B G Giraud

R Peschanski

Profile of a Galactic Spherical Cloud of Self-Gravitating Fermions

Keywords: numbers: 95, 35, +d 95, 30, Cq 98, 52, Wz

The field which binds a thermal fermionic cloud is defined as a Hartree integral upon its density. In turn, the density results from the field via a Thomas-Fermi occupation of the local phase space. This defines a complete theory of all properties and observables for the cloud. As an application to dark matter halos, comparisons with astronomic data on dwarf spheroidal galaxies are provided and discussed. Estimates of the elementary fermion mass are obtained, serving as a phase-space bound on fermionic dark matter.

I INTRODUCTION

While there is strong evidence for the existence of dark matter, the absence of its direct detection triggers many questions, such as i) are we talking about light or heavy particles? ii) are they fermions, such as, for instance, sterile neutrinos, or a new kind of BSM fermions, or are they some bosons? iii) are they self-binding into balls by gravitation only or is there a contribution from some additional interaction between the dark particles? iv) what could be the mass of a dark particle? and so on.

The present work deals with the questions of selfbinding and elementary mass, under very simple hypotheses, namely a fermionic nature of the particles, in non relativistic thermal equilibrium, and a self-consistent gravitational field. Recently, such a hypothesis has been explored for dark matter made of sub-keV degenerate fermions [1,[START_REF] Alexander | Gravitationally bound BCS state as dark matter[END_REF] (see also [START_REF] Randall | Cores in Dwarf Galaxies from Fermi Repulsion[END_REF] which considers a degenerate core surrounded by a thermal envelope). In the following we consider the case of quasi-degenerate fermions, meaning that the temperature to be considered is low enough such that the quantum effects have to be taken into account together with the temperature. For the same reason we are led to consider fermions presently in a nonrelativistic thermal state. The comparison with data is made for fermionic clouds of dark matter supposed to be associated with the known Dwarf Spheroidal Galaxies (DSG) in the vicinity of the Milky Way [START_REF] Simon | The Kinematics of the Ultra-Faint Milky Way Satellites: Solving the Missing Satellite Problem[END_REF][START_REF] Gilmore | The Observed properties of Dark Matter on small spatial scales[END_REF].

More generally, and independently from this precise hypothesis, the determination of the properties of thermal fermionic galactic clouds proved to be useful to determine model-independent phase-space lower bounds on the mass of fermionic dark matter, starting with the pioneering Ref. [START_REF] Tremaine | Dynamical Role of Light Neutral Leptons in Cosmology[END_REF] which considered an initial thermal relativistic stage of the fermion cloud. The bound is due to the existence of a maximal phase-space density which stays valid after further collision-less and dissipation-less evolution, leading to a quite model-independent lower mass bound for the fermion. Various phase-space bounds related to this method have been discussed and evaluated by different ways since then, noticeably in Refs. [START_REF] Dalcanton | Halo cores and phase space densities: Observational constraints on dark matter physics and structure formation[END_REF][START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF][START_REF] Destri | Fermionic warm dark matter produces galaxy cores in the observed scales because of quantum mechanics[END_REF][START_REF] Shao | The phase space density of fermionic dark matter haloes[END_REF].

In our scheme, since our calculations define a complete theory of all properties and observables for the cloud, depending on its temperature, the determination (or the lower bound) of an allowed elementary fermion mass from the cloud finds a natural application to the problem of the elementary fermion mass using DSG data.

We will here define the gravitational field by a simple Hartree convolution integral between the basic Newton attraction and the dark matter density. In turn, this density will result from the field according to a Thomas-Fermi approximation. This allows a closing of the algebra into a self-consistent equation for the field, or a self-consistent equation for the density. Such equations are reminiscent of the Poisson equation and can be easily solved numerically. We thus obtain a complete description of the profile of a dark object, and of properties estimated by astronomic observations, such as velocity dispersion and radial extension. Our description is parametrized by five parameters, namely Planck's constant h, the Newton gravitational constant G, the elementary dark particle mass m, a temperature T and a scale parameter, that turns out to be a non-relativistic velocity v 0 . Then the simple structure of our theory provides a formula that calculates the elementary mass m in terms of the other parameters and of astronomic observations.

The equations obeyed by a self-gravitating fermionic cloud at a temperature T are displayed in Section II. They provide a rich zoology of density profiles and related observables, via both analytical and numerical approaches. Section III is a numerical application of our results to astronomic cases. It shows somewhat realistic numbers, on the galactic scale. We will even find a glimpse of an answer to the question of the elementary masses, leading to an evaluation of a lower mass bound on a dark fermion, when combining the data on the various DSG's. Finally Sections IV and V offer a discussion and conclusion.

II BASIC EQUATIONS AND FORMAL CONSEQUENCES

Spherical symmetry is assumed throughout this paper. In a first part, corresponding to degenerate fermions (T=0), we also assume (and indeed check) that the dark matter cloud has a finite radius R. The dark particles, assumed to be spin 1 2 fermions with mass m, are bound by their self-consistent potential field φ. Let r mean the position coordinates and set r ≡ | r|. Then,

φ(r) = -G d 3 r ′ ρ(r ′ ) | r -r ′ | , (1)
also reads,

1 4π G φ(r) = - 1 r r 0 dr ′ r ′ 2 ρ(r ′ ) - R r dr ′ r ′ ρ(r ′ ) . (2)
Here, G is the gravitational constant and ρ(r ′ ) is the mass density of the dark matter inside the spherical shell contained between radii r ′ and r ′ + dr ′ . The first term in the right hand side of Eq.( 2) accounts for the mass internal to the sphere with radius r. The second term accounts for the mass in the "corona", between r and R.

At the surface, the field reaches the value, φ(R) = -GM/R, where M is the total mass, M ≡ M (R) = 4π R 0 dr ′ r ′ 2 ρ(r ′ ). Above the surface, φ continues as, φ(r) = -GM/r, but, in the following, we shall be concerned with internal properties only, namely 0 ≤ r ≤ R.

