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external leads and thus cannot easily be converted into a large spin current, at least not in linear response. 

Fig. 1 gives a sketch of the system we focus on. We consider a two-dimensional chaotic ballistic quantum dot connected via reflectionless transport terminals to two external electron reservoirs. The reservoirs are biased and an electric current flows through the dot. Assum- ing that SOI is strong enough that the spin-orbit time is shorter than the mean electronic dwell time in the dot, τ$_{so}$ ≪ τ$_{D}$ , spin rotational symmetry is totally broken in- side the dot. Together with chaotic boundary scattering, this leads to spin relaxation. Does the current flowing through this SOI coupled, spin-relaxing system generate a spin polarization ? On one hand, the relative positions of the contacts to the external electrodes determine an average electronic drift. In that sense, the situation is very similar to that considered by Edelstein [2], and one would accordingly argue that a current-induced spin po- larization should emerge. On the other hand, however, electronic distribution functions are not well defined lo- cally in such a system where momentum and energy relax on the scale of the system itself. One might thus wonder how local spin-dependent potentials can be consistently defined. 

Here, we circumvent this difficulty by introducing an additional mobile terminal whose purpose it is to probe 

## B. The spin-probe model 

the local chemical potentials on the dot. We assume that spin currents are well defined in the probe as well as in the transport terminals, which presupposes that spin re- laxation processes are frozen there. In particular, SOI is present only in the dot. The probe is assumed to carry a single-transport channel. It is connected at one end to a macroscopic external reservoir, which defines its electro- chemical and spin-dependent chemical potentials unam- biguously, while its other end is contacted to the quan- tum dot via a tunnel barrier of transparency 0 ≤ Γ$_{P}$ ≤ 1. When Γ$_{P}$ ≪ 1, the probe does not perturb the dot and we can identify the spin-dependent chemical potentials on the dot with the corresponding probe potentials de- fined by the condition that neither charge nor spin cur- rents flow between the probe and the dot [26]. The probe being mobile, it can scan the whole dot area, and being single-channeled, it has maximal resolution. It can thus map the spin-polarization in the dot with maximal accu- racy. 

The spin-probe model becomes a powerful theoretical tool to calculate spin accumulations once electric and spin currents in the probe and the two transport termi- nals are related to chemical potentials via the scattering approach to transport [27, 28]. When the probe car- ries a single transport channel, time-reversal symmetry and the unitarity of the scattering matrix ensure that the reflection coefficients from the probe back to itself are diagonal in spin indices, independently of Γ$_{P}$ . This allows to directly express the electrochemical and spin- dependent chemical potentials of the probe in terms of spin-dependent transmission coefficients from the trans- port leads to the probe, and the chemical potentials at those leads. This is done below in Eq. (9). We addition- ally find that, for a single-channel probe, the dependence that transmission coefficients have on Γ$_{P}$ factorizes. This has the important, and somewhat surprising consequence that the chemical potentials ensuring the vanishing of the currents through the probe, being given by the ratio of transmission coefficients, do not depend on Γ$_{P}$ . This is an exact result, valid as long as one has a single-channel probe, and we stress that it is true not only on average nor to some leading order. Consequently, we are able to obtain the value of the spin-dependent potentials on the probe in the limit Γ$_{P}$ → 0, where it is justified to identify the potentials on the dot with those on the probe, while performing the calculation for the analytically simpler case Γ$_{P}$ = 1. 

Our scattering theory of spin accumulations can be ap- plied to both diffusive and ballistic mesoscopic systems in the linear response regime. Because current-induced spin polarization has not been considered in these sys- tems so far, we specialize here to ballistic systems, and accordingly apply random matrix theory (RMT) [24, 25] to calculate average and fluctuations of the spin-resolved 

## C. Summary of main results 

