二維磁性系統(tǒng)和準(zhǔn)一維磁性納米管的內(nèi)能

米斌周，馮翠菊，祁云平，丁 東，薛永紅（.華北科技學(xué)院基礎(chǔ)部，北京 060；.西北師范大學(xué)物電學(xué)院，甘肅蘭州 730070）

文章編號：1001?246X（2015）01?0086?07

二維磁性系統(tǒng)和準(zhǔn)一維磁性納米管的內(nèi)能

米斌周1，馮翠菊1，祁云平2，丁 東1，薛永紅1
（1.華北科技學(xué)院基礎(chǔ)部，北京 101601；2.西北師范大學(xué)物電學(xué)院，甘肅蘭州 730070）

采用量子統(tǒng)計理論的多體格林函數(shù)法計算二維單離子各向異性海森伯鐵磁體、反鐵磁體以及單壁鐵磁納米管的內(nèi)能，對比鐵磁體和反鐵磁體的結(jié)果.在相同的參量下，反鐵磁能量總是低于相應(yīng)的鐵磁能量（相變點(diǎn)除外）.由于反鐵磁能量隨溫度上升的速度較鐵磁能量快，當(dāng)溫度升高到居里點(diǎn)TC和奈爾點(diǎn)TN時（TC＝TN），鐵磁能量和反鐵磁能量相等.橫向關(guān)聯(lián)效應(yīng)對系統(tǒng)內(nèi)能的影響較大，不能忽略.

海森伯模型；二維系統(tǒng)；磁性納米管；內(nèi)能；多體格林函數(shù)法

0 引言

近年來，在納米科技蓬勃發(fā)展的背景下，低維磁性材料研究引起了科學(xué)家們的極大興趣.為了研究二維和一維磁性系統(tǒng)的磁化行為和物理機(jī)制，人們發(fā)展了不同的理論技術(shù)方法.文獻(xiàn)［1］采用自旋動力學(xué)方法研究了磁偶極相互作用表現(xiàn)的邊界效應(yīng)對小尺寸正方形鐵磁薄膜的磁化翻轉(zhuǎn)過程的影響，文獻(xiàn)［2］采用Monte Carlo模擬對自旋為1/2的一維鉆石鏈反鐵磁Ising系統(tǒng)的磁化進(jìn)行研究，文獻(xiàn)［3］采用變分法研究了具有反鐵磁界面交換耦合的鐵磁/反鐵磁雙層膜系統(tǒng)中交換偏置場和矯頑力場隨冷卻場的變化等.在眾多的理論方法中，涉及多體格林函數(shù)法的研究相對較少.由于多體格林函數(shù)法考慮了量子漲落效應(yīng)［4－12］并且可以從低溫到高溫統(tǒng)一地處理鐵磁理論問題，此方法運(yùn)用于處理海森伯磁性系統(tǒng)被證明是非常成功的［6－9，12－19］，因而具有更加誘人的吸引力.對于一個海森伯磁性系統(tǒng)，人們成功地運(yùn)用量子統(tǒng)計理論的多體格林函數(shù)方法求出磁化強(qiáng)度隨溫度的變化，有一個對于任意自旋量子數(shù)S都適用的普遍公式［4－6］.內(nèi)能是熱力學(xué)系統(tǒng)的基本物理量.在量子統(tǒng)計力學(xué)中，內(nèi)能就是哈密頓量的系綜平均值.文獻(xiàn)［7］嚴(yán)格給出了S＝1/2，1時鐵磁體的內(nèi)能公式.文獻(xiàn)［8］給出了一個鐵磁系統(tǒng)對于任意自旋量子數(shù)都適用的考慮橫向關(guān)聯(lián)函數(shù)時比較嚴(yán)格的內(nèi)能表達(dá)式.文獻(xiàn)［9］利用多體格林函數(shù)方法計算了單壁鐵磁納米管的內(nèi)能，并與二維平面的情況做了比較.其