The centripetal force reads, upon derivating Eq.( 2),

- dφ dr = - GM (r) r 2 , M (r) = 4π r 0 dr ′ r ′ 2 ρ(r ′ ). ( 3
)
The field φ, an increasing function of r, is negative ∀r.

It can be noticed, incidentally, that the definition of φ from Eq.(1) makes φ be a Hartree potential [11]. In principle, one should also consider a Fock potential [12], the result of exchange terms in the fermionic gravitational interaction. This Fock term is neglected here.

With the well known, Thomas-Fermi approximation [13], the fermionic population of each volume element of this large system is assumed, at zero temperature, to make a Fermi sphere in momentum space, with some radius p F (r). Accordingly, the mass density reads,

ρ(r) = m 8π 3h 3 [p F (r)] 3 , ( 4
)
where h is Planck's constant and m is the dark matter particle mass. At position r the Fermi momentum, p F (r), is defined from a maximal energy level, m µ, available for occupation,

p 2 F (r) 2m + m φ(r) = m µ, (5)
where the product, m µ, defines a chemical potential. That radius R, where φ equates µ, induces p F (R) = 0, hence the "local Fermi sphere" shrinks to zero and the Thomas-Fermi approximation experiences difficulties with a turning point. In practice for the present theory, the density, ρ(R), vanishes according to Eq.( 4) since then p F = 0. This defines the surface as we see it. Accordingly, we set µ = -GM/R.

Let us now consider a cloud in a thermal equilibrium, the likely result of a cosmological period when dark matter was not yet decoupled. For a finite temperature T the Fermi sphere at position r is partly filled only, and there are "compensating" fermions above the Fermi surface, both facts described by the well known Fermi occupation formula,

ρ = 8πm h 3 ∞ 0 p 2 dp 1 1 + exp p 2 2m + mψ /(kT ) , (6)
with k the Boltzmann constant and ψ ≡ φ-µ. We might still define the surface by the condition,

ψ(R) = 0 , (7)
but, above that surface, namely when ψ becomes positive, the formula, Eq.( 6), allows for residual fermions. Their probability in momentum space will become exponentially small, but a density tail occurs in coordinate space. It is easy to verify that, actually, the obtained solutions tell that ψ(r) is of order log r and ρ(r) of order r -2 , when r → ∞. Hence a divergence occurs for the total mass, another concern with the validity of a Thomas-Fermi approach if one uses it too brutally. For the sake of caution and rigor, we shall present in the following, when necessary, two sets of results at least, namely i) a set where integrals upon r obtain from a cut-off at R as defined by Eq.( 7), then ii) a set where integrals are extended until 1.5 R, for example, or more, to show tail effects. (One might also argue that a momentum cut-off is necessary to avoid relativistic situations, but we will stay in the non-relativistic regime.) It is now convenient to introduce a velocity v 0 such that

ψ 0 ≡ ψ(0) = -v 2 0 /2, then the scalings, p = m v 0 q and ψ(r) = v 2 0 χ(r)/2
, where q and χ become dimensionless. Then χ 0 ≡ χ(0) = -1, χ(R) = 0 and Eq.( 6) reads,

ρ(r) = 8π m 4 v 3 0 h 3 ∞ 0 q 2 dq 1 + exp{ [q 2 + χ(r)]/η } , (8)
where η = k T /(m v 2 0 /2). In the following, we shall only consider cases where η < 1, for the obvious reason that quantum effects might be washed out if the thermal energy order of magnitude, kT , exceeds the kinetic energy order of magnitude, mv 2 0 /2. The density also reads,

ρ(r) = - 2 (π η) 3 2 m 4 v 3 0 h 3 Li 3 2 , -e -χ(r)/η , ( 9
)
where Li is the polylogarithm function. Define now,

L 3 (χ, η) ≡ 3/4 √ π η 3 2 Li 3 2 , -e -χ/η . (10)
Notice that,

lim η→0 L 3 (χ, η) = -(-χ) 3 2 , (11)
making easier the transition with the degenerate fermion case. Moreover we find

ρ(r) = - 8 π m 4 v 3 0 3 h 3 L 3 (χ, η). ( 12
)
The second of Eqs.(3) then becomes,

M (r) = - 32 π 2 m 4 v 3 0 3 h 3 r 0 dr ′ r ′ 2 L 3 [χ(r ′ ), η] . (13)
Since dφ/dr and dψ/dr are trivially equal, because µ is a constant, the first of Eqs.(3) reads,

- dχ dr = 64 π 2 m 4 v 0 G 3 h 3 r 2 r 0 dr ′ r ′ 2 L 3 [χ(r ′ ), η] , (14)
where we again used the scaling relating ψ and χ. A further derivative yields,

1 r 2 d dr r 2 dχ dr = -K 0 L 3 [χ(r), η] , (15)
with

K 0 = 64 π 2 m 4 v 0 G 3 h 3 . ( 16
)
This Eq.( 15) is actually the Poisson equation for the present problem. Define now a length scale r 0 by the condition, r 2 0 K 0 = 1, namely

r 0 = √ 3 h 3 2 8 π m 2 √ v 0 G . (17)
Then implement the scalings, r = r 0 x, R = r 0 X, and χ(r) = ξ(r/r 0 ). This yields,

1 x 2 d dx x 2 dξ dx = -L 3 [ξ(x), η] . (18)
This simplifies into, 1

x 2 d dx x 2 dξ dx = [-ξ(x)] 3 2
, for η = 0. We show in Figure (1) the solutions ξ(x) numerically obtained for η = 0, .1, .3., .5, respectively. We used the following boundary conditions, ξ(0) = -1 and dξ/dx| x=0 = 0. The latter means that the field has a smooth minimum at the center of the dark ball.

