In Chapter 2, we investigate the characteristics of a spin-dependent pumping in the low-dimensional system. We propose and demonstrate theoretically that resonant inelastic scattering (RIS) can play an important role in dc spin current generation. The RIS makes it possible to generate dc spin current via a simple gate configuration: a single finger gate that locates atop and orients transversely to a quantum channel in the presence of Rashba spin-orbit interaction. The ac-biased finger gate gives rise to a time variation in the Rashba coupling parameter, which causes spin-resolved RIS and, subsequently, contributes to the dc spin current. The spin current depends on both the static and the dynamic parts in the Rashba coupling parameter. The proposed gate configuration has the added advantage that no dc charge current is generated. Our study also shows that the spin current generation can be enhanced significantly in a double finger-gate configuration. In double finger-gate with the finite phase difference φ, it is also show that the spin current and the charge current are generated by the double ac-biased finger-gate with a finite phase difference φ. We also explore the robustness of such dc spin current generation against elastic scattering in the RQC. The effect of backscattering is studied by introducing two kinds of scattering potentials in the transverse dimension. These two kinds of scattering potentials are divided into type (A): full static barrier and type (B):
small scatterer in the transverse dimension of a RQC. The modulations of spin currents depend on the forms of scattering potentials.
with the spin densities. Here, we employ the nonequilibrium Green’s functions to calculate all diffusion equations and take the suitable orders into account. The restrictions of boundary conditions are given by spin currents. In particularly, spin currents turn out to vanish for hard-wall boundaries. In our cases, we consider the hard-wall boundaries in a 2D strip.
In Chapter 4, the intrinsic spin Hall effect (SHE) on spin accumulation and electric conductance in a diffusive regime has been studied for a 2D strip with a finite width d. It is found that the spin polarization near the edges of the strip exhibits damped oscillations as a function of the width and strength of the Dresselhaus spin-orbit interaction (SOI) while an electric current is applied in the longitudinal direction. Cubic terms of Dresselhaus SOI are crucial for spin accumulation near the edges. As expected, no effect on the spin accumulation and electric conductance have been found in the case of Rashba SOI.
At the same time, the conventional electric current can be correlated by the SHE. This correlation is associated with the magnitude of the spin accumulations on the edges.
In Chapter 5, we studied the intrinsic spin-Hall effect (SHE) induced by a driving electric field E ˆx, in the presence of an in-plane magnetic field Bk = Bxx + Bˆ yy on a 2Dˆ strip. In the diffusive regime, the spatial distribution of the spin density Si (i = x, y, z) is calculated from a spin diffusion equation derived from the nonequilibrium Green’s function. In the presence of the in-plane magnetic field, the z-component spin density Sz normal to the 2D strip remains zero with or without Bk field for the case of Rashba spin-orbit interaction (SOI). For the case of Dresselhaus SOI, the spatial distribution of spin density show either symmetric or asymmetric features which depend on the direction of the in-plane magnetic field. By applying the longitudinal magnetic field Bx, the spatial distributions of spin densities Sx and Sz show the even parity in Bx but Sy shows the odd parity in Bx. The asymmetric property of Sz versus By is demonstrated for the intrinsic SHE in case of Dresselhaus SOI. The extrinsic SHE experimentally performed the symmetric behavior of Sz at boundaries by applying in-plane magnetic field By. These
elastic scatterers by the intrinsic spin-Hall effect in the two-dimensional electron gas (2DEG) subject to the Rashba spin-orbit interaction. The spin polarization normal to the 2DEG can be calculated in the diffusive regime around the elastic scatterer. It is found that there is the finite spin polarization around each impurity. However, the macroscopic spin density turns out to vanish by averaging of individual spin dipole distribution over impurities for a hard wall boundary. At the same time, the spin density is finite near the boundary of 2DEG for a soft-wall boundary.
Finally, we present our conclusion and future works in Chapter 7.