實，文獻(xiàn)［7－9］都是將海森伯交換相互作用項中的縱向和橫向部分放在一起處理的，計算過程較復(fù)雜，而且只適用于鐵磁系統(tǒng).文獻(xiàn)［7］的近似更嚴(yán)格，只給出了S＝1/2，1時鐵磁體的內(nèi)能，對于大的自旋量子數(shù)則會涉及到更高階的格林函數(shù)，而且不同自旋量子數(shù)的內(nèi)能公式不統(tǒng)一.本文的做法是將海森伯交換相互作用項中的縱向和橫向部分分開成兩項分別計算.對于縱向部分（1/2）J，采用無規(guī)相近似：≈，（i≠j）［4－5］，相應(yīng)的能量稱為縱向平均場能；對于橫向部分（1/2）J，利用譜定理［5－6］做嚴(yán)格的計算，相應(yīng)的能量稱為橫向關(guān)聯(lián)能.對于簡單格子和復(fù)式格子都可以計算.既適用于鐵磁系統(tǒng)，也適用于反鐵磁系統(tǒng)和亞鐵磁系統(tǒng)，還適用于單壁磁性納米管.本文數(shù)值計算二維單離子各向異性海森伯鐵磁體和反鐵磁體的內(nèi)能.對于鐵磁體，包括準(zhǔn)一維的單壁鐵磁納米管，將數(shù)值結(jié)果與文獻(xiàn)［8－9］的結(jié)果做了比較.自旋量子數(shù)較小時（S＝1，3/2），接近零溫時，文獻(xiàn)［8－9］的內(nèi)能曲線更低.當(dāng)溫度升高時，本文的內(nèi)能曲線更低.自旋量子數(shù)較大時（S＝2，5/2，3），在接近零溫的低溫區(qū)，本文與文獻(xiàn)［8－9］的計算結(jié)果較為一致；溫度升高時，文獻(xiàn)［8－9］的內(nèi)能曲線較低.在接近居里點(diǎn)的高溫區(qū)，本文的內(nèi)能曲線較低.

1 模型、方法和計算公式

描述一個海森伯磁性體的哈密頓量為

其中，第一項是海森伯交換相互作用項，下標(biāo)i、j表示格點(diǎn)，Jij表示格點(diǎn)i、j之間的交換積分．本文只考慮最近鄰交換作用，求和遍及所有最近鄰格點(diǎn)，設(shè)最近鄰格點(diǎn)間的交換積分Jij＝J．J＜0表示鐵磁交換，J＞0則表示反鐵磁交換．最近鄰格點(diǎn)間的距離設(shè)為a．第二項是單離子各向異性項，D是各向異性強(qiáng)度，也稱為各向異性場.由于各向同性的海森伯模型在二維和一維情況下是沒有自發(fā)磁化的［10－11］，要加上各向異性項才會出現(xiàn)自發(fā)磁化，即低于三維情形時（1）式中的D必須不為零．一般認(rèn)為D比 |J| 小兩個數(shù)量級．第三項是有外磁場時的能量．本文設(shè)玻爾茲曼常數(shù)kB＝1，則交換強(qiáng)度J、各向異性強(qiáng)度D、外磁場Bz和溫度T等參量都是無量綱的量．本文的數(shù)值計算中?。簗J |＝100，D＝1，2，5．

采用多體格林函數(shù)法計算二維正方格子單離子各向異性海森伯鐵磁體（FM）、反鐵磁體（AFM）和單壁鐵磁納米管的內(nèi)能.要計算該磁性系統(tǒng)的內(nèi)能，需要先計算出磁化強(qiáng)度.計算磁化強(qiáng)度時，要對高階格林函數(shù)做分解近似，對于海森伯交換相互作用項和單離子各向異性項分別采用了無規(guī)項近似［5，15－16］（RPA）和Anderson?Callen（AC）分解近似［14，20－21］.內(nèi)能是哈密頓量的系綜平均值.整個晶體有N個格點(diǎn).考慮到晶格的平移周期性和對稱性，只需要計算出平均每個自旋的能量，以下簡稱鐵磁（或反鐵磁）能量，記為E，即

式（2）中的能量包含了四項，其中前兩項分別是縱向平均場能EMF和橫向關(guān)聯(lián)能ETC，后兩項分別是各向異性能EA和外磁場能EB.為了簡單起見，本文只計算不加外磁場（Bz＝0）時的內(nèi)能，令EB≡0.為了討論方便，我們在后面的數(shù)值計算中將縱向平均場能、各向異性能之和稱為經(jīng)典能量，計為EC＝EMF＋EA.用S表示自旋量子數(shù)，〈Sz〉表示鐵磁體的z分量磁化強(qiáng)度，〈Szμ〉（μ＝1，2）分別表示反鐵磁體兩個子晶格的z分量磁化強(qiáng)度.下面分別給出鐵磁、反鐵磁能量的計算公式.