Then we show, with a sequence of dots in Figure ( 2), the evolution in terms of η of the "scaled radius" X where ξ vanishes. This is of interest if the physical scale of the ball radius is defined as, R = r 0 X. It is seen that X diminishes when η increases. The same Fig. shows (long dashes) that the density at the origin increases, and (short dashes) that the radius, Y , where the density is half the density at center, ρ(Y ) = ρ(0)/2, increases, but slightly only. Notice that such conclusions, and most forthcoming ones, are valid only for a fixed value of v 0 when η changes. Cases with different values of v 0 cannot be so trivially compared, since, given m, the present model of a dark ball is driven by two independent physical parameters, namely v 0 and the temperature, T = (mv 2 0 /2)η/k. Then T occurs in η only but v 0 influences both η and r 0 . In Figure (3), we compare densities. The comparison is done for normalized profiles, ρ n (x) ≡ ρ(x)/ρ(0). The full line corresponds to η = 0 and the long, moderate and short dashes to η = .1, .3, .5, respectively. Tails grow when η increases.

As is seen in Figure ( 4), the trend of the total mass is not the same whether one calculates it with integrals limited to X or extended to 1.5 X. In the first case, M first increases, then decreases as a function of η. The second case cannot return a mass including tail effects for η = 0, since there is no tail. A systematic increase is then found for finite values of η.

Of special interest for our subject is the traditional observation [START_REF] Binney | Galactic Dynamics[END_REF] of the radial velocity dispersion for stars and the cosmological hypothesis that the star dispersion is equal to that of the dark matter particles. Since in our Thomas-Fermi scheme the phase space density reads,

ρ ps ( r, p) d r d p = 2 h -3 d r d p 1 + exp{[ p 2 2m + mψ]/(kT )} , (19)
the radial velocity dispersion for the dark particles is, inside a ball of radius R,

p 2 3m 2 = 32π 2 3h 3 m 2 R 0 ∞ 0 r 2 dr p 4 dp 1 + exp{[ p 2 2m + mψ]/(kT )} , (20)
namely, after now familiar scalings, p = q m v 0 , r = x r 0 , R = X r 0 and

ψ = ξ(r/r 0 ) v 2 0 /2, p 2 3m 2 = -K v 2 0 X 0 x 2 dx Li 5 2 , -exp -ξ(x) η .
(21) with K = 4 (π η) 5 2 (r 0 m v 0 /h) 3 . Actually, it may be more interesting to use here the function,

L 5 (ξ, η) ≡ 15 √ π η 5 2 8 Li 5 2 , -e -ξ η , (22)
with the advantage that lim η→0 L 5 (ξ, η) = -(-ξ)

5
2 . Then Eq.(21) becomes

p 2 3m 2 = - 32 π 2 (mv 0 r 0 ) 3 v 2 0 15 h 3 X 0 x 2 dx L 5 [ξ(x), η]. (23)
Note that the particle number reads,

N = 32 π 2 h 3 R 0 ∞ 0 r 2 dr p 2 dp 1 + exp{[ p 2 2m + mψ]/(kT )} , (24)
or, as well,

N = - 32 π 2 (mv 0 r 0 ) 3 3 h 3 X 0 x 2 dx L 3 [ξ(x), η]. (25)
At T = 0, Eq.(20) reduces to,

p 2 3m 2 = 128 √ 2 π 2 m 3 15 h 3 R 0 r 2 dr ψ(r) 5 2 , (26)
hence, after scaling,

p 2 3m 2 = 32 π 2 (mv 0 r 0 ) 3 v 2 0 15 h 3 X 0 x 2 dx ξ(x) 5
2 , X ≃ 3.6537.

(27) Similarly, at T = 0,

N = 32 π 2 (mv 0 r 0 ) 3 3 h 3 X 0 x 2 dx ξ(x) 3 2 , X ≃ 3.6537. ( 28
)
For each value of r, or of the scaled radius x, we may define, from the previous formulas, a local average radial dispersion, L 5 /L 3 , where we discarded all inessential coefficients. We show in Figure [START_REF] Gilmore | The Observed properties of Dark Matter on small spatial scales[END_REF], from top to bottom, the corresponding curves for η = .7, .5, .3, .1, respectively. The length of dashes increases as η diminishes. The full line in Figure [START_REF] Gilmore | The Observed properties of Dark Matter on small spatial scales[END_REF] shows the limit ratio, -ξ, corresponding to T = 0. The line stops beyond the sharp surface radius, X = 3.6537, naturally. Note how all finite temperature curves seem to stabilize when x exceeds that critical value, 3.6537. This might help defining empirically a "surface" for the ball, as a transition between a "core" and a "tail", typically at a radius of order, R ∼ 4r 0 . From Eq.( 23) and Eq.( 25), the radial dispersion, when averaged over the whole ball, reads

∆ ≡ σ 2 ≡ p 2 3m 2 /N = v 2 0 5 X 0 x 2 dxL 5 (ξ, η) X 0 x 2 dxL 3 (ξ, η) , (29)
where, for each η, we take X as the root of ξ or extend X to 1.5 that root. For the case, T = 0, L 5 and L 3 are replaced by their now familiar limits and, obviously, there is no "extended integral" result. In Figure [START_REF] Tremaine | Dynamical Role of Light Neutral Leptons in Cosmology[END_REF], where the coefficient, v 2 0 /5, is voluntarily omitted, the results of this averaging of the radial velocity dispersion over the ball volume are shown by a full line for integrals limited by the field roots and by dots for "extended integrals". It will be noted that tail effects seem to be small for the present observable. But, as might be expected, the dispersion increases as a function of η.

Another concept used in galactic astronomy [START_REF] Binney | Galactic Dynamics[END_REF] is the surface density, ρ(s), defined as follows. Consider an equatorial plane of the dark ball and a (positive or negative) height z measured from that plane. Coordinates inside the plane may be chosen as polar coordinates, {s, θ} and the triplet, {s, θ, z}, make a simple set of cylindrical coordinates. Actually, the angle, θ, will be irrelevant in the following. The radius from the ball center reads, obviously, r = √ s 2 + z 2 . Given a radius R for the ball, the surface density is then defined as,

ρ(s) = 2 √ R 2 -s 2 0 dz ρ s 2 + z 2 . ( 30
)
Upon taking advantage of Eq.( 9), this becomes, after an obvious scaling of the form, R = r 0 X, z = r 0 ζ and s = r 0 σ, where

ρ(r 0 σ) = - 8 π m 4 v 3 0 r 0 3 h 3 Z(σ) 0 dζ L 3 ξ σ 2 + ζ 2 , η , (31)
Z(σ) = √ X 2 -σ 2 .
Here again we shall compare situations where X is the root of ξ or is increased by 50% and even 100%. The results are shown in Figures [START_REF] Dalcanton | Halo cores and phase space densities: Observational constraints on dark matter physics and structure formation[END_REF], [START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF] and [START_REF] Destri | Fermionic warm dark matter produces galaxy cores in the observed scales because of quantum mechanics[END_REF]. respectively. At low temperatures, a saturation of tail effects is seen in Figure [START_REF] Shao | The phase space density of fermionic dark matter haloes[END_REF], where Figs. [START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF] and (9) are fused.

