DMI-induced DW moment in the same direction as the field, the DW energy reads (with h = H x /H K0 [35]):

σ = σ 00 [ 1 -h 2 -(h + 2 π D D c0 ) arccos h]. (7)
This falls to zero at h ≈ 1 -D/D c0 . For other in-plane angles between field and DW normal, the zero crossing takes place at larger fields, reaching H K0 when the field is along the DW. This appears to be a unique feature of the Dzyaloshinskii DW.

In conclusion, the nucleation of reversed domains in Pt/Co/AlO x microstructures was observed to be chiral, and could be explained by the presence of a strong DM interaction, already identified as being responsible for the chiral texture of domain walls, observed in some noncentrosymmetric systems. Asymmetric nucleation measurements constitute a straightforward way to determine the sign of D and therefore the chirality of Néel walls. Note that edge nucleation of reversed domains bypasses the topological problem of nucleating skyrmions inside a sample [36,37].

A droplet model including DMI gives quantitative agreement with the measurements, even when assuming a complete rigidity of the magnetic structures. A full treatment of the domain wall profile shows that for large DMI and in-plane field much lower than the anisotropy field the DDW energy becomes negative, a feature hitherto unnoticed. Although all consequences of this specific feature of the Dzyaloshinskii domain walls need to be explored, we already observed nucleation of reversed domains under the application of the sole H x field.

This work was supported by the Agence Nationale de la Recherche, project ANR 11 BS10 008 ESPERADO and by the Nanoscience Foundation, project MIDWEST. SR and AT thank Jacques Miltat for discussions. SP acknowledges the support of E. Wagner and of the Nanofab team at the Institut Néel.

Supplemental Materials

EXPERIMENTAL SETUP AND PROCEDURE

Measurements were carried out using a wide-field Magneto-Optical Kerr Effect microscope in polar geometry using a 630 nm LED. Figure 5(a) shows the variation of the nucleation field H z,n at the edge of the sample, measured for two values of the applied H z pulse length, 20 ms and 500 ms. For a fixed H x field, we have (arbitrarily) defined as the nucleation field the H z field for which 10-15 domains nucleate systematically at the edge. The experimental procedure was the following: with the H x field switched on, a H z pulse of fixed length was applied. The H x field was then switched off, and an H z field pulse was applied to allow the created domain to enlarge, via domain wall propagation, to a size larger than the spatial resolution of the Kerr microscope. In order to account for the unavoidable misalignment of the H x electromagnet that can introduce a weak H z component adding to the H z pulse, the procedure was carried out twice, starting from positive and negative magnetic saturation, and the presented values are the average of the two measurements.

Note that a larger error bar is associated to H z,n corresponding to the smallest H x values. This is due to the fact that when the nucleation field for domains within the film becomes smaller than that at the edges, the observation of domain nucleation at the edge is hindered by the expansion of the domains within the film. The criterium which we have used to define H z,n is more difficult to obtain, making the estimation of this field subject to larger errors. It should be noted that several samples with the same magnetic properties have been measured using the same experimental procedure. The minimum field for which the H z,n field is obtained depends on the number and position of the defects within the film.

We have shown in the manuscript that the nucleation field depends on the length of the H z pulse through the coefficient 1/p where p = ln(τ /τ 0 ). For the two H z pulses of 20 ms and 500 ms, p is respectively 16.8 and 20 when assuming τ 0 = 1 ns, as usually found in the literature. Figure 5(b) shows the H z,n values multiplied by the respective p values. The invariance of the results with respect to p, within the experimental error bars, is in agreement with the theoretical model.

ZERO TEMPERATURE MODEL

We consider a semi-infinite 1D model to describe the magnetization profile in the vicinity of an edge of the sample. Let x be the coordinate normal to the edge, and z the easy magnetization axis. As the field is applied in the (x, z) plane, and as DMI forces a magnetization rotation in this plane too, the single unknown is the angle θ(x) of the magnetization with the easy axis. The energy density reads then

E = A dθ dx 2 -D dθ dx +K 0 sin 2 θ-µ 0 M s (H x sin θ + H z cos θ) (8) where K 0 = K u -µ 0 M 2
s /2 is the effective anisotropy constant including the shape anisotropy. Writing down the Lagrange-Euler equations for the minimization of the total energy and integrating back gives the following first integral

A dθ dx 2 = K 0 sin 2 θ -µ 0 M s (H x sin θ + H z cos θ) + C st (9)
Infinitely inside the sample, the angle is constant (θ 0 in the following) so that the energy reduces to a macrospin one:

K 0 e(θ) = K 0 sin 2 θ -µ 0 M s (H x sin θ + H z cos θ) (10)
The angle θ 0 minimizes this energy, which is solved by the standard Stoner-Wohlfarth problem under a general field [1,2]. Thus, Eq. ( 9) can be re-expressed as

A dθ dx 2 = K 0 [e(θ) -e(θ 0 )] (11)
the right-hand side being indeed positive as θ 0 minimizes e(θ).

At the edge of the sample, the micromagnetic boundary condition with DMI reads [28] 2A

dθ dx = D (12)
Let us call θ 1 the magnetization angle at the edge. From Eq. ( 11) and Eq. ( 12) we have

e(θ 1 ) -e(θ 0 ) = D 2 4AK 0 ≡ 2 π D D c0 2 ( 13
)
This shows that the problem reduces to a two-macrospin problem. The first macrospin θ 0 is independently solved and an analytical solution can be found [1,2]. Concerning the second macrospin (edge magnetization), Eq. ( 13) shows that its energy is larger than the previous one which lets expect a switching at smaller fields than for θ 0 , or in other words, an easier magnetization switching at the edges than inside the film. Around the metastable solution for θ 0 , two solutions for θ 1 can be found which correspond to the two edges, θ 1 -θ 0 being of the sign of D on the right edge (see Eq. 12). If the symmetry is broken (H arbitrarily oriented in space), the two solutions will disappear (which corresponds to a nucleation event) at different applied field magnitudes, revealing the chirality of edge nucleation process. Note that the easiest branches are iso-barrier curves of the Stoner-Wohlfarth problem. These have already been investigated, when studying the switching field of single-domain nanoparticles at non-zero temperatures, both experimentally through single particle measurements using the micro SQUID technique [4], and theoretically in the macrospin model [5]. These curves are simply shrunk astroids. The solutions for D/D c0 = 0 and 0.5 are shown in Fig. 6.

This model neglects dipolar couplings (only the shape anisotropy is taken into account), as generally done for ultrathin films. However, the comparison to full micromagnetic simulations (2D mesh, modified OOMMF code) that include the demagnetizing field exactly, is also shown in Fig. 6. Small differences are seen, especially for small in-plane fields because of the magnetostatic charge that appears at the sample edge reduces the edge magnetization tilt [3]. (b) Sketch of the calculation using Eq. 13 for two different field magnitudes, and for a DMI energy scale indicated by the distance between the two dotted lines. The solutions for θ1 are the intersection between the upper dotted line and the full curves (indicated by the arrows). For the lower field magnitude (in black), two solutions for θ1 are found corresponding to both edges. For the higher one (in red), one solution disappears indicating that one edge magnetization is unstable and that nucleation occurs.

