稀有金属(英文版) 2015,34(01),45-50

收稿日期：8 October 2012

基金：financially supported by the National Natural Science Foundation of China(Nos.11204001,10804110,and 11174004);Anhui University Scientific Research Fund(Nos.06060283,2009QN006A,and 32030028);‘‘211 Project’’of Anhui University,Anhui Provincial Natural Science Foundation(Nos.1208085QA07 and 1308085MA04);the Major Project ofEducation Department in Anhui(No.06070241);the Educational Commission(No.KJ2013A031);

First-principles calculations on spin-polarized transport properties of Mn₄O₄ cluster

Zhen-Xiang Dai Gan-Hong Zheng Bing Wang Wei-Wei Wang Yong-Qing Ma Zhi Zeng

School of Physics and Materials Science, Anhui University

Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences

Abstract：

Taking the Mn4O4 cluster as a model system,the spin-polarized transport properties of the small cluster system were systematically probed. The theoretical investigations are based on density-functional theory and nonequilibrium Green's functional method. The equilibrium transport mechanism is illustrated by the band structure of the electrode, the electronic structure of the Mn4O4 cluster and the coupling between the cluster and the electrodes. To well understand the non-equilibrium mechanisms, one straightforward and simple band-matching model was proposed. Moreover, such a band-matching model can be extended to well illustrate the transport properties of other nano-scale systems.

Keyword：

Cluster; First-principles; Band-matching model; Transport mechanism;

Author： Zhen-Xiang Dai,e-mail: physdai@ahu.edu.cn;

Received： 8 October 2012

1 Introduction

In the past two decades, quantum transport properties ofnano-scale systems have been received great attentionsince these systems present the ultimate size limit offunctional electronic devices and have great potentialutilities in molecular electronic devices [1]. Being animportant kind of low-dimensional systems, the clustersmay provide suitable building blocks for the nanodevices[2]. Therefore, besides the geometrical and electricalstructure properties etc., it is also of fundamental andpractical importance to probe the quantum transport properties of clusters both theoretically and experimentally.

Up to now, some research works have been performed toinvestigate the transport properties of the clusters. Forexample, Yu et al. [3] probed the electronic transportacross the S₉cluster via first-principles calculation. Saitoet al. also reported the current–voltage characteristics carbon nanostructures [4]. At the same time, our previousworks calculated the transport properties of the Si₄cluster[5, 6], Ge₇cluster [7], and Au₃₂cluster [8] via the investigations of the effect of contact geometry and gate voltage.However, these earlier researches were primarily concentrated on the spin un-polarized electronic transport properties; so far, less attention was focused on the spinpolarized transport properties of the clusters. However, themolecular-scale systems including clusters show greatpotential applications in the future spintronics devices [9,10]. Therefore, it is very necessary and interesting toillustrate the spin-polarized transport mechanisms of smallclusters theoretically and experimentally.

Moreover, in recent years, single-molecule magnets(SMMs) [11, 12] have also attracted much attention due totheir potential candidates for future applications in ultradensity data storage devices or quantum computing. Themost widely investigated class of SMMs is the Mn₁₂group[11, 12]. The ligated Mn₁₂O₁₂clusters were recently foundto have high spins and exhibit magnetic bistability. Herefour Mn^4?(S = 3/2) form a central tetrahedron surroundedby eight Mn^3?(S = 2). The spins at Mn^4?sites point upwhile the spins at Mn^3?sites point down. The net magneticmoment of the SMM Mn₁₂O₁₂is, therefore, 20 lb. Aspointed out in the Ref. [13], being the core of the SMMMn₁₂O₁₂, the Mn₄O₄model system is very important sinceit has the geometric structure of a distorted cube and hasthe magnetic moment 20 lb similar to Mn₁₂O₁₂. Therefore,taking the Mn₄O₄cluster as a model system and focusingon understanding the transport mechanism, calculationswere performed on the spin-polarized transport propertiesof such a small cluster. Our results suggest that, sandwiched in between two gold nanowire electrodes, theMn₄O₄cluster exhibits the equilibrium conductance of1.64 G₀. To well understand the non-equilibrium transportmechanism of the small system, one simple and straightforward band-matching model was proposed. Based on thespecific band structure of the adopted electrodes etc., sucha model can be easily extended to well understand thetransport behaviors of other systems.