A frequent observable used in astronomy is that ra- 8), but now with integration ranges twice the roots of ξ. dius, r c ≡ r 0 τ , where the surface density reaches half the surface density at center of the equatorial plane, namely,

Z(τ ) 0 dζ L 3 ξ τ 2 + ζ 2 , η = X 0 dζ L 3 [ξ(ζ), η] /2 .
(32) Clearly, this r c is not observed directly. Rather, it is estimated from luminosity, decreasing from the center of the galaxy towards its edge. When the luminosity has diminished by 50%, it is assumed that the same is true for the underlying dark matter surface density.

We show in Figure (11) four sets of solutions ρ(τ ; η) of Eq.( 32). In all four sets, the mid-density increases as a function of η. The first set, where the ranges of integrals are defined by the nodes of the ξ's, is illustrated by 8 dots, corresponding to η = 0, .1, .2, ..., .7, respectively. It indicates that τ first increases then decreases. The second set results from integration ranges extended by 100%. It contains 7 points, with η = .1, .2, ..., .7, and is illustrated by squares. It hints again a lack of monotonicity for τ , but much less pronounced. The third set, illustrated by 7 diamonds, corresponds again to η = .1, .2, ..., .7. It is obtained with ranges 5 times those of the first set and exhibits a monotonic trend for τ . For the fourth set, shown by triangles, with the same 7 values of η, the range extension factor is 10. The set does not differ much from the previous one and a saturation of tail effects can be expected. Notice how the four patterns converge when η diminishes. Actually, the lowest point of the first pattern (η = 0, no extension) may be considered as belonging to all patterns as well. We list here the 7 values of τ obtained, when η = .1, .2, . . . , .7, and the factor of integration range extension is 10 : {1.62, 1.73, 1.85, 1.95, 2.02, 2.07, 2.11}. Now that we have at our disposal both quantities, σ 2 = p 2 /(3m 2 N ) and τ , we can return to realistic, unscaled quantities and consider the ratio, which has the advantage of being independent from v 0 and m. For η = 0, with integral ranges confined inside R = 3.6537 r 0 when we calculate surface densities, we find, numerically, R ≃ 1.60. For η = .1, . 11), because we want to take full tail effects into account. It is obvious that all results are compatible with a value R ≃ 1.55 ± .05. This privileges the King radius [START_REF] Binney | Galactic Dynamics[END_REF] found in the literature. Somewhat disappointing is the fact that our results for R show little dependence on η therefore, cannot suggest a measure of the temperature of the dark matter cloud. But independence from parameters, on the other hand, may lead to solid experimental testing. Assume that the integration range X is large enough to saturate tail effects. Then, according to Eq.( 29), v 2 0 = 5 σ 2 F (η), where, F , in a schematic notation, reads

R ≡ σ 2 G ρ 0 r 2 c = σ 2 G ρ 0 r 2 0 τ 2 = 8π L 5 5 L 3 (-1, η) τ 2 L 3 , (33)
F (η) = L 3 / L 5 .
It is easy to tabulate. As well, according to Eq.(32), τ is a function G(η), for which we listed seven values just a few lines above. Now, according to Eq.( 17), we can invert logics and obtain a formula for the mass,

m 4 = 3 h 3 64 π 2 v 0 G r 2 0 = 3 h 3 [G(η)] 2 64 √ 5 π 2 σ F (η) G r 2 c . (34)
With the amplification of the integration range taken as 10, the list of values for F (η), when η = .1, .2, . . . , .7, reads, {1.19, 1.02, .85, .71, .61, .53, .47}. With explicit numerical constants, Eq.(34) also reads, in eV units for mc 2 , then km/s for σ and pc (parsec) for r c , When reduced to its 1/4-th power, this coefficient does not seem to be very influential.

III REALISTIC ILLUSTRATIONS

Standard values in SI units for physical constants read, k = 1.38 10 -23 , hence k = 8.6 10 -5 eV K -1 , h = 6.6 10 -34 , G = 6.7 10 -11 , c = 3.0 10 8 . For the mass of dark matter particles, we shall first consider that a tolerable guess is, m ≃ 3.6 10 -34 , hence, mc 2 ≃ 200 eV . If v 0 ≃ 10 -4 c, a non relativistic situation, then r 0 ≃ 6.4 10 18 , i.e. of order 210 pc. Recall that, X ≃ 3.5 for T = 0, and that, for positive temperatures, tails may more than double the qualitative estimates of the radius. Then the cloud size may reach the kpc range. Incidentally, with such estimates for m and v 0 , the coefficient, γ ≡ .5 m v 2 0 /k, which converts η into a temperature T , is of order ≃ 0.01. We are here, therefore, in a situation of very low temperatures if η is kept ≤ 1.

With different parametric assumptions, such as a lighter mass, m ≃ 2.0 10 -34 ≃ 110 eV /c 2 , and a higher reference velocity, v 0 ≃ 2 10 -3 c, we obtain r 0 ≃ 4.6 10 18 , about 150 pc, and the temperature coefficient becomes, γ ≃ 2.6. This value of γ brings that range of η values, studied in this paper, able to induce temperatures much closer to the CMB temperature order of magnitude.