1.1 鐵磁能量的計算公式

對于鐵磁系統(tǒng)，（2）式中的縱向平均場能為

其中，

式（2）中的橫向關(guān)聯(lián)能為

其中，

式（6）、（8）中的自旋波能譜為

這里波矢k＝（p，q），晶格常數(shù)為a.在方程（9）中，

式（2）中的各向異性能為

其中

1.2 反鐵磁能量的計算公式

對于反鐵磁系統(tǒng)，晶格分成了兩個子晶格（復(fù)式格子）.整個晶體有N個格點(diǎn)，其中兩個子晶格各有N/2個格點(diǎn).無外磁場時，兩個子晶格的自發(fā)磁化強(qiáng)度大小相等，方向相反，〈〉＝－〈〉＝－〈Sz〉．式（2）中的縱向平均場能為

其中

式（17）、（18）中的自旋波能譜分為兩支

其中

晶格常數(shù)b1＝b2＝2a．式（2）中的橫向關(guān)聯(lián)能為

其中

式（2）中的各向異性能為

其中

當(dāng)S＝1/2時，（Sz）2＝1/4，即使加上式（1）等號右邊的第二項，即單離子各向異性項，也不顯現(xiàn)各向異性［8－9，12］.因此，計算S＝1，3/2，2，5/2，3時的情況.

2 數(shù)值結(jié)果和討論

不加外磁場時，將鐵磁（或反鐵磁）能量分成了兩部分：經(jīng)典能量EC＝EMF＋EA和橫向關(guān)聯(lián)能ETC.圖1 （a）、（b）、（c）給出了幾個不同自旋量子數(shù)（S＝1，3/2，2，5/2，3）時鐵磁體、反鐵磁體的內(nèi)能隨溫度的變化，并將二者做了比較.其中，圖1（a）是鐵磁能量，圖1（b）是反鐵磁能量，圖1（c）將二者對比.可以看出，若自旋量子數(shù)一定，在零溫時，鐵磁能量高于反鐵磁能量.溫度升高，鐵磁能量和反鐵磁能量都上升.反鐵磁能量隨溫度上升的速度較鐵磁能量更快，當(dāng)溫度升高到居里點(diǎn)TC和奈爾點(diǎn)TN時（這里TC＝TN），鐵磁能量和反鐵磁能量相等.在零溫和相變點(diǎn)之間，包括零溫，反鐵磁能量總是低于鐵磁能量.

圖1 不同自旋量子數(shù)（S＝1，3/2，2，5/2，3）時：（a）鐵磁體（FM）、（b）反鐵磁體（AFM）的內(nèi)能隨溫度的變化；（c）鐵磁能量和反鐵磁能量；（d）本文和文獻(xiàn)［8］（原文圖4）的鐵磁能量Fig.1 Intrinsic energy versus temperature：（a）FM energy，（b）AFM energy，（c）FM and AFM energies，and（d）FM energies（Spin quantum number S＝1，3/2，2，5/2，3.）

此外，我們利用本文的近似方法計算了幾個不同自旋量子數(shù)（S＝1，3/2，2，5/2，3）和管徑［9］m＝20時鐵磁納米管（FM?nanotube）的內(nèi)能隨溫度的變化，并與文獻(xiàn)［9］的近似作了比較，如圖2所示.本文與文獻(xiàn)［9］結(jié)果差別的主要原因在于能量表達(dá)式中近似項的多少，本文的近似項數(shù)較少，得到的內(nèi)能數(shù)值更準(zhǔn)確一些.在這里需要指出和強(qiáng)調(diào)的一點(diǎn)是，本文計算內(nèi)能的近似方法對于簡單格子和復(fù)式格子都適用.既適用于鐵磁系統(tǒng)，也適用于反鐵磁系統(tǒng)和亞鐵磁系統(tǒng)，還適用于準(zhǔn)一維單壁磁性納米管.關(guān)于亞鐵磁體的內(nèi)能，將會在后續(xù)工作中做詳細(xì)計算.

