Spin Accumulation in a Quantum Wire with Rashba Spin-Orbit Coupling

Abstract

We investigate theoretically the spin accumulation in a Rashba spin-orbit coupling quantum wire. Using the scattering matrix approach within the effective free-electron approximation, we have demonstrated the three components of spin polarization. It is found that by a few numerical examples, the two peaks for the out-of-plane spin accumulation ⟨𝑆_𝑧⟩ shift to the edges of quantum wire with the increase of propagation modes. The period of intrinsic oscillations ⟨𝑆_𝑥/𝑦⟩ inversely proportions to the Rashba SOC strength. This effect may be used to differentiate the intrinsic spin accumulation from the extrinsic one.

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Spintronics is a rapidly developing field of research in recent years because spin-based devices are believed to be the new generation of electronic devices [1–3]. Among the research area of spintronics, the spin-orbit coupling (SOC) creates another way to manipulate spins by means of an electric field [3], which has great potential applications in the future. The Rashba spin-orbit coupling (RSOC) effect is found to be very pronounced in some semiconductor layer (e.g., InAs or InGaAs) and its strength can be controlled by gate voltage [4].

The spin polarization in two-dimensional electron gas (2DEG) systems with SOC has been attracting much attention [5–9]. Recently, the experimental observations [10, 11] of spin Hall effect (SHE) in 2DEG with SOC have given evidence of the possibility to control electron spins by an electric field. Wang and Sheng [12] have investigated the current induced local spin polarization due to weak RSOC in a narrow strip. Furthermore, Nikolić et al. [13] have demonstrated that the flow of a longitudinal unpolarized current through a ballistic 2DEG with RSOC will induce a nonequilibrium spin accumulation along the lateral edges. More recently, the out-of-plane spin Hall accumulation effect has been observed in a Rashba bar [14]. Using the nonequilibrium Green's function approach, Wang et al. [15] have studied the intrinsic oscillation of spin accumulation induced by RSOC in a quantum wire (QW) system. However, the work in [12, 14, 15] has only considered the hard-wall potential confined Rashba QWs.

In this paper, using the scattering matrix method within the effective free-electron approximation, we study theoretically the spin accumulation in a parabolic potential confined Rashba QW connected to two normal leads (without SOC) nonadiabatically, and the results are discussed and compared with the previous works [12–15]. The system under consideration is a wide-narrow-wide (WNW) geometry of structure, and the number of propagation modes in the narrow QW and wide leads is controlled by the different strength of parabolic confining potentials. The transverse direction corresponds to 𝑥-axis and the longitudinal direction to 𝑦-axis. Therefore, the effective mass single-electron Hamiltonian in the QW is 𝐻=𝐻₀+𝐻_𝑠𝑜, (1)where𝐻₀ℏ=−²2𝑚^∗(𝜕²𝜕𝑥²+𝜕²𝜕𝑦²)+𝑉(𝑥) (2)is the uncoupled QW Hamiltonian while the RSOC Hamiltonian [14, 15] reads𝐻_𝑠𝑜=𝑖𝛼_𝑅(𝜎_𝑦𝜕𝜕𝑥−𝜎_𝑥𝜕𝜕𝑦). (3)In the above Hamiltonians, 𝑉(𝑥)=𝑚^∗𝜔²𝑥²/2 is the lateral confining potential with 𝑚^∗, and 𝜔 the effective electron mass and oscillator frequency, while 𝛼_𝑅 denotes the Rashba strength, and 𝜎_𝑥/𝑦 the 𝑥/𝑦 component of pauli matrix.

Since the Hamiltonian in the leads has no RSOC term, we assume that an unpolarized electron wave is injected from the left lead into the QW. Due to the translation invariance in the longitudinal 𝑦-direction, the electron wave function in the leads can be decomposed into two components of a plane wave and a transversally confined wave function for the 𝑁th mode 𝜙_𝑁↑↓𝑒(𝛼𝑥)=^{−(𝛼𝑥)2/2}2^𝑁𝑁!√𝜋𝐻_𝑁(𝛼𝑥)(𝑁=1,2,…), (4)where ↑(↓) represents spin-up (-down) state, 𝛼=(𝑚^∗𝜔/ℏ)^1/2, and 𝐻_𝑁(𝛼𝑥) is the Hermite polynomial. Since no RSOC in the leads, the total electron energy for spin-up or -down electron is same, 𝐸=ℏ²𝐾²_𝑁/2𝑚^∗+𝜀_𝑁, where the lateral sublevel 𝜀_𝑁=(𝑁+1/2)ℏ𝜔 and the longitudinal wavevector 𝐾_𝑁=2𝑚^∗(𝐸−𝜀_𝑁)/ℏ.

