Int J Theor Phys (2013) 52:1908–1913DOI 10.1007/s10773-012-1308-2

Study on Spin-Dependent Shot Noise in Parabolic-WellMagnetic Tunneling Junction

Zheng Huang · Shui-jie Qin · Chao-yun Long

Received: 3 July 2012 / Accepted: 14 August 2012 / Published online: 16 September 2012© Springer Science+Business Media, LLC 2012

Abstract On the basis of the Landauer-Büttiker scattering formalism and transfer matrixmethod, we investigated the spin-dependent shot noise in parabolic-well with two ferromag-netic contacts (F/PW/F). The quantum size and Rashba spin-orbit interaction are discussedsimultaneously. The results indicate that the shot noise is periodic function of the parabolic-well width. The oscillation frequencies of the shot noise decrease with the increasing ofthe parabolic-well depth, and increase with the increasing of the Rashba spin-orbit couplingstrength. The amplitude and peak to valley ratio of the shot noise are strongly dependent onthe magnetization configuration of the junction.

Keywords Spin-dependent shot noise · Rashba spin-orbit coupling · Spin-polarizedtransport

1 Introduction

Since the seminal proposal of a spin transistor by Datta and Das [1] based on spin pre-cession controlled by an external electrical field via spin-orbit coupling [2], electron spininjection from ferromagnetic metals (F) into semiconductors (S) has attracted much interest[3]. Schapters et al. [4] pointed out that an enhanced spin signal can be attained in an F/S/Ftransistor when quantum interference is considered. Mireles and Kirczenow [5, 6] had ex-amined coherent quantum transport in the same system within the Landauer framework ofballistic transport, and indicated that quantum spin valve effect occurs even in the absenceof a net spin-polarized current flowing through the nanostructure. Guo et al. [7, 8] developedthe scattering approach and obtained a general formula of the shot noise in DBRDs with theDresselhaus SOC effect.

In recent years, it is well known that molecular-beam epitaxy (MBE) readily fabricatedthe parabolic quantum-well structures [9–11]. Because they have the special nature and

Z. Huang (�) · S.-j. Qin · C.-y. LongKey Laboratory for Photoelectric Technology and Application, Guizhou University, Guiyang 550025,Chinae-mail: huangz888@163.com

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Fig. 1 A schematic of F/PW/Fmagnetic tunneling junction

various applications, and they are different from the square potential wells, the parabolicquantum-well structures have received increased interest. De Liu et al. [12, 13] investigatedthe spin-dependent transmission coefficient and tunneling magnetoresistance in F/SPW/Fjunction. However, the effects of parabolic-well magnetic tunneling junction on the shotnoise have not been discovered to report. In this paper, we investigated the shot noise inparabolic well with two ferromagnetic contacts (F/PW/F). The effects of the width and depthof the parabolic well and the Rashba spin-orbit coupling on the spin-dependent shot noiseare studied.

2 Model and Formulas

We Consider spin transport through F/PW/F junction. F indicates the ferromagnet, PW in-dicates the parabolic-well. In the ferromagnetic electrodes, a Stoner-Wohlfarth-like [14,15] model of the magnetization is assumed such that the spin-up and spin-down band en-ergies offset is set by an exchange splitting Δ. We assume the semiconductor region tohave a quasi-one-dimensional wave-guide shape which laterally confines the electrons inthe direction transverse to transport, which is assumed to be normal to the interface andalong the y-axis. In the semiconductor channel, a Rashba spin-orbit coupling will be con-sidered, and neglect intersubband mixing which is permissible. It is assumed that the elec-trons traversal through the F/PW/F structure is from the left (y < 0) to the right (y > 0)

as shown in Fig. 1. The magnetization is chosen along z-direction, parallel to the interface.PW is compositional symmetrical parabolic potential-well made of semiconductor materialGaAs/AlxGa1−xAs/GaAs. The well depth V0 determined by x (x is the Al concentration).