Dc spin current generation in a Rashba-type ballistic quantum channel
In this chapter, we investigate the characteristics of a spin-dependent pumping in the low-dimensional system. We propose and demonstrate theoretically that resonant inelastic scattering (RIS) can play an important role in dc spin current generation. The RIS makes it possible to generate dc spin current via a simple gate configuration: a single finger gate that locates atop and orients transversely to a quantum channel in the presence of Rashba spin-orbit interaction. The ac-biased finger gate gives rise to a time variation in the Rashba coupling parameter, which causes spin-resolved RIS and, subsequently, contributes to the dc spin current. The spin current depends on both the static and the dynamic parts in the Rashba coupling parameter, α0 and α1, respectively, and is proportional to α0α21. The proposed gate configuration has the added advantage that no dc charge current (CC) is generated. Our study also shows that the spin current generation can be enhanced significantly in a double finger-gate configuration. In a double finger-gate with a finite phase difference φ, it is also show that the spin current and the CC are generated by a double ac-biased finger-gate with a finite phase difference φ. We
subband mixing is studied by introducing a static partial-barrier (type B) that is spatially localized and non-uniform in the transverse dimension. In addition, we compare the cases of attractive and repulsive partial-barriers. It is found that attractive partial-barrier gives rise to additional DC spin current structures due to resonant inter-subband and inter-sideband transition to quasi-bound states formed just beneath subband thresholds.
2.1 Introduction
Quantum charge pumping (QPC) has attracted a lot of interest in recent years [61–
64]. The dc CC can be generated across an unbiased mesoscopic structure by time-dependent periodic deformation of two structure parameters. Original proposal of QCP, was suggested [53, 54] in the adiabatic regime. They considered the current generated by a slowly varying travelling wave in an isolated one-dimensional system. The number of electrons transported per period was found to be quantized if the Fermi energy lies in the gap of the spectrum of the instantaneous Hamiltonian. This quantized charge pumping has great potential for the direct-current standard [65]. The requirement of the adiabatic pumping is either the Fermi energy εF À Ω in a continuum mesoscopic system (ex: quantum wires) or the discrete level spacing ∆E À Ω in the quantized system (ex: quantum dots), where Ω is the oscillating frequency. In above cases, the frequency Ω of a time-modulation structure parameters is restricted to be much smaller than an energy scalar in the considered system such that the charge evolves with time adiabatically. Beyond the regime of adiabatic QPC, the non-adiabatic QCP becomes applicable and interesting in a quantum system without strict restriction of Ω. The
mechanically in non-adiabatic QCP respecting to the semiclassical adiabatic QCP.
More recently, the spintronics has become an emerging field because of in both ap-plication and foundation arenas [1, 32, 44, 66]. The recent key issue of great interest is the generation of dc spin current (SC) without charge current. Various dc SC genera-tion schemes have been proposed, involving static magnetic field [67–69], ferromagnetic material [70], or ac magnetic field [47]. More recently, Rashba-type spin-orbit interaction (SOI) in two dimension electron gas (2DEG) [12, 15, 56] has inspired attractive proposals for nonmagnetic dc SC generation [22, 23, 71]. Of these recent proposals, including a time-modulated quantum dot with a static spin-orbit coupling [71], and time modula-tions of a barrier and the spin-orbit coupling parameter in two spatially separated regions [22], the working principle is basically adiabatic quantum pumping. Hence, simultaneous generation of both dc spin and charge current is the norm. The condition of zero dc CC, however, is met only for some judicious choices for the values of the system parameters.
It is known, on the other hand, that quantum transport in a narrow channel exhibits resonant inelastic scattering (RIS) features when it is acted upon by a spatially localized time-modulated potential [72, 73]. This RIS is coherent inelastic scattering, but with resonance at work, when the traversing electrons can make transitions to their subband threshold by emitting m~Ω [72, 73]. Should this RIS become spin resolved in a Rashba-type quantum channel (RQC), of which its Rashba coupling parameter is time modulated locally, we will have a simpler route to the nonmagnetic generation of dc SC. Thus, we opt to study, in this work, the RIS features in a RQC. This requires us to go beyond the adiabatic regime and into the regime when either µ or µn ∼ ~Ω. We solve the time-dependent spin-orbit scattering (SOS) for all possible incident electron energies and obtain large RIS contribution. In the adiabatic regime, however, with µ, µn À ~Ω, we find that the dc spin-pumping effect from a sole SOI time-modulated region is small [22].
