Physics Letters A 374 (2010) 1522–1526

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Physics Letters A

www.elsevier.com/locate/pla

Magnetic quantum wire as a spin filter: An exact study

Moumita Dey a, Santanu K. Maiti a,b,∗, S.N. Karmakar a

a Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata-700 064, Indiab Department of Physics, Narasinha Dutt College, 129 Belilious Road, Howrah-711 101, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 January 2010Accepted 25 January 2010Available online 2 February 2010Communicated by R. Wu

Keywords:Magnetic quantum wireSpin filterConductanceDOPI–V characteristic

We propose that a magnetic quantum wire composed of magnetic and non-magnetic atomic sites can beused as a spin filter for a wide range of applied bias voltage. We adopt a simple tight-binding Hamil-tonian to describe the model where the quantum wire is attached to two semi-infinite one-dimensionalnon-magnetic electrodes. Based on single particle Green’s function formalism all the calculations whichdescribe two-terminal conductance and current through the wire are performed numerically. Our exactresults may be helpful in fabricating mesoscopic or nano-scale spin filter.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Within the last few decades spin polarized transport phenom-ena [1–3] in low-dimensional systems have drawn much attentiondue to its potential application in the field of nanoscience andnanotechnology [4,5]. Discovery of GMR effect [6] in Fe/Cr mag-netic multilayer in 1980’s has led to the development of a newbranch in Condensed Matter Physics – Spintronics, which dealswith the key idea of exploiting electron spin in transport phe-nomena. The central idea of spintronic applications involves threebasic steps [7,8], which are injection of spin through interfaces,transmission of spin through matter, and finally detection of spin.Having considerably larger spin coherence time, quantum confinednanostructures such as quantum dots and molecules are thereforeideal candidates to study spin dependent transmission which playsa significant role for further development in magnetic data stor-age and device processing applications and quantum computationtechniques. With the increasing interest in generating pure spincurrent for technological purposes, modeling of spin filter is ofhigh importance today.

Till date many theoretical [9–16] and experimental efforts [17,18] are made to design spin filter and increase the efficiency ofspin polarization significantly. In 2004, Rokhinson et al. prepareda spin filter device using GaAs, by atomic-force microscopy with

* Corresponding author at: Theoretical Condensed Matter Physics Division, SahaInstitute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata-700 064, India.

E-mail address: santanu.maiti@saha.ac.in (S.K. Maiti).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2010.01.055

local anodic oxidation and molecular beam epitaxy methods. Theywere able to separate charge carriers depending on their spin stateusing the idea of spin–orbit interaction with a weak magnetic field.Generation of pure spin current in mesoscopic systems is a majorchallenge to us for further advancement in quantum computation.A more or less common trend [19,20] to develop a spin filter the-oretically is by using ferromagnetic leads or by external magneticfields. But experimental realization of these proposals is somewhatdifficult. For the first case, spin injection from ferromagnetic leadsbecomes difficult due to large resistivity mismatch and for the sec-ond one the difficulty is to confine a very strong magnetic fieldto a small region like a quantum dot (QD). Therefore, attention isbeing paid for modeling of spin filter device using the intrinsicproperties of quantum dots [21–26], such as spin–orbit interactionor voltage bias. Ring shaped or Aharonov–Bohm (AB) type geome-tries can achieve high degree of spin polarization using Rashbaspin–orbit interaction, which lifts the spin degeneracy. This hasalso been achieved by using an AB ring having periodic magneticmodulation.

The aim of the present Letter is to study spin dependent trans-mission through a magnetic quantum wire which is an array ofatomic sites. This system, composed of alternately placed mag-netic and non-magnetic atoms, is attached symmetrically to twonon-magnetic (NM) semi-infinite one-dimensional (1D) electrodes.A simple tight-binding Hamiltonian is used to describe the systemwhere all the calculations are done by using single particle Green’sfunction formalism [27–33]. With the help of Landauer formulaspin dependent conductance is obtained, and the current–voltagecharacteristics are computed from the Landauer–Büttiker formal-