The wave function for the weak Rashba QW is a two-components spinor 𝜓(𝑥,𝑦)={𝜓_↑(𝑥,𝑦)𝜓_↓(𝑥,𝑦)} and 𝜓_↑↓(𝑥,𝑦) can be also decomposed into two components for spin-up (↑) and spin-down (↓) electron, respectively. The corresponding transverse (𝑥 direction) wave functions in the perturbation regime [16] can be solved as𝜑_𝑛↑↓=𝑒(𝛼𝑥)_{−(𝛼𝑥)2/2}2^𝑛𝑛!√𝜋{𝐻_𝑛𝑙(𝛼𝑥)±^𝜔𝑙_𝛼𝑅[𝑛𝐻_𝑛−11(𝛼𝑥)+2𝐻_𝑛+1(𝛼𝑥)]}, (5)where 𝑙_𝛼𝑅=ℏ²/(2𝑚^∗𝛼_𝑅) and 𝑙_𝜔=1/𝛼 the character length related to the oscillator frequency, and the sign “+/−" corresponding to “↑/↓" state. The electron energy in the QW is 𝐸_↑↓=ℏ²𝑘²_𝑛↑↓/2𝑚^∗+𝜀_𝑛↑↓, where the lateral sublevel of the 𝑛th mode is 𝜀_𝑛↑=(𝑛+1/2)ℏ𝜔+𝛼_𝑅𝑘_𝑛↑ and 𝜀_𝑛↓=(𝑛+1/2)ℏ𝜔−𝛼_𝑅𝑘_𝑛↓, respectively. Therefore, the longitudinal wavevectors can be expressed as𝑘^±_𝑛↑(𝐸_↑𝑚)=^∗ℏ²[−𝛼_𝑅±𝛼²_𝑅+2ℏ²𝐸_↑−(𝑛+1/2)ℏ𝜔𝑚^∗𝑘],_±𝑛↓(𝐸_↓𝑚)=^∗ℏ²[𝛼_𝑅±𝛼²_𝑅+2ℏ²𝐸_↓−(𝑛+1/2)ℏ𝜔𝑚^∗] (6)for spin-up and -down electron, respectively.