The system configuration considered is based on a RQC that forms out of a 2DEG in an asymmetric quantum well by the split-gate technique. As is depicted in Fig. 2.1 (a), a finger gate (FG) is positioned above while separated from the RQC by an insulating
2DEG
Figure 2.1: (a) Top-view schematic illustration of the RQC. The ac-biased FG, of width l, is indicated by the gray area; (b) the electron dispersion relation of an unperturbed
layer. A local time variation in the Rashba coupling parameter α(r, t) can be induced by ac biasing the FG [22, 23]. The Hamiltonian is given by H = p2/2m + Hso(r, t) + Vc(y) where the Rashba term
Hso(r, t) = M · 1
2[α(r, t)p + pα(r, t)]. (2.1)
Here, M = ˆz × σ is normal to the 2DEG, σ is the vector of Pauli spin matrices, and Vc(y) is the confinement potential. The unperturbed Rashba coupling parameter α(r, t) is α0 throughout the RQC, but becomes α0+ α1cos(Ωt) in the region underneath the ac-biased FG. In principle, the time-modulating potential can also modulate the electron density but one can applying a backgate to compensate the fluctuation of electron density [56].
The Dresselhaus term is neglected for the case of a narrow-gap semiconductor system [74].
We also investigate the effect of elastic scattering on the dc SC generation in a single FG configuration. The method of approach is time-dependent scattering matrix method [52, 75] with a static potential V (x, y) in a RQC. The backscattering effect can be studied via a static full-barrier locating either inside or outside of the AC-biased FG. Strong barrier position-dependent effect on the dc SC generation is found in our theoretical calculation.
The elastic scattering effect is further studied by considering a repulsive or attractive partial-barrier. The partial-barrier introduces intersubband scattering to the system due to the fact that it covers only part of the transverse dimension of the quantum channel. We have studied the barrier position dependence of the dc SC generation. For an attractive partial-barrier, the intersubband transition into a quasi-bound state formed just beneath the subband bottom causes the SC to have an additional structure at m below the second subband bottom. In all the above elastic scattering effect on the dc SC, as long as the barrier breaks the longitudinal symmetry of the configuration, the CC will become nonzero.
To demonstrate the pumping mechanism, we consider a narrow RQC in which its subband energy spacing is much greater than the Rashba-induced subband mixing. As such, the unperturbed Hamiltonian, in its dimensionless form, is H0 = −∇2+ α0σy(i∂/∂x) + Vc(y).
Appropriate units have been used such that all physical quantities presented here, and henceforth, are dimensionless. In particular, α is in unit of vF∗/2 , and spin in unit of
~/2. The right-going (R) eigenstate of H0, in the nth subband, is φn(y)ψnσ(x), where ψnσ(x) = exp[iknRσ (x)]χσ. The wave vector knRσ = √
µn+ ησα0/2, while ησ = ±1 denotes the eigenvalue of χσ to the operator σy. µn is the energy measured from the nth subband threshold such that the energy of the eigentstate is E = µn+ εn− α20/4, for εn = (nπ/d)2. This dispersion relation is shown in Fig. 2.1 (b). The subband with µn∼ ~Ω is found to contribute most to the RIS-enhanced spin pumping. It is of import to note that right-going electrons have |kR↑| > |kR↓| and that, at the subband threshold kR↑(↓)= k↑(↓)L .
The physical origin of the dc SC generation can be understood from two perspectives.
A weak pumping regime result is then obtained for an explicit confirmation of our physical reasoning. The first perspective is associated with the vector potential. In the ac-biased region, H = Hx+ Hy, the transverse part Hy = −∂2/∂y2 + Vc(y), and the longitudinal part
Hx(t) = µ
−i ∂
∂x + α(x, t) 2 M · ˆx
¶2
− α(x, t)2
4 (2.2)
The form of Eq. (2.2) suggests an effective vector potential, A(t) = 12α(x, t)M · ˆx, which depends on the spin and gives rise to a spin-resolved driving electric field E = −∂A/∂t.
turns out that the full term linear in A, given by −i(∂/∂x)ˆx · A, manages to give rise to nontrivial spin-resolved transmissions. By the perturbation concept, this term becomes k↑(↓)R Ax, for the case of a right-going electron incident upon a spatially uniform α(t).
