Controllable persistent spin-polarized charge current in a Rashba ring

    Corresponding author: Feng Liang, bd10583@163.com
  • 1. School of Mathematics and Physics, Huaiyin Institute of Technology, Huai'an 223003, China
  • 2. Department of Physic and Siyuan Laboratory, Jinan University, Guangzhou 510632, China

Key words: spin currentRashba spin-orbit couplingRashba ring

Abstract: We theoretically predict the appearance of a persistent charge current in a Rashba ring with a normal and a ferromagnetic lead under no external bias. This charge current is the result of the breaking of the time inversion symmetry in the original persistent pure spin current induced by the Rashba spin-orbit coupling (RSOC) in the ring due to the existence of the ferromagnetic lead. With the Keldysh Green's function technique, we find that not only the magnitude and sign but also the spin polarization of the generated charge current is determined by the system parameters such as the magnetization direction of the ferromagnetic lead, the tunneling coefficient, the strength of the RSOC and the exchange energy of the ferromagnetic lead, which are all tunable in experiments, that is, a controllable persistent spin-polarized charge current can be obtained in such a device.

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1.   Introduction
  • Since the giant magnetoresistance effect was discovered two decades ago, spintronics which aims to utilize the spin degree of freedom of an electron in solid state structures to carry and transform information has developed into one of the most active subfields in condensed matter physics. This is because spintronics has potential applications in information industry and also because it is directly related to some fundamental physics of electron spin. With the advancements of spintronics, a good generation, manipulation and detection of a spin-polarized current in nanostructures have become three important issues and attracted enormous attention in the research community.

    In order to harness the problem of the effective generation of a spin-polarized current, many magnetic and electric methods[1-7] have been proposed. The major magnetic means is spin pumping[1-3], which is usually attainable in a magnetic system or with an alternating magnetic field. For example, the work of a spin battery proposed by Brataas et al.[1] is based on the generation of a pure spin current pumped by processing ferromagnets. Electric methods have also been put forward[4-7]. In these methods, the spin-Hall effect[4, 5] is the most prominent one in which a longitudinal electric field can induce a transverse dissipationless spin current in the spin-orbit coupled systems.

    In fact, the spin current generation can also be realized in the ring or ring-like systems containing RSOC so long as the system dimension is within the phase-coherent length of an electron[8-13]. For example, Sun et al.[12, 13] have demonstrated that a persistent spin current can exist in a mesoscopic Rashba ring which stems from the RSOC induced spin precession phase. Such a persistent spin current is a pure one without any accompanying charge current. However, a spin current accompanied with a charge current, that is, a spin-polarized charge current especially one whose spin-polarization degree can be controlled is also usually required in spintronics applications. Hence, in this work, we will propose a new scheme to produce a controllable spin-polarized charge current based on the manipulation of the persistent spin current in a Rashba ring by introducing a ferromagnetic lead to connect with the ring. The physical origin for the spin-polarized charge current is that the introduction of the ferromagnetic lead can break the time inversion symmetry in the original persistent spin current so that in addition to the spin current a charge current is also generated in the Rashba ring. By means of the Keldysh Green's function technique, we will study the properties of the generated spin-polarized charge current and reveal the controllability of the charge current.

2.   Model and formula
  • The device we investigate is depicted in Fig. 1. The main part of the device is a one-dimensional mesoscopic ring with a uniform RSOC in it, which lies in the $x$-$y$ plane. In addition to the ring, there also exist two leads, one (we say the left one) is normally metallic and another one is ferromagnetic. The two metallic leads are both one-dimensional and connected with the ring. Then the Hamiltonian of the two-terminal Rashba ring system can be described as

    with

    where $H_{\rm L(R)} $ represents the Hamiltonian of the left (right) lead, $\varepsilon_{k} $ is the single spin-degenerate electron energy, $\sigma h$ is the Zeeman splitting energy where $\sigma =\pm 1;\uparrow, \downarrow $ and $h$ is the exchange energy of the ferromagnetic lead. $C_{k\alpha \sigma }^{+} $ and $C_{k\alpha \sigma } (\alpha =$ L, R) denote the creation and annihilation operator of an electron in lead $\alpha $ respectively. $H_{\rm T} $ describes the coupling between the two leads. $V_{{\rm LR}\sigma }^{1} $ and $V_{{\rm LR}\sigma }^{2} $ are the tunneling coefficients via two different paths of the ring which are