In order to calculate the scattering matrix separately for spin-up and -down electron, one needs to match the wave functions at the two interfaces of wire-lead connection. For a symmetric system, one can only calculate the scattering matrix for one interface. The total scattering matrix for the system can be obtained by the symmetry property of the system [17]. Here, we first consider the left interface at 𝑦=0 as if without the right interface (𝑦=𝐿) for electron incident from left to right. Because of the degeneracy, in the left lead (𝑦<0), the wave functions for spin-up and -down electron can be written as𝜓_↑↓(𝑥,𝑦)=𝜙_𝑁↑↓(𝛼𝑥)𝑒^{𝑖𝐾𝑁𝑦}+_𝑁′𝑟_{↑↓𝑁′𝑁}𝜙_{𝑁′↑↓}(𝛼𝑥)𝑒^{−𝑖𝐾𝑁′𝑦}, (7)where 𝑟_{↑↓𝑁′𝑁} are the coefficients of incident mode 𝑁 reflecting to mode 𝑁^′. In the QW (𝑦>0), the wave functions for spin-up and -down electron are nondegenerate𝜓_↑↓(𝑥,𝑦)=_𝑛𝑡^′_{↑↓𝑛𝑁}𝜑_𝑛↑↓(𝛼𝑥)𝑒^{𝑖𝑘+𝑛↑↓𝑦}, (8)where 𝜑_𝑛↑↓ are given in (5) and 𝑡^′_{↑↓𝑛𝑁} the coefficients of incident mode 𝑁, transmitting to mode 𝑛. The continuity of the wave functions at left interface 𝑦=0 gives 𝜙_𝑁↑↓(𝛼𝑥)+_𝑁′𝑟_{↑↓𝑁′𝑁}𝜙_{𝑁′↑↓}(𝛼𝑥)=_𝑛𝑡^′_{↑↓𝑛𝑁}𝜑_𝑛↑↓(𝛼𝑥). (9)The continuity of the derivative of the wave functions may not hold due to the existence of RSOC [16]. However, the current conservation for the system still holds [18, 19]:ℏ−𝑖𝑚^∗𝜕0ℏ𝜕𝑦0−𝑖𝑚^∗𝜕𝜓𝜕𝑦^↑𝜓(𝑥,𝑦<0)_↓=ℏ(𝑥,𝑦<0)−𝑖𝑚^∗𝜕𝛼𝜕𝑦ℏ^𝑅𝛼_𝑅ℏℏ−𝑖𝑚^∗𝜕𝜓𝜕𝑦^↑𝜓(𝑥,𝑦≥0)_↓,(𝑥,𝑦≥0) (10)hence we can obtain𝐾_𝑁𝜙_𝑁↑↓(𝛼𝑥)−_𝑁′𝑟_{↑↓𝑁′𝑁}𝐾_𝑁′𝜙_{𝑁′↑↓}(𝛼𝑥)=_𝑛𝑡^′_{↑↓𝑛𝑁}[𝑘⁺_𝑛↑↓𝜑_𝑛↑↓(𝛼𝑥)+𝜑_𝑛↓↑(𝛼𝑥)/2𝑊], (11)where 𝑊=𝑙_𝛼𝑅=ℏ²/(2𝑚^∗𝛼_𝑅) is the effective width of the QW. Multiplying 𝜙_{𝑁′↑↓} and 𝜑_{𝑛′↑↓} with (9) and (11), respectively, and integrating over the range of −𝑊/2<𝑥<𝑊, one obtains the matrices of reflection 𝑟_↑↓ and transmission 𝑡^′_↑↓ at the 𝑦=0 interface as𝑟_↑↓=𝜆_↑↓𝑡^′_↑↓−1, (12)with𝑡^′_↑=[𝜒_↑⁻¹(𝜆^𝑇_↑𝐾𝜆_↑+𝑝_↑𝑘⁺_↑)−(𝜆^𝑇_↓𝐾𝜆_↓+𝑝_↓𝑘⁺_↓)⁻¹𝜒_↓]⁻¹×[2𝜒_↑⁻¹𝜆^𝑇_↑𝐾−2(𝜆^𝑇_↓𝐾𝜆_↓+𝑝_↓𝑘⁺_↓)⁻¹𝜆^𝑇_↓𝑡𝐾],_′↓=[𝜒_↓⁻¹(𝜆^𝑇_↓𝐾𝜆_↓+𝑝_↓𝑘⁺_↓)−(𝜆^𝑇_↑𝐾𝜆_↑+𝑝_↑𝑘⁺_↑)⁻¹𝜒_↑]⁻¹×[2𝜒_↓⁻¹𝜆^𝑇_↓𝐾−2(𝜆^𝑇_↑𝐾𝜆_↑+𝑝_↑𝑘⁺_↑)⁻¹𝜆^𝑇_↑𝐾], (13)where the corresponding matrices elements are as follows 𝑝_{𝑛𝑛′↑(↓)}=∫^𝑊/2_−𝑊/2𝜑_𝑛↑(↓)(𝛼𝑥)𝜑_{𝑛′↑(↓)}(𝛼𝑥)𝑑𝑥, 𝜒_{𝑛𝑛′↑(↓)}=(𝑚^∗𝛼_𝑅)/ℏ²×∫^𝑊/2_−𝑊/2𝜑_𝑛↑(↓)(𝛼𝑥)𝜑_{𝑛′↓(↑)}(𝛼𝑥)𝑑𝑥, 𝜆_{𝑁𝑛↑(↓)}=∫^𝑊/2_−𝑊/2𝜙_𝑁↑(↓)(𝛼𝑥)×𝜑_𝑛↑(↓)(𝛼𝑥)𝑑𝑥, 𝑘_{𝑛𝑛′↑(↓)}=𝑘_𝑛↑(↓)𝛿_𝑛𝑛′, and 𝐾_𝑁𝑁′=𝐾_𝑁𝛿_𝑁𝑁′, respectively. Furthermore, an analogous calculation for an incident wave from the right side to the left one at 𝑦=0 interface gives matrices 𝑟^′_↑↓ and 𝑡_↑↓. Thus, the scattering matrix 𝑆⁽¹⁾_↑↓ at 𝑦=0 interface can be expressed as 𝑆⁽¹⁾_↑↓=^{0𝑥002𝑑𝑐}𝑟𝑡_{𝑁𝑁′↑↓}^{0𝑥002𝑑𝑐}_{𝑁𝑛↑↓0𝑥002𝑑𝑐}𝑡𝑟^′_{𝑛𝑁↑↓}^{0𝑥002𝑑𝑐′}_{𝑛𝑛′↑↓}, (14) where the normalized wave amplitudes ^0𝑥0𝑡^{02𝑑𝑐′}_{𝑛𝑁↑↓}=(𝑘⁺_𝑛↑↓/𝐾_𝑁)^1/2×𝑡^′_{𝑛𝑁↑↓}, ^0𝑥0𝑡^02𝑑𝑐_{𝑁𝑛↑↓}=(𝐾_𝑁/𝑘⁻_𝑛↑↓)^1/2𝑡_{𝑁𝑛↑↓}, ^0𝑥0𝑟^02𝑑𝑐_{𝑁𝑁′↑↓}=(𝐾_𝑁/𝐾_𝑁′)^1/2𝑟_{𝑁𝑁′↑↓}, and ^0𝑥0𝑟^{02𝑑𝑐′}_{𝑛𝑛′↑↓}=(𝑘⁻_𝑛↑↓/𝑘⁺_𝑛↑↓)^1/2𝑟^′_{𝑛𝑛′↑↓}. Due to the symmetry of the system, [17] the scattering matrix 𝑆⁽²⁾_↑↓ at 𝑦=𝐿 interface is𝑆⁽²⁾_↑↓=^{0𝑥002𝑑𝑐}𝑟𝑡^′_𝑛𝑛^′_↑↓^{0𝑥002𝑑𝑐′}_{𝑛𝑁↑↓0𝑥002𝑑𝑐}𝑡𝑟_{𝑁𝑛↑↓}^{0𝑥002𝑑𝑐}_{𝑁𝑁′↑↓}. (15)Therefore, we can obtain the total wave functions in the QW as=𝜓Ψ(𝑥,𝑦)_↑𝜓(𝑥,𝑦)_↓(𝑥,𝑦)=_𝑛^{𝑁0𝑥002𝑑𝑐}𝑡^′_𝑛𝑁↑𝑒^{𝑖𝑘+𝑛↑𝑦}𝜑_𝑛↑(𝑥)+𝑟_𝑛′^{0𝑥002𝑑𝑐′}_{𝑛𝑛′↑}𝑒^{−𝑖𝑘+𝑛↑(𝐿−𝑦)}𝜑_𝑛↑(𝑥)_{𝑁0𝑥002𝑑𝑐}𝑡^′_𝑛𝑁↓𝑒^{𝑖𝑘+𝑛↓𝑦}𝜑_𝑛↓(𝑥)+𝑟_𝑛′^{0𝑥002𝑑𝑐′}_{𝑛𝑛′↓}𝑒^{−𝑖𝑘+𝑛↓(𝐿−𝑦)}𝜑_𝑛↓,(𝑥) (16)and the local spin polarization [12, 14]⟨𝑆_𝑖ℏ(𝑥,𝑦)⟩=2Ψ^†(𝑥,𝑦)𝜎_𝑖Ψ(𝑥,𝑦). (17)