M. Dey et al. / Physics Letters A 374 (2010) 1522–1526 1523

Fig. 1. (Color online.) A magnetic quantum wire (framed region) of N atomic sitesattached to two semi-infinite 1D NM electrodes, namely, source and drain. Filledgreen circles correspond to NM atomic sites, while the filled blue and brown circlesrepresent atomic sites having unequal magnetic moments.

ism [34–36]. We explore various features of spin transport usingthis simple geometry. Quite interestingly, we see that for a certainenergy range, transmission probability of up spin electron dropsto zero, whereas for down spin electrons it becomes non-zero andvice-versa. Therefore, tuning the Fermi energy (E F ) of the system,the magnetic quantum wire can be used as a spin filter for a widerange of applied bias voltage depending on the strength of local-ized magnetic moments in the wire.

The scheme of the Letter is as follow. With a brief introduction(Section 1), in Section 2, we describe the model and theoreticalformulations for the calculation. Section 3 explores the significantresults which explain the filtering action, and finally, we concludeour study in Section 4.

2. Model and synopsis of the theoretical background

The schematic representation of our model is depicted in Fig. 1.In this figure we illustrate the nanostructure through which spindependent transport is investigated. We study spin transmissionthrough a quantum wire of N atomic sites composed of alternatelyplaced magnetic and non-magnetic atoms. The wire is attachedsymmetrically to two non-magnetic semi-infinite 1D metallic elec-trodes termed as source and drain. The atomic sites forming thedevice are of 3 different types. One of them being non-magneticand the other two being magnetic of types A and B, having twodifferent values of localized magnetic moments, hA and hB , as-sociated with them. The orientation of the local moments asso-ciated with each magnetic site is specified by angles θn and φn

(n denotes the n-th site) in spherical polar coordinate system. Thetwo metallic electrodes consist of infinite number of non-magneticatoms labeled as 0,−1,−2, . . . ,−∞ for the left electrode and(N + 1), (N + 2), (N + 3), . . . ,∞ for the right one.

For the whole system (source–wire–drain) we can write theHamiltonian as

H = HW + HL + H R + HLW + HW R (1)

where HW corresponds to the Hamiltonian of the wire, HL(R) rep-resents the Hamiltonian for the left (right) electrode, and H LW (W R)

is the Hamiltonian representing the wire–electrode coupling.The spin polarized Hamiltonian for the quantum wire can be

written in effective one-electron approximation, within the tight-binding formalism in Wannier basis, using nearest-neighbor ap-proximation as

HW =N∑

c†n(ε0 − �hn. �σ )cn +

N∑(c†

i tci+1 + h.c.)

(2)

n=1 i=1

where

c†n =

(c†

n↑ c†n↓

)

cn =(

cn↑cn↓

)

ε0 =(

ε0 00 ε0

)

t = t

(1 00 1

)

�hn. �σ = hn

(cos θn sin θne−iφn

sin θneiφn − cos θn

)

The first term of Eq. (2) represents the effective on-site energiesof the atomic sites in the wire. ε0’s are the site energies, whilethe �hn. �σ refers to the interaction of the spin (σ ) of the injectedelectron with the localized on site magnetic moments. This termis responsible for spin flipping at the sites. The second term de-scribes the nearest-neighbor hopping strength between the sitesof the quantum wire.

Similarly, the Hamiltonian HL(R) can be written as

HL(R) =∑

i

c†i εL(R)ci +

∑i

(c†

i tL(R)ci+1 + h.c.)

(3)

where εL(R) ’s are the site energies of the electrodes and tL(R) isthe hopping strength between the nearest-neighbor sites of the left(right) electrode.

Here also,

εL(R) =(

εL(R) 00 εL(R)

)

tL(R) =(

tL(R) 00 tL(R)

)

The wire–electrode coupling Hamiltonian is described by

HLW (W R) = (c†

0(N)tLW(WR)c1(N+1) + h.c.)

(4)

where tLW (W R) being the wire–electrode coupling strength.In order to calculate the spin dependent transmission probabil-

ities and the current through the magnetic quantum wire we usesingle particle Green’s function technique. Within the regime of co-herent transport and for non-interacting systems this formalism iswell applied.