From astronomic observations there might be some hope for measuring the center density, ρ 0 ≡ ρ(0). Typically, ρ 0 ∼ 10 -21 kg/m 3 . From Eq.( 4) and again a guess of m ≃ 110 eV /c 2 , such a value of ρ 0 would return,

p 0 ≡ p F (0) ∼ 5.6 10 -23 (ρ 0 ) 1 3 ∼ 5.6 10 -30 . (36)
This gives v 0 ≃ 2.8 10 4 m/s ≃ 9. 10 -5 c, lower than, but not too far from the guesses made a few lines above. More realistically, let us directly use Eq.( 35) and introduce orders of magnitude σ = 10 4 m/s and r c = 400 pc, as suggested by astronomic observations. Then we obtain the string of estimates, mc 2 = {195, 206, 217, 228, 237, 244, 250} eV , according to η.

From [START_REF] Simon | The Kinematics of the Ultra-Faint Milky Way Satellites: Solving the Missing Satellite Problem[END_REF], we extract the following table of astronomic data and corresponding results, For each astronomic object, we calculated the resulting elementary mass, seen in the right-hand-side column of the table, from Eq.(35) with the "η coefficient" taken as 4.5, an average value. Let ∆r c and ∆σ be the absolute errors for r c and σ, respectively. Then, according to Eq.(35), we estimated the relative error for each estimated m as, ∆m/m = 4(∆r c /r c ) 2 + (∆σ/σ) 2 /4. As seen in the table, this elementary mass spreads between 200 and 370 eV , with error bars also spreading, between ±30 and ± 50 eV . The pattern of values and error bars is illustrated by Figure (12), their representation in increasing order being purely a convention. 12. Masses and error bars deduced from the astronomic data in [START_REF] Simon | The Kinematics of the Ultra-Faint Milky Way Satellites: Solving the Missing Satellite Problem[END_REF]. Each integer abscissa means rank in Table I.

To compensate for this lack of precision, we implemented a χ 2 fit for the obtained m i , m i -∆m i and m i + ∆m i , i = 1, ..., 9. This gives, mc 2 = 290 ± 30 eV , with χ 2 = 2.5 for the central value.

A similar set of data, taken from [START_REF] Gilmore | The Observed properties of Dark Matter on small spatial scales[END_REF], reads, with the resulting estimates for the elementary mass again from Eq.( 35) and again the "η coefficient" taken as 4. The estimates and error bars are shown in Figure (13). After the χ 2 processing of such 8 results, we obtain the following estimate of the mass, mc 2 = 263 ± 29 eV . It occurs with a disappointing χ 2 = 8.9, however.

We stress, finally, that a compromise value, mc 2 ≃ Abscissa is rank in Table II.

275 eV , reasonably sits in the overlap of the final error bars, ± 30 eV , observed from the two distinct sets of data.

IV DISCUSSION

Despite rather large uncertainties in astronomic data on DSG's, we have reached, however, a reasonable estimate of the "elementary dark mass", mc 2 = 275 ± 30 eV . This value, while a bit larger, is in a reasonable agreement with those obtained with different methods in Refs. [1][START_REF] Alexander | Gravitationally bound BCS state as dark matter[END_REF][START_REF] Randall | Cores in Dwarf Galaxies from Fermi Repulsion[END_REF], taking into account the experimental error bars.

Concerning the role of temperature in the range we consider, it may modify the resulting DSG-related masses by a maximum of an extra 15% amount only.

A few words of caution are in order, despite this agreement. While our theory is consistent, we found that the dispersion of obtained mass estimates between various clouds remains somewhat large, with resulting large values of χ 2 's. Our agreement, therefore, tends to allow for the possibility that some DSG's could be made of quasi degenerate fermions, but it should not be taken as an absolute proof of such a description for all DSG's. Another option is to turn to a different interpretation, in terms of the determination of a lower fermionic mass bound.

Indeed, assuming a thermal equilibrium for an initial stage of the formation of a dark matter cloud, then Dwarf Spheroidal Galaxies provide a rather model-independent, mass lower bound for a possible elementary dark matter fermion. Assume an initial thermal equilibrium state, like in Refs. [START_REF] Dalcanton | Halo cores and phase space densities: Observational constraints on dark matter physics and structure formation[END_REF][START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF][START_REF] Destri | Fermionic warm dark matter produces galaxy cores in the observed scales because of quantum mechanics[END_REF][START_REF] Shao | The phase space density of fermionic dark matter haloes[END_REF] that we referred to in the introduction, but with a temperature low enough to allow neglecting relativistic corrections. Then let us list some distinct features that we obtain. As discussed by the authors of Ref. [START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF], the determination of a lower bound, from the sequence of mass determinations for the DSG's of tables I and II, is complicated by various dynamical indeterminacies. Following the choice made by [START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF], and for the sake of a comparison, we shall consider the central value obtained from the DSG LeoIV as a typical lower bound. This leads us to a mean value, mc 2 550 eV (see Table II). This is compatible with, while somewhat larger than, the result of the relativistic case (mc 2 480 eV ) quoted in [START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF]. Note that we also find a consistency with the simulation-based determination, mc 2 gtrsim500 eV , Ref. [START_REF] Shao | The phase space density of fermionic dark matter haloes[END_REF]. Interestimgly, the agreement is even valid for each of our individual mass determinations for DSG's in Tables I andII. All these lie within 1 standard deviation from those quoted in [START_REF] Shao | The phase space density of fermionic dark matter haloes[END_REF].

The results quoted in Refs. [START_REF] Dalcanton | Halo cores and phase space densities: Observational constraints on dark matter physics and structure formation[END_REF][START_REF] Destri | Fermionic warm dark matter produces galaxy cores in the observed scales because of quantum mechanics[END_REF] are somewhat different and larger, but a comparison is then not easy, because the DSG's which are considered by [START_REF] Dalcanton | Halo cores and phase space densities: Observational constraints on dark matter physics and structure formation[END_REF][START_REF] Destri | Fermionic warm dark matter produces galaxy cores in the observed scales because of quantum mechanics[END_REF] are not the same as those that we and Refs. [START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF][START_REF] Shao | The phase space density of fermionic dark matter haloes[END_REF] have used.