圖3給出了自旋量子數(shù)S＝2時，鐵磁體和反鐵磁體的經(jīng)典能量、橫向關(guān)聯(lián)能和內(nèi)能隨溫度的變化.其中，圖3（a）是鐵磁體，圖3（b）是反鐵磁體.可以看出，經(jīng)典能量EC和內(nèi)能E都隨溫度的升高而升高，橫向關(guān)聯(lián)能ETC隨溫度的升高而降低．三者之間的關(guān)系是E＝EC＋ETC．在某一有限溫度處，橫向關(guān)聯(lián)能等于經(jīng)典能量.對于鐵磁體，零溫時，橫向關(guān)聯(lián)能為零，鐵磁能量和經(jīng)典能量相同.這是因為，在零溫時，不存在熱力學(xué)漲落，對于鐵磁系統(tǒng)，也不存在量子漲落.對于反鐵磁系統(tǒng)，情況大不一樣，零溫時，橫向關(guān)聯(lián)能不為零，鐵磁能量低于經(jīng)典能量.這是因為，在零溫時，反鐵磁系統(tǒng)存在量子漲落.溫度升高，量子漲落和熱力學(xué)漲落效應(yīng)出現(xiàn)并逐漸增強(qiáng)，磁化強(qiáng)度減小，鐵磁能量和反鐵磁能量上升.溫度升高到相變點(diǎn)時，對于鐵磁系統(tǒng)和反鐵磁系統(tǒng)，經(jīng)典能量都非常接近零，鐵磁能量和反鐵磁能量分別等于相應(yīng)的橫向關(guān)聯(lián)能.在零溫和相變點(diǎn)之間，鐵磁能量、反鐵磁能量總是低于相應(yīng)的經(jīng)典能量，這表明橫向關(guān)聯(lián)效應(yīng)是很重要的，不能忽略.此外，圖3表明，在居里點(diǎn)TC和奈爾點(diǎn)TN，鐵磁能量和反鐵磁能量的數(shù)值都不為零.這是因為，在鐵磁系統(tǒng)和反鐵磁系統(tǒng)中，由于橫向關(guān)聯(lián)效應(yīng)，體系具有某種程度的短程序，這樣，在居里點(diǎn)TC和奈爾點(diǎn)TN，即使磁化強(qiáng)度為零，鐵磁能量和反鐵磁能量也是非零的.

圖2 不同自旋量子數(shù)（S＝1，3/2，2，5/2，3）時鐵磁納米管的內(nèi)能隨溫度的變化Fig.2 Intrinsic energy of FM nannotubes versustemperature at several S

圖3 （a）鐵磁體（FM）、（b）反鐵磁體（AFM）的經(jīng)典能量、橫向關(guān)聯(lián)能和內(nèi)能隨溫度的變化Fig.3 Longitudinalmean?field energy，transverse correlation energy，and intrinsic energy versus temperature：（a）FM，（b）AFM

圖4 不同自旋量子數(shù)（S＝1，3/2，2，5/2，3）時：（a）鐵磁體（FM）、（b）反鐵磁體（AFM）的橫向關(guān)聯(lián)能隨溫度的變化Fig.4 Transverse correlation energy versus temperature at several S：（a）FM and（b）AFM

圖5 不同各向異性強(qiáng)度（D＝1，2，5）時鐵磁體（FM）、反鐵磁體（AFM）的橫向關(guān)聯(lián)能隨溫度的變化Fig.5 Transverse correlation energy versus temperature （a）FM，（b）AFM

圖5給出了幾個不同各向異性強(qiáng)度（D＝1，2，5）時鐵磁體和反鐵磁體的橫向關(guān)聯(lián)能隨溫度的變化.參量都相同時，在零溫和相變點(diǎn)之間，包括零溫，反鐵磁體比鐵磁體的橫向關(guān)聯(lián)能更低.在相同的參量下，鐵磁體的橫向關(guān)聯(lián)能隨溫度減小的更快，這樣到達(dá)相變點(diǎn)（TC＝TN）時，鐵磁體和反鐵磁體的橫向關(guān)聯(lián)能相同.可以看出，對于鐵磁體在有限溫度下和反鐵磁體在全溫區(qū)，橫向關(guān)聯(lián)能的絕對值隨著各向異性強(qiáng)度D的增大而減小.

