Spin Polarized Transport in an AC-Driven Quantum Curved Nanowire

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.

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]𝐻₁=−ℏ²2𝑚^∗𝜕²𝜕𝑆²−𝑖𝛼𝜎_𝑏1𝜕𝜕𝑆+𝑉^𝑑+𝑒𝑉_𝑔,𝑆∈(−∞,0),(1)

where 𝑚^∗ is the effective mass of the electron, 𝑉_𝑑 is the barrier height at the interface with the leads, 𝑉_𝑔 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 by𝐻₂=−ℏ²2𝑚^∗𝑎²𝜕²𝜕𝜃²−ℏ²8𝑚^∗𝑎²−𝑖𝛼𝑎𝜎^𝑏2𝜕𝜃∈0,𝜕𝜃𝜃_𝜔−21𝜎^𝑡2,.(2)

Also, the third part of the curved nanowire is𝐻₃=−ℏ²2𝑚^∗𝜕²𝜕𝑆²−𝑖𝛼𝜎_𝑏3𝜕𝜕𝑆+𝑒𝑉^𝑎𝑐cos𝜔𝑡,𝑆∈𝜃_𝜔𝑎,∞.(3)

In (1), (2), and (3), the parameter 𝑆 represents the arc length along the curved part of the nanowire, 𝜃 is the polar angle, 𝜃_𝜔 is the angle between two rectilinear parts of the nanowire and 𝑉_𝑎𝑐 is the amplitude of the induced photons with frequency 𝜔. The spin operators in (1), (2), and (3) are represented as follows [17]:𝜎_𝑏2𝜎_𝑏1=−𝜎_𝑥,𝜎=−𝜎_𝑥cos𝜃−𝜎_𝑦sin𝜃,_𝑏3𝜎_𝑡2=−𝜎_𝑥cos𝜃_𝜔−𝜎_𝑦sin𝜃_𝜔,=−𝜎_𝑥sin𝜃+𝜎_𝑦cos𝜃.(4)

The energy spectrum and the unnormalized eigenstates for the two parts of the straight line of the nanowire are given by [18–20]𝐸^𝜇=ℏ²𝑘²2𝑚^∗Φ−𝜇𝛼𝑘,(5)_𝜇1,𝜆Φ_𝜇3,𝜆=𝑒^{𝑖𝜆𝑘𝑠}⋅𝜒₁,=𝑒^{𝑖𝜆𝑘𝑠}⋅𝜒₃,(6)

where the symbol 𝜇 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]Φ^𝜇2,𝜆=𝑒^{𝑖𝜆𝑛′𝜃}⋅𝜒₂,(7)

where 𝑛′ is the orbit quantum number𝐸^𝜇2,𝜆=Ω⎡⎢⎢⎣𝑛^+12𝜆²−𝜇𝑛^+12𝜆1+𝜔^2SocΩ²⎤⎥⎥⎦.(8)

The spinors 𝜒₁,𝜒₂, and 𝜒₃ in (6), (10), and (7) are given by 𝜒^±1=⎛⎜⎜⎜⎜⎝√2±√22𝜒₊₂2⎞⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎝𝑒_𝑖𝜃sin2𝜉cos𝜉𝜒₋₂2⎞⎟⎟⎟⎠,=⎛⎜⎜⎜⎝−𝑒co_𝑖𝜃s2𝜉sin𝜉𝜒_±32⎞⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎝√2±√222𝑒^𝑖𝜃𝜔⎞⎟⎟⎟⎟⎠.(9)

