We present fluctuation relations that connect spin-polarized current and
noise in mesoscopic conductors. In linear response, these relations are
equivalent to the fluctuation-dissipation theorem that relates equilibrium
current--current correlations to the linear conductance. More interestingly, in
the weakly nonlinear regime of transport, these relations establish a
connection between the leading-order rectification spin conductance, the spin
noise susceptibility and the third cumulant of spin current fluctuations at
equilibrium. Our results are valid even for systems in the presence of magnetic
fields and coupled to ferromagnetic electrodes.Comment: Submitted to the Proceedings of the 31st ICP

Lim, Jong Soo

Lopez, Rosa

Sanchez, David

English

arXiv

Proceedings of the 31st International Trabajo presentado en la Conference on the Physics of Semiconductors (ICPS) 2012, celebrada en Zurich del 29 de julio al 3 de agosto de 2012.We present fluctuation relations that connect spin-polarized current and noise in mesoscopic conductors. In linear response, these relations are equivalent to the fluctuation-dissipation theorem that relates equilibrium current-current correlations to the linear conductance. More interestingly, in the weakly nonlinear regime of transport, these relations establish a connection between the leading-order rectification spin conductance, the spin noise susceptibility and the third cumulant of spin current fluctuations at equilibrium. Our results are valid even for systems in the presence of magnetic fields and coupled to ferromagnetic electrodes.Work supported by the MINECO Grants No. IS2011-23526 and CSD2007-00042 (CPAN).N