MICROMAGNETIC SIMULATIONS

We consider a magnetic ultrathin film grown on a substrate with a capping layer in a different material so that the inversion symmetry is broken along the vertical axis (z). The magnetization is oriented out-of-plane with a strong perpendicular anisotropy. In addition to the standard micromagnetic energy density which includes the exchange, the magnetocrystalline anisotropy, the Zeeman and the demagnetizing energy, we add the Dzyaloshinskii-Moriya contribution (DM) that reads in a continuous form [6]:

E DM = D[m z ∂m x ∂x -m x ∂m z ∂x + id.(x → y)] (14)
Micromagnetic simulations are based on the Landau-Lifshitz-Gilbert equation integration:

∂m ∂t = -γ 0 [m × H ef f ] + α(m × ∂m ∂t ) (15)
3D micromagnetic simulations are performed using the homemade micromagnetic solver Micro3D [7]. The following material parameters have been used: exchange stiffness A = 1 × 10 -11 J/m, saturation magnetization M s = 1.09 × 10 6 A/m, uniaxial magnetocrystalline anisotropy constant K = 1.25 × 10 6 J/m 3 (along the out-of-plane direction z), Dzyaloshinskii-Moriya constant D = 2 mJ/m 2 , Gilbert damping parameter α = 0.5. They differ slightly from those considered in the main text (mainly by the lower A and higher anisotropy H K0 = 0.92 T), but correspond to a previous work [8].

The thermal fluctuations were neglected and the size of the sample was 100 nm × 1024 nm × 0.6 nm. Starting from positive out-of-plane remanent state, two types of situation were analyzed: i) simultaneous application of an in plane field H x (parallel to the short edge of the sample) and an easy axis H z field (perpendicular to the film plane). The Dzyaloshinskii-Moriya constant was first set to D = 0 J/m 2 and after to D = 2 mJ/m 2 ; ii) application of one single in-plane H x field with D = 2 mJ/m 2 .

Figure 7(a-b) summarizes the results of the simulations carried out applying µ 0 H x = +0.1 T and µ 0 H z = -0.7 T for D = 0 J/m 2 . Starting from out of plane remanent state at t = 0 ps, the magnetization reverses by nucleation of a domain in the center of the sample, where the demagnetizing field is the largest. The magnetization reversal follows a symmetric mechanism. Figure 7(a) shows the time evolution of the in-plane m x component of the magnetization across the short edge of the sample. Snapshots at different moments of time are shown on Fig. 7(b): the color is related to the out-of-plane magnetization and the arrows give the in-plane magnetization vector projection. The magnetization reversal starts in the middle of the sample and propagates to the edges.

Figure 7(c-d) shows that the magnetization reversal changes drastically in the presence of DMI. Before the application of the field, the magnetization at the sides of the sample is tilted towards the center, in opposite directions on either side of the sample. The effect of the x-field is to tilt the magnetization in the direction of the applied field; with a positive H x , the m x component of the magnetization will have a maximum value on the left side of the sample. Magnetization reversal is then triggered by nucleation of reversed domains at this edge, where the m x component of magnetization is initially parallel to the H x field.

Figure 8 (a) shows the effect of an in-plane field µ 0 H x = +0.4 T applied at time t = 0 ps, for the case where = 2 mJ/m 2 . For this value of the x-field, the magnetization simply tilts in the direction of the applied field. The situation is very different for a larger H x field of +0.5 T (Fig. 8(b)). In this case, a domain wall is created starting from the left edge of the sample, and its position stabilizes around the center of the sample due to demagnetizing effects. The creation of this domain wall in the presence of an in-plane field agrees with the results of the model developed in the main text of this Letter, which shows that in the presence of a large DMI, the domain wall energy becomes negative beyond a threshold in-plane field, therefore favoring its formation. Figure 8(c) gives a 2D representation of the dynamics of the magnetization and its evolution towards equilibrium, showing the initial tilt of the magnetization at one edge of the sample, the formation of the domain wall and its propagation away from the edge.

ESTIMATION OF THE MAGNETIC PARAMETERS OF THE SAMPLE

The various micromagnetic parameters of the sample are not exactly known: some of them are directly obtained from characterizations, but some others are more difficult to measure, so that one has to see whether or not, with reasonable values assumed for them, the experimental results can be reproduced. In the case at hand, the relevant parameters are: spontaneous magnetization M s , exchange constant A, effective anisotropy K 0 (the sum of the volumic, interfacial and shape anisotropies), thickness averaged interfacial DMI D, and p the number of thermal energy quanta k B T involved in the Arrhenius process, given by τ = τ 0 e p with τ 0 the attempt time and τ the duration of field application for the domain nucleation. We assume for the calculations that τ 0 = 1 ns. The value 0.1 ns is often quoted but, as in the Brown theory the attempt time is linked to the damping constant α, and because from the DW dynamics experiments [9] a value of α about 10 times larger than for bulk cobalt was deduced, we took this larger value. Note also that taking τ 0 = 0.1 ns increases p by 2.3 only, for a typical value of 18 that corresponds to τ = 66 ms. Finally, the DW energy reduction at the defect in the center of the film is given by the factor .

For M s , we have taken the value 1.1 MA/m. The bulk Co value is 1.4 MA/m. SQUID measurements, with large error bars as the sample thickness is only 3 atomic layers (t = 0.6 nm), are consistent with a lower value of 1.1 MA/m, in agreement with literature.

The effective anisotropy field H K0 = 2K 0 /(µ 0 M s ) was directly measured by extraordinary Hall effect under inplane field. We found µ 0 H K0 = 0.7 T.

The exchange constant was assumed to amount to 16 pJ/m. For bulk Co, the literature contains values ranging from 10 to 30 pJ/m. The value used here corresponds to an estimate based on the Curie temperature of the ultrathin films [9]. We will discuss below what changes if a lower value of 9 pJ/m is taken.

With these parameters, the primitive DW energy density (without DMI, and under no applied in-plane field) is σ 00 = 4 (AK 0 ) 1/2 = 9.92 mJ/m 2 , thus the critical DMI for destabilization of the uniform state is D c0 = σ 00 /π = 3.16 mJ/m 2 . The DW width parameter is ∆ = (A/K 0 ) 1/2 = 6.4 nm. From these values, we can evaluate the scaling z field of the problem, given by

H z00 = πσ 2 00 t 2µ 0 M s pk B T = 4πt pk B T AH K0 (16)
and find µ 0 H z00 = 1172 mT. In order to estimate the value of the DMI parameter D, we have systematically varied the D/D c0 ratio and computed the nucleation fields in the film and at the edge, keeping = 1 for the moment so that the film nucleation Finally, in order to reproduce the experimental value of the nucleation field in the film, we need = 0.42. From Eq. ( 16), we see that the value of the nucleation field in the film will be given by the product A 2 . Thus, with a lower value A = 9 pJ/m, a value = 0.57 would be required. This would however shift the value of the in-plane field where the two nucleation fields are equal, from 105 to 56 mT, whereas experimentally it is about 125 mT.

A word of caution is in order here about the physical meaning of this defect parameter . It was originally introduced in order to model the effect of the defect on the nucleation field [10], in the case of a sole easy axis field, through a reduction of the DW energy. Here, when applying an in-plane field, the DW energy becomes orientation dependent so that the validity of an isotropic reduction of its value by the factor should be investigated in more detail. We leave this as a subject of future work.

FIG. 1 .