Consider the one-dimensional symmetrized version of the Rashba Hamiltonian, in theone-band effective-mass approximation, The Hamiltonian for the regions F and S are:

Hf = 1

2py

1

m∗f

py + 1

2ΔσZ, (1)

and

HS = 1

2py

1

m∗S

py + 1

2�σZ(pyαR + αRpy) + δEC + V (y) (2)

Here, m∗f is the effective mass in the metal, py is the electron momentum operator, Δ is

the exchange splitting energy in the ferromagnets, σZ denotes the spin Pauli matrices, αR is

1910 Int J Theor Phys (2013) 52:1908–1913

the spin-orbit Rashba parameter, and δEC is the conduction band mismatch between thesemiconductor and the ferromagnets.

The potential V (y) is

V (y) ={

4V0y2/L2, 0 ≤ y ≤ L

0, y < 0, y > L(3)

The effective mass of the electron in AlxGa1−xAs PW is given [16]

m∗S = (0.0665 + 0.0835x)me (4)

The eigenstates in the ferromagnetic region have the form

Ψ υσ (y) = Aυ

σ eikυσ y + Bυ

σ e−ikυσ y (5)

where σ = ↑,↓ indicates the spin state of the split band, υ = L,R denotes the left and rightferromagnets, and kυ

σ is the Fermi wave vector with spin state σ in the υ ferromagnet. Inthe semiconductor region, due to the structure of potential is a function, the wave functionis no longer the plane wave. In order to solve the problem, we separate this region intoa series of reasonably narrow layers and ignore the state hopping as the wave functionspropagate through the layers. In each slice, the wave function can be regarded as a planewave approximately with different wave vectors. The wave function can be written as

ψS↑ (y) = C↑e

ikS↑(y)y + D↑e−ikS↓(y)y (6)

and

ψS↓ (y) = C↓e

ikS↓(y)y + D↓e−ikS↑(y)y (7)

Here kSσ (y) denotes the Fermi wave in the semiconductor with spin σ . The relation between

the wave number and the energy are [8, 9]

Eυσ = �

2

2m∗f

(kυ

σ

)2 + 1

2λσ Δ (8)

and

ESσ = �

2

2m∗S

{(kS

σ

)2 + [(kS

σ

)′y]2 + 2kS

σ

((kS

σ

)′)y + 2i(kS

σ )′ + i(kSσ )′′y

}

+ λσ αR

[kS

σ + (kS

σ

)′y] + δEC + V (y) (9)

where λ↑,↓ = ±1, (kSσ )′ and (kS

σ )′′ are the first and second order derivative of the wave vector,respectively.

We use the matching conditions of the wave function and current conservation, and usethe transfer matrix method.

The expression of the transmission coefficient as follows [9, 10]:

Tσ = (k

↑R|AR↑|2 + k

↓R|AR↓|2)/(kσ

L|ALσ |2) (10)

According to the Landauer-Büttiker formula [17], the conductance and shot noise throughthe nanostructure are:

Gσ = e2

hTσ (11)

and

Sσ = 2e3V

hTσ (1 − Tσ ) (12)

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Fig. 2 Shot noise for theparallel (a) and antiparallel (b)configurations of the magneticmoments of the twoferromagnetic electrodes againstthe L. The solid line and dashedlines correspond to T↑ and T↓ ,respectively

Fig. 3 Shot noise for theparallel (a) and antiparallel (b)configurations of the magneticmoments of the twoferromagnetic electrodes againstthe depth V0. The solid line anddashed lines correspond to T↑and T↓, respectively

3 Results and Analyses

In this section, we use the formulas given above to investigate spin-dependent shot noiseand transport properties in the F/PW/F magnetic tunneling junction in the presence of theRashba spin-orbit interaction. We assume m∗

f = me, where me is the free electron mass. Theexchange splitting energy in the ferromagnets has been set to be Δ = 3.46 eV, the bandmismatch δEC = 2.4 eV, the Fermi energy Ef = 2.47 eV. The spin-orbit coupling strengthwill be characterized as kR/k0 by a Rashba spin-orbit wave vector kR = m∗

