8 Our starting point is Eq. (20) for the scattering matrix, which we rewrite here, Appendix A S = 1 − 2 πiW $^{†}$( E$_{F}$ − H + iπWW $^{†}$)− $^{1}$W . (23) The matrix WW † = V V † + PP † projects onto the sys- tem’s states coupled to the transport leads ( V V $^{†}$) and the probe ( PP $^{†}$). We can write a perturbative series in PP † for the elements of the scattering matrix connect- ing the leads to the probe ( j$_{I}$ is a channel index on the I = L, R leads) We demonstrate that the potentials ensuring zero cur- rent through a single-channel probe do not depend on the transparency of the contact between the probe and the system. According to Eq. (9), the chemical poten- tials in an external probe connected to a voltage-biased two-terminal cavity are given by the ratio of the differ- ence and the sum of transmission coefficients between the transport leads and the probe. We show that the Γ$_{P}$ -dependence of the coefficients T ( αβ ) PI factorizes when N$_{P}$ = 1 so that Γ$_{P}$ does not affect the chemical potential. S$_{P,j}$$_{I}$ = − 2 πi ∑ x,x ′ P ∗ $_{xP}$V$_{x}$ ′ j$_{I}$ 〈 x | ( E$_{F}$ − H + iπWW $^{†}$)− $^{1}$| x $^{′}$〉 = − 2 πi ∑ x,x ′ P ∗ $_{xP}$V$_{x}$ ′ j$_{I}$ [ 〈 x | ( E$_{F}$ − H + iπV V $^{†}$)− $^{1}$| x $^{′}$〉 − iπ 〈 x | ( E$_{F}$ − H + iπV V $^{†}$)− $^{1}$PP $^{†}$( E$_{F}$ − H + iπV V $^{†}$)− $^{1}$| x $^{′}$〉 + . . . ] . (24) The transparency of the contact between probe and system is contained exclusively in the matrix P . We are free to choose any orthogonal basis {| x 〉} for the system and pick one where the M × 1 coupling matrix between probe and system is P$_{xP}$ = ω$_{P}$ δ$_{xP}$ . One then has for the M × M matrix 〈 x | PP $^{†}$| x $^{′}$〉 = 〈 P | PP $^{†}$| P 〉 δ$_{xP}$ δ$_{x}$ ′ P = | ω$_{P}$ | $^{2}$δ$_{xP}$ δ$_{x}$ ′ $_{P}$, with | ω$_{P}$ | 2 = ( Mδ/π $^{2}$Γ$_{P}$ )(2 − Γ$_{P}$ − $_{2}$√$_{1}$ − Γ$_{P}$ ) [35]. One gets S$_{Pj}$$_{I}$ = − 2 πiω ∗ p ∞ ∑ n =0 ( − iπ | ω$_{P}$ | 2 〈 P | ( E$_{F}$ − H + iπV V $^{†}$)− $^{1}$| P 〉 ) n × ∑ x ′ V$_{x}$ ′ j$_{I}$ 〈 P | ( E$_{F}$ − H + iπV V $^{†}$)− $^{1}$| x $^{′}$〉 = f (Γ$_{P}$ ) × ∑ x ′ V$_{x}$ ′ j$_{I}$ 〈 P | ( E$_{F}$ − H + iπV V $^{†}$)− $^{1}$| x $^{′}$〉 . (25) One sees that, because the probe is single-channeled, the Γ$_{P}$ -dependent part of S$_{Pj}$$_{I}$ factorizes, and the same obviously holds for transmission coefficients T (00) PI = ∑ j$_{I}$ | S$_{Pj}$$_{I}$ | $^{2}$. This is not the case for a multichannel probe, in which case | P 〉 is no longer unique. Thus, for N$_{P}$ = 1, the electrochemical potential in Eq. (10) do not depend on the strength of the tunnel barrier. The argument is straightforwardly extended to the spin- dependent chemical potentials in Eq. (9), because for a single-channel probe, the reflection coefficient T ( αβ ) PP is di- agonal in spin-indices. [1] F.T. Vasko and N.A. Prima, Sov. Phys. Solid State 21 , 994 (1979); L.S. Levitov, Y.V. Nazarov, and G.M. Eliash- berg, Sov. Phys. JETP 61 , 133 (1985); A.G. Aronov and Yu.B. Lyanda-Geller, JETP Lett. 50 , 431 (1989). [2] V. M. Edelstein, Solid State Comm. 73 , 233 (1990). [3] M.I. Dyakonov and V.I. Perel, Sov. Phys. JETP Lett. 13 , 467 (1971). [4] For a review on the spin Hall effect, see: H.-A. Engel, E.I. Rashba, and B.I. Halperin, Theory of Spin Hall Ef- fects in Semiconductors , Handbook of Magnetism and Advanced Magnetic Materials, Wiley and Sons (2007); arXiv:cond-mat/0603306. [5] Y.K. Kato, R.C. Myers, A. C. Gossard, and D. D. [10] E.G. Mishchenko, A.V. Shytov, and B.I. Halperin, Phys. Rev. Lett. 93 , 226602 (2004). [6] J. Wunderlich, B. K¨ astner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94 , 047204 (2005). [7] S.O. Valenzuela and M. Tinkham, Nature 442 , 176 (2006). [8] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88 , 182509 (2006); T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98 , 156601 (2007); 98 , 249901(E) (2007). [9] H. Zhao, E.J. Loren, H.M. van Driel, and A.L. Smirl, Phys. Rev. Lett. 96, 246601 (2006). Awschalom, Science 306 , 1910 (2004).