In the following, we present some numerical examples of the calculated spin polarization (in units of ℏ=1) in the QW at low temperatures. It should be pointed out that we only consider the weak RSOC case [16–19], that is, 𝑙_𝜔0/𝑙_𝛼𝑅=2𝑚^∗𝛼_𝑅/ℏ²𝛼₀=0.1 with 𝑙_𝜔0=1/𝛼₀=(ℏ/𝑚^∗𝜔₀)^1/2, and the width of two leads supporting number of propagating modes up to 𝑁=6. For the case of 𝛼_𝑅=0.5×10⁻¹¹eVm [7], the incident electron energy is taken as 𝐸=𝐸_↑↓=7ℏ𝜔₀=0.0367eV (for 𝛼_𝑅=1.0×10⁻¹¹eVm, 𝐸=0.147eV). Our system is a WNW QW (𝑛<𝑁); however, the actual width of QW is chosen as 𝑊=𝑙_𝛼𝑅 and the parabolic confining potential controls the number of propagation modes 𝑛 in the QW with the corresponding character oscillator frequency length 𝑙_𝜔<𝑊. The electron effective mass is taken as that for InGaAs quantum well 𝑚^∗=0.037𝑚_𝑒 [8] and the length of QW is chosen as 𝐿=1200nm.

Figure 1 shows the 2D contour of spin polarization components ⟨𝑆_𝑖⟩(𝑖=𝑥,𝑦,𝑧) for the cases of three and five propagation modes in the QW. It is noted that the components ⟨𝑆_𝑥⟩ and ⟨𝑆_𝑦⟩ present periodic oscillations along the longitudinal direction which comes from Rashba spin precession [15] while no oscillations presented for ⟨𝑆_𝑧⟩. This is the main difference from the results of [14] due to the different confining potential.

Figure 1 

The 2D contour of spin polarization components ⟨𝑆𝑖⟩(𝑖=𝑥,𝑦,𝑧) as functions of transverse position 𝑢 and length 𝐿 for 𝑛=3 (a), (b), (c)) and 𝑛=5 (d), (e), (f)) propagation modes QW with 𝛼𝑅=0.5×10−11eVm.

The polarization components ⟨𝑆_𝑖⟩ versus the transverse position 𝑢=𝑥/𝑊 for different propagation modes in the QW are demonstrated in Figure 2, with the longitudinal positions for ⟨𝑆_𝑥⟩ and ⟨𝑆_𝑦/𝑧⟩ as 0 and 300nm, respectively. More peaks and extended polarization width toward two edges of the QW have been shown with the increasing modes 𝑛. The component ⟨𝑆_𝑧⟩ presents two inverse-direction peaks (shifting to the two edges of the QW), consequently, produces the out-of-plane spin accumulation in the QW. The spin accumulation is formed with the spin distribution polarized along the 𝑧 direction due to the interaction between the pseudomagnetic field arising from RSOC and the accelerating electrons [20, 21]. Moreover, the net spin polarization rather than in-plane spin accumulation is illustrated for ⟨𝑆_𝑥/𝑦⟩ while the present results of ⟨𝑆_𝑧⟩ are qualitatively consistent with [12, 14].

(a)

(b)

(c)

Figure 2 

The spin polarization components ⟨𝑆𝑖⟩(𝑖=𝑥,𝑦,𝑧) as a function of the transverse position 𝑢. The Rashba strength 𝛼𝑅=0.5×10−11eVm and ⟨𝑆𝑧⟩ indicates the out-of-plane spin accumulation.

Furthermore, in Figure 3 we plot the longitudinal profile of spin polarization components ⟨𝑆_𝑥⟩ and ⟨𝑆_𝑦⟩ at a fixed transverse position 𝑢=0.15. As can be seen, due to the Rashba spin precession along the longitudinal position of QW, the period of the oscillations (with 𝜋/2 phase separation) for Rashba strength 𝛼_𝑅=1.0×10⁻¹¹eVm is half that of the 0.5×10⁻¹¹eVm case. One notes that the Rashba precession lengths 𝐿_SO1 and 𝐿_SO2 are 600 nm and 300 nm, respectively, which are consistent with the results (𝐿_SO=𝜋ℏ²/(2𝑚^∗𝛼_𝑅)) in [14, 22]. It should be pointed out that the presented results may be observable experimentally to differentiate the intrinsic and extrinsic spin accumulation.

(a)

(b)

Figure 3 

The intrinsic oscillations of spin polarization components ⟨𝑆𝑥⟩ and ⟨𝑆𝑦⟩ for 𝛼𝑅=0.5×10−11eVm (dashed line) and 𝛼𝑅=1.0×10−11eVm (solid line) with 𝑛=4 and the transverse position 𝑢=0.15.

In conclusion, we have studied the spin polarization in a Rashba QW confined by a parabolic potential. It is found that the two inverse direction peaks of the out-of-plane spin accumulation ⟨𝑆_𝑧⟩ shift to the two edges of the QW with the increasing propagation modes. Furthermore, the intrinsic oscillation of spin polarization due to Rashba spin precession can be used to identify the intrinsic spin accumulation by changing the RSOC strength.

Acknowledgments

This work is supported by the Nature Science Foundation of China under Grant no. 10574042, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant no. 20060542002, and the Hunan Provincial Natural Science Foundation of China under Grant no. 06JJ2097.