 1 FIG. 1. Kerr images showing the chiral nucleation of domains at one edge of the pad of the Pt/Co/AlOx microstructure, by application of an out-of-plane field pulse. (a)-(d): magnetization is initially saturated ↑ and Bx=0, +160, +215 and +260 mT; (e-f) magnetization is initially saturated ↑ and Bx =-160 mT and -260 mT; (g)-(h) magnetization is initially saturated ↓ and Bx is +160 mT and +260 mT. The width of the pad is 70 µm. The dotted lines highlight the left and right edges of the pad and the arrows show the side of the sample where nucleation takes place.

FIG. 2 .

 2 FIG. 2. (color online). Nucleation fields measured as a function of in-plane field Hx for the reversed domain in the middle (squares) and at the left edge of the sample (circles). Note the difference of scales between the two axes.

Figure 3 (

 3 a) sketches the effect of H x and H z on the micromagnetic configurations.

FIG. 3 .

 3 FIG. 3. (color online). a) Sketch of the micromagnetic configuration within a microstructure with DMI in zero applied field (i), under an x field (ii), under an additional negative z field (iii), and after reversal, with a domain wall of magnetization parallel to the x field (iv). b) results of a 1D calculation showing the reversal field for D/Dc0 = 0 (dashes) and 0.5 (lines). For D = 0 an easy and a hard branch develop, corresponding to the reversal at the two edges of the microstructure. Inset: complete astroids.

FIG. 4 .

 4 FIG. 4. (color online). Nucleation field vsHx for a domain at the edge and in the film calculated using the rigid droplet model (full and dotted line) and the relaxed model (dots and square symbols). Insets: calculated droplet shapes for Hx/HK0 = 0 (lines) and 0.21 (dots).

FIG. 5 .

 5 FIG. 5. (a) nucleation fields Hz,n measured as a function of Hx for two values of the applied Hz pulse length, 20 ms and 500 ms. (b) the data in (a) have been multiplied by p, respectively 16.8 and 20.

s y e d g e h a r d e d g e e a s y e d g e w i t h f u l l d i p o l a r c o u p l i nFIG. 6 .

 6 FIG. 6. (a) Nucleation field Hz versus Hx at the edges of the sample, obtained for D = 0 and D = Dc0/2 from Eq. 13. For D = 0 two branches are obtained, depending on the considered edge orientation. The dots correspond to the micromagnetic simulation including complete dipolar coupling (it has been checked that, in the local demagnetizing field approximation, simulations are in perfect agreement with the model).(b) Sketch of the calculation using Eq. 13 for two different field magnitudes, and for a DMI energy scale indicated by the distance between the two dotted lines. The solutions for θ1 are the intersection between the upper dotted line and the full curves (indicated by the arrows). For the lower field magnitude (in black), two solutions for θ1 are found corresponding to both edges. For the higher one (in red), one solution disappears indicating that one edge magnetization is unstable and that nucleation occurs.

FIG. 7 .

 7 FIG. 7. Micromagnetic simulations showing the modification of the micromagnetic structure of the sample under the effect of: (a) µ0Hz= -0.7 T and µ0Hx=+0.1 T for D = 0 J/m 2 -the reversal occurs starting from the center of the sample; (b) µ0Hz= -0.5 T and µ0Hx=+0.1 T for D= 2 mJ/m 2 -the reversal starts from the left edge of the sample.

FIG. 8 .FIG. 9 .

 89 FIG. 8. Micromagnetic simulations showing the modification of the micromagnetic structure of the sample under the effect of an in-plane Hx field applied at time t = 0 ps for D = 2 mJ/m 2 . (a) for µ0Hx = +0.4 T the magnetization tilts in the direction of the applied field; (b) for µ0Hx = +0.5 T a domain wall is formed at the left edge of the sample and stabilizes away from the edge due to magnetostatic effects; (c) snapshots of the magnetization illustrating the evolution towards equilibrium of the domain wall formed under µ0Hx = +0.5 T.


Thierry Mayer
email: tmayer@univ-paris1.fr

MARKET POTENTIAL AND DEVELOPMENT

Keywords: JEL Classification: F15 and O11 Keywords: development, gravity, international trade and market potential

. The empirical part evaluates this market potential for all countries in the world with available trade data over the 1960-2003 period and relates it to income per capita. Overall results show that market potential is a powerful driver of increases in income per capita.

Introduction

This paper provides evidence on the long-term impact of market potential on economic development. Providing explanations for cross-country differences in development levels is perhaps one of the most important question in economics. A large number of alternative frameworks have been proposed, and the literature has recently focused on whether physical geography, culture or institutions matter most in the long term economic performance of countries (Acemoglu et al. 2005 provide a nice summary of the different theories in competition, arguing strongly in favor of the institutions' view). I focus here on a different explanation, where economic geography, synthesized and measured though a market potential index is key in economic development. The paper derives from the New Economic Geography literature a structural estimation where the level of factors' income of a country is related to its export capacity, labeled Market Access (MA) by [START_REF] Redding | Economic geography and International Inequality[END_REF], or Real Market Potential (RMP) by [START_REF] Head | Market Potential and the Location of Japanese Investment in the European Union[END_REF]. The empirical part evaluates this market potential for all countries in the world with available trade data over the 1960-2003 period and relates it to income per capita.

Overall results show that market potential is a powerful driver of increases in income per capita and average wages.

This paper extends our knowledge on how market potential affects development in several dimensions. First and most important, I show that the cross-sectional striking success of the economic geography to predict income per capita in [START_REF] Redding | Economic geography and International Inequality[END_REF] holds when considering panel data. This reinforces their finding strongly, and confirms other recent panel data results, mostly done on an infra-national basis. Second, the results are robust to an instrumentation strategy intended to capture omitted variable bias that would survive the introduction of country-level fixed effects. Third, I allow for a larger set of trade costs variables, notably border effects, colonial preferences and regional agreements, all of them have a time-varying effect in my specification. This enables to do simulations regarding policy changes and how those would affect income per capita through market potential, which is the fourth value added of this paper. The remainder of the paper is as follows: Section 2 spells out the theoretical foundations of my exercise. Section 3 describes the data used, while Sections 4 and 5 present respectively econometric results for the gravity estimates that help build the market potential and the economic development regressions themselves. Section 6 concludes.

2 Theory [START_REF] Redding | Economic geography and International Inequality[END_REF] and [START_REF] Hanson | Market potential, increasing returns and geographic concentration[END_REF] were the first contributions to emphasize the implications of the economic geography model in terms of wage differentials and to apply it empirically to wages in US counties for Hanson, and to the income per capita levels in the world for Redding and Venables. The relationship uncovered explains the level of factor incomes in a country i (wages if labor is the only factor) by a weighted sum of expenditures of all countries in the world. The weights are bilateral trade costs from i to each of the destination countries for i's exports. The resulting term is labeled Market Access (MA) by [START_REF] Redding | Economic geography and International Inequality[END_REF], Market Potential by [START_REF] Hanson | Market potential, increasing returns and geographic concentration[END_REF] or Real Market Potential (RMP) by [START_REF] Head | Market Potential and the Location of Japanese Investment in the European Union[END_REF], the "real" aspect being detailed below. Here I will adopt the market potential terminology in order no to introduce confusion since market access is also often used for describing the level of tariffs and other barriers faced by a country.