SαR/�2, where

k0 = 1.0 × 105 cm−1.In Fig. 2, we show the shot noise as a function of the width of the parabolic-well. The

Rashba parameter is kept the same as kR = 5.0k0, with a parabolic-well depth V0 = 0.8 eV.The shot noise is periodic function of the parabolic-well width. When the two magnetic mo-ments are parallel (θ = 0) [see Fig. 2(a)], the shot noise for spin-up and spin-down electronshas the same phase, and the oscillation amplitudes for spin-down electrons are always notbelow the spin-up ones, especially, at the peaks and valleys of the oscillation, their differ-ence are the largest. When the magnetic moments in the two ferromagnetic electrodes areantiparallel (θ = π) [see Fig. 2(b)], the shot noise for spin-up electrons is the same as that forspin down ones. When the magnetizations of the ferromagnetic contacts in a F/PW/F variesbetween the parallel (P) and antiparallel (AP) configurations, the peak to valley ratio of theshot noise increases with the increasing of the angles between the two magnetic momentsof the ferromagnets for spin-up electrons. That is decreases for spin-down electrons fromFig. 2. These characteristics may result from the band structure in F and AlxGa1−xAs/GaAs[18] and the dependence of the two spin electronics on the magnetization direction.

Figure 3 shows the shot noise as a function of the depth of a parabolic-well. The Rashbaparameter is kept the same as kR = 5.0k0, with a fixed length of the semiconductor layer(L = 1.0 µm). The period of the shot noise for spin-up and spin-down electrons increase

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Fig. 4 Shot noise for theparallel (a) and antiparallel (b)configurations of the magneticmoments of the twoferromagnetic electrodes againstthe kR/k0. The solid line anddashed lines correspond to T↑and T↓, respectively

with the increasing of depth. The shot noise makes Quasi-periodic oscillation with the in-creasing of depth. Its peak to valley ratio of the shot noise is almost constant. The rea-son is that the number of bound states increases in semiconductor with the increasing ofdepth, and corresponding resonant energy levels decrease, so the two spin electrons makeQuasi-periodic oscillations. These phenomena can be interpreted as the effects of a nonlinearparabolic potential structure on transport properties of magnetic tunnel junctions. The shotnoise for spin-up electrons is always smaller than those for spin-down electrons in Fig. 3.

In Fig. 4, we plot the spin-dependent shot noise as a function of the spin-orbit strengthwith a fixed well width (L = 1.0 µm) and a fixed well depth (V0 = 0.8 eV). From Fig. 4, wecan see that the periods of the shot noise become more and more shorter. These oscillationscan be ascribed to the combined influence of quantum well effect and Rashba spin-orbitcoupling effect in semiconductor layers. In Fig. 4, the shot noise makes Quasi-periodic os-cillation with the increasing of kR . The amplitude is almost constant, and its peak to valleyratio is decreased. It is the typical properties of resonant tunneling. This means that whenthe Rashha spin-orbit coupling strength increases, the number of electrons reflected in semi-conductors is increased, tunneling is more difficult, and the oscillation frequency is alsoincreased. It also means that the spin coupling strength can be measured by the shot noise.

4 Conclusions

In summary, based on the Landauer-Büttiker scattering formalism and transfer matrixmethod, the spin-dependent shot noise is investigated for magnetic junctions consisting oftwo ferromagnetic electrodes separated by a parabolic well. We find that the shot noise isperiodic function of the parabolic-well width. The oscillation frequencies of the shot noisedecrease with the increasing of the parabolic-well depth, and increase with the increasing ofthe Rashba spin-orbit coupling strength. The amplitude and peak to valley ratio of the shotnoise are strongly dependent on the magnetization configuration of the junction. It exhibitsuseful instructions for the design of spin electronic devices.

Acknowledgements This project was supported by NSFC (No. 10865003).

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