V CONCLUSION

The present theory allows the calculation of basically all properties of a dark cloud, in terms of three parameters, namely an elementary particle mass m, a velocity scale v 0 and a temperature scale η. It suffers from a weak divergence due to density tail effects, but we have seen that many properties are not sensitive to the divergence. In fact, astronomy related observables such as the core radius r c , and the radial velocity dispersion σ, (assumed here to be given by the luminous content of the DSG's) are shown to be free of divergence.

Note that the theoretical estimates might also suffer from an insufficient treatment of exchange terms in this mean field theory, but exchange potentials are unlikely to reach astronomic ranges.

We have seen, towards the end of Section II, that the theory provides values of the ratio, R = σ 2 /(G ρ 0 r 2 c ), that are almost parameter independent, in particular from the temperature, and in agreement with the King radius. This may lead to some trust in the theoretical treatment.

Obviously, the particle mass m is not really a parameter. Rather, it should emerge as the same physical result for all observations. Our theory easily allows, see Eq.( 35), to estimate m from the two traditional observables, σ and r c . It will be noted that the "slightly debatable" functions, F (η) and G(η), contained in Eq.(34), have values not far from O(1), and vary smoothly only. With the former and the latter functions taken to powers 1/8 and 1/2, respectively, their influence remains small.

All being considered, our results can be expressed in terms of two different fermionic mass scales, depending on the physical interpretation of the stage at which the fermionic cloud is considered to be in a non-relativistic thermal equilibrium. If one assumes that thermal equilibrium describes the present observable stage of the DSG's, then one obtains a fermion mass estimate of mc 2 = 275 ± 30 eV. If one rather assumes that the equilibrium corresponds to an initial stage only, followed by a collision-less and dissipation-less evolution, one is led to an estimated lower bound, 550 eV m c 2 , according to the reasoning of Refs. [START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF][START_REF] Shao | The phase space density of fermionic dark matter haloes[END_REF]. The compatibility, or possible tension, of such determinations with other sources of astronomic observables is beyond the focus of the present work. But it obviously deserves to be studied.

As a final remark on our theory, we have not been able to design a precise way to deduce a dark cloud temperature from astronomic data, because our main result, Eq.(35), does not seem to make the elementary mass to depend on the "η parameter" strongly enough. This problem will remain under our consideration.

2 ΞFIG. 1 .FIG. 2 .

 212 FIG.1. Scaled profiles ξ. Full line for η = 0, long dashes for η = .1, moderate dashes for η = .3, short dashes for η = .5.

FIG. 3 .FIG. 4 .

 34 FIG.3. Renormalized densities ρn(x). Full line, η = 0. Long, moderate, short dashes for η = .1, .3, .5. respectively.

FIG. 5 .ΔFIG. 6 .

 56 FIG. 5. Profiles of mean square radial velocities. Full line, η = 0. Dashes shorten as η = .1, .3, .5, .7.

FIG. 7 .

 7 FIG.7. Surface densities when the radii are defined by the roots of ξ. Full line, η = 0. Dashes shorten as η = .1, .3, .5, .7.

FIG. 8 .

 8 FIG.8. Surface densities with integration ranges increased by 50%. Dashes shorten as η = .1, .3, .5, .7.

FIG. 9 .

 9 FIG. 9. Same as Fig.[START_REF] Boyarsky | A Lower bound on the mass of Dark Matter particles[END_REF], but now with integration ranges twice the roots of ξ.

FIG. 10 .

 10 FIG. 10. Comparison of the surface densities shown in Figs. (8) and (9).

FIG. 11 .

 11 FIG. 11. Evolutions of the halved surface density and its position.

mc 2 100 eV 4 =

 4 96 • {2.4, 3.0, 3.7, 4.5, 5.2, 5.9, 6.5} σ/(10 kms -1 ) [r c /(100 pc)] 2 , (35) where we use the typical observational scales. The string in the numerator shows the η dependent coefficient, [G(η)] 2 / F (η), with our usual set η = .1, .2, . . . , .7.

  FIG.12. Masses and error bars deduced from the astronomic data in[START_REF] Simon | The Kinematics of the Ultra-Faint Milky Way Satellites: Solving the Missing Satellite Problem[END_REF]. Each integer abscissa means rank in TableI.

FIG. 13 .

 13 FIG.13. Masses and error bars, astronomic data from[START_REF] Gilmore | The Observed properties of Dark Matter on small spatial scales[END_REF]. Abscissa is rank in TableII.

  2, ..., .7 we find the following, respective values, 1.59, 1.55, 1.53, 1.52, 1.51, 1.51, 1.51. These are obtained with integral ranges extended by a factor 10, see the sequence of triangles in Fig.(

  5,

	Object	rc (pc) v (km/s) mc 2 (eV )
	CanVenat I	564 ± 36 7.6 ± 2.2 206 ± 16
	UrsMaj I	318 ± 45 7.6 ± 2.4 274 ± 29
	Hercules	330 ± 63 5.1 ± 2.4 297 ± 45
	LeoT	178 ± 39 7.5 ± 2.7 367 ± 52
	UrsMaj II	140 ± 25 6.7 ± 2.6 426 ± 56
	LeoIV	116 ± 30 3.3 ± 2.8 558 ± 139
	ComaBeren	77 ± 10 4.6 ± 2.3 630 ± 89
	CanesVenat II 74 ± 12 4.6 ± 2.4 644 ± 99
	TABLE II. Data from [5] and resulting masses.

ACKNOWLEDGEMENTS

We thank Patrick Valageas for remarks and suggestions on the manuscript.


Emmanuel Lhuillier
email: emmanuel.lhuillier@insp.upmc.fr

Jean-Francois Dayen

Daniel O Thomas

Adrien Robin

Sandrine Ithurria

Herve Aubin

Benoit Dubertret

Phototransport in colloidal Nanoplatelets Array

Keywords: semiconductor nanoparticles, quantum dot, nanoplatelets, photodetection, electrolyte gating, noise

Colloidal nanocrystals are promising materials for achieving low cost optoelectronic devices. In this paper, we focus on the transport and photo transport properties of 2D nanoplatelet thin films and their use for photodetection. We present evidence that improved performances relies on good trap passivation as well as overcoming the inherent large exciton binding energy of the 2D NPL. This can be achieved using a phototransistor configuration with transport at the single particle scale.