本文雖然沒有計算加外磁場時的內(nèi)能，從哈密頓量可以看出，加上外磁場后能量會降低.我們可以推測出，對于鐵磁體在有限溫度下和反鐵磁體在全溫區(qū)，橫向關(guān)聯(lián)能的絕對值隨著外磁場Bz的增大而減小.

3 結(jié)論

采用量子統(tǒng)計理論的多體格林函數(shù)法計算了二維單離子各向異性海森伯磁性體的經(jīng)典能量、橫向關(guān)聯(lián)能和內(nèi)能，將鐵磁體和反鐵磁體的結(jié)果作了比較.對于鐵磁體的內(nèi)能，包括準(zhǔn)一維的單壁鐵磁納米管，將本文的計算結(jié)果與文獻(xiàn)［8－9］的結(jié)果做了比較.主要結(jié)論如下：

1）溫度升高，鐵磁能量和反鐵磁能量都上升.參量相同時，在零溫和相變點(diǎn)之間，包括零溫，反鐵磁能量總是低于鐵磁能量.由于反鐵磁能量隨溫度上升的速度較鐵磁能量更快，當(dāng)溫度升高到居里點(diǎn)TC和奈爾點(diǎn)TN時（TC＝TN），鐵磁能量和反鐵磁能量相等.

2）對于鐵磁體的內(nèi)能，包括單壁鐵磁納米管，我們將本文的數(shù)值結(jié)果與之前文獻(xiàn)的結(jié)果做了比較.本文與文獻(xiàn)［8－9］結(jié)果差別的主要原因在于能量表達(dá)式中近似項的多少.由于本文的近似項數(shù)較少，因此得到的內(nèi)能數(shù)值更準(zhǔn)確.

3）橫向關(guān)聯(lián)效應(yīng)對系統(tǒng)內(nèi)能的影響較大，不能忽略.對于鐵磁體在有限溫下和反鐵磁體在全溫區(qū)，橫向關(guān)聯(lián)能（數(shù)值為負(fù)）的絕對值隨著自旋量子數(shù)S、各向異性場D和外磁場Bz的增大而減小.

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Article ID：1001?246X（2015）01?0093?08

Abstract： Magnetic properties of single?wall ZnSnanotubes（NTs）doped with Fe atoms are studied with first?principles calculations. Formation energies of doped NTs are smaller than that of pristine one，which indicating that doping is an exothermic reaction. Monodoped NTs has atom?likemagnetic momentsmainly due to 3d componentof Fe atoms.It indicates that Fe?doped ZnSNTs tend to adopt antiferromagnetic（AFM）configurations.To obtain room temperature ferromagnetism，we replaced an S atom by a C atom and found that C atom prefers to substitute Satom connecting two Fe atoms.Ferromagnetic（FM）state energy is lower than that of AFM state by 164 meV.It implies that room temperature ferromagnetism is expected in these systems.

Key words： codoping；nanotube；magnetic properties；density functional theory

0 Introduction

ZnS is an important II?VIcompound semiconductorwith potentialapplications in electronics and optoelectronics due to itswide direct band gaps，3.77 eV for the wurtzite structure［1］and 3.72 eV for the zinc?blende structure［2］.Nanodimensional ZnS is found to exhibit excellent optical and optoelectronic properties that are remarkably different from the bulk［3－5］，making it a versatile building block for nanoscale electronic and photonic devices.

Over the past few years，considerable efforts have been placed on the synthesis of ZnS tubular structures［6－10］.Magnetic NTs are interesting and technologically important objects of physical researches with many promising applications of future generations of nanoelectronic devices［11－13］. Previous computations have made on magnetic properties of Mn?doped ZnO NTs［14］and transition atom（TM）doped GaN NTs［15］.Several theoretical researches have made on the magnetic properties of Mn?doped ZnS surfaces［16－17］，TM doped ZnSbulk materials［18－20］and ZnSNWs［21］. Our previous work on ZnS NTs focused on the magnetic property of monodoped ZnS NTs［22］.To obtained room temperature ferromagnetism，interaction between two TM atomsmust be considered.