The relationship between factor incomes and market potential has been labeled the wage equation. The founding contributions use the now classical Dixit-Stiglitz type of monopolistic competition combined with iceberg trade costs. One might argue that this is maybe not the most relevant framework for developing economies, at least in some industries. It however seems that this prediction is somehow more general than what was originally thought. The main elements for the wage equation to emerge seem to be a gravity structure of bilateral trade combined with some rigidity in the distribution of output shares of different countries in the world. I will here build on [START_REF] Head | Structuring Gravity[END_REF] and try to keep as general as possible.

Gravity and the wage equation

The derivation makes use of the gravity equation that explain the pattern of bilateral trade flows. Gravity involves two important constraints: budget allocation for the importer and market-clearing for the exporter. Consider an exporter country i and an importer country j. Budget allocation considers X j the total expenditure of j to be allocated between exporting countries, and Π ij the proportion of income allocated to country i. By definition:

X ij = Π ij X j , (1)
where i Π ij = 1 and i X ij = X j . The important step to derive a gravity equation from ( 1) is to show that Π ij can be expressed in the following multiplicatively separable form:

Π ij = A i φ ij Φ j . (2)
Loosely speaking, A i represents "capabilities" of exporter i, 0 ≤ φ ij ≤ 1 represents the ease of access of market j to exporters in i, and Φ j measures the set of opportunities of consumers in j or, equivalently, the degree of competition in that market. A wide range of different micro-foundations yield the crucial requirement of equation (2). Those include Dixit-Stiglitz monopolistic competition, Anderson and van Wincoop (2003) model based on national product differentiation, but also comparative advantage models such as [START_REF] Eaton | Technology, Geography, and Trade[END_REF] and more recently models incorporating firms' heterogeneity such as [START_REF] Chaney | Distorted Gravity: the Intensive and Extensive Margins of International Trade[END_REF]. All those models have their budget allocation rule imply a gravity equation for bilateral trade which takes a simple multiplicative form:

X ij = A i × φ ij × X j /Φ j , (3)
and Φ j = h φ hj A h , with different definitions of A i and φ ij depending naturally on the specific structure of the model.

As a second accounting identity, it has to be that the sum of i's shipments to all destinations-including itself-equals the total value of i's production, noted Q i .

Q i = j X ij = A i j φ ij X j Φ j . ( 4
)
If B i is country i's trade balance, we have Q i ≡ X i + B i .
At the world level, j B j = 0, and therefore production must be equal to expenditure, Q = X.

If we have data on both expenditures X j and production, Q i , then the market-clearing condition tell us something about the unobserved attribute of the exporter, A i . To see this define s X j = X j /X = X j /Q as country j's share of world expenditure (and production). Next, define the following term:

Φ * i = h φ ih s X h Φ h . ( 5
)
This term is central in what follows. It is an index of market potential or market access (the same as in [START_REF] Redding | Economic geography and International Inequality[END_REF][START_REF] Head | Market Potential and the Location of Japanese Investment in the European Union[END_REF][START_REF] Hanson | Market potential, increasing returns and geographic concentration[END_REF].

Relative access to individual markets is measured as φ ih /Φ h . Hence, Φ * i is an expenditureweighted average of relative access.

Hence, using ( 5) and ( 4), market-clearing conditions yields a very simple relationship between the exporter's capabilities A i , its share of production s Q i ≡ Q i /Q and its market potential index Φ * i :

A i = s Q i (Φ * i ) -1 or s Q i = A i Φ * i . or A -1 i s Q i = Φ * i (6)
This relationship is very general since it relies only on the gravity assumptions, namely the multiplicative budget allocation rule, and market clearing. The last formulation is particularly illustrative of the forces at work in an economic geography model. An exogenous increase in market potential Φ * i , can translate in two different effects: one on s Q i , one on 1/A i . A i in all models involves a negative function of prices charged by i firms, and therefore also their production costs. Suppose for a minute that those are constant.

A rise in market potential is therefore beneficial to firms located in i, and attracts firms there, raising s Q i . This is the mechanism behind the home market effect, as detailed in [START_REF] Head | Regional Wage and Employment Responses to Market Potential in the EU[END_REF]. On the contrary, suppose that the country's share of production is left unchanged. The rise in market potential will have to be entirely absorbed through a decrease in A i , that is an increase in prices, and therefore wages practised in i. This is the source of the wage equation, which therefore should hold for this class of model when the production structure across countries exhibits some rigidity, which is the case in Dixit-Stiglitz, and in Anderson van Wincoop (2003) notably. In practice, both effects can naturally enter into play, their respective size depending on how rigid are the mobility of factors and wages respectively. In what follows, I develop the wage equation part of this fundamental relationship, and show that it is quite successful empirically, as in the existing literature.

The precise derivation of an estimable wage equation involves to specify two things more precisely: A i and s Q i . In some models, and in particular in Dixit-Stiglitz-Krugman, s Q i is in fact a constant. Indeed in this model, all firms are symmetric and the zero profit condition imposes a uniform firm-level production of q * . The exporter's capabilities being

A i = N i p 1-σ i , we obtain p 1-σ i = N i p i q * N i Q (Φ * i ) -1 ⇒ p i = κ 1 (Φ * i ) 1/σ , (7)
where

κ 1 = q * Q 1/σ
is a constant. This equation means that firms in i faced with a good access to world markets (a high Φ * i ) can increase their price accordingly. 1 To simplify suppose that the production process in i involves an immobile composite factor, labor with share β, price w i and productivity z i . Other factors of production (with share α are supposed to have a constant price over countries, r. Since producer prices in the DSK model are a simple markup over marginal costs, p i = σ σ-1 r α w β i z i , we obtain the wage equation:

w i = κ 2 × z 1/β i × (Φ * i ) 1/(βσ) , (8)
where κ 2 = σ-1 σr α κ 1 1/β is again a constant. Wages in i will be a positive function of the productivity of workers there (z i ) and market potential of the country (Φ * i ). In the empirical part of the paper, I will consider that productivity of an economy i is a positive function of the average years of schooling of its working age population. The empirical counterpart of Φ * i is more complex and deserves its own subsection.

Market Potential computation

In the Dixit-Stiglitz model of trade, the set of alternatives to consumers in h, Φ h is inversely related to the CES price index of accessing all varieties from this country 1/Φ h = P 1-σ h . Market potential can be re-expressed as

Φ * i = 1/Q × h φ ih X h P 1-σ h , (9)
and bilateral trade as

X ij = A i φ ij X j /Φ j = N i p 1-σ i φ ij X j P 1-σ j , (10)
This equation can be estimated using a bilateral trade dataset, taking logs, specifying a vector if trade costs composing ln φ ij and absorbing ln(N i p 1-σ i ) as a fixed effect for the exporter country (FE i ) and ln(X j P 1-σ j ) as a fixed effect for the importer country (FE j ). The market potential can therefore be re-constructed as

Φ * i Q = h φ ih exp( FE j ). ( 11
)
The second stage wage equation in logs becomes then

ln w i = κ 3 + 1 β ln z i + 1 βσ ln RMP i , (12)
where RMP i ≡ Φ * i , and the constant is now κ 3 = ln κ 2 + 1 σ ln Q. As in [START_REF] Redding | Economic geography and International Inequality[END_REF], I consider income per capita to be a natural proxy for the price of immobile factors in i, w i . The real market potential RMP is therefore an element explaining income per capita of the country. An empirical issue with RMP is that it contains own income X i , causing evident endogeneity issue. This problem is all the more important that local trade costs are lower than international trade costs, a well documented fact, known as the border effect, which I will estimate below. A solution that has been proposed by the literature is to calculate a

FMP i ≡ h =i φ ih exp( FE j ),
which does not include own demand of the country. This alleviates the endogeneity problem although it does not constitute an ideal solution as will be clear below.