INTRODUCTION

Colloidal nanocrystals have raised great hope towards achieving low cost optoelectronic devices, 1 however this goal requires fine control over both optical and transport properties of nanocrystal arrays. Over the past two decades, great progress has been obtained on the synthetic procedure of colloidal quantum dots which can now be achieved for a large range of materials with band gap spanning from UV to mid IR 2,3 . Improved synthetic procedures now allow achieving nano-objects with controlled dimensionality as well as those based on complex core shell heterostructures. Among these colloidal materials, 2D nanoplatelets 4,5 (NPL) based on cadmium chalcogenides 6 offer particularly interesting structural and optical properties. Thanks to their large lateral extension above the Bohr radius, the quantum confinement only occurs in one direction. Moreover the thickness is perfectly controlled at the atomic scale (i.e. there is no roughness). This results in extremely narrow optical features (Figure 1b) due to the absence of inhomogeneous broadening. Moreover, the 2D aspect of the nanoparticles leads to a high exciton binding energy 7,8 (250meV) and a fast photoluminescence lifetime. While the spectroscopic properties have attracted huge interest in particular for stimulated emission, the integration of NPL 9 into conductive devices remains far less investigated 10,11 . Nevertheless, NPL look promising because their sharp optical features can be used to develop narrow emission electroluminescence devices 12 or photodetectors with finely controlled cutoff wavelengths. In this article, we review some of the recent results obtained by our group in the integration of NPL for visible photodetection.

METHODS

CdSe core

In a 100ml three neck flask, 240mg of cadmium myristate, 25mg of selenium powder and 30mL of octadecene are mixed and degassed under vacuum at 80°C for 30min. The temperature of the flask is then set at 240°C. Once the content of the flask reached 190°C, 160mg of cadmium acetate are quickly added to the flask while removing the septum. The solution is then further heated at 240°C for 12min before 1mL of oleic acid is injected to quench the solution. The flask is cooled and its content precipitated by addition of ethanol. After centrifugation the formed pellet is redispersed in fresh hexane.

CdSe/CdS core shell synthesis

The growth of a CdS shell is performed at room temperature using the c-ALD procedure 13 . The CdSe NPL cores are successively exposed to cadmium and sulfide precursor and the excess precursor is removed by beaning at each step. A total of 3.5 layers of CdS is grown.

Ligand exchange

NPL originally dispersed in hexane are mixed with a mixture of sodium sulfide in N-methyl formamide (NMFA). After strong sonication, the particles get phase transferred. The polar phase is then cleaned three times by addition of hexane. The clear top phase is discarded each time. Ethanol is added to precipitate the particle. The formed pellet is eventually dispersed in pure NMFA.

Electrode fabrication

A Si/SiO2 substrate is cleaned by sonication in acetone for 5min. The wafer is then rinsed using isopropanol. Once dried, the wafer is further cleaned using O2 plasma for 5min. AZ5214E lithography resist is then spin-coated on the wafer and the solvent dried by cooking the wafer on a hot plate at 110°C for 90s. The wafer is then illuminated through a shadow mask for 2s, before being cooked on a hot plate at 125°C for 2min. A flood UV exposure of the wafer is performed for 40s. The patterns are then developed in AZ326 for 32s. Then 3nm Cr and 40nm gold are thermally evaporated. Finally the wafer is dipped overnight in an acetone bath and eventually rinsed with isopropanol before being dried.

Nanotrenches are prepared using two steps of the previously described 14 procedure. The first step defines the source while the second evaporation defining the drain is performed under a 60° tilt.

Electrolyte preparation

In an air free glove box, 50mg of LiClO4 are mixed in 230mg of polyethylene glycol for two hours on a hot plate at 170°C.

Thin film preparation

In an air free glove box, the nanoparticles are drop-casted onto the electrodes on a hot plate at 100°C. Once the film is dried, the melted electrolyte is brushed directly on the film and a metallic grid is deposited on the top of the electrolyte to be used as top gate.

RESULTS AND DISCUSSION

Photoconductive properties

As a first step in investigating the photoconductive properties of NPL, we focus on the transport in a NPL thin film connected with µm spaced interdigitated electrodes (Figure 1a). While the conductance of the film is very low under dark condition, we observe a significant current modulation under illumination, 15,16 whose photon energy is above the band gap, see Figure 1c. This photocurrent has a fast time response 16 (f3db >100kHz) and a linear dependence with the photon flux. However, the overall photoresponse is weak (few µA.W -1 ). In such devices, the transport occurs through a hopping process as the nanoparticle size remains far smaller than the electrode spacing. At each hopping step, the photocarrier has to overcome a tunnel barrier and may get trapped on surface defects. Ligand exchange 17 from long alkane chain to atomically short (S 2-) has been conducted to improve the overall mobility, however it is insufficient to achieve a large photoresponse.

