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.
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