In thiswork，we carried outa systematic computational study on Fe?doped ZnSNTs.The paper is organized as follows.We first describe computational details in Sec.1 and then present results and discussions in Sec.2.Finally，conclusions are given in Sec.3.

1 Theoreticalmethod and com putational details

The calculations were performed by using spin?polarized density functional theory（DFT）asimplemented in the DMOL package［23－24］.All electron treatments and double numerical basis set including d?polarization functions（DND）were chosen when thismethod was used.The exchange?correlation interaction was treated using generalized gradient approximation（GGA）with the functional parameterized by Perdew?Burke?Ernzerhof correction（PBE）［25］.SCF calculations were performed with a convergence criterion of 10－6hartree on total energy.All structures were fully optimized with no symmetry constraint，with a convergence criterion of0.002 hartree·?－1for forces and 0.005?for displacement.Mulliken population analysis［26］was performed to determine charge transfer and magneticmoment on each atomic site.

2 Results and discussions

2.1 Pristine ZnS tube

The pristine ZnSnanotube（P?ZnS）with a 1.00 nm diameter is cut from a bulk ZnSalong the ［0001］direction.The relaxed structures are shown in Fig.1.Distinct surface relaxation occurred on the facets after geometry optimization.Zn atoms move inward whereas S atoms move outward，forming a buckle of Zn?S dimer.

Fig.1 （a）Top?view and（b）side view of P?ZnSNT（The small and big balls represent Zn and Satoms，respectively.）

2.2 Femonodoped ZnS tube

Since Fe atoms adopt divalent ionic states，they substitute readily for divalent cations. Therefore，only substitutional doping was considered.This scenario has been confirmed by previous experiments［27－28］.To achieve realistic experimental doping concentrations（approximately 2% －3%），a super cell that consisted of 36 Zn and 36 S atoms was used，in which one Zn atom is replaced by a Fe atom，named M?ZnS.This corresponds to a doping concentration of 2.8%.A structural optimization was subsequently performed for P?ZnS.The electronic properties of P?ZnS had been calculated in our previous work［22］.It has been shown that P?ZnS is a direct band gap semiconductor with a band gap（3.52 eV）larger than that of the bulk ZnS（2.23 eV）due to quantum confinement effects.M?ZnS was further optimized in this investigation.The optimized structures are plotted in Fig.2（a）.We observedminor changes in lattice constants and bond lengths after structural optimization due to the different ionic radii of Zn and Fe atoms.This indicates that Fe atom doping does not change the crystal structure of ZnS NTs，which is in agreement with experimental results［27－28］.

Fig.2 （a）Side?view of Femonodoped and（b）to（d）bidoped ZnSNTs（Small，middle，and big balls represent Zn，S，and Fe atom，respectively.）

To compare energy stability of doped ZnSNTs，the formation energy Efwas calculated Ef＝Etot－EP－ZnS－n1EFe－n2EC＋n3EZn＋n4ES，where Etot，EP－ZnS，EFe，EC，EZn，and ESare the total energy of doped NT，the energy of P?ZnS，the energy of Fe，C，Zn，and Satom，respectively.n1，n2，n3and n4are numbers of Fe，C，Zn，and S atoms in the doped NTs，respectively.It is known that themore negative the Efis，themore stable a NTwould be.The calculated results are shown in Table 1.The formation energy of doped NT is lower than thatof P?ZnSNT，which indicated that the Fe?doped NT is exothermic.

Table 1 Formation energy Ef，band gap Ep，local charge QFeand localmagnetic momentμFeof Fe atom，and total magnetic m omentμtotof M?ZnS，magnetic moments contributed by the nearest neighboring S and Zn atom sμS，μZn

Electronic properties of M?ZnSwere calculated.The band gap is shown in Table 1.Clearly，the band gap of M?ZnS is smaller than that of P?ZnS（3.52 eV）.The band structures of M?ZnS alongΓ?Z direction are shown in Fig.3.It can be seen that certain localized states existed near Fermi levels，which would make the band gap narrower than that of P?ZnS.In addition，M?ZnS shows semiconducting characteristic with indirect band gap.These indicate that Fe?doping changes significantly electronic properties of P?ZnS.

