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Spin-dependent current noises in transport through coupled quantum dots

Current fluctuations can provide additional insight into quantum transport in mesoscopic systems. The present work is carried out for the fluctuation properties of transport through a pair of coupled quantum dots which are connected with ferromagnetic electrodes. Based on an efficient particle-number-resolved master equation approach, we are concerned with not only fluctuations of the total charge and spin currents, but also of each individual spin-dependent component. As a result of competition among the spin polarization, Coulomb interaction, and dot-dot tunnel coupling, rich behaviors are found for the self- and mutual-correlation functions of the spin-dependent currents.


I. INTRODUCTION
Noise of transport current through mesoscopic system can provide useful information beyond the average current [1,2]. Typical examples include quantum shuttles [3,4,5,6], spin valves [7,8,9], double-dot structures [10,11,12,13], as well as qubit measurement [14,15,16,17]. These studies were focused largely on the total charge current. A relatively less exploited subject, but now gaining increasing attention, is the spin-dependent current fluctuations [18,19,20,21]. A number of interesting examples are listed as follows: (i) It was shown in [18] that the shot noise of spin current is closely related to the spin unit of quasiparticles. (ii) In the absence of charge current, the magnetically pumped spin current noise through quantum dots in the Coulomb blockade regime can serve as sensitive probe to spin decoherence [19]. (iii) For a quantum dot strongly coupled to cavity field, shot noise of the spin current exhibits clear signature of the discrete nature of the photon states [20].
(iv) Spin-dependent shot noise of unpolarized currents can be used to detect attractive or repulsive correlations [21].
In this work we study the spin-dependent current noises for transport through a pair of coupled quantum dots connected with ferromagnetic (FM) electrodes. Owing to the additional modulation of the dot-dot coupling, rich noise behaviors are found as a consequence of interplay between the Coulomb correlation and the spin polarization of the FM electrodes.
Methodologically, we employ an efficient particle-number-resolved master equation approach [15,22,23], with proper extension by including the spin degrees of freedom. This approach, which can be considered as a finite temperature and voltage extension of the "n"-resolved quantum Bloch-type equation proposed by Gurvitz and Prager [24], has the advantages in its simplicity of treating properly a broad range of dissipation, as well as its transparency in the involving dynamics and many-body interactions.
The paper is organized as follows. We begin in the next section with the model setup of double dots connected with FM electrodes. We then present in Sec. III a general formalism for both the spin-dependent currents and their noises. Numerical results are presented in sections IV and V, and then followed by the conclusion in Sec. VI.

II. MODEL DESCRIPTION
The system under study is schematically shown in Fig. 1. The left and right quantum dots are labeled by "L" and "R", respectively. The total Hamiltonian contains three parts: H B = α=L,R k,σ ε αk c † αkσ c αkσ describe the left and right electrodes, which are magnetically polarized. c αkσ (c † αkσ ) is the electron annihilation (creation) operator of the electrode α = L or R, with spin σ = ↑ or ↓. The FM electrodes are characterized by the spin-dependent density of states g ασ (ω). Moreover, the densities of the states are assumed to be energy independent, i.e. g ασ (ω) = g ασ . Then, the FM polarization of the electrodes is simply determined by a parameter p α = (g α↑ − g α↓ )/(g α↑ + g α↓ ), which satisfies −1 p α 1.
The second Hamiltonian is for the coupled dots: Heren ασ = d † ασ d ασ andn α = σn ασ , with d ασ (d † ασ ) the electron annihilation (creation) operator in the quantum dot α and with spin σ. Note that here α labels the left and right quantum dots rather than the electrodes. Each quantum dot is assumed to have only one spin-degenerate energy level E L/R . U 0 and U ′ are the intra-dot and inter-dots Coulomb interactions; and Ω is the coupling strength of the two dots. The typical value of intra-dot interaction U 0 is estimated to be 4∼5meV, and U ′ ≈ U 0 /2 [25].
The third Hamiltonian describes the coupling of the central dots to the electrodes, in terms ofĤ ′ = αkσ t αk c † αkσ d ασ +H.c. . Note that, due to spin polarization of the electrodes, the tunnel coupling strength would become spin-dependent, i.e. with Γ α↑ = (1 + p α )Γ α /2 is the total coupling strength regardless the spin orientation, and is estimated to be 10 9 ∼ 10 10 s −1 [26].
In this work, corresponding to different bias voltages, we will effectively consider the  [23].

III. FORMALISM
In this section we outline the master equation approach for the calculation of the spindependent shot noise. The main point is to obtain the equation of motion for the conditional reduced density matrix of the central quantum dots, which is conditioned not only on the transmitted electron numbers, but also on their spins.