2 Computation methods and model

The calculations were performed by using the program ATK[14], which is based on the combination of density-functional theory (DFT) [15] (as implemented in the well-testedSIESTA method [16]) with the non-equilibrium Green’sfunction [17, 18] technique. The ATK package is able tomodel the electrical properties of nano-scale devices, whichare usually referred to the two-probe system as shown inFig. 1. The two-probe system is pided into three parts: theleft electrode, the right electrode, and the central scatteringregion. In the calculations, one Mn₄O₄cluster is symmetrically sandwiched with two atomic-scale Au(111) nanowire[19, 20] electrodes. Moreover, a large enough vacuum layeris included in the x and y directions so that the device has nointeraction with its mirror images, and in the z direction threeatomic layers were chosen for the electrode cell. The contactstructure between the cluster and electrodes is determined asfollows. First, the vertical distance is the equilibrium one,corresponding to the minimum total energy. Second, withthe electrode atoms fixed, the central region is relaxed. Thecurrent–voltage (I–V) characteristics are obtained fromLandauer–Bu¨ ttiker formula [21]

where l_rand l_lare the electrochemical potentials of theleft and right electrodes, respectively, E is the electronenergy. For an applied bias v_b,

For simplicity, l_le0T and l_re0T are set to zero. The energyregion between ev_b=2 and ev_b=2, contributing to thecurrent integral above, is referred to as the bias window.T eE; v_bT is the transmission coefficient at energy E and biasvoltage v_b. In this investigation, only valence electrons areconsidered, and the atomic cores are described by normconserving pseudopotentials [22]. Valence wave functionsare expanded in a SIESTA-localized basis set (numericallocalized pseudo-atomic orbitals) [23]. The generalizedgradient approximation to the exchange–correlationpotential, especially the functional of the revised Perdew–Burke–Ernzerhof parametrization [24], was used. Theconvergence criterion for the Hamiltonian is 1 9 10^-5viathe mixture of the Hamiltonian.

Fig. 1 Two-probe structure model. Gold, gray, and red spherespresenting Au, Mn, and O atoms, respectively

Fig. 2 Equilibrium transmission spectrum and MPSH levels. Fermilevel being set to zero, as shown by the vertical line

3 Results and discussion

Figure 2 presents the equilibrium transmission spectrum.In the probed energy region, the whole transmissionspectrum of this cluster can be pided into four parts. Thefirst energy region is from -3.00 to -1.15 e V, the secondone is a large transmission gap region between -1.15 and-0.50 e V, the third one is from -0.50 to 2.62 e V, and thelast one is also a transmission gap region above 2.62 e V. Inthe first one, the whole spectrum consists of many sharppeaks in both the majority-spin and the minority-spinchannels. In the third one, there are several transmissionpeaks below Fermi level, in particular just around theFermi level. At the Fermi level, the total conductance ofthis cluster is 1.64 G0 (G0 is conductance quantum).Above the Fermi level, in the majority-spin channel, itexhibits a largely broadened transmission peak within theenergy region from 0.0 to 1.2 e V. In the case of minorityspin channel, it shows some transmission peaks in onelarger energy range from 0 to 1.9 e V, and there also existsone largely broadened peak around 2.3 e V.

These futures of the equilibrium transmission can befirst understood via the combination of the density of state(DOS) of the isolated Mn₄O₄cluster, the band structure ofthe gold electrode, and the coupling between the clusterand two electrodes. As shown in Fig. 3a, for the goldelectrode, there exist Bloch states from -3.00 to-1.15 e V. For the isolated Mn₄O₄cluster, as shown inFig. 3c, there are also electronic state distributions in thisenergy region in both the majority and minority channel.Thus, the electrons in the electrodes can transmit fromleft to right via these states of the Mn₄O₄cluster. Consequently, in the first energy region, there are transmission values in both majority-spin and minority-spinchannel. In the second region, the transmission gap isdirectly determined by the band gap of the nanowireelectrode. Around the Fermi level, there exist some sharppeaks in both spin channels, which can be understood asfollows. For the majority-spin channel, the original electronic states around the Fermi level directly contribute tothe transmission. For the minority-spin channel, it isfound that there are no corresponding states in this regionas shown in Fig. 3c. However, as well known, when onemolecular-scale system is sandwiched between two electrodes, owing to the couple with the electrode, thosemolecular levels will be broadened and shifted, especiallythose levels around the Fermi level. Therefore, owing tosuch a coupling effect, the levels in the vicinity of theFermi energy of the Mn₄O₄cluster are well-broadenedand shifted. Consequently, there also exists transmissionin the minority-spin channel in this region. For the energyregion above about 1.90 e V, the transmission in themajority-spin channel is smaller; however, the transmission in the minority-spin channel is larger. This resultsfrom there being more states in the minority-spin channelthan the majority-spin channel. The transmission gapbetween 2.62 and 3.00 e V is also directly due to the bandgap of the gold nanowire in this energy region. Second,the equilibrium transmission is also further illustrated viathe molecular projected self-consistent Hamiltonian(MPSH) [25] of the cluster. The MPSH states areobtained by diagonalizing the molecular part of the fullself-consistent Hamiltonian. In the calculations, theMPSH states are considered via projecting the wholescattering region including the surface atoms and Mn₄O₄cluster so that the coupling between electrode and clusteris also observed directly. In Fig. 2, there are dense MPSHlevel distributions in the first energy region, contributingto those sharp transmission peaks in this energy range.Although there are some MPSH states from -1.15 and-0.50 e V, the band gap of the gold nanowire determinesthe transmission gap. Around the Fermi level, like thedense transmission peaks, there are also dense MPSHlevels. In a word, the transmission peaks are generallyassociated to the corresponding MPSH levels, in particular the dense transmission peaks correspond to the denseMPSH levels distribution. Therefore, the transmissionproperties are determined not only by both the electrodeand the cluster itself, but also by the coupling betweenelectrode and cluster.

