# Spin Polarized Transport in an AC-Driven Quantum Curved Nanowire

Authors: Walid A. Zein; Nabil A. Ibrahim; Adel H. Phillips
Journal: Physics Research International
 (2011)

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## Abstract

Using the effective-mass approximation method, and Floquet theory, we study the spin transport characteristics through a curved quantum nanowire. The spin polarization,P
, and the tunneling magnetoresistance, TMR, are deduced under the effect of microwave and infrared radiations of wide range of frequencies. The results show an oscillatory behavior of both the spin polarization and the tunneling magnetoresistance. This is due to Fano-type resonance and the interplay between the strength of spin-orbit coupling and the photons in the subbands of the one-dimensional nanowire. The present results show that this investigation is very important, and the present device might be used to be a sensor for small strain in semiconductor nanostructures and photodetector.


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## Body

## 1. Introduction

In the rapidly growing field of semiconductor spintronics, the spin degree of freedom is used for information processing [1, 2]. Devices concepts have been proposed which offer lower power consumption and a higher degree of functionality [3]. Among the research area of spintronics, the spin-orbit coupling (SOC) creates another way to manipulate spins by means of an electric field [4]. The Rashba spin-orbit coupling effect [4] is found to be very pronounced in semiconductor heterostructure, for example, quantum dots, quantum wires, and quantum rings [5], and its strength can be controlled by gate voltage. The spin polarization in two-dimensional electron gas (2DEG) systems with spin-orbit coupling (SOC) has been attracted extensive attention by many authors [6–9]. Spin-orbit coupling (SOC) has been investigated in parallel quantum wires [10, 11], where universal conductance fluctuations are suppressed. More recently, ballistic spin resonance due to an intrinsically oscillating spin-orbit field has been realized experimentally in a quantum wire [12]. The observation of the one-dimensional spin-orbit gap in quantum wires has also been reported in [13] and also recently in [14].The aim of the present paper is to investigate the spin transport characteristics through a curved quantum nanowire under the effect of microwave (MW) and infrared (IR) radiations. The main difference between the straight nanowire and curved one is that the spin rotation is characterized by certain angles, as it will be shown below.


## 2. The Model

We will now derive an expression for both the spin polarization for spin injection current in the curved nanowire and the corresponding tunneling magnetoresistance (TMR) for a curved nanowire under the effect of induced photons of wide range of frequencies. This nanowire is connected to two metallic leads. The effective Hamiltonian for spin-injected electrons through one-dimensional nanowire can be written as three parts according the geometrical design of the curved nanowire as [15, 16](1)Ĥ1=-ℏ22m*∂2∂S2-iασb1∂∂S+Vd+eVg,S∈(-∞,0),wherem* is the effective mass of the electron, Vd is the barrier height at the interface with the leads, Vg is the gate voltage, and S is the arc length along the curve. The second part of the curved nanowire including the Rashba spin-orbit coupling effect is given by(2)Ĥ2=-ℏ22m*a2∂2∂θ2-ℏ28m*a2-iαa[σb2∂∂θ-12σt2],θ∈(0,θω).Also, the third part of the curved nanowire is(3)Ĥ3=-ℏ22m*∂2∂S2-iασb3∂∂S+eVaccos⁡⁡ωt,S∈(θωa,∞).In (1), (2), and (3), the parameter S represents the arc length along the curved part of the nanowire, θ is the polar angle, θω is the angle between two rectilinear parts of the nanowire and Vac is the amplitude of the induced photons with frequency ω. The spin operators in (1),  (2), and (3) are represented as follows [17]:(4)σb1=-σx,σb2=-σxcos⁡⁡θ-σysin⁡θ,σb3=-σxcos⁡⁡θω-σysin⁡θω,σt2=-σxsin⁡θ+σycos⁡⁡θ.The energy spectrum and the unnormalized eigenstates for the two parts of the straight line of the nanowire are given by [18–20](5)Eμ=ℏ2k22m*-μαk,(6)Φ1,λμ=eiλks⋅χ1,Φ3,λμ=eiλks⋅χ3,where the symbolμ corresponds to the spin up ↑ and spin down ↓, and λ=± which corresponds to the direction of motion along the nanowire. The eigenstate function and the energy eigenvalue for the curved section of the nanowire are given as [18–20](7)Φ2,λμ=eiλn′θ⋅χ2,wheren’ is the orbit quantum number(8)E2,λμ=Ω[(n′+12λ)2-μ(n′+12λ)1+ωSoc2Ω2].The spinorsχ1,χ2, and χ3 in (6), (10), and (7) are given by
(9)χ1±=(22±22),χ2+=(sin⁡ξ2eiθcos⁡⁡ξ2),χ2-=(cos⁡⁡ξ2-eiθsin⁡ξ2),χ3±=(22±22eiθω).In (8), we have the frequency associated with spin-orbit coupling ωSoc and the parameter Ω which are defined as(10)ωSoc=αaℏ,Ω=ℏ22m*a2,
where α is the strength of the spin-orbit coupling and a is the radius of curvature.The eigenfunctions corresponding to the spin transport through a curved wire are given by [18–20]
(11)ψ1=∑n=-∞∞∑μJn(eVacℏω)×[cos⁡⁡(ϕ)Φ1,+μ+sin⁡(ϕ)Φ1,+μ+R↑Φ1,-μ+R↓Φ1,-μ],ψ2=∑μ∑n=-∞∞Jn(eVacℏω)[C1Φ2,+μ+C2Φ2,-μ],ψ3=∑μ∑n=-∞∞Jn(eVacℏω)[Γ↑Φ3,++Γ↓Φ3,-],
where Jn(eVac/ℏω) is the nth order Bessel function. The solutions of (11) must be generated by the presence of different subbands, n, in a quantum nanowire, which come with phase factor exp(-inωt), where ω is the frequency of the induced photons. Now, the tunneling probability |Γμwithphoton(E)|2 could be obtained by applying Griffith boundary conditions [20]. Accordingly, therefore, the expressions for the tunneling probabilities corresponding to spin-up and spin-down electrons, respectively, are given by(12)|Γ↑μwithphoton(E)|2=∑nJn2(eVacℏω)×[8A2B2(cos⁡2⁡ξ+cos⁡⁡ϕcos⁡⁡(ϕ-2ξ)cos⁡[(1+2γ)θ+sin2⁡(ϕ-ξ)])(A+B)4+(A-B)4-2(A2-B2)2cos⁡⁡(2βθ)].And that for spin-down as(13)|Γ↓μwithphoton(E)|2=∑nJn2(eVacℏω)[8A2B2(cos⁡2⁡(ϕ-ξ)-cos⁡⁡ϕcos⁡⁡(ϕ-2ξ)cos⁡⁡[(1+2γ)θ]+sin2⁡ξ)(A+B)4+(A-B)4-2(A2-B2)2cos⁡⁡(2βθ)].where the parameters, in (12) and (13), A,B,β,and  γ,are expressed as
(14)A=(αℏ)2+2(EF+Vd+eVg+nℏω)m*,B=(αℏ)2+2(EF+(Ω/4)+Vd+eVg+nℏω)m*,β=Bℏ2aΩ,γ=-12+121+ωSoc2Ω2,tan⁡ϕ=-ωSocΩ.
So, the spin polarization of the tunneled electrons [21] is
(15)P=|Γ↑μwithphoton(E)|2-|Γ↓μwithphoton(E)|2|Γ↑μwithphoton(E)|2+|Γ↓μwithphoton(E)|2,
In order to investigate the spin injection tunneling through the curved nanowire, we could calculate the tunneling magnetoresistance (TMR) which is related to the spin polarization (15) as [21–23]:
(16)TMR=P21-P2+ΓS,
where ΓS is the relaxation parameter and is given by [21–23]:
(17)ΓS=e2N(0)RTAlτS,
where N(0) is the normal-state density of electrons calculated for both spin-up and spin-down distribution function fσ(E), which is expressed as [21–23](18)fσ(E)≅f0(E)-(∂f0∂E)⋅μδf,
where δf is the shift of the chemical potential, τS is the spin relaxation time, A is the cross-sectional area of the nanowire, and RT is the resistance at the interface of the tunnel junction.