This renders the effective longitudinal driving field to become spin dependent, through the factor k↑(↓)R . The difference in the current transmissions, for spin-up and spin down cases, is proportional to the difference in k↑(↓)R , or α0, and is found to be amplified by RIS. This breaking of the longitudinal symmetry in the effective driving field by α0 leads to the generation of dc spin current in a FG-RQC structure that has but an apparent longitudinal configuration symmetry, and with zero source-drain bias. No dc CC will be generated, however, in such a structure.
An alternate perspective for the understanding of the origin of the spin-resolved cur-rent transmission is associated with unitary transformation. By introducing the unitary transformation Ψσ(x, t) = exp[(iησ/2)Rx
−l/2α (x0, t) dx0]ψσ(x, t), the Schr¨odinger equation [Eq. (2.2)] becomes
·
− ∂2
∂x2 + U1(t) + U2σ(t)
¸
ψσ(x, t) = i∂
∂tψσ(x, t) (2.3)
of which the two time-dependent potentials are U1(t) = −α(x, t)2/4 and U2σ(t) = (Ωα1/2)(x+
l/2)cos(Ωt/ + ησπ/2). Even though only U2σ depends on spin, both the term in U1(t) that oscillates with frequency Ω and U2σ together constitute a pair of quantum pumping po-tential that pump SC. This is our major finding in this work: that spin pumping nature is built-in even in a single FG configuration.
Next, we can write down the total wave functions in the different region for the one-FG configuration in Fig. 2.2. For convenience, the region of the ac-biased FG is located from x = −l/2 to x = l/2 and the channel width is d. The Appendix A shows the derivation of x-dependent wave function in the region (II) via a transformation Ψσ(x, t) = exp(ησαΩ1 sin (Ωt)∂x∂)ψσ(x, t). The wave function Ψσ satisfies Hx(t)Ψσ(x, t) = i∂Ψσ(x, t)/∂t and one can rewrite the wave function in the Bessel’s function form. The
inelastic and spin-dependent scattering processes due to the time-modulation FG in region (II). In summary, we can express the scattering wave function in x direction as following
to the right-(left-) moving electron in the nth subband, m0th sideband, and with kinetic energy µmn0. The reflection amplitude rn,LLm,σ indicates that an incident electron is injected from the left-hand side and scattered into the left-hand side with energy µmn in region (I).
The transmission amplitude tm,σn,RL indicates that an incident electron is injected from the left-hand side and scattered into the right-hand side with energy µmn in region (III). The coefficients Amn,RL0,σ and Bn,LLm0,σ corresponding to the amplitude of right-going and left-going wave functions have an energy µmn0 and the spin state σ in the region (II), respectively.
Furthermore, the total scattering wave functions can be written as Ψσ(x, t)ϕn(y), where ϕn(y) =p
2/dsin(nπy/d) is the nth subband wave function for the hard-wall confinement with the channel width d.
Our aim is to solve the reflection and transmission coefficients by the imposed bound-ary conditions: (i) wave functions continuous at x = ±l/2 and (ii) the slope of wave
FG
(I) (II) (III)
x= − l/2 x= l/2 FG
(I) (II) (III)
x= − l/2 x= l/2
Figure 2.2: The wave functions can be separated by three different regions (I)(x < −l/2), (II) (−l/2 < x < l/2), and (III) (x > l/2). The region (II) includes the static and dynamic Rashba spin-orbit coupling constant.
satisfy
Ψ(I)σ (x = −l/2, t) = Ψ(II)σ (x = −l/2, t) Ψ(II)σ (x = l/2, t) = Ψ(III)σ (x = l/2, t)
. (2.5)
For the continuity of the wave function’s slope, the Eq. (2.4) satisfy
−∂x∂ Ψσ¯
¯x=−l2 + ∂x∂ Ψσ¯
¯x=−2l − 2iησα1cos (Ωt) Ψσ|x=−l
2 = 0
−∂x∂Ψσ¯
¯x=2l + ∂x∂ Ψσ¯
¯x=2l +2iησα1cos (Ωt) Ψσ|x=l
2 = 0.
. (2.6)
Essentially, all unknown variables can be calculated from Eqs. (2.5) and (2.6) by cutting off the large enough sideband index m (m0) in the exactly numerical sense. (Appendix B) The charge transport generates a CC and the spin transport generates a spin current (SC). The CC is a good physical quantity due to the conservation of the total charges.
