Spin-orbit interaction induced anisotropic property in interacting quantum wires
We investigate theoretically the ground state and transport property of electrons in interacting quantum wires (QWs) oriented along different crystallographic directions in (001) and (110) planes in the presence of the Rashba spin-orbit interaction (RSOI) and Dresselhaus SOI (DSOI). The electron ground state can cross over different phases, e.g., spin density wave, charge density wave, singlet superconductivity, and metamagnetism, by changing the strengths of the SOIs and the crystallographic orientation of the QW. The interplay between the SOIs and Coulomb interaction leads to the anisotropic dc transport property of QW which provides us a possible way to detect the strengths of the RSOI and DSOI.
PACS numbers: 73.63.Nm, 71.10.Pm, 73.23.-b, 71.70.Ej
KeywordsCoulomb Interaction Electron Ground State Charge Density Wave Spin Density Wave Luttinger Liquid
Dresselhaus spin-orbit interaction
Rashba spin-orbit interaction.
All-electrical manipulation of spin degree of freedom is one of the central issues and the ultimate goal of spintronics field. The spin-orbit interaction (SOI) is a manifestation of special relativity. An electric field in the laboratory frame can transform into an effective magnetic field in the moving frame of electron and consequently leads to electron spin splitting. Therefore, the SOI provides us an efficient way to control electron spin electrically and has attracted tremendous interest because of its potential application in all-electrical spintronic devices [1, 2]. The spin degeneracy can be lifted by applying magnetic field to break the time-reversal symmetry and/or by applying electric fields to break the spatial inversion symmetry. However, the latter could be more easily realized in spintronics devices. In semiconductors the spatial inversion symmetry can be broken by the structural inversion asymmetry and bulk crystal inversion asymmetry, named, Rashba SOI (RSOI) and Dresselhaus SOI (DSOI), respectively [3, 4]. Usually, the RSOI in semiconductor quantum well is much stronger than that of DSOI, and therefore, most of the previous theoretical and experimental studies focused on the RSOI and its consequence on the spin transport properties in two-dimensional electron gas (2DEG) [5, 6, 7]. In thin quantum wells, the strength of the DSOI is comparable to that of the RSOI since the strength of the DSOI depends significantly on the thickness of quantum wells. The interplay between the RSOI and DSOI leads to interesting phenomena in 2DEG, e.g., the anisotropic photogalvanic effect , and the persistent spin helix . However, the interplay between the RSOI and DSOI in quantum wires (QWs) remains relatively unexplored.
Very recently, the anisotropic behavior of transport property in semiconductor QWs was proposed to detect the relative strength between the RSOI and DSOI in a quasi-one-dimensional (Q1D) semiconductor QW system [10, 11]. However, the effect of the Coulomb interaction on the transport property is not addressed. Since the Coulomb interaction becomes very strong in Q1 D electron systems where electrons are strongly correlated, and therefore the conventional Fermi liquid theory breaks down. There are no fermionic quasi-particle in Q1 D electron gas, and the elementary excitations are bosonic collective charge and spin fluctuations with different propagating velocities. The Luttinger liquid (LL) theory  is of fundamental importance because it is one of a very few strongly correlated non-Fermi liquid systems that can be solved analytically. The LL displays very unique properties, e.g., the spin and charge separation and the power-law behavior of the correlation functions. The unique behavior was observed experimentally in many Q1 D systems, for instance, narrow QW formed in semiconductor heterostructures , carbon nanotube , graphene nanoribbon , as well as the edge states of the fractional Quantum Hall liquid . Recent studies have found that the RSOI would lead to the mixing between the spin and charge excitations in QW with the RSOI alone [17, 18, 19, 20]. It is interesting to study the interplay between the RSOI and DSOI on the ground state and transport property of QWs in the presence of the Coulomb interaction.
In this study, based on the LL theory, we study the effect of the interplay between the Coulomb interaction and SOIs on the electron ground state and transport property of an interacting QW oriented along different crystallographic directions in different planes. The electron ground state can display the transitions among the different phases, e.g., the spin density wave (SDW), charge density wave (CDW), singlet superconductivity (SS), and metamagnetism (MM), by tuning the crystallographic plane and orientation of the QW, the strengths of SOIs and the Coulomb interaction. The anisotropy of the dc conductivity of interacting QW is induced by the interplay between the Coulomb interaction and SOIs, which could be used for detecting the strengths of the RSOI and DSOI.