Data

The needed data for the empirical exercise is fairly standard. The first stage is a fixed effect gravity equation that require bilateral trade flows over a long time period, obtained from IMF DOTS, and a vector of trade impediments, obtained from CEPII. 2 The second stage involves factor incomes on the left hand side, and productivity on the right hand side, combined with the first stage market potential estimate. Following Redding and Venables, I consider income per capita of the country to be a good measure of immobile factor incomes.3 Skill measures come from Barro and Lee.

I will present below RMP estimated from two different methods. They differ in one dimension which is how the effect of national borders is considered. More precisely, in RMP i = h φ ih exp( FE j ), there is an issue about the measurement of φ ii . In addition to having shorter distance, self-trade has a preferential dimension, that has been widely documented in the border effect literature (see [START_REF] Anderson | Trade Costs[END_REF] for a survey of the evidence). [START_REF] Redding | Economic geography and International Inequality[END_REF] deal with this by an adjustment on the distance coefficient, which they divide by two for self-trade in their preferred specification. [START_REF] Head | Market Potential and the Location of Japanese Investment in the European Union[END_REF] adopt a different approach by estimating those border effects in the first step. This method involves measuring self trade for all countries in the world over the period. At the industry level, this is fairly easy, one just has to take global production of an industry, and retrieve total exports to obtain "exports" to self. For aggregate trade, this is a little bit more subtle, since one needs to retrieve total exports from the value of production that is actually tradable in the country. I follow Wei (1996) initial method here and consider the non-service part of the country's GDP to be its tradable part. In what follows, the two methods will be referred to as RV04 and HM04 respectively.

Gravity results

The first step estimates a gravity-type relationship where bilateral trade is regressed each year on a set of importer and exporter dummies and on a vector of trade impediments that is larger than the one used by [START_REF] Redding | Economic geography and International Inequality[END_REF], who focus on distance and contiguity only. The components of φ ij include distance and contiguity, but also common language, colonial links, dummies for common membership of a regional trade agreement (RTA), a currency union (CU) or GATT/ WTO. Summarizing results from the HM04 method estimation, including border effects, the average fit is .73, with an average number of observations around 13000. The average coefficients on trade costs are very much in line with existing findings. The coefficients for distance is very close to -1 and common language, RTA and GATT membership have comparable mean effects around .4.

I present figures of the resulting coefficients over time. The most interesting and puzzling result is the increasing coefficient of distance on trade flows over time in panel (a) of figure 1. This trend is not isolated in the literature. [START_REF] Disdier | The Puzzling Persistence of the Distance Effect on Bilateral Trade[END_REF] report such an evolution in their meta-analysis of distance coefficients in gravity equations. In what is perhaps the most comparable set of results in terms of estimation method, [START_REF] Redding | Distance, Skill Deepening and Development: Will Peripheral Countries Ever Get Rich[END_REF] show in their Table 1, that the coefficient on distance starts at -1.18 in 1970 and rises gradually to end at -1.49 in 1995 (they only include contiguity in the regression as a control for trade costs, which might explain the slightly lower impact of distance in their case in all years).4

Panel (b) of figure 1 shows a more expected result, namely that the impact of national borders decreases over time. Note however that the estimated negative impact of crossing a national border on trade flows remains considerable in 2003, with a dividing factor around 50. This figure naturally aggregates very different situations, and is probably driven by developing countries that are usually estimated to have much larger border effects. Figure 2 present the schedule of estimated coefficients for colonial linkages and common RTA membership across time. The preferential trading relationship between excolonies and their ex-hegemon has a striking downward trend. While the effect remains strongly positive in the early 2000s, the relative deterioration of historical preferences is extremely clear, and should have important consequences for the market potential of 7 the ex-colonies, which are usually small markets located near to other small markets. I will return to that point in the next section. The evolution of the RTA coefficient seems to be strongly influenced by changes in the composition of the main agreements. The effect drops massively around 1973 and 1986 which are dates of significant entries into the European Community (UK, Ireland and Denmark in the first case, Spain and Portugal in the second). Entries of countries into an RTA tends to initially lower the statistical estimate of its effect quite naturally. The effect is also present in 1994, when Mexico adds to the already free trade area between the USA and Canada to form NAFTA.

5 Market potential results

Graphical representation

The above summarized gravity equations enable a computation of market potential indices, RMP i and FMP i , along the lines described in section 2.2, for all countries with available trade data over the 1960-2003 period. This will allow to replicate and much further understand the income per capita / market potential relationship uncovered by [START_REF] Redding | Economic geography and International Inequality[END_REF]. I start by replicating one of their most interesting figure, in which GDP per capita in i is graphed against RMP i and FMP i . I express both in relative terms to the USA in 2003, in order to ease the reading of the axes on figure 3. The existence of a tight relationship between market potential and income per capita is quite clear. Larger and /or more centrally located countries are much richer than countries characterized by a small local market and few or also small neighbors. The case of Belgium and Netherlands is of course very interesting: with the exception of Hong-Kong and Singapour5 , Belgium and the Netherlands are the two top market potential countries in terms of RMP. Looking at panel (b) shows that this comes in great part from their advantageous location, as for Switzerland. Opposed to the case of those countries are the United States and Japan. Both are among the top RMP economies, but that comes almost entirely from their internal demand, since in terms of FMP, panel (b) shows a quite weak position. China and Thailand are similar cases for the developing world. Both have a quite high RMP (which should warrant higher average wages, according to panel a) but a fairly average FMP. Moving away from cross-section, one can exploit the new dimension of my market potential estimates to evaluate whether this tight relationship has had some persistence over time. Figure 4 confirms that this is the case. In 1970, a year where the United States were still the richest economy in the world, or in 1985 for instance, the statistical association of GDP per capita with RMP is obvious.