The single particle Green’s function representing the full systemfor an electron with spin σ is defined as [27–29],

Gσ = (E − Hσ )−1 (5)

where

E = (ε + iη)I (6)

ε being the energy of the electron passing through the system.iη is a small imaginary term added to make the Green’s function(Gσ ) non-Hermitian.

Now Hσ and Gσ representing the Hamiltonian and the Green’sfunction for the full system can be partitioned like [27–29],

Hσ =⎛⎝

HLσ HLWσ 0

H†LWσ HWσ HWRσ

0 H†WRσ HRσ

⎞⎠ (7)

Gσ =⎛⎝

GLσ GLWσ 0

G†LWσ GWσ GWRσ

0 G† G

⎞⎠ (8)

WRσ Rσ

1524 M. Dey et al. / Physics Letters A 374 (2010) 1522–1526

where HLσ , HRσ , and HWσ represent the Hamiltonians (in ma-trix form) for the left electrode (source), quantum wire and rightelectrode (drain), respectively. HLWσ and HWRσ are the matri-ces for the Hamiltonians representing the wire–electrode couplingstrength. Assuming that there is no coupling between the elec-trodes themselves, the corner elements of the matrices are zero.Similar definition goes for the Green’s function matrix Gσ as well.

Our first goal is to determine GWσ (Green’s function for thewire only) which defines all physical quantities of interest. Follow-ing Eq. (5) and using the block matrix form of Hσ and Gσ the formof GWσ can be expressed as [27–29],

GWσ = (E − HWσ − �Lσ − �Rσ )−1 (9)

where �Lσ and �Rσ represent the contact self-energies introducedto incorporate the effects of semi-infinite electrodes coupled to thesystem, and, they are expressed by the relations [27–29],

�Lσ = H†LWσ GLσ HLWσ

�Rσ = H†WRσ GRσ HWRσ (10)

Thus the form of self-energies are independent of the nanostruc-ture itself through which transmission is studied and they com-pletely describe the influence of electrodes attached to the system.Now, the transmission probability (Tσ ) of an electron with spin σis related to the Green’s function as [27–29],

Tσ = Tr[�Lσ Gr

Wσ �Rσ GaWσ

](11)

where GrWσ and Ga

Wσ are the retarded and advanced single particleGreen’s functions (for the device only) for an electron with spin σ .�Lσ and �Rσ are the coupling matrices, representing the couplingof the magnetic quantum wire to the left and right electrodes, re-spectively, and they are defined by the relation [27–29],

�Lσ (Rσ ) = i[�r

Lσ (Rσ ) − �aLσ (Rσ )

](12)

Here �rLσ (Rσ ) and �a

Lσ (Rσ ) are the retarded and advanced self-energies, respectively, and they are conjugate to each other. It isshown in literature by Datta et al. that the self-energy can be ex-pressed as a linear combination of a real and imaginary parts inthe form,

�rLσ (Rσ ) = �Lσ (Rσ ) − i�Lσ (Rσ ) (13)

The real part of self-energy describes the shift of the energy levelsand the imaginary part corresponds to broadening of the levels.The finite imaginary part appears due to incorporation of the semi-infinite electrodes having continuous energy spectrum. Therefore,the coupling matrices can be easily obtained from the self-energyexpression and is expressed as

�Lσ (Rσ ) = −2 Im(�Lσ (Rσ )) (14)

Considering linear transport regime, conductance (gσ ) is obtainedusing Landauer formula [27–29],

gσ = e2

hTσ (15)

Knowing the transmission probability (T ) of an electron withspin σ , the current (Iσ ) through the system is obtained usingLandauer–Büttiker formalism. It is written in the form [27–29],

Iσ (V ) = e

h

+∞∫−∞

[f L(E) − f R(E)

]Tσ (E)dE (16)

where f L(R) = f (E −μL(R)) gives the Fermi distribution function ofthe two electrodes having chemical potentials μL(R) = E F ± eV /2.E F is the equilibrium Fermi energy.