Lim, Jong-Soo

Sánchez, David

López, Rosa

Digital.CSIC

arXiv:1208.4746v1  [cond-mat.mes-hall]  23 Aug 2012Spin-current noise from fluctuation relationsJong Soo Lim∗, David Sánchez∗,† and Rosa López∗,†∗Institut de Física Interdisciplinària i Sistemes Complexos IFISC (UIB-CSIC), E-07122 Palma de Mallorca,Spain†Departement de Física, Universitat de les Illes Balears, E-07122 Palma de Mallorca, SpainAbstract. We present fluctuation relations that connect spin-polarized current and noise in mesoscopic conductors. Inlinear response, these relations are equivalent to the fluctuation-dissipation theorem that relates equilibrium current–currentcorrelations to the linear conductance. More interestingly, in the weakly nonlinear regime of transport, these relations establisha connection between the leading-order rectification spin conductance, the spin noise susceptibility and the third cumulant ofspin current fluctuations at equilibrium. Our results are valid even for systems in the presence of magnetic fields and coupledto ferromagnetic electrodes.Keywords: fluctuation relations, spintronics, mesoscopic transportPACS: 72.10.Bg, 05.40.–a, 72.25.–b, 85.75.–dWe have recently derived fluctuation relations validbeyond the linear response regime of mesoscopic trans-port fully taking into account the spin degree of freedomand magnetic interactions [1]. These relations representhigher-order fluctuation-dissipation relations [2] and aresatisfied even if microreversibility is broken due to anexternally applied magnetic field B [3, 4].Consider a generic multiterminal conductor coupledto reservoirs α,β . . . with voltages Vα ,Vβ . . . and back-ground temperature T (see Fig. 1). We assume that thereexists a spin quantization axis common to all terminalsand thus denote the spin index with s,s′ . . ., where s =+(s = −) for electrons with spin ↑ (↓). Additionally, theleads can be magnetized with a polarization p, allow-ing for spin-dependent transport across the sample. Weallow for possible spin biases present in the lead volt-ages: Vαs,Vβ s′ . . . Then, the current 〈Iαs〉 flowing throughterminal α associated with electronic spins s can beexpanded in powers of voltage around the equilibriumpoint:〈Iαs〉= ∑β s′G(1)αs,β s′Vβ s′ +12 ∑β s′,γs′′ G(2)αs,β s′γs′′Vβ s′Vγs′′+O(V 3) , (1a)〈Sαsβ s′〉= S(0)αsβ s′ +∑γs′′S(1)αsβ s′,γs′′Vγs′′ +O(V2) , (1b)〈Cαsβ s′γs′′〉=C(0)αsβ s′γs′′ +O(V) , (1c)where G(1)αs,β s′ is a linear conductance coefficient andG(2)αs,β s′γs′′ is the leading-order rectification term. Both co-efficients are evaluated at equilibrium but their functionaldependence on the scattering properties of the conductoris very different [5]. In Eqs. (1b) and (1c) we have alsoBI αsVαs Vβs’Vγ s’’I γ s’’I βs’FIGURE 1. Sketch of a mesoscopic conductor in the pres-ence of magnetic field B and attached to multiple terminals withapplied voltages Vαs,Vβ s′ . . .. We indicate spin currents flowingthrough the system.expanded the noise S (current–current correlations) andthe third cumulant C, for which (0) label equilibrium re-sponses.Let fαs± = fαs(B, p)± fα s¯(−B,−p) be the symmet-ric (+) and antisymmetric (−) combinations of a giventransport coefficient (G, S or C). In Ref. [1] we show thatthe fluctuation relations for spintronic systems readS(0)αsβ s′± = kBT(G(1)αs,β s′±+G(1)β s′,αs±), (2a)C(0)αsβ s′γs′′± = kBT[(S(1)αsβ s′,γs′′±+ S(1)αsγs′′,β s′±+ S(1)β s′γs′′,αs±)−kBT(G(2)αs,β s′γs′′±+G(2)β s′,αsγs′′±+G(2)γs′′,αsβ s′±)].(2b)These expressions are to be supplemented withS(0)αsβ s′(B, p) = S(0)α s¯β s¯′(−B,−p) and C(0)αsβ s′γs′′(B, p) =−C(0)α s¯β s¯′γ s¯′′(−B,−p), which state that the even (odd)equilibrium cumulants respond symmetrically (anti-symmetrically) to simultaneous reversals of the spindirection, the magnetic field and the lead magnetization.Using the symmetry of S(0)αsβ s′ in Eq. (2a) we recover theOnsager-Casimir reciprocity relations, valid in the linearresponse regime, i.e., G(1)αs,β s′(B, p) = G(1)β s¯′,α s¯(−B,−p).Importantly, Eq. (2) is valid not only for arbitraryCoulomb interactions but also for magnetic couplingssuch as spin-flip relaxation processes [1].Now, in spintronics it is convenient to work with spin-polarized current and noises. In fact, spin-resolved shotnoise can provide valuable information about the role ofinteractions in mesoscopic systems [6]. Therefore, wedefine the spin-polarized (SP) quantities〈ISPα 〉= 〈(Iα↑− Iα↓)〉= 〈Iα↑〉− 〈Iα↓〉= ∑ss〈Iαs〉 ,(3a)〈SSPαβ 〉= 〈(Iα↑− Iα↓)(Iβ↑− Iβ↓)〉= ∑s,s′ss′〈Sαsβ s′〉 ,(3b)〈CSPαβ γ〉= 〈(Iα↑− Iα↓)(Iβ↑− Iβ↓)(Iγ↑− Iγ↓)〉= ∑s,s′,s′′ss′s′′〈Cαsβ s′γs′′〉 , (3c)and, similarly to Eq. (1), expand Eq. (3) in powers ofvoltage:〈ISPα 〉= ∑β s′G˜(1)α ,β s′Vβ s′ +12 ∑β s′,γs′′ G˜(2)α ,β s′γs′′Vβ s′Vγs′′+O(V 3) , (4a)〈SSPαβ 〉= S˜(0)αβ +∑γs′′S˜(1)αβ ,γs′′Vγs′′ +O(V2) , (4b)〈CSPαβ γ〉= C˜(0)αβ γ +O(V) . (4c)Our goal is to find the fluctuation relations obeyed bythe response coefficients G˜, S˜ and C˜. We substitute Eq.(1) in Eq. (3) and use the fluctuation relations expressedin Eq. (2). Then, we findS˜(0)αβ± = kBT ∑ss(G˜(1)α ,β s±+ G˜(1)β ,αs±)(5a)C˜(0)αβ γ± = kBT ∑ss(S˜(1)αβ ,γs±+ S˜(1)αγ,β s±+ S˜(1)β γ,αs±)− (kBT )2 ∑s,s′ss′(G˜(2)α ,β sγs′±+ G˜(2)β ,αsγs′±+ G˜(2)γ,αsβ s′±)(5b)together with S˜(0)αβ− = 0 and C˜(0)αβ γ+ = 0 (the formeris consistent with the Onsager-Casimir relations). Tworemarks are in order. First, Eq. (5a) establishes a re-lation between the spin-resolved noise at equilibrium(i.e., in the absence of bias voltage) and the polarizedlinear conductance. It represents an extension of thefluctuation-dissipation theorem valid for spintronic sys-tems in the presence of magnetic fields and ferromag-netic electrodes. Second, Eq. (5b) describes a nontrivialconnection between the third cumulant at equilibrium,the spin-resolved noise susceptibility [S(1)] and the po-larized nonlinear conductance to leading order. Clearly,the differential conductance need not obey the Onsager-Casimir symmetry relations [7, 8]. This is due to break-ings of microreversibility away from equilibrium. How-ever, Eq. (5b) connect the different asymmetries of thenoise susceptibility and the nonlinear conductance evenin systems for which C˜(0)αβ γ+ = 0.In conclusion, we have formulated novel fluctuationrelations applied to spin-dependent current and noisesfor multiterminal systems in the presence of Coulomb in-teractions, magnetic fields and ferromagnetic electrodes.Our formalism is based on rather broad assumptions:equilibrium microreversibility, global detailed balanceand probability conservation [1]. We emphasize that ourresuls hold even if a spin bias is present in the systemdue to possible spin accumulations formed at the bound-ary between the leads and the scattering region. There-fore, our central result [Eq. (5)] is an excellent tool to in-vestigate purely spintronic effects in out-of-equilibriumsituations.ACKNOWLEDGMENTSWork supported by the MINECO Grants No. FIS2011-23526 and CSD2007-00042 (CPAN).REFERENCES1. R. López, J.S. Lim, and D. Sánchez, Phys. Rev. Lett. 108,246603 (2012).2. M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod.Phys. 81, 1665 (2009).3. H. Förster and M. Büttiker, Phys. Rev. Lett. 101, 136805(2008).4. D. Sánchez, Phys. Rev. B 79, 045305 (2009).5. T. Christen and M. Büttiker, EPL 35, 523 (1996).6. O. Sauret and D. Feinberg, Phys. Rev. Lett. 92, 106601(2004).7. D. Sánchez and M. Büttiker, Phys. Rev. Lett. 93, 106802(2004); Phys. Rev. B 72, 201308(R) (2005).8. B. Spivak and A. Zyuzin, Phys. Rev. Lett. 93, 226801(2004).

Spin-current noise from fluctuation relations

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