I continue the illustration with maps. The preceding graphs show an interesting correlation between RMP and income, but makes it hard to detect what is core in the concept .5 1 1.5 real market potential / USA of market potential, the spatial correlation of the forces behind economic development. Indeed, the theory of market potential tells us that being near large markets makes a country richer, and therefore itself a large market. This suggests that in equilibrium, "spatial clubs" of development will form. It will be very hard for a country surrounded by a small and poor economies to reach a high level of income per capita, and inversely, the proximity of large and wealthy countries is a strong advantage in this economic geography world. The maps contained in figures 5 and 6 represent the levels of RMP and FMP in each country in the world, expressed again relative to the United States in 2003. Those are still based on the HM04 methodology. Those figures indeed show evidence of spatial correlation in RMP and even more in FMP. Western Europe, North America and to a lesser extent East Asia are places were the spatial proximity of high GDP countries fuels each other's market potential and therefore income. The case of the United States and its immediate neighbors is illustrative of the problems raised by FMP. While the RMP figure in 2003 predicts the USA to have a much higher income per capita than Canada and Mexico, the reverse is true for FMP. One can also see in the FMP map the extent to which high demand zones exert a positive influence on their neighbors. The "pull-factor" of Western Europe is particularly visible in Eastern Europe and Northern Africa, while central America is clearly benefiting from being close to NAFTA countries in terms of FMP. Both figures, and in particular the Foreign Market Potential one makes very apparent the existence of market potential clubs of countries geographically proximate and having similar rates of high or low income growth that fuel each other market potential, and therefore income growth. East Asian countries are characterized by a very fast growing Figure 6: FMP 2003 market potential during the period, while most if not all African countries are faced with neighbors receding in the worldwide hierarchy of market potential, which dampens their possibilities of economic expansion. In Latin and South America, there seem to be a clear gradient, where proximity to the Northern part of the continent helps the growth of market potential. Note also that Eastern Europe suffers from a low growth of the overall market potential during this period, despite a high growth of their FMP, driven by increased access to Western European markets. Particularly striking is the strong performance of three emerging countries over that period in terms of RMP: Mexico, Turkey and Malaysia. The performance of Turkey is particularly remarkable since figure 8 reveals that its FMP, that is the dynamism of its neighbors, actually decreased during that period. On the contrary, Mexico and Malaysia benefited largely from a very dynamic geographic environment.

Baseline results

As stated above, I have several specifications of the RMP variable: one that follows the [START_REF] Redding | Economic geography and International Inequality[END_REF] approach to internal trade, which simply divides the distance coefficient by half for internal trade. The other following [START_REF] Head | Market Potential and the Location of Japanese Investment in the European Union[END_REF], who estimate instead border effects in the gravity equation, that is the privileged access of producers to their own market from the data, rather than assuming a functional form of distance on it. In order to provide comparison with existing results, I start with the RV04 specification in Table 1. Column (1) mimics RV04, providing results for a cross section of 182 countries in 1995 (they use 101 countries in 1996, but the skills data I use later is only available every five year, including 1995). The coefficient is .58, slightly larger than the .395 they obtain. I interpret the difference, as well as the respectable but lower fit of .445, as the consequence of the much larger sample of countries used. Column (2) pools over the whole set of years available for our countries, and column (3) presents results with country fixed effects, which are to our knowledge the first within estimates of this type of equation. The within results are particularly interesting. Market potential can potentially be correlated with a vast number of other variables relevant to the level and growth of income per capita. This is the rationale behind Table 2 of [START_REF] Redding | Economic geography and International Inequality[END_REF], that includes a large number of controls draw from the development literature. Those include primary resource endowments, other features of physical geography, but also measures of property rights protection, and a dummy if the country was under socialist rule between 1960 and 1985. Most of those controls offer variance that is mostly or exclusively cross-sectional. The use of panel data with country fixed effects permits to control for those and all other factors constant over time, that can affect the level of income per capita. As expected, the coefficients on market potential drops but stays very significant and within a range comparable to the literature on this type of estimates.

The last three columns report coefficients using foreign market potential only. Recall that this is a way to alleviate the endogeneity problem, but not a perfect one. Theory requires own market size to affect the level of factors' income of a country, since those sales to domestic consumers often represent a large part of overall sales. Coefficients are somehow surprisingly larger than for the complete market potential, and the fit is lower (more expectedly). Note that this is also the case in [START_REF] Redding | Economic geography and International Inequality[END_REF]. Once again, the use of panel data reduces the estimated impact of market potential, but leaves it strongly positive and significant. 1995 1960-2003 1960-2003 1995 1960-2003 1960-2003 2), (3), ( 5) and ( 6). Those columns also include a full set of year dummies. Within R 2 reported in columns (3) and ( 6).

The RV04 results are therefore robust to panel data estimation, which is the first important and comforting finding of this paper. The impact of economic geography (market access) on income per capita is not driven by some fixed omitted variable in the cross-sectional regression. The within impact is smaller than in the cross-sectional one, as expected but remains economically large in magnitude. Pushing further the inspection of the impact of market potential, one can naturally be worried that some time varying factor might be omitted from the regression. The first such factor of concern is of course the evolution of average skills in the population. Theory and dozens of empirical paper tells us that education should enter this equation, and might possibly have a relationship with market potential, for instance if the incentives to accumulate human capital are larger in large/central markets. Note that the original [START_REF] Redding | Economic geography and International Inequality[END_REF] paper did not control for skills, although another paper co-authored by Steve Redding shows that indeed the level of skills in a country is related to its market access [START_REF] Redding | Distance, Skill Deepening and Development: Will Peripheral Countries Ever Get Rich[END_REF]. More recently, some papers have included education levels as controls: [START_REF] Head | Regional Wage and Employment Responses to Market Potential in the EU[END_REF]) on a regional level basis, [START_REF] Fally | Economic Geography and Wages in Brazil: Evidence from Micro-Data[END_REF] and Hering and Poncet (forthcoming) at the individual level for Brazilian and Chinese workers respectively.6

Table 2 includes the Barro and Lee measure of average years of schooling among the more than 25 years old in the population of the country. The cross-sectional and pooled results of market potential in columns 1, 2, 4 and 5 are lowered as expected. The preferred within specification however maintains a very significant and high coefficient on both the complete and foreign measures of market potential. 1995 1960-2003 1960-2003 1995 1960-2003 1960 2), (3), ( 5) and ( 6). Those columns also include a full set of year dummies. Within R 2 reported in columns (3) and (6).

Tables 3 and4 replicate the regressions of Tables 1 and2, with the HM04 method of estimating market potential, which introduces border effects directly, rather than through a differential effect of internal distance. Results are very comparable, with a slightly better fit in general, and larger coefficients for market potential variables. Note also the very high coefficients on the skills variable in the non-within specifications. The lower values obtained when using the country fixed effects reinforce the attractiveness of those specifications: the estimates averaging around .10 are now more comparable to what has been found in the above quoted literature (Head andMayer, 2006, Fally et al., 2008, and[START_REF] Hering | Market access impact on individual wages: evidence from China[END_REF]Poncet, forthcoming).

In the following, I stick to the HM04 methodology, including the Barro-Lee control for skills in the country, which is the specification that seem to yield the most interesting results. It is interesting to quantify a little bit more precisely those results, going further than statistical significance. Consider the following experiment: In 2003, take a country with a low RMP, say the Congo Democratic Republic, and one with a large RMP, say Thailand, which in 2003 has an RMP 66 times larger than CDR. Using the 0.37 estimate of column (2) in Table 4, raising the RMP of CDR to the one of Thailand is predicted and [START_REF] Mion | Urbanization Externalities, Market Potential and Spatial Sorting of Skills and Firms[END_REF] for Italy, had all already shown (in specifications less grounded in economic geography theory) that geographic wage differentials are largely influenced by skill differences. 1995 1960-2003 1960-2003 1995 1960-2003 1960-2003 2), (3), ( 5) and ( 6). Those columns also include a full set of year dummies. Within R 2 reported in columns ( 3) and ( 6). 1995 1960-2003 1960-2003 1995 1960-2003 1960-2003 2), (3), ( 5) and ( 6). Those columns also include a full set of year dummies. Within R 2 reported in columns (3) and ( 6).

to increase its GDP per capita by a factor of around 24, while the real ratio in 2003 is around 22. Part of this increase is in fact tautological since own GDP enters RMP as stated above. Another interesting experiment is to raise FMP of a country, which does not include own GDP. Still in 2003, I observe Brazil to be in the tenth percentile of the lowest FMP countries, while Mexico is ranked 18th in terms of FMP, among the top ten percent countries. column (5) estimate, the model predicts that based on a 900% difference in FMP, Mexico should have a GDP per capita around five times higher than Brazil, the real factor being 2.24. Last, one wants to evaluate the size of the market potential impact based on within variance alone. Over the last ten years of our sample (1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003) the average growth of RMP is 111%, and the corresponding figure for FMP is 161%. Using estimates from columns (3) and ( 6), this corresponds to a predicted income per capita growth of 61% and 105% respectively. In addition to the very strong fit of the model, and the very high precision of market potential coefficients, the economic magnitude implied by the estimates is therefore quite large.