## 3. Result and Discussion

The nanowire is the semiconductor heterostructure InAs-InGaAs with characteristic valuesm*=0.023me, EF=11.13 meV, |θ|≤π [14, 16, 17]. The features of our present results are the following.(i) Figure1 shows the variation of the spin polarization with the strength of the spin-orbit coupling, α,  at different values of the radius of the curvature of the nanowire. The results show periodic oscillations of the polarization. Also, the peak heights vary in a quantized form for the two values of a (radius of curvature).Figure 1
The variation of the spin polarization with the strength of the spin-orbit coupling,α, at different values of the radius of curvature of the nanowire.(ii) Figure2 shows the variation of the tunneling magnetoresistance (TMR) with the strength of the spin-orbit coupling, α. As in Figure 1, periodic oscillations of the tunneling magnetoresistance (TMR) are observed. Also, peak heights vary in a quantized form for the two values of the radius of the curvature for nanowire. Such results show that the spin transport through curved nanowire is very sensitive to the geometrical shape of the nanowire. The strength of the spin-orbit coupling, α, can be controlled by the gate voltage, the energy of the induced photons and the geometrical shape. Such results are found to be concordant with those in the literature [7, 11, 14].Figure 2
The variation of the tunneling magnetoresistance (TMR) with the strength of the spin-orbit coupling,α, at different values of the radius of curvature of the nanowire.(iii) Figure3, shows the variation of polarization with the photon energy at different values of the strength of the spin-orbit coupling, α. An oscillatory behavior of the polarization is observed. This is due to Fano-type resonance [24–27].Figure 3
The variation of the spin polarization with the photon energy at different values of the radius of curvature of the nanowire.(iv) Figure4 shows the variations of the tunneling magnetoresistance (TMR) with the photon energy at different values of the strength of SOC. Oscillations are observed as in the case of the spin polarization (Figure 3). These results show a good concordant with those in the literature [24–27]. These results show that the location and line shape of Fano-type resonance can be controlled by both the frequency of the induced photons and the strength of the spin-orbit coupling.Figure 4
The variation of the tunneling magnetoresistance (TMR) with the photon energy at different values of the strength of SOC.We can conclude that the present investigation is very important for devising a mesoscopic nanowire with controllable curvature. By this device, we can determine very minute strain in semiconductor heterostructure solids [28]. Also, this nanowire can be used as a photodetector [29].



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