3. Numerical results and discussion

All the essential features of spin transport through a mag-netic quantum wire are studied by performing several numericalcalculations. Here we assume that the non-magnetic 1D metallicelectrodes are made from identical materials, therefore, the siteenergies and the nearest-neighbor hopping strengths are chosento be identical for the two electrodes. Let us begin our discussionby mentioning the values of different parameters used for numer-ical calculations. We choose the quantum wire to be made up of64 (N = 64) sites. The on-site energies (ε0) in the quantum wireare chosen to be 0. Magnitudes of the two different local magneticmoments hA and hB associated with two types of magnetic sitesA and B are set as 1.9 and 0.1. The hopping strength between thenearest-neighbor sites of the magnetic quantum wire and for thetwo NM electrodes are set at t = 3 and tL = tR = 4, respectively.The site energies (εL(R)) of all the sites in the two electrodes arefixed at 0. For the sake of simplicity, we choose the unit whereh = c = e = 1. Throughout the analysis we study the basic featuresof spin transport for two distinct regimes of electrode-to-magneticquantum wire coupling.

Case 1: Weak-coupling limit.

This regime is defined by the criterion tLW (W R) � t . In this case,we choose the values as tLW = tW R = 0.5.

Case 2: Strong-coupling limit.

This limit is described by the condition tLW (W R) ∼ t . In this regimewe choose the values of hopping strengths as tLW = tW R = 2.5.

3.1. Conductance–energy spectrum

To explain all the relevant features of spin transport we startwith the conductance–energy characteristics. As illustrative exam-ples, first in Fig. 2 we plot the variation of conductance g withrespect to the energy E for up (↑) and down (↓) spin electronsseparately in the limit of weak wire–electrode coupling strength.Also, the variation of average density of states (ADOS) are super-imposed in each case. The mathematical description for the ADOS(symbolized as ρ(E)) of the magnetic quantum wire including theeffect of the two electrodes for an electron with spin σ is ex-pressed as

ρσ (E) = − 1

NπIm

[Tr[GWσ ]] (17)

It is observed from the conductance spectra that up and downspin electrons follow entirely two different channels while passing

Fig. 2. (Color online.) g–E and ρ–E (thick solid lines) curves in the limit of weakcoupling for a magnetic quantum wire with N = 64. Upper and lower panels cor-respond to the results of up and down spin electrons, respectively. Dotted linerepresents the location of Fermi energy E F of the wire.

M. Dey et al. / Physics Letters A 374 (2010) 1522–1526 1525

Fig. 3. (Color online.) g–E and ρ–E (thick solid lines) curves in the limit of strong-coupling for a magnetic quantum wire with N = 64. Upper and lower panels cor-respond to the results of up and down spin electrons, respectively. Dotted linerepresents the location of Fermi energy E F of the conductor.

through the wire. This splitting of up and down conduction chan-nels is responsible for spin filtering action. Quite interestingly wesee that for a certain range of energy for which the transmissionprobability (T↑) and hence the conductance (g↑) for up spin elec-tron drops to zero value, shows non-zero transmission probability(T↓) as well as conductance (g↓) due to down spin electron. Thepresence of sharp resonant peaks in the conductance spectrum areassociated with the energy eigenvalues of the full system. In thiscase due to large system size, N = 64, the sharp resonant peaksget closely spaced to form a quasi-band as shown by the greenand blue regions.

In the limit of strong wire–electrode coupling as shown inFig. 3, transmission probability (Tσ ) almost reaches to the valueunity and the sharp conductance peaks acquire some broaden-ing which is quantified by the imaginary part of the self-energyexpression, incorporated to include the effect of semi-infinite elec-trodes. The feature of broadening is not very clear from this fig-ure due to large system size. Apart from increase in transmissionprobability and broadening of conductance peaks, all the other fea-tures, as observed in the weak-coupling case, remain the samee.g., formation of quasi-bands and position of band gaps. There-fore, increase in electrode–wire coupling strength does not changethe position of global gaps in the conductance spectrum, but thecoupling has a strong influence in the study of current–voltagecharacteristics as discussed clearly in Refs. [30–33].