    Here, we assume that the magnitudes of the two coefficients are both equal to $V$. The phase $\sigma \phi /2$ is the RSOC induced spin precession phase[8] that a $\sigma $-spin electron picks up when it tunnels from the left lead to the right one via path 1. Because the RSOC in the ring is uniform and it is also assumed that the two paths are symmetric, the spin precession phase that the $\sigma $-spin electron tunneling from the left lead to the right one via path 2 picks up is $-\sigma \phi /2$ which appears in $V_{{\rm LR}\sigma }^{2} $. Then the phase $\phi $ is the total spin precession phase the $\sigma $-spin electron acquires after a circulation in the ring and $\phi =\gamma mL/\hbar^{2}$ with $\gamma $ the RSOC strength, $m$ the effective mass of the electron and $L$ the perimeter of the Rashba ring. Since the RSOC strength $\gamma $ can be readily varied in experiments by an electric field or a gate voltage[14], the spin precession phase is electrically tunable.

    Now we proceed to calculate the charge and spin currents in the ring when no external bias is applied between the leads. The electronic current per spin channel flowing from the left lead to the right one can be obtained from the time evolution of the occupation number for electrons with the corresponding spin in the left lead, that is,

    with

    where $[\cdots]$ and $\langle \cdots \rangle$ denote operator commutation and thermal average respectively, $C_{i\sigma }^{+} (C_{i\sigma } )$ describes the creation (annihilation) of an $\sigma $-spin electron at site $i$ in the left lead. With the Keldysh Green's function method, the spin-dependent electronic current flowing in path 1 and path 2 can be obtained

    with the lesser Green's function $G_{\alpha \sigma, \beta \sigma }^{<} (t, t{'})$ defined as

    Then the charge current in the two paths can be given by

    Here Tr denotes the trace over spin indices. $V_{\alpha \beta }^{1(2)} $ and $G_{\alpha \beta }^{<} $ are the matrix of the tunneling coefficients and the lesser Green's function in spin space. After applying Fourier transformation to the above equation, we arrive at

    Following the similar procedure, the spin current in the ring can also be worked out

    where $\sigma_{ s} $ describes the three Pauli matrices with spin index $s = x, y, z$ denoting the spin polarization direction of the spin current. Since no bias is applied to the system considered by us and thus the system is in equilibrium, we can evaluate the lesser Green's functions in Eqs. (12) and (13) by the well-known relations

    where $G^{r}$ and $G^{a}$ are the retarded and advanced Green's functions of the system respectively, and $f(E)$ is the Fermi function of the system. As is well-known, it is convenient to solve the retarded and advanced Green's functions by adopting the Dyson equations

    with

    where $G_{\alpha \alpha }^{r(a)} $ and $g_{\alpha \alpha }^{r(a)} $ are the retarded (advanced) Green's function and surface Green's function of lead $\alpha $ respectively. For the system being in equilibrium state, the Green's function $G_{\rm LL}^{r(a)} $ can be evaluated by the direct matrix inversion[15]

    where the surface Green's functions $g_{\rm LL}^{r(a)} $ and $g_{\rm RR}^{r(a)} $ are given by[15]

    and

    Here $a$ is the lattice constant of the two leads; $t$ is the hopping energy of electron; $k$ is the wave length of electrons in the normal metal lead; and $k_{\uparrow } $ and $k_{\downarrow } $ are respectively the wave lengths of the up-spin electron and down-spin electron in the ferromagnetic lead. Eq. (19) is only valid when the ferromagnetic lead is magnetized along the $z$-direction. If the magnetization direction is along the direction determined by the azimuth angles $(\alpha, \beta )$, the Green's function $g_{\rm RR}^{r(a)} $ needs to be transformed to

    with the unitary matrix

    With these preparations, we can straightforward work out the charge and spin currents in Eqs. (12) and (13)