From photoconductor to phototransistor

In order to improve the ability of this material to convert light into current, we propose to build a phototransistor. Gating of colloidal nanocrystals can be obtained through a conventional solid state approach using a thin dielectric. However, with such gating the carrier density modulation remains low (<10 13 cm -2 ), which is generally less than 1 carrier per dot. Larger carrier density can be obtained while switching to electrolyte gating. In this case, the sheet capacitance is indeed no longer given by the ratio of the dielectric constant by the dielectric thickness but rather by the ratio of the dielectric constant by the ion size. The latter can be as small as 100pm in the case of Li + , which allows carrier density above 10 14 cm -2 . Liquid electrolytes are generally used to gate nanocrystals film, but this strategy remains poorly integrable for real optoelectronic devices since electrochemical cells are bulky and leaky. We develop an ion gel approach for NPL thin film gating 18 which combines several advantages: (i) gating is solid and compact, (ii) the obtained devices are air operable and (iii) the method is also able to gate thick films (µm). In practice, LiClO4 dissolved in a polyethylene glycol matrix is melted and deposited solvent free on the nanoparticle film. This approach leads to an electrolytically gated transistor with exceptional performance 19 . The on/off ratio reaches nearly 9 decades, while the subthreshold slope can be as small as 80mV/decade, see Figure 2a. The driving gate bias remains moreover limited to a few volts. As a consequence, we can take advantage of this very sharp change of the current with the carrier density to build a phototransistor. By operating the transistor close to its turn on threshold, we bring the system in a regime where the current dependence with the carrier density is no longer linear but rather exponential. Thus, by shining light on the sample, the excess photocarriers are expected to bring a dramatic current modulation. This was indeed experimentally observed 16 : The typical photocurrent enhancement under gate bias ranges from 100 to 1000. As a positive gate bias is applied, cations from the electrolyte start penetrating the nanoparticles film. As a consequence, the electrodes inject electrons in the NPL. Since NPL are intrinsically undoped, the Fermi level is expected to be close to the middle of the band gap. As electrons get injected, the Fermi level moves toward the conduction band and start filling the unpassivated trap states likely introduced by the c-ALD growth process. The filling of these trap states increases the photocarriers lifetime in the conduction band, since they can no longer be trapped after their photogeneration. This leads to an increased overall gain and explains the improved photoresponse 16 . However, this increased gain comes at the price of reduced bandwidth which drops at the same ratio as the gain increases. This gating approach can also be used to build a bicolor nanocrystal based detector. Such a device is of particular interest once operated in the infrared. Indeed, bicolor detection not only provides thermal contrast but can also discriminate objects with different emissivity. To do so we combine a CdSe/CdS NPL phototransistor with a 630nm cut-off wavelength in parallel with a photoconductive sample made from HgTe colloidal quantum dots (λcut-off=2.1µm). In absence of gate bias, the narrow band gap material dominates the transport because of the larger number of thermally activated carriers. Thus, only the photoresponse of the HgTe thin film is observed. As a result, the system is responsive both in the visible and near IR, see Figure 2b. Once the CdSe/CdS NPL film is gated, its conductance rises and can even overcome the conduction of the HgTe film. In this situation, the infrared photoresponse is washed out and only the signal originating from visible illumination is measured, see Figure 2b.

Photoconduction with optimized transport at the nanoscale

While electrolyte gating is an efficient strategy to get rid of surface traps, it does not address the large exciton binding energy which is inherent to 2D systems 7,8 . With an exciton binding energy significantly larger than the thermal energy or the electric field drop per particle, it is very unlikely that the photogenerated electron-hole pairs get efficiently dissociated. To overcome this issue, a larger electric field needs to be applied, which can be obtained by shrinking the device size. Switching to smaller devices is also very appealing in order to get rid of µm scale morphology defects inside the NPL film. We use a simple method based on a two steps optical lithography to design nanotrenches with short electrode spacing (20 to 100nm) but large aspect ratio 14 . A picture of the device functionalized with CdSe/CdS NPL is given on Figure 3a. In such nanotrench devices, the photoresponse is dramatically boosted 20 compared to the same material operated in the hopping regime, and responsivity as large as 1kA.W -1 has been demonstrated. This value is the record value obtained for CdSe nanomaterial based devices. It is worth noting that similar responsivity have already been reported in PbS nanocrystals film 21 , in which the large dielectric constant of the material allows the photocarriers to behave almost as free carriers. In our case, the large applied electric field fully compensates the binding energy. In addition to this increased charge dissociation, the 8 decades boost of the photoresponse compared to µm scale devices is also the result of a large gain, corresponding to 3000 generated electrons per absorbed photon. 22 Small electrode spacing indeed results in short transit time and thus large gain. The last element which contributes to the improved performance is evidenced by the root square power dependence of the photoresponse (IαP -1/2 , see Figure 3b) which is consistent with current limited by electron-hole pair recombination 23 rather than by trapping. Contrary to hopping transport where the photocarriers will very likely experience trap states regardless of the path they take, in nanotrench devices the paths limited by trap states are simply ineffective and do not participate in the transport. In this geometry, the transport is actually limited by the charge carrier injection at the contact and careful contact engineering will be required to obtain even higher performance 20 .

CONCLUSION AND PERSPECTIVES

We demonstrate that nanoplatelets are a promising material to achieve low cost photodetectors as long as we can take care of their specific properties such as their large exciton binding energy. Further improvements are needed to increase the optical absorption in nanotrench devices and to reduce the Schottky barrier at the contacts. Another aspect needing to be addressed is the electrical noise which remains limited by the 1/f contribution 24 in disordered nanocrystals arrays.

Figure 1 a

 1 Figure 1 a. scheme of a photoconductive device based on Nanoplatelets thin film connected to interdigitated electrodes. The inset is a Transmission Electron Microscopy image of CdTe NPL. b absorption and photoluminescence spectra of CdSe NPL. C current as a function of the applied bias for a CdSe NPL thin film under dark condition and under illumination (50mW of a 405nm laser).

Figure 2

 2 Figure 2 Drain current and photoresponse as a function of gate bias for a CdSe/CdS NPL thin film gated with an ion gel electrolyte. The inset is a scheme of the device. For b and c the device is made of CdSe/CdS NPL film gated with electrolyte in parallel with a non gated HgTe nanocristal film. b (resp. c) represents the current as a function of time during an illumination pulse by a near IR LED (980nm) followed by a blue Laser pulse (405nm) under 0V gate bias (resp. 1.1V gate bias).

Figure 3 a

 3 Figure 3 a is an SEM image of a nanotrench decorated with CdSe/CdS NPL. b. Photoresponse as a function of incident light power for a nanotrench based device. A fit with a power law has been added.

ACKNOWLEDGEMENTS

The authors thank Xiang Zhen Xu for her help with TEM imaging. We thank Agence National de la Recherche for funding through grants SNAP and H2DH. This work has been supported by the Region Ile-de-France in the framework of DIM Nano-K.