Fig.3 Band structures ofmonodoped ZnSNTs

We performed Mulliken population analysis to determine charge transfer and magnetic moment on each atomic site.Themagnetic properties of doped NT are presented in Table 1.3d electrons of Fe atom followed Hund's rule and maximized the magnetic moment.Our results indicate that the magneticmoment is very close to those of free atom，which suggesting that the defect behaved like an isolated Fe atom at this doping level.This aspect of DMS has been observed previously and explained alongwith the large ferromagnetic exchange splitting of the impurity's defectband［29－30］.

An important quantity characterizing the delocalization ofmagnetic moments around Fe atom is induced magnetic moments in surrounding atoms of the host semiconductor.The hybridization between Fe and S atoms plays an important role in the formation of induced magnetic moments. Their magnetic moments have same direction，indicating that Fe atom induces FM interactions between surrounding S atoms（Table 1）.Substitutionally doping a Fe atom at a Zn site in ZnSNT changes the number of spin?majority or spin?minority states in the valence band of ZnS NT.The spin?majority states of Fe atom are occupied，and the spin?minority states are partially occupied. Thus，the spin?majority states in S atoms becomemore occupied than the spin?minority states，and the induced magneticmoments in them are parallel to those in Fe atom.Zn atoms interacte with S atoms in the same way as Fe atoms.

2.3 Fe bidoped ZnS tube

In this section，two Zn atoms in a super cell substituted by two Fe atoms is studied formagnetic interaction.We considered three possible configurationswith different spatial positioning of Fe atoms （B1，B2，and B3），respectively.Each was fully relaxed before the magnetic moments were calculated.The optimized structures are plotted in Figs.2（b）－2（d）.The distances between two Fe atoms are listed in Table 2.The formation energy and the relative energyΔEr＝EFM（AFM）－Egroundstatebetween the AFM （FM）states and the ground state are presented in Table 2.Energy differencesΔE＝（EFM－EAFM）between AFM and FM states are also listed in Table 2.

Magnetic properties of all bidoped structures were computed and presented in Table 3.Again，F(xiàn)e atoms induced FM interactions between surrounding Satoms.

For isomers B3 with a large distance over 5?，AFM and FM states are degenerate in energy，and have same geometry，local charge，and magnetic moment except that the AFM state has a slightly larger band gap.It means that magnetic coupling between two Fe atoms is short?ranged，which has been confirmed by previous researches［31－34］.

Table 2 Distances between two Fe atom s dFe，formation energy Ef，band gap Ep，relative stabilityΔEr，and relative energiesΔE＝（EFM－EAFM）of Fe?bidoped ZnSNTs

Table 3 Local charge QFe1and QFe2and magnetic mom entμFe1andμFe2of two Fe atom s，magneticmoment of thenearest neighboring S atomsμS，and totalmagnetic momentμtot，of Fe?bidoped ZnSNTs

The lowest energy configuration B1 is an AFM state where Fe atoms replace two nearest Zn atoms，which implies that Fe atoms have a tendency to form clusters around Ssites.For isomers B1 and B2，the states with AFM coupling are lower in energy than the states with FM coupling.It indicates that Fe?doped ZnS NTs tends to adopt AFM configuration.How can we obtain FM coulping？

2.4 Fe/C codoped ZnS tube

Additional hole doping may further promote ferromagnetism.A similar idea was proposed by Huang et al［35］who showed that an acceptor like N codoping in Mn doped ZnSe nanocrystal can induce ferromagnetism.This is because the holesmediate Mn?Mn spins.Sharma et al［36］found that C codoping in Mn doped ZnO has room temperature ferromagnetism due to carriers introduced by oxygen vacancies and the substitution of C atO sites.Here，we replaced a Satom by a C atom in Fe doped ZnSNTs.We only considered the configuration that C atom replace a S atom between Fe atoms，named CB1 and CB2.The optimized structures are plotted in Fig.4.After relaxation，the local structure around C dopant shrinks slightly，with Fe atomsmoving closer to C atom.