    with

    and

    where $k$ and $k_{\sigma } $ satisfy the following dispersion relations

    As Eqs. (22)-(27) show, a charge current can emerge and circulate in the ring accompanied with a spin current which is polarized along the $z$-direction. Since the Fermi distribution function is in the integration, all the electrons below the Fermi surface have contributions to the two currents. The spin current is just the persistent spin current in the Rashba ring[12, 13]. It results from the RSOC induced spin precession phase, when the RSOC is absent, that is, $\phi =0$, it vanishes as shown in Eqs. (24) and (26). In addition to the persistent spin current, a persistent charge current also exists in the Rashba ring. This charge current may be attributed to the existence of the ferromagnetic lead. It is just the ferromagnetic lead induced breaking of the time inversion symmetry of the system that destroys the balance between the up-spin electronic flow and the down-spin one in the original persistent spin current and thus generate the charge current. Therefore, the ferromagnetic lead and the persistent spin current are two prerequisites for the generation of the charge current, so that either $k_{\uparrow } -k_{\downarrow } $ or $\sin \phi $ is 0, and the charge current in Eq. (22) disappears. From Eq. (25), it is found that the magnetization direction also has a huge effect on the generated charge current, to be exact, the charge current is proportional to $\cos \alpha $, which indicates that only the magnetization component along the $z$-direction contributes to the generation of the charge current in the ring.

3.   Results and discussions
  • In the following, we perform numerical calculations of the induced spin-polarized charge current at zero temperature. During our calculations, 1 meV is set as the energy unit and the hopping energy of electrons is always set as $t=1$. In order to investigate the effects of the system parameters on the spin polarization of the spin-polarized charge current, we define the inversion of the spin polarization degree as $\eta =I_{\rm Lc}^{1} /J_{\rm Lz}^{1} $. In Fig. 2, the persistent charge current $I_{\rm Lc} =I_{\rm Lc}^{1} $ in the Rashba ring is plotted as a function of the tunneling coefficient $V$ for different $\phi $. As we can see when $V$ increases from zero, the magnitude of $I_{\rm Lc} $ increases quickly and after the magnitude arrives at its peak value, it begins to decrease and approaches zero gradually. It is clear that the height and location of the peak are both affected by the spin precession phase. In addition, the spin precession phase can also influence the sign of the induced charge current, while the tunneling coefficient cannot.

    Since the RSOC strength which is directly related to the spin precession phase can be tuned in a large extent by an external electric field or a gate voltage in experiments, we present the induced charge current as a function of the spin precession phase $\phi $ for different $V$ in Fig. 3. From Fig. 3 we see that the induced charge current oscillates with the spin precession phase. It means the magnitude and sign of the induced charge current can both be modulated by the RSOC strength. From Fig. 3 it is also found that the behavior of the induced charge current is correlated with the tunneling coefficient. For a small tunneling coefficient, the induced charge current exhibits a strict sinusoidal behavior, which in physics can be seen as a result of linear response theory; while for a big tunneling coefficient the behavior of the induced charge current can deviate from a sinusoidal one obviously, which can be regarded as a nonlinear result due to the strong coupling between the leads.

    Besides the spin-polarized charge current itself, the spin-polarization of the current is also important for spintronics applications. So in the following, we will discuss the dependence of $\eta $ on the system parameters. Fig. 4 displays $\eta $ versus the tunneling coefficient $V$ for different spin precession phase. As $V$ increases the magnitude of $\eta $ decreases firstly and after passing through a valley the value of the magnitude of $\eta $ increases with the further increase of $V$. The location of the valley in the $\eta $-$V$ curve is determined by other system parameters such as the spin precession phase. More importantly, when $V$ is chosen approximately $\eta $ has the chance to be one, which indicates that a fully spin-polarized charge current can be obtained by adjusting the tunneling coefficient. Finally, the effect on the spin-polarization of the current exerted by the exchange energy $h$ is also investigated. As shown by Fig. 5, $\eta $ increases monotonously with the increase of $h$. This phenomenon may be attributed to that a stronger ferromagnetic field can break the balance between the up-spin and down-spin currents in the original persistent spin current more heavily and then gives rise to a stronger charge current. Furthermore, at some choices of $h$ the spin-polarization degree of the current can also reach 100%, that is, a fully spin-polarized charge current forms in the ring.

4.   Conclusion
  • In conclusion, we have studied a Rashba ring system in the presence of a ferromagnetic lead and found that a persistent spin-polarized charge current can circulate in the ring without any applied bias. Here the spin-polarized charge current is induced due to the breaking of time inversion symmetry of the system originated from the ferromagnetic field, which manipulates the up-spin and down-spin currents in the original persistent spin current driven by the RSOC induced spin precession phase. More interestingly, the spin-polarized charge current can be controlled by the system parameters in its magnitude, sign and even spin-polarization. Hence, with such a device, a controllable spin-polarized charge current can be achieved.

Figure (5)  Reference (15) Relative (20)

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