Noise and fluctuation relations of a spin diode
© Lim et al.; licensee Springer. 2013
Received: 19 April 2013
Accepted: 6 May 2013
Published: 20 May 2013
We consider fluctuation relations between the transport coefficients of a spintronic system where magnetic interactions play a crucial role. We investigate a prototypical spintronic device - a spin-diode - which consists of an interacting resonant level coupled to two ferromagnetic electrodes. We thereby obtain the cumulant generating function for the spin transport in the sequential tunnelling regime. We demonstrate the fulfilment of the nonlinear fluctuation relations when up and down spin currents are correlated in the presence of both spin-flip processes and external magnetic fields.
KeywordsSpin noise Spin diode Fluctuation relations
Nonequilibrium fluctuation relations overcome the limitations of linear response theory and yield a complete set of relations that connect different transport coefficients out of equilibrium using higher-order response functions [1–7]. Even in the presence of symmetry-breaking fields, it is possible to derive nonlinear fluctuation relations from the microreversibility principle applied to the scattering matrix at equilibrium . A possible source of time-reversal symmetry breaking are magnetized leads. Then, it is necessary to include in the general formulation the spin degree of freedom, which is an essential ingredient in spintronic applications  such as spin-filters  and spin-diodes [10–17].
We recently proved nonequilibrium fluctuation relations valid for spintronic systems , fully taking into account spin-polarized leads, magnetic fields, and spin-flip processes. Here, we investigate a spin diode system and explicitly demonstrate that the spintronic fluctuation relations are satisfied. Furthermore, we calculate the spin noise (correlations of the spin-polarized currents) and discuss its main properties.
where , , and f±(ε)=1/[ exp(±ε/k B T)+1]. Here, V α σ is a spin-dependent voltage bias, and μ i σ is the dot electrochemical potential to be determined from the electrostatic model. i=0,1 is an index that takes into account the charge state of the dot. Then, the cumulant generating function in the long time limit is given by , where λ0(χ) denotes the minimum eigenvalue of that develops adiabatically from 0 with χ. From the generating function, all transport cumulants are obtained .
We consider a gauge-invariant electrostatic model that treats interactions within a mean-field approach . For the geometry sketched in Figure 1b, we employ the discrete Poisson equations for the charges Q ↑ and Q ↓ : Q ↑ =Cu 1(ϕ ↑ −V L ↑ )+Cu 2(ϕ ↑ −V L ↓ )+Cu 3(ϕ ↑ −V R ↑ )+Cu 4(ϕ ↑ −V R ↓ )+C(ϕ ↑ −ϕ ↓ ) and Q ↓ =Cd 1(ϕ ↓ −V L ↑ )+Cd 2(ϕ ↓ −V L ↓ )+Cd 3(ϕ ↓ −V R ↑ )+Cd 4(ϕ ↓ −V R ↓ )+C(ϕ ↓ −ϕ ↑ ), where C ℓ i represent capacitance couplings for ℓ=u/d and i=1⋯4. We then find the potential energies for both spin orientations, , N σ being the excess electrons in the dot. For an empty dot, i.e., N ↑ =N ↓ =0, its electrochemical potential for the spin ↑ or ↓ level can be written as μ0σ=ε σ +U σ (1,0)−U σ (0,0). This is the energy required to add one electron into the spin ↑ or ↓ level when both spin levels are empty.
Importantly, our results are gauge invariant since they depend on potential differences () only. When the dot is charged, then N ↑ =1 or N ↓ =1, and we find , with and .
Results and discussion
Nonlinear fluctuation relations
Notably, the Fano factor is always sub-Poissonian whenever ε eff lies inside the transport window. This is due to correlations induced by Coulomb interactions .
Nonequilibrium fluctuation relations nicely connect nonlinear conductances with noise susceptibilities. We have derived spintronic fluctuation relations for a prototypical spintronic system: a spin diode consisting of a quantum dot attached to two ferromagnetic contacts. We have additionally investigated the fulfilment of such relations when both spin-flip processes inside the dot and an external magnetic field are present in the sample. We have also inferred exact analytical expressions for the spin noise current correlations and the Fano factor. Further extensions of our work might consider noncollinear magnetizations and energy dependent tunneling rates.
This work was supported by MINECO Grants No. FIS2011-2352 and CSD2007–00042 (CPAN), CAIB and FEDER.