A. Electron-number-resolved master equation
To achieve the description of spin-dependent transport, we will first extend the "n"resolved quantum master equation [15,22,23] to include spin degrees of freedom. Let us start with the reduced density matrix, defined as ρ(t) ≡ Tr B [ρ T (t)], i.e. tracing over the electrode states from the total density matrix. Assuming the tunneling HamiltonianĤ ′ is weak, the second-order cumulant expansion leads to the master equation [27], , where n ασ is the number of electrons with spin-σ arrived at the electrode α (α=L, or R). Partially tracing over states in each subspace leads to the spin-resolved quantum master equation for the reduced density matrix, Here, A ασσ ′ (t)e ±iLt . The involved bath correlation functions are defined as C standing for the usual meaning of thermal bath average.

B. Spin-dependent currents
With the knowledge of the above conditional state, the joint probability distribution function is obtained as P [( , where Tr(· · · ) denotes the trace over the system states. It allows us to evaluate the spin-σ dependent current through the junction α by . Based on Eq. (4), we obtain This leads to where ρ(t) is the unconditional density matrix that satisfies Eq. (3), oṙ

C. Spin-dependent noises
Note that the total charge and spin currents through junction α are I ch The total charge current noise and spin current noise can then be expressed as where the individual spin-dependent noise spectrum is defined as In the following, fluctuation between the same or opposite spin currents is referred to as self-or mutual-correlation shot noise, respectively. Based on the MacDonald's formula [6,28,29], it follows that whereĪ σ α is the stationary current obtained from Eq. (8), and M σσ ′ αα ′ (t) ≡ e 2 n L↑ ,n L↓ ,n R↑ ,n R↓ n ασ n α ′ σ ′ P ( n L↑ , n R↑ n L↓ , n R↓ ), t . Using the spin-resolved master equation (4), one finds where N σ α (t) can be found from Eq. (6), and ρ st is the stationary solution of the master equation (8). Specially, the zero-frequency spin-dependent noise which is of most interest reads [6,10,19] So far we have outlined a compact formalism for spin-dependent transport through mesoscopic systems. As will show in the following, this approach can be efficiently used to study the spin-dependent transport phenomena, say, not only the spin-dependent currents, but γ d ≡ 4Γ L + Γ R are the corresponding total effective tunneling rates for these three cases.
also their noise properties. The major approximation involved in our formalism is the standard second-order Born approximation with respect to the perturbative expansion of the tunneling Hamiltonian. Apparently, this approach is applicable under the Markov-type condition for relatively high temperatures, say, higher than the tunneling width [30]. In this case, the formalism can be safely applied for arbitrary bias voltages, including those with Fermi level of electrode in near resonance with the energy level of the central (transport) system. Also, the present formalism is applicable to low temperature under large bias voltage, i.e. the electrode's Fermi surface is further away from the central system's energy level than the level broadening. Similar statement holds also for the widely applied rate equation approach at zero temperature [24]. Finally, at low temperature and under low bias voltage (near-resonance transport), an improved self-consistent Born approximation [31] or exact hierarchical equations of motion approach [32] can be satisfactorily employed. to p = 1, ρ L↓ decreases rapidly, while Γ L↑ increases linearly. Then, a turnover behavior for the spin-up current is formed as shown by the solid curve in Fig. 2(a). This is also the basic reason for the unique features of the charge and spin currents shown by the solid curves in Fig. 2(c) and (d).

V. NOISE CHARACTERISTICS
In this section, instead of using the circuit noise [8,13,23], we will investigate various fluctuations of spin currents through left or right junction, i.e. the auto-correlations S σσ ′ αα with α = L or R. For the cross-correlations, they simply satisfy the relation S σσ ′ LR = −S σσ ′ αα in the present two-terminal case, as we have checked. It should be noted that for a threeterminal structure such a simple relation generally does not take place (see, for instances, Ref. [36,37,38]).

A. Self-correlation shot noise
We now turn to the noise characteristics, and first to the self-correlation shot noise. Note that there is a symmetry between S ↑↑ αα and S ↓↓ αα with respect to the spin polarization, i.e.,