However, the spin current is not conserved due to the flip of spin during the scattering processes. In our case, the SC conservation is maintained by the suppression of subband mixing and the associated spin-flipping in a RQC. The SC expression for a state Ψσ is
∂x 2
The density operator ˆjxy describes the electron moving along x-direction with the y-component spin polarization. For a scattering state Ψσ, the SC can be expressed in terms of the transmission coefficients. More specifically, the ratio between the time-averaged transmitted and the incident SC gives the spin-resolved current transmission Tβασ , where α, β, are, respectively, the incident and the transmitting lead. Summing over contributions from all states in reservoirs R and L, the SC is
Is= I↑− I↓, (2.8)
where
Iσ = Z
dEf (E) [TRLσ − TLRσ ] (2.9)
and Iσ is the number current due to electrons with spin from both reservoirs that are under zero source-drain bias condition. Here TRLσ = P
n
P
m(µmn>0)Tn,RLm,σ and f (E) is the Fermi-Dirac distribution. The transmission coefficient Tn,RLm,σ = ¯
¯tm,σn,RL¯
¯2p
µmn/µn denotes the current transmission that an electron incident from terminal L in the spin channel σ, subband n, energy E, is scattered into terminal R, sideband m, with kinetic energy µmn = µn+ mΩ. The reflection coefficient is calculated by Rm,σn,LL=¯
¯rn,LLm,σ ¯
¯2p
µmn/µn. The net CC is given by Ic= I↑+I↓. In a symmetric FG configuration, we have TLRσ = T=RL−σ , so that the net spin current is Is = 2R
dEf (E)
³
TRL↑ − TRL↓
´
and the net CC is identically zero. Our numerical results have to check the conservation of the particle flux to satisfy
for the nth subband.
2.3 One-sideband approximation of the single ac-biased FG in the weak pumping regime
For the case of a single ac-biased FG, we can employ the one-sideband approximation to estimate the transmission coefficient Tn,RL(LR)m,σ with m = 0, ±1, and the SC in the weak pumping (WP) regime. In the WP regime, when α1 is small, we can demonstrate analytically, and most unequivocally, that spin-dependent reflection arises merely from the aforementioned linear A term in Hx(t). We outline the derivation here while leaving the detail in Appendix C. Tracing up to the first order in α1, our derivation retains the reflection amplitudes to m = ±1 sideband and drops that to the m = 0 sideband. Contri-bution to the total reflection includes thus reflection at either the left or the right edges of the time-modulated region. For an electron incident from terminal L with wave vec-tor kσn,R(E), the reflection at the left edge is obtained from the wave-function continuous condition and the boundary condition
− ∂
In the time-modulated region, the wave function Ψσ consists of one-sideband terms, given by the form eikσn,R(E±Ω)xe−i(E±Ω)t, and eikσn,R(E)xe−iEt£
1 + ησ/ (2Ω) α1kn,Rσ (E)¡
eiΩt− e−iΩt¢¤
is given by m = 0 term. The extra Ωt dependence in the m = 0 term is resulted from the time-dependent driving effect of A, which is obviated by the weighting factor that involves α1kn,Rσ . The reflection amplitude rLm,σ, at the left edge is obtained
rLm,σ = sgn(m)ησ
(left-) moving electron in the nth subband, mth sideband, and with kinetic energy µmn. It is clear then that wave-vector differences in both the numerator and the denominator of rLm,σ, are spin independent. Hence, the spin dependence arises solely from the α1kn,Rσ factor in the first term of the numerator in Eq. (2.12), or from A. This confirms our understanding of the physical origin of the dc SC generation.
Including the reflection at the right edge, we obtain the total reflection amplitude
rn,LLm,σ = h
1 − ei(kn,Rσ −kn,Lm,σ)l i
rLm,σ (2.13)
We note that the spin dependence of this total reflection amplitude is associated with α0. In fact, it turns out that the SC is proportional to α0. The SC is related to the
We note that the spin dependence of this total reflection amplitude is associated with α0. In fact, it turns out that the SC is proportional to α0. The SC is related to the