Spin-orbit interaction induced anisotropic property in interacting quantum wires
© Cheng et al; licensee Springer. 2011
Received: 7 August 2010
Accepted: 11 March 2011
Published: 11 March 2011
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
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) [5–7]. 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 [17–20]. 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 , where ; (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 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 , where . 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 axis, i.e., θ = 3π/4 When α = β the dc conductivity becomes σ ρ = 2K ρ e 2/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 k F = 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.
Dresselhaus spin-orbit interaction
Rashba spin-orbit interaction.
- žutić I, Fabian J, Sarma SD: Spintronics: Fundamentals and applications. Rev Mod Phys 2004, 76: 323.View Article
- Winkler R: Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems. Springer Tracts in Modern Physics. Berlin: Springer; 2003. and the references therein and the references thereinView Article
- Rashba EI: Properties of semiconductors with an extermum loop .1. syclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop. Sov Phys Solid State 1960, 2: 1109. Rashba EI, Sherman EYa: Spin-orbital band splitting in symmetric quantum wells, Phys Lett A 1988, 129:175 Rashba EI, Sherman EYa: Spin-orbital band splitting in symmetric quantum wells, Phys Lett A 1988,129:175
- Dresselhaus G: Spin-Orbit Coupling Effects in Zinc Blende Structures. Phys Rev 1955, 100: 580. 10.1103/PhysRev.100.580View Article
- Luo J, Munekata H, Fang FF, Stiles PJ: Effects of inversion asymmetry on electron energy band structures in GaSb/InAs/GaSb quantum wells. Phys Rev B 1990, 41: 7685. 10.1103/PhysRevB.41.7685View Article
- Hu CM, Nitta J, Akazaki T, Takayanagai H, Osaka J, Pfeffer P, Zawadzki W: Zero-field spin splitting in an inverted In 0.53 Ga 0.47 As/In 0.52 Al 0.48 As heterostructure: Band nonparabolicity influence and the subband dependence. Phys Rev B 1999, 60: 7736. 10.1103/PhysRevB.60.7736View Article
- Koga T, Nitta J, Akazaki T, Takayanagi H: Rashba Spin-Orbit Coupling Probed by the Weak Antilocalization Analysis in InAlAs/InGaAs/InAlAs Quantum Wells as a Function of Quantum Well Asymmetry. Phys Rev Lett 2002, 89: 046801. 10.1103/PhysRevLett.89.046801View Article
- Ganichev SD, Bel'kov VV, Golub LE, Ivchenko EL, Schneider P, Giglberger S, Eroms J, De Boeck J, Borghs G, Wegscheider W, Weiss D, Prettl W: Experimental Separation of Rashba and Dresselhaus Spin Splittings in Semiconductor Quantum Wells. Phys Rev Lett 2004, 92: 256601. 10.1103/PhysRevLett.92.256601View Article
- Koralek JD, Weber CP, Orenstein J, Bernevig BA, Zhang SC, Mack S, Awschalom DD: Emergence of the persistent spin helix in semiconductor quantum wells. Nature 2009, 458: 610. 10.1038/nature07871View Article
- Scheid M, Kohda M, Kunihashi Y, Richter K, Nitta J: All-Electrical Detection of the Relative Strength of Rashba and Dresselhaus Spin-Orbit Interaction in Quantum Wires. Phys Rev Lett 2008, 101: 266401. 10.1103/PhysRevLett.101.266401View Article
- Wang M, Chang K, Wang LG, Dai N, Peeters FM: Crystallographic plane tuning of charge and spin transport in semiconductor quantum wires. Nanotechnology 2009, 20: 365202. 10.1088/0957-4484/20/36/365202View Article
- Luttinger JM: An Exactly Soluble Model of a Many-Fermion System. J Math Phys 1963, 4: 1154. 10.1063/1.1704046View Article