- Esposito M, Harbola U, Mukamel S: Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev Mod Phys 2009, 81: 1665. 10.1103/RevModPhys.81.1665View ArticleGoogle Scholar
- Tobiska J, Nazarov YV: Inelastic interaction corrections and universal relations for full counting statistics in a quantum contact. Phys Rev B 2005, 72: 235328.View ArticleGoogle Scholar
- Astumian RD: Reciprocal relations for nonlinear coupled transport. Phys Rev Lett 2008, 101: 046802.View ArticleGoogle Scholar
- Saito K, Utsumi Y: Symmetry in full counting statistics, fluctuation theorem, and relations among non-linear transport coefficients in the presence of a magnetic field. Phys Rev B 2008, 78: 115429.View ArticleGoogle Scholar
- Förster H, Büttiker M: Fluctuation relations without microreversibility in nonlinear transport. Phys Rev Lett 2008, 101: 136805.View ArticleGoogle Scholar
- Sánchez D: Magnetoasymmetric current fluctuations of single-electron tunneling. Phys Rev B 2009, 79: 045305.View ArticleGoogle Scholar
- Sánchez R, López R, Sánchez D, Büttiker M: Mesoscopic Coulomb drag, broken detailed balance, and fluctuation relations. Phys Rev Lett 2010, 104: 076801.View ArticleGoogle Scholar
- žutić I, Fabian J, Das Sarma S: Spintronics: fundamentals and applications. Rev Mod Phys 2004, 76: 323. 10.1103/RevModPhys.76.323View ArticleGoogle Scholar
- Recher P, Sukhorukov EV, Loss D: Quantum dot as spin filter and spin memory. Phys Rev Lett 2000, 85: 1962. 10.1103/PhysRevLett.85.1962View ArticleGoogle Scholar
- Cottet A, Belzig W, Bruder C: Positive cross correlations in a three-terminal quantum dot with ferromagnetic contacts. Phys Rev Lett 2004, 92: 206801.View ArticleGoogle Scholar
- Souza FM, Egues JC, Jauho AP: Quantum dot as a spin-current diode: a master-equation approach. Phys Rev B 2007, 75: 165303.View ArticleGoogle Scholar
- Feng C, Yan L, Lianliang S: Tunable spin-diode with a quantum dot coupled to leads. J Semiconductors 2010, 31: 062002. 10.1088/1674-4926/31/6/062002View ArticleGoogle Scholar
- Cottet A, Belzig W, Bruder C: Positive cross-correlations due to dynamical channel blockade in a three-terminal quantum dot. Phys Rev B 2004, 70: 115315.View ArticleGoogle Scholar
- Bułka BR: Current and power spectrum in a magnetic tunnel device with an atomic-size spacer. Phys Rev B 2000, 62: 1186–1192. 10.1103/PhysRevB.62.1186View ArticleGoogle Scholar
- Wang RQ, Sheng L, Hu LB, Wang B, Xing DY: Coexistence of super-Poissonian mechanisms in quantum dots with ferromagnetic leads. Phys Rev B 2011, 84: 115304.View ArticleGoogle Scholar
- Braun M, König J, Martinek J: Frequency-dependent current noise through quantum-dot spin valves. Phys Rev B 2006, 74: 075328.View ArticleGoogle Scholar
- Weymann I, Barnaś J: Shot noise and tunnel magnetoresistance in multilevel quantum dots: effects of cotunneling. Phys Rev B 2008, 77: 075305.View ArticleGoogle Scholar
- López R Lim J S, Sánchez D: Fluctuation relations for spintronics. Phys Rev Lett 2012, 108: 246603.View ArticleGoogle Scholar
- Sánchez D, Büttiker M: Chirality in Coulomb-blockaded quantum dots. Phys Rev B 2005, 72: 201308.View ArticleGoogle Scholar
- Büttiker M: Scattering theory of current and intensity noise correlations in conductors and wave guides. Phys Rev B 1992, 46: 12485–12507. 10.1103/PhysRevB.46.12485View ArticleGoogle Scholar
- Sauret O, Feinberg D: Spin-current shot noise as a probe of interactions in mesoscopic systems. Phys Rev Lett 2004, 92: 106601.View ArticleGoogle Scholar
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