Instrumented results

As stated above, substituting FMP to RMP helps to solve partially the endogeneity problem, since own income does not appear any more in the explanation of income per capita. However, it is a significant departure from the theory. Returning to maps helps clarify the point. Comparing figures 5 and 6, some striking differences appear. One of them is for the United States. While the United States has a much larger RMP than Canada and Mexico, it has a much lower FMP than both. If foreign demand was the only driver of factor incomes in the NEG model, Canada and Mexico should both be richer than the USA. On the contrary, the NEG model predicts that the United States should be richer than its two neighbors precisely because it has a large internal demand that makes it a more profitable location for firms. The same paradox of FMP is very clearly appearing for Brazil. Hence FMP has nice features, but is clearly not ideal as an instrument for RMP. What is preferable is an instrument that does not use the income information altogether, but keeps the measures of trade costs, including trade costs to self. We look for an exogenous source of variance of RMP that would come from trade costs, in the cross-section and if possible in the time dimension as well. Geographic centrality of i ( j d -1 ij ) is a good candidate that has been used in the literature, but that does not vary over time. A related instrument that does vary over time is j φ ijt , that is the complete measure of trade costs, including RTAs, currency unions... which vary in membership. Note also that my first-step gravity regression estimates trade costs coefficients (on distance, common language...) for each year. This is another source of variance of the φ ijt over time that can be exploited. Table 5 reports results. The first stage F-test shows that the two proposed instruments are quite powerful determinants of RMP either in the cross-section or in the temporal dimensions. Column (4) is the most demanding, instrumenting while including the full sets of country and year dummies. The first stage regression exhibits an unreported coefficient of .77 on j φ ijt explaining RMP in the pure within dimension, with a t-stat of nearly 13. The second stage result show both a very significant effect of RMP, and a more reasonable coefficient of schooling near .10. Combined with the set of results on FMP, this instrumentation strategy leaves me quite confident that endogeneity, while a potentially serious issue in this type of regression, is not seriously biasing results here. 3) and (4). Those columns also include a full set of year dummies. Within R 2 reported in columns (4).

Simulated policy changes

An application of the wage equation that has not been used before is to evaluate what would be the market potential effect and therefore the impact on wage / income per capita in the country of a trade policy change. For instance, if a country enters a regional trading arrangement (RTA), its access to demand from other members of the RTA improves, which raises its RMP. the wage equation theoretical framework tells us that this should impact the average wage of the country. By how much depends on a certain number of things: The size and locations of the other members of the RTA (which will determine the RMP boost), and the elasticity of wages to RMP. Theory (equation 12) tells us that this elasticity is σβ, the product of the constant elasticity of substitution and of the share of labor in the production function. Let us take reasonable values for those two parameters such that σ = 5 and β = 0.2. Then I run the following experiment: Suppose that in 2003, all RTAs in the world were ended, everything else staying unchanged. What would be the predicted loss of wage / income per capita predicted by the economic geography model? I do the experiment for both RTAs and the WTO and report results for the 50 biggest drops in Table 6. This table reports wage fall in percent of the benchmark wage estimated from the market potential graphed in figure 3. In 2003, the first stage gravity equation reveals that the average RTA raises bilateral trade by exp(0.538) -1 = 71% and that two members of the GATT/WTO have their bilateral trade increased by exp(0.592)-1 = 80%. The global effect then depends naturally on the size and locations of your partners in those agreements.

In a world with no RTA, the countries that would be notably poorer are a group of mostly small but relatively rich economies. The small EU countries would notably lose (Ireland has a predicted loss of 20%), but also Canada and Mexico. The low income countries would not in fact lose a lot in terms of market potential, since the RTAs involving them do not count large and/or rich economies among them, Mali for instance is predicted to lose only 1.7 % of its wage level. The picture is radically changed however if the no-WTO world is considered. The most important losses in this case are for the poorest economies in the world, that we have seen have a very low "local" RMP, and depend very much on demand from far away larger markets of WTO members. For instance, the Malian loss would now be 36%. South-South RTAs might be important for other reasons (for instance because of their pacifying effects, see Martin et al. 2008), but in terms of the market potential sources of income per capita differentials, multilateral trade liberalization seems to be much more important. Those simulations should naturally be taken with great caution. The everything-else-equal assumption of the thought experiment might be more reasonable for some countries than for others, and the results are of course sensitive to the assumed value of σβ and to estimates of trade effects of RTAs and the WTO.

Conclusion

This paper provides evidence that access to markets, measured here as a theory-based index of market potential is an important factor in development. I generalize the theoretical and empirical finding of [START_REF] Redding | Economic geography and International Inequality[END_REF] in many directions, and find very robust evidence that the economic geography of countries matter greatly in their income per capita trajectory. To illustrate, my results show that in 2003, bringing the market potential of the Congo Democratic Republic to the one of Thailand is predicted to increase its GDP per capita by a factor of around 24. The average growth of market potential due to neighbor countries between 1993 and 2003 in our sample is estimated to have raised income per capita by around 105%. (column 2) or WTO (column 4) were to be abandoned. The simulation supposes σ = 5 and β = 0.2 in equation ( 12).

Figure 1 :

 1 Figure 1: The effects of distance and national borders on trade

Figure 2 :

 2 Figure 2: The effects of colonial linkages and regional agreements on trade

Figure 3 :

 3 Figure 3: Market Potential and development in 2003

  Figure 4: Market Potential and development over time

Figure

  Figure 5: RMP 2003

Figure 8 :

 8 Figure 7: RMP rank evolution 1970-2003

  Acemoglu, D., S. Johnson, and J. Robinson , 2005, "Institutions as the Fundamental Cause of Long-Run Growth"in Philippe Aghion and Stephen Durlauf, eds., Handbook of Economic Growth (North Holland). Anderson, J.E. and E. van Wincoop, 2003, "Gravity With Gravitas: A Solution to the Border Puzzle," American Economic Review, 93(1), 170-192.