Fig. 3(a) versus (b) and (d) versus (e)
. Therefore, it needs only to consider either S ↑↑ αα or S ↓↓ αα . For S ↑↑ αα in Fig. 3(a) and (d), it is found that both the Coulomb correlation in the dots and the spin polarization of the electrodes will enhance the Fano factor. The shot noise is of NINT < SDCB < DDCB, consistent with the Coulomb correlation strength. The only exception in terms of spin polarization is that shown by the solid curve in Fig. 3(a), for the weak dot-dot tunnel coupling and in the absence of many-body Coulomb interaction. In this case the Fano factor does not monotonically increase with the polarization degree of the electrodes. This behavior originates from the same reason as that leading to the shoulder behavior of the charge current in Fig. 2(c).
Remarkably, a profound, strong super-Poisson behavior can be developed by increasing the dot-dot tunnel coupling for both the SDCB and DDCB cases, provided the electrodes are properly spin polarized, see the dashed and dotted curves in Fig. 3(d). This novel behavior can be understood in terms of the so-called dynamical spin blockade mechanism. Take S ↑↑ αα again for illustration. If the electrode is sufficiently spin-up polarized, the spin-up electrons can more easily pass through the two dots than the spin-down electrons. Furthermore, the SDCB (DDCB) does not allow double occupation in single dot (double dots). Then a mechanism of fast-to-slow channels is developed, which results in a bunching behavior and also the super-Poisson statistics of the transport electrons.
Note that the strong Coulomb interaction is essential to the bunching behavior of tunneling events, otherwise the spin-up and spin-down electrons will transport independently, and will not result in the super-Poisson statistics at all [1]. Also, a relatively strong dot-dot coupling is required for the super-poisson noise in the coupled double dots system. If the dot-dot coupling is weak, electron will attempt to stay longer in the left dot. This will weaken the bunching behavior, thus lead all the noise components to sub-Poisson, see the dashed and dotted curves in Fig. 3(a) and (b).
The mechanism of dynamic spin blockade plays also an essential role in a three-terminal quantum dot [37,38], and in an interacting quantum dot with intradot spin-flip scattering connected to FM electrodes [39]. Meanwhile, a similar mechanism called dynamic charge blockade is responsible for the super-Poisson noise in interacting two-channel systems [40,41,45], as well as double quantum dots structures [42,43].

B. Mutual spin correlation shot noise
We now turn to the mutual spin correlation shot noise (S ↑↓ αα or S ↑↓ αα ), which are symmetric to the polarization, as shown in Fig. 3(c) and (f). For noninteracting dots, the mutual correlation between opposite spin currents are zero, as shown by the solid curves in Fig. 3(c) and (f). This is simply because the spin-up and spin-down currents are uncorrelated in the absence of Coulomb interaction, despite the electrodes are spin polarized. Similar behavior was also found in [21], where the mutual correlation is zero for noninteracting single dot connected with normal electrodes.
For interacting dots, i.e., the SDCB and DDCB shown in Fig. 3  In this context, we notice that the super-Poissonian self-correlation does not necessarily imply a positive mutual correlation, as shown by the dashed curve in Fig. 3(f) for the SDCB, where the mutual correlation, despite the polarization degree of the electrodes, is fully negative. A simple and unambiguous understanding for the above positive and negative mutual correlation is yet to be found. It seems more subtle than the super-Poisson statistics of the self-correlation of current.

C. Total charge and spin current noises
The total charge and spin current noises are nothing but the combination of those components of the spin-dependent noises, according to Eq. (9). The results are displayed in Fig. 4, where most features such as the sub-Poisson and super-Poisson behaviors can be accordingly 2eĪ d understood in terms of the above interpretation to the partial noises.
A noticeable result is the spin current noise resulting from the DDCB. We find that it is constantly Poissonian, regardless the polarization degree and dot-dot coupling, as shown by the dotted curves in Fig. 4(b) and (d). Similar Poissonian spin current noise was also found in [21], for single quantum dot in the Coulomb blockade regime.
To complete this section, we consider all kinds of noises discussed above in the limit of p = 0. In this unpolarized situation, simple analytic results can be obtained, as shown in table II, which serves as a complement to the numerical results in Fig. 3 and Fig. 4. The total charge current noise in the NINT case differs from the ones in [10,12,44] by an overall factor of 2 (implied inĪ 0 , see table I). This factor originates from the spin degree of freedom, which was ignored there. For interacting cases, under symmetric coupling (Γ L = Γ R ) and resonant levels (E L = E R ), the total charge current noises are found here to be sub-Poissonian, in either the SDCB or the DDCB cases. Very interestingly, however, if E L = E R and Γ L > Γ R , the interplay of quantum coherence and the DDCB would develop a dynamical charge blockade mechanism, which then result in a remarkable super-Poissonian noise. Our result in the DDCB case slightly differs from the ones in [10,42]. Again, the difference arises from the spin degree of freedom, which was neglected in the previous work.

VI. CONCLUSION
To summarize, based on an efficient particle-number resolved quantum master equation with inclusion of the spin degrees of freedom, we investigated the spin-dependent noises in transport through a pair of coupled quantum dots connected with ferromagnetic electrodes.
The modulation of the dot-dot coupling, and the interplay between Coulomb interactions and this work were carried out at zero temperature. At finite temperatures, backward processes are possible, which will in general reduce the current, but enhance the noise, as clearly demonstrated in [45].
Finally, we briefly discuss the measurement of spin-dependent noise. It was shown in [46] that in hybrid ferromagnetic-normal metal structures, the spin current noise exerts a fluctuating spin torque on the magnetization vector, which causes an observable magnetization noise. By measuring this magnetization noise, which has been realized experimentally [47], one then obtains the spin current noise.