3.2. Degree of polarization vs. energy spectrum

Next, in Fig. 4 we show the variation of degree of polarization(DOP) with respect to the energy of the injected electrons. For aparticular energy value E , the DOP is defined in terms of transmis-sion probabilities Tσ in the following way,

DOP(E) =∣∣∣∣ T↑(E) − T↓(E)

T↑(E) + T↓(E)

∣∣∣∣ (18)

DOP gives a quantitative measurement of the spin polarizationachieved. It is observed from Fig. 4 that the degree of polarizationis significantly enhanced with the increase in wire-to-electrodecoupling strength and for wide range of energies it (DOP) almostreaches to the value unity or 100% as expressed conventionally inmost of the literatures. Thus our proposed model quantum systemis a very good example for designing a spin filter.

3.3. Current–voltage characteristics

The spin filtering action becomes clearer in the current–voltage(I–V ) characteristics presented in Fig. 5. Current across the mag-

Fig. 4. (Color online.) Degree of polarization as a function of energy for a magneticquantum wire with N = 64. (a) Weak-coupling limit and (b) strong-coupling limit.

Fig. 5. (Color online.) Current I as a function of bias voltage V in the limit of strong-coupling for a magnetic quantum wire with N = 64. The blue and orange curvescorrespond to the currents for up and down spin electrons, respectively.

netic quantum wire is obtained by integrating over the transmis-sion function following Landauer–Büttiker formalism for a constantFermi energy (E F ). In this case we set the value of E F at −0.4(dotted line in Fig. 3). As this value of E F falls in the energy gapregion of the up spin conductance spectrum, therefore, non-zerovalue of up spin current is obtained after overcoming a finite valueof the applied bias voltage, the so-called threshold voltage (V th).On the other hand, for any given bias voltage V , non-zero value ofdown spin current is observed. Thus spin filtering takes place upto the bias voltage (V th), when up spin current is totally blockedand only down spin current is obtained. In an exactly similar way,if we set the Fermi energy at some value of down spin energy gapregion, down spin current can be blocked totally by passing theup spin current only. Here we plot the current–voltage character-istics only in the strong-coupling limit. Exactly a similar filteringaction is also observed in the case of weak-coupling limit. Butthe point is that in the limit of strong-coupling, current ampli-tude gets magnified significantly compared to the weak-couplinglimit. The plateau like structures in the current–voltage charac-teristics are observed due to the presence of global gaps in theconductance spectrum. The value of V th depends on the differ-

1526 M. Dey et al. / Physics Letters A 374 (2010) 1522–1526

ence between the localized magnetic moments associated with themagnetic atoms. Therefore, changing the atoms the magnitude ofV th can be tuned.

4. Concluding remarks

To conclude, in the present Letter we have investigated spintransport through a magnetic quantum wire using single parti-cle Green’s function formalism. We have adopted a simple tight-binding framework to illustrate the system, which is a quantumwire formed by magnetic and non-magnetic atomic sites and con-nected symmetrically to source and drain. We have shown thevariation of conductance as a function of injecting electron en-ergy for up and down spin electrons separately for two differ-ent strengths of wire-to-electrode coupling. Conductance spectrumclearly depicts the splitting of up and down spin conduction chan-nels which is the key idea behind the modeling of a spin fil-ter. Larger the difference between the local magnetic momentsof the two types of magnetic atoms, smaller is the overlap be-tween the up and down conduction channels. Also we have plottedthe variation of degree of polarization with respect to energy forboth the coupling regimes. It shows that enhancement of cou-pling strength increases the degree of polarization i.e., improvesthe quality of filtration significantly. Finally, we have obtained thecurrent passing through the device using Landauer–Büttiker formu-lation. The feature of spin filtering is visualized more prominentlyin the current–voltage characteristics. Tuning the Fermi energy toa particular value the device can act as a spin filter i.e., up to acertain range of bias voltage only up or down spin current is ob-tained.

In this work we have calculated all these results by ignoringthe effects of temperature, spin–orbit interaction, electron–electroncorrelation, electron–phonon interaction, disorder, etc. Here we fixthe temperature at 0 K, but the basic features will not change sig-nificantly even in non-zero finite (low) temperature region as longas thermal energy (kB T ) is less than the average energy spacing ofthe energy levels of the magnetic quantum wire. In this model it isalso assumed that the two side-attached non-magnetic electrodeshave negligible resistance.

All these predicted results using such a simple geometry maybe useful in designing a spin polarized source.

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