- Tarucha S, Honda T, Saku T: Reduction of quantized conductance at low temperatures observed in 2 to 10 m-long quantum wires. Solid State Commun 1995, 94: 413. Levy E, Tsukernik A, Karpovski M, Palevski A, Dwir B, Pelucchi E, Rudra A, Kapon E, Oreg Y: Luttinger-Liquid Behavior in Weakly Disordered Quantum Wires, Phys Rev Lett 2006, 97:196802 Levy E, Tsukernik A, Karpovski M, Palevski A, Dwir B, Pelucchi E, Rudra A, Kapon E, Oreg Y: Luttinger-Liquid Behavior in Weakly Disordered Quantum Wires, Phys Rev Lett 2006, 97:196802 10.1016/0038-1098(95)00102-6View Article
- Yacoby A, Stormer HL, Wingreen NS, Pfeiffer LN, Baldwin KW, West KW: Nonuniversal Conductance Quantization in Quantum Wires. Phys Rev Lett 1996, 77: 4612. 10.1103/PhysRevLett.77.4612View Article
- Zarea M, Sandler N: Nonuniversal Conductance Quantization in Quantum Wires. Phys Rev Lett 2007, 99: 256804. 10.1103/PhysRevLett.99.256804View Article
- Chang AM, Pfeiffer LN, West KW: Observation of Chiral Luttinger Behavior in Electron Tunneling into Fractional Quantum Hall Edges. Phys Rev Lett 1996, 77: 2538. 10.1103/PhysRevLett.77.2538View Article
- Moroz AV, Samokhin KV, Barnes CHW: Theory of quasi-one-dimensional electron liquids with spin-orbit coupling. Phys Rev B 2000, 62: 16900. 10.1103/PhysRevB.62.16900View Article
- Iucci A: Correlation functions for one-dimensional interacting fermions with spin-orbit coupling. Phys Rev B 2003, 68: 075107. 10.1103/PhysRevB.68.075107View Article
- Gritsev V, Japaridze GI, Pletyukhov M, Baeriswyl D: Competing Effects of Interactions and Spin-Orbit Coupling in a Quantum Wire. Phys Rev Lett 2005, 94: 137207. 10.1103/PhysRevLett.94.137207View Article
- Yu Y, Wen YC, Li JB, Su ZB, Chui ST: Luttinger liquid with strong spin-orbital coupling and Zeeman splitting in quantum wires. Phys Rev B 2004, 69: 153307. 10.1103/PhysRevB.69.153307View Article
- Gogolin AO, Nersesyan AA, Tsvelik AM: Bosonization and Strongly Correlated Systems. Cambridge: Cambridge University Pres; 1998.
- Dolcini F, Trauzettel B, Safi I, Grabert H: Transport properties of single-channel quantum wires with an impurity: Influence of finite length and temperature on average current and noise. Phys Rev B 2005, 71: 165309. 10.1103/PhysRevB.71.165309View Article
- Cheng F, Zhou GH: Transport properties for a Luttinger liquid wire in the presence of a time-dependent impurity. Phys Rev B 2006, 73: 125335. 10.1103/PhysRevB.73.125335View Article
- Eckert D, Ruck K, Wolf M, Krabbes G, Müller K-H: Magnetic behavior of the low-dimensional compounds Ba 2 Cu 3 O 4 Cl 2 and Ba 2 Cu 2 O 4 Cl 2 . J Appl Phys 1998, 83: 7240. 10.1063/1.367613View Article
- Gerhardt C, Mütter K-H, Kröger H: Metamagnetism in the XXZ model with next-to-nearest-neighbor coupling. Phys Rev B 1998, 57: 11504. 10.1103/PhysRevB.57.11504View Article
- Levy E, Tsukernik A, Karpovski M, Palevski A, Dwir B, Pelucchi E, Rudra A, Kapon E, Oreg Y: Luttinger-Liquid Behavior in Weakly Disordered Quantum Wires. Phys Rev Lett 2006, 97: 196802. 10.1103/PhysRevLett.97.196802View Article
- Steinberg H, Barak G, Yacoby A, Pfeiffer LN, West KW, Halperin BI, Le Hur K: Charge fractionalization in quantum wires. Nat Phys 2008, 4: 116. and references therein and references therein 10.1038/nphys810View Article
- Yang W, Chang K: Nonlinear Rashba model and spin relaxation in quantum wells. Phys Rev B 2006, 74: 193314. 10.1103/PhysRevB.74.193314View Article
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.