Fig. 3 Band structure of gold electrode nanowire a, transmission spectrum of pure gold nanowire b, and DOS plot of isolated Mn4O4 clusterc. Fermi level being relative to zero, shown by the horizontal line in a and b, and the vertical dash line in c

Fig. 4 Current–voltage curve of Mn4O4cluster. Open red squaremeaning majority-spin current, filled green square presenting minor

Fig. 4 Current–voltage curve of Mn4O4cluster. Open red squaremeaning majority-spin current, filled green square presenting minority-spin current, and open green circle representing total current

The current–voltage properties are shown in Fig. 4.Taking the positive case as an example, the non-equilibrium properties are illustrated. In the small voltage, the I–V curve is almost linear, and the cluster exhibits metallictransport behavior. At the same time, the minority-spincurrent is smaller than the majority-spin current. With thevoltage increasing, the minority-spin current increasessteadily; however, the majority-spin current increases evenmore. As a result, the majority-spin current is still largerthan the down current, and the total current is determinedmostly by the majority-spin channel. When the bias is0.5 e V, the majority-spin current reaches maximum value.The ratio between these channels currents also reachesmaximum 54. As the bias increases further, the majorityspin current decreases gradually. Since the minority-spincurrent is suppressed to a certain extent and the total current is mainly contributed by the majority-spin channel.Therefore, when the bias is 0.5, the majority-spin currentbegins to decrease, and thus the total current also decreases. As the voltage increases from 1.2 e V, although themajority-spin current decreases further, the minority-spincurrent increases more. Then the total current is decided bythe minority-spin channel. As a result, the total currentincreases, consequently the NDR appears in this bias range.

The non-equilibrium properties are first understood bythe evolution of transmission spectrum, as shown in Fig. 5.When the voltage is below 0.7 V, similar to the case ofzero voltage, the whole transmission spectrum can bepided into four similar parts from the low-energy regionto high-energy region. In the first energy region ranges,from -3.00 to about -1.15 e V, the average transmissionthrough majority-spin and minority-spin channel is about1.0. The second gap one is [-1.15, -0.50] e V. The thirdone is [-0.50, 2.62] e V. In such an energy region, thetransmission through the majority-spin channel is largerthan that via minority-spin channel around the Fermi level.This is mainly owing to the coupling between the electrodeand cluster, then the electrons can transmit from theseshifted and broadened energy levels below the Fermi level.The fourth one is gap region correspondingly.

Fig. 5 Non-equilibrium transmission spectrum as function of energyunder different voltages. Up panel a being for the majority-spin case,and low panel b being for the case of minority spin. Two green linesindicating bias window. Energy being relative to average Fermi level

With the voltage increasing, the second and fourth gapregions are both increased, and the first and third transmission regions are correspondingly decreased. Especiallyas the voltage increases from 0 to 0.5 V, compared with theminority-spin channel, the well-broadened majority-spintransmission peak above Fermi level gradually moves intothe bias window, and then contributes to the majority-spincurrent. Therefore, with the rise of the voltage in such avarying range, the majority-spin current increases morethan the minority current. When the bias is above 0.5 V,the second gap region begins to move into the bias window.Therefore, the current begins to decrease correspondingly,especially the majority-spin current. As a result, the currentexhibits maximum value at 0.5 V. When the voltage isgreater than or equal to 0.8 V, one new transmission regionbegins to appear in the second gap region, and the originsecond gap region is split into two new gap regions.Moreover, with the rise of the voltage, the new transmission region and those two new gap regions all increase, butthe third transmission region always decrease. As a result,the current in both spin channels are decreased. However,when the voltage increases up to 1.2 V, as for the minorityspin current the new transmission region begins to moveinto the bias window to contribute to the current. Moreover, with the combination of the well-broadened transmission peak around 2.0 e V, more and more transmissionin the bias window begins to contribute to the current.Consequently, the minority-spin current dominates the totalcurrent, and the total current also begins to increase correspondingly. As a result, the Mn4O4cluster exhibits NDRbehavior.