In (8), we have the frequency associated with spin-orbit coupling 𝜔_Soc and the parameter Ω which are defined as𝜔_Soc=𝛼𝑎ℏ,Ω=2𝑚ℏ^2∗𝑎²,(10) 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] 𝜓₁=^∞_𝑛_=−∞_𝜇𝐽_𝑛×cos(𝜙)Φ𝑒_𝜇1,+ℏ𝑉𝜔^𝑎𝑐𝜓₂+sin(𝜙)Φ^𝜇1,++𝑅_↑Φ^𝜇1,−+𝑅_↓Φ^𝜇1,−,=_𝜇∞𝑛_=−∞𝐽_𝑛𝑒ℏ𝑉𝜔^𝑎𝑐𝐶¹𝜓₃Φ^𝜇2,++𝐶₂Φ^𝜇2,−,=_𝜇∞𝑛_=−∞𝐽_𝑛𝑒ℏ𝑉𝜔^𝑎𝑐Γ^↑Φ_3,++Γ_↓Φ_3,−,(11) where 𝐽_𝑛(𝑒𝑉_𝑎𝑐/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 |Γ_{𝜇withphoton}(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|||Γ_{↑𝜇withphoton}(𝐸)|||²=_𝑛𝐽^2𝑛𝑒ℏ𝑉𝜔^𝑎𝑐×8𝐴²𝐵²cos²𝜉+cos𝜙cos(𝜙−2𝜉)cos(1+2𝛾)𝜃+sin²(𝜙−𝜉)(𝐴+𝐵)⁴+(𝐴−𝐵)⁴−2𝐴²−𝐵²²cos(2𝛽𝜃)(12.)And that for spin-down as|||Γ_{↓𝜇withphoton}(𝐸)|||²=_𝑛𝐽^2𝑛𝑒ℏ𝑉𝜔^𝑎𝑐8𝐴²𝐵²cos²(𝜙−𝜉)−cos𝜙cos(𝜙−2𝜉)cos[(1+2𝛾)𝜃]+sin²𝜉(𝐴+𝐵)⁴+(𝐴−𝐵)⁴−2𝐴²−𝐵²²cos(2𝛽𝜃)(1.3)where the parameters, in (12) and (13), 𝐴, 𝐵, 𝛽, and  𝛾, are expressed as 𝐴=𝛼ℏ²+2𝐸^𝐹+𝑉_𝑑+𝑒𝑉_𝑔+𝑛ℏ𝜔𝑚^∗𝐵=,ℏ𝛼²+2𝐸^𝐹+(Ω/4)+𝑉_𝑑+𝑒𝑉_𝑔+𝑛ℏ𝜔𝑚^∗𝛽=𝐵ℏ,2𝑎Ω,𝛾=−12+211+𝜔^2SocΩ²tan𝜙=−𝜔_Soc,Ω.(14) So, the spin polarization of the tunneled electrons [21] is 𝑃=|||Γ^{↑𝜇withphoton}(𝐸)|||²−|||Γ_{↓𝜇withphoton}(𝐸)|||²|||Γ_{↑𝜇withphoton}(𝐸)|||²+|||Γ_{↓𝜇withphoton}(𝐸)|||²,(15) 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]: TMR=𝑃²1−𝑃²+Γ_𝑆,(16) where Γ_𝑆 is the relaxation parameter and is given by [21–23]: Γ_𝑆=𝑒²𝑁(0)𝑅_𝑇𝐴ℓ𝜏_𝑆,(17) where 𝑁(0) is the normal-state density of electrons calculated for both spin-up and spin-down distribution function 𝑓_𝜎(𝐸), which is expressed as [21–23]𝑓_𝜎(𝐸)≅𝑓₀(𝐸)−𝜕𝑓⁰𝜕𝐸⋅𝜇𝛿𝑓,(18) where 𝛿𝑓 is the shift of the chemical potential, 𝜏_𝑆 is the spin relaxation time, 𝐴 is the cross-sectional area of the nanowire, and 𝑅_𝑇 is the resistance at the interface of the tunnel junction.

3. Result and Discussion

The nanowire is the semiconductor heterostructure InAs-InGaAs with characteristic values 𝑚^∗=0.023𝑚_𝑒, 𝐸_𝐹=11.13 meV, |𝜃|≤𝜋 [14, 16, 17]. The features of our present results are the following.

(i) Figure 1 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) Figure 2 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) Figure 3, 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) Figure 4 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].