Fig.4 Side?view of Fe/C codoped ZnSNTs（The small，middle，big，and huge balls represent S，Zn，F(xiàn)e，and C atom，respectively.）

Calculated formation energies，relative energies，and energy differences between AFM and FM states are listed in Table 4.After C codoping，all doped NTs show FM coupling.For CB1 configuration，the formation energy of FM state is lower than that of AFM state by 164 meV.Such energy differences imply that room temperature ferromagnetism may be expected in these systems.

Table 4 Totalmagnetic momentsμtot，local charges QFe1，QFe2and localmagnetic momentsμFe1，μFe2andμCof Fe and C atoms，formation energies Ef，relative energiesΔEr，energy differencesΔE of Fe/C codoped ZnSNTs

Mulliken population analysis for charge transfer and magnetic moment on each atomic site of Fe/C codoped ZnS NTs are presented in Table 4.Due to hybridization with C atom，the local magnetic moments of Fe atoms decrease.For CB1 and CB2 configurations，totalmagnetic moments are only 4μBfor FM states.The localmagnetic moment of C atom is always antiparallel to those of neighboring Fe atoms.

3 Conclusions

Structural，electronic，and magnetic properties of ZnS NTs doped with Fe atoms were systematically studied by using first?principles calculations.The formation energies of doped NTs are smaller than that of P?ZnSNT，which indicating that the process for forming the Fe?doped NTs is exothermic.Mulliken population analysis shows that all monodoped ZnS NTs have atom?like magnetic moments，which ismainly contributed to the 3d component of Fe atoms.The hybridization between Fe atoms and S atoms playes an important role in the formation of induced magnetic moments.Fe atoms induced FM interactions between surrounding S atoms of the host semiconductor.To obtain ferromagnetism，we replaced a Satom by a C atom.The FM state energies are lower than those of AFM states by 0.164 eV.Such energy differences imply that room temperature ferromagnetism could be expected in these systems.

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Intrinsic Energies of Two?dimensional Heisenberg M agnets and Ferromagnetic Single?walled Nanotubes

MIBinzhou1，F(xiàn)ENG Cuiju1，QIYunping2，DING Dong1，XUE Yonghong1
（1.Department of Basic Curriculum，North China Institute ofScience and Technology，Beijing 101601，China；2.College of Physics and Electronic Engineering，Northwest Normal University，Lanzhou 730070，China）

Intrinsic energy of two?dimensional square lattice single?ion anisotropic Heisenberg ferromagnets，antiferromagnets，and ferromagnetic single?walled nanotubes are calculated withmany?body Green's functionmethod in quantum statistical theory.Calculated results of ferromagnets and antiferromagnets are compared.Between zero temperature and phase transition point，including zero temperature，anti?ferromagnetic energy is always lower than that of ferromagnetic energy.Calculationmethod of intrinsic energy in this paper is applicable not only to ferromagnetic system，butalso suitable for antiferromagnetic system and ferrimagnetic system，aswellas ferromagnetic single?walled nanotubes.Intrinsic energies are greatly lower than classical energies，which shows that transverse correlation effect is important.

Heisenbergmodel；two?dimensional systems；magnetic nanotubes；intrinsic energy；many?body Green's functionmethod

M agnetic Properties of Single?wall ZnS Nanotubes Doped w ith Fe Atom s

XIE Jianming，CHEN Hongxia

（College of Physical Science and Electronic Techniques，Yancheng Teachers University，Yancheng 224002，China）

O469 Document code：A

O469

A

2013－12－28；

2014－06－03

收稿日期：2014－01－06；修回日期：2014－06－17

中央高?；究蒲袠I(yè)務(wù)費(fèi)專項資金（3142012017）、河北省教育廳科技基金（A2013024）和河北省高等學(xué)?？茖W(xué)技術(shù)研究項目（QN2014330）資助項目

米斌周（1980－），男，碩士，講師，博士生，主要利用多體格林函數(shù)法進(jìn)行磁性材料的理論計算，E?mail：mbzfjerry2008＠126.com

Received date： 2014－01－06；Revised date： 2014－06－17

Received date：2013－12－28；Revised date：2014－06－03

Foundation item s：Supported by Natural Science Foundation of China（11247235，11404279）and Qinglan Project of Jiangsu Province

Biography：Xie Jianming（1976－），male，major in first?principles calculations of structure and property ofmaterials，E?mail：dtxiejianming＠sina.com