Where ϑ ρ and ϑ σ are the phase fields for the charge and spin degrees of freedom, respectively, and Π ρ and Πσ are the corresponding conjugate momenta. v ρ and v σ are the propagation velocities of the decoupled charge and spin-collective modes in the absence of the SOIs (δv = 0), respectively.
where the parameters α, β, and θ are the same as in Equation (1). The dominant differences are (1) the Fermi velocity is different Open image in new window , where Open image in new window ; (2) the effective magnetic field induced by the DSOI is perpendicular to (110) plane, which could generate an out-of-plane spin-polarized current; and (3) the anisotropy is mainly determined by the DSOI. These differences would lead to the distinct anisotropic behavior of QWs embedded in (110) plane. For (111) plane, the DSOI shows the same formulism as the RSOI; the Hamiltonian is Open image in new window which does not contain any θ-dependent term, and therefore the anisotropic behavior disappear. This isotropic character can be easily seen from the constant energy surface (see Figure 1b). We will not discuss the (111) plane any more.
are the propagation velocities of coupled collective modes that depend on the crystallographic orientation θ. From the above equations, one can see that the interplay between the RSOI and DSOI will lead to anisotropy, i.e., the dependence on the crystallographic direction. This is the dominant difference between this work and the previous study . Interestingly, for a QW in (001) plane, the spin and charge excitations can be decoupled again when θ = 3π/4 and α = β, θ = π/4 and α = -β, since the coupling between the spin and charge excitation disappear when δv = 0 (see the third term in Equation (4)).
Numerical results and discussions
For an interacting QWs embedded in (110) plane, the anisotropy of the dc conductivity becomes different (see Figure 4c,d): (1) The RSOI becomes less important for the anisotropy of the dc conductivity comparing with that in (001) plane, the increase of the strength of RSOI only affects the anisotropy slightly (see the dashed green line and dotted blue line in Figure 4c), since the anisotropy is mainly determined by the DSOI. This can be understood from the formulism of the Fermi velocities Open image in new window , where Open image in new window . It is noted that for (001) plane, the anisotropy of dc conductivity will disappear with the RSOI or DSOI alone, while for (110) plane, the anisotropy will disappear only when the DSOI is absent, which can be understood from the expression of δv. (2) The effect of the Coulomb interaction on the anisotropy of dc conductivity is weakened significantly when compared to that in (001) plane (see Figure 4b,d).
Finally, we discuss how to detect the relative strength of the SOIs utilizing the anisotropic dc conductivity. We consider a QW embedded in (001) plane. One can tune the strength of RSOI by adjusting the gate voltage. For a QW oriented along the Open image in new window axis, i.e., θ = 3π/4 When α = β the dc conductivity becomes σ ρ = 2K ρ e2/h, where the Coulomb interaction parameter K ρ can be deduced from the experiments which ranges from 0.4 to 0.7 in semiconductor QWs [26, 27]. Therefore, one determines the relative strength of the RSOI and DSOI from the dc conductivity. In our calculation, we take the electron effective mass m = 0.067m e , the strengths of SOIs α and β are about 1 × 10-11 eVm which is the typical strength of SOI in conventional semiconductors . Electrons only occupy the lowest subband in GaAs QW assuming the Fermi wavevector is kF = 0.01 nm-1.
In conclusion, we investigate theoretically the effect of the interplay between the Coulomb interaction and the SOIs on the electron ground state and charge transport property of interacting QWs oriented along different crystallographic directions in different planes. We find that the ground state of electrons in the QWs can transit among the different phases, e.g., the SDW, CDW, SS, and MM, by tuning the plane and orientation of the QW, the strengths of SOIs and the Coulomb interaction. The anisotropy of the dc conductivity in an interacting QW is induced by the interplay between the Coulomb interaction and SOIs. This anisotropy enables us to detect the strengths of RSOI and DSOI, which are very important for the comprehensive understanding of spin decoherence and constructing all-electrical spintronic device.
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