Fig. 6 Band match model between these two electrodes under three voltages V = 0, 0.5, and 1.0 V. Letters L, R, and M presenting left and rightelectrode, and middle cluster, respectively, and letters A, B, and C presenting different energy regions of band structure. Blue-filled energyregions in which the electrons can transmit being labeled by ‘‘trans1,’’ ‘‘trans2,’’ ‘‘trans3,’’ and ‘‘trans12;’’ and ‘‘gap1,’’ ‘‘gap2,’’ ‘‘gap21,’’ and‘‘gap22’’ all representing transmission gap regions

To further understanding the transport mechanism, asschematically shown in Fig. 6, one straightforward andsimple band-matching model was proposed. This model isdescribed as follows. First, if there are matching bandsbetween the left and right electrode, it is possible for theelectrons of one electrode to transmit to the other electrodevia the broadened and shifted levels of the cluster. Second,the bands of these two electrodes will be shifted up ordown with the variation of the applied voltage. Third, therelative shift of the bands of these two electrodes results inthe variation of the band matching of these two electrodes,leading to the change of the electron transmission. Asdescribed below, such a simple and straightforward can beeasily further extended to understand the quantum transportproperties of other systems, based on the specific electrodeband and the molecule.

In the case of 0 V, the band structure of the perfect goldnanowire can be pided into three parts labeled, respectively, as A, B, and C, as presented in Fig. 6. Betweenthese three parts, there are also two band gaps labeled by‘‘gap1’’and‘‘gap2’’in the probed energy region. In theenergy region labeled by‘‘trans1,’’‘‘trans2,’’and‘‘trans3,’’the electrons can transmit from left to right electrode viathe broadened and shifted energy levels of the cluster.These energy levels are represented by the wavy lines. Thehighest energy of the‘‘trans1’’region (or part‘‘A’’) is-1.15 e V, the‘‘trans2’’region (or part B) is [-0.5, 2.6]e V, and the lowest energy of the‘‘trans3’’ (or part‘‘C’’) is3.4 e V.

The features of the non-equilibrium properties can befurther understood as follows. When the bias is increasedup to 0.5 V, the band of the left electrode is shifted up0.25 V, and the band in the right electrode is shifted down0.25 V. Therefore, the band mismatching region betweenthe left and right electrode is increased, eventually thesetwo‘‘gap1’’and‘‘gap2’’regions are both increased. At thesame time, the transmitting region‘‘trans2’’between thesetwo gaps is decreased correspondingly. Therefore, asshown in Fig. 5, the increase of the band mismatch of theleft and right electrode directly results in the increase oftransmission gap when the bias gradually increases from 0to 0.5 V. At the same time, the third transmission region isalso gradually decreased. When the applied voltage is0.7 V, the left band is shifted up 0.35 e V, and the right isshifted down 0.35 e V. As a result, the‘‘trans2’’regiondecreases further, and‘‘gap1’’and‘‘gap2’’also increase.Particularly, there exists very small band overlap about0.05 e V between the part‘‘A’’of left electrode and the part‘‘B’’of right electrode. With the bias being 0.8 V, thereexists an obvious band overlap about 0.15 e V between thepart‘‘A’’of left electrode and the part‘‘B’’of right electrode. These new band-overlapping region is labeled as‘‘trans12.’’The electrons can transmit from this new bandmatching region‘‘trans12.’’This is the reason that one newobvious transmission region in the second gap region in thecase of 0.8 V, as shown in Fig. 5. The original gap region‘‘gap2’’is, respectively, changed into two gap regions. Forclarity, these four new gap regions are labeled as‘‘gap1,’’‘‘gap 2,’’‘‘gap21,’’and‘‘gap22.’’As the voltage isincreased further, the band of left electrode is shifted upand the band of the right electrode will also be shifteddown further, and eventually, the‘‘trans’’region betweenthe‘‘gap1’’and‘‘gap21’’region decreases more and more.However, the‘‘trans12’’region between the‘‘gap21’’and‘‘gap22’’is increased. Therefore, such a band-matching isvery helpful to understand those variation features of the I–V curve presented in Fig. 5.

4 Conclusion

In summary, the spin-polarized transport properties of theMn₄O₄cluster sandwiched between two atomic-scale goldnanowires were theoretically investigated. Focusing on themechanism understanding, the transport properties areillustrated via the band structure of gold nanowire, electronic structure of cluster, and the coupling between thecluster and the electrodes. In particular, one straightforward and simple band-matching model is proposed to wellanalyze the I–V characteristics of the cluster. Such a bandmatching model can be easily extended to illustrate thequantum transport properties of other molecular-scalesystems.