Table 1 :

 1 Market Potential and GDP/cap RV method

		Dependent Variable: ln GDP/cap
	(1)	(2)	(3)	(4)	(5)	(6)
	ln RMP (RV04) 0.58 a	0.56 a	0.47 a
	(0.05)	(0.04)	(0.01)
	ln FMP (RV04)			0.88 a	0.88 a	0.57 a
				(0.11)	(0.10)	(0.02)
	Time Frame

Table 2 :

 2 Market Potential and GDP/cap RV method -with skills control

			Dependent Variable: ln GDP/cap
		(1)	(2)	(3)	(4)	(5)	(6)
	Average years of schooling 0.38 a	0.28 a	0.03	0.42 a	0.36 a	0.12 a
		(0.04)	(0.03)	(0.03)	(0.03)	(0.02)	(0.03)
	ln RMP (RV04)	0.28 a	0.31 a	0.50 a
		(0.06)	(0.05)	(0.03)
	ln FMP (RV04)				0.42 a	0.39 a	0.60 a
					(0.09)	(0.06)	(0.06)
	Time Frame

Table 3 :

 3 Market Potential and GDP/cap HM method

		Dependent Variable: ln GDP/cap
	(1)	(2)	(3)	(4)	(5)	(6)
	ln RMP (HM04) 0.80 a	0.70 a	0.59 a
	(0.06)	(0.05)	(0.02)
	ln FMP (HM04)			0.88 a	0.88 a	0.58 a
				(0.11)	(0.10)	(0.02)
	Time Frame

  Standard errors in parentheses. Levels of statistical significance are signaled by a (1%), b (5%), and c (10%). Robust standard errors clustered by country in columns (

	Country FE	No	No	Yes	No	No	Yes
	Observations	180	6245	6245	180	6245	6245
	R 2	0.521	0.547	0.748	0.280	0.318	0.711
	Note:

Table 4 :

 4 Market Potential and GDP/cap HM method -with skills control

			Dependent Variable: ln GDP/cap
		(1)	(2)	(3)	(4)	(5)	(6)
	Average years of schooling 0.37 a	0.29 a	0.08 b	0.42 a	0.36 a	0.12 a
		(0.03)	(0.03)	(0.03)	(0.03)	(0.02)	(0.03)
	ln RMP (HM04)	0.41 a	0.37 a	0.55 a
		(0.06)	(0.05)	(0.04)
	ln FMP (HM04)				0.42 a	0.39 a	0.65 a
					(0.09)	(0.06)	(0.06)
	Time Frame

Table 5 :

 5 Market Potential and GDP/cap HM method -with skills control and IV

		Dependent Variable: ln GDP/cap
		(1)	(2)	(3)	(4)
	ln RMP (HM04)	0.40 a	0.40 a	0.30 b	0.35 a
		(0.12)	(0.10)	(0.12)	(0.10)
	Average years of schooling 0.37 a	0.28 a	0.31 a	0.10 a
		(0.05)	(0.03)	(0.04)	(0.03)
	Time Frame	1995	1960-2003 1960-2003 1960-2003
	Country FE	No	No	No	Yes
	IV	j d -1 ij	j d -1 ij	j φ ijt	j φ ijt
	First stage F	31.83	23.21	12.65	127.59
	Observations	108	866	866	855
	R 2	0.809	0.791	0.789	0.797

Note: Standard errors in parentheses. Levels of statistical significance are signaled by a (1%), b (5%), and c (10%). Robust standard errors clustered by country in columns (2), (

Table 6 :

 6 Wage loss predicted from RTA or WTO absence The table reports predicted wage falls if all RTAs

	World with no RTA	World with no WTO
	Country % wage fall Country	% wage fall
	SVK	30.521	KNA	43.867
	CAN	29.48	NAM	40.936
	AUT	26.572	NER	40.246
	BGR	22.272	CAF	40.229
	CHE	21.251	MNG	40.022
	IRL	20.555	TCD	39.699
	CZE	20.341	MMR	39.274
	ROM	20.201	BWA	38.68
	MEX	16.997	ZAR	38.276
	HUN	16.94	PNG	37.565
	NOR	16.488	MOZ	36.971
	MYS	16.35	LSO	36.895
	POL	16.312	MDV	35.868
	SWE	15.425	MLI	35.759
	DNK	12.768	SLB	35.697
	FIN	12.531	BFA	35.003
	BOL	11.784	SVK	34.98
	FRA	9.136	BOL	34.708
	URY	8.394	MRT	33.699
	PRT	7.844	CMR	33.504
	PRY	7.41	SUR	32.971
	BEL	7.211	CAN	32.796
	HND	6.535	ZMB	32.179
	NIC	6.087	BLZ	31.363
	ARG	5.605	MWI	31.226
	ESP	5.511	TGO	30.687
	DEU	4.875	GNB	30.56
	ITA	4.386	MDA	30.209
	IDN	3.728	KGZ	30.158
	CHL	3.613	TZA	30.025
	VNM	3.463	GIN	29.736
	SLV	3.447	ALB	29.47
	VEN	3.414	AUT	29.439
	NLD	3.35	MKD	29.185
	GRC	3.194	BGR	28.758
	COL	3.15	CUB	28.518
	PER	3.005	SWZ	28.494
	GBR	2.671	BEN	28.443
	GTM	2.537	GAB	28.069
	ZMB	2.259	UGA	26.918
	GAB	2.128	ZWE	26.337
	PHL	1.94	SLE	25.458
	MWI	1.814	TUN	25.415
	ECU	1.732	HRV	25.316
	MLI	1.713	BDI	24.893
	MRT	1.657	GEO	24.888
	BFA	1.612	ROM	24.672
	SVN	1.359	PAK	24.319
	SDN	1.244	RWA	24.262
	CRI	1.16	COG	24.228
	Note:

In fact, firms in this model must increase their price if the zero profit condition is to be satisfied, since such an increase reduces the demand they face.

http://www.cepii.fr/anglaisgraph/bdd/distances.htm

It is possible to go into deeper industry level detail, where the LHS variable becomes average wage in the industry.[START_REF] Head | Regional Wage and Employment Responses to Market Potential in the EU[END_REF] do this for a European sample,[START_REF] Paillacar | An empirical study of the changing world economic geography of manufacturing industries: 1980-2003[END_REF] provide data and analysis for a much larger sample of countries and years.

This puzzling increase in the impact of distance might be due to several statistical artifacts. For instance, increased regionalism combined with the end of colonialism in this period. I do however control for those two factors here. Also the increase in the number of trade partners in the database, mostly from small and remote countries might cause this trend. I restricted the sample to the set of country pairs with positive trade for at least 25 years and obtained very similar results.

As can be seen from comparing both panels of figure3, the very high RMP of Hong-Kong and Singapour comes mostly from the internal part. This comes from the fact that a fairly large expenditure is located in both cases on an extremely small territory. The precise internal distance assumed plays a role in such special cases.

[START_REF] Duranton | Mind the Gaps: The Evolution of Regional Earnings Inequalities in the U. K., 1982-1997[END_REF] for the UK,[START_REF] Combes | Spatial wage disparities: Sorting matters![END_REF] for France,

This Discussion Paper is issued under the auspices of the Centre's research programme in INTERNATIONAL TRADE AND REGIONAL ECONOMICS. Any opinions expressed here are those of the author(s) and not those of the Centre for Economic Policy Research. Research disseminated by CEPR may include views on policy, but the Centre itself takes no institutional policy positions.

The Centre for Economic Policy Research was established in 1983 as a private educational charity, to promote independent analysis and public discussion of open economies and the relations among them. It is pluralist and non-partisan, bringing economic research to bear on the analysis of medium-and long-run policy questions. Institutional (core) finance for the Centre has been provided through major grants from the Economic and Social Research Council, under which an ESRC Resource Centre operates within CEPR; the Esmée Fairbairn Charitable Trust; and the Bank of England. These organizations do not give prior review to the Centre's publications, nor do they necessarily endorse the views expressed therein.