BaTi2Bi2O的電子結構與磁性

王廣濤，張 琳，張會平，劉 暢

（河南師范大學物理與電子工程學院，河南新鄉(xiāng) 453007）

文章編號：1001?246X（2015）01?0107?08

BaTi2Bi2O的電子結構與磁性

王廣濤，張 琳，張會平，劉 暢

（河南師范大學物理與電子工程學院，河南新鄉(xiāng) 453007）

采用第一性原理方法，對BaTi2Bi2O的電子結構和磁性進行計算.非磁性態(tài)的計算結果顯示：費米能級處的態(tài)密度主要來自dz2，dx2－y2和dxy三個軌道，同時費米面也主要有三部分組成，并且將其沿著矢量q1＝（π/a，0，0）和q2＝（0，π/a，0）平移時，第三部分費米面（沿著X?R連線）與第一部分費米面（M?A連線）嵌套明顯，計算得出磁化系數(shù)χ0（q）在X點出現(xiàn)峰值，與峰值出現(xiàn)在M點的FeAs基超導體不同.上述磁化率峰值可以誘導產(chǎn)生自旋密度波，使得BaTi2Bi2O材料的磁性基態(tài)是bi?collinear antiferromagnetism（AF3）與blocked checkerboard antiferromagnetism （AF4）的二度簡并態(tài).隨著空穴摻雜，χ0（q）的峰值降低，而電子摻雜則導致峰值變大.當自旋漲落被完全壓制時，超導出現(xiàn)，這可以解釋為什么超導只出現(xiàn)在空穴摻雜型化合物而非電子摻雜型.

第一性原理；費米面嵌套；超導

0 引言

高溫鐵基超導體［1］的發(fā)現(xiàn)，引發(fā)了一輪新的探尋層狀超導體（SC）的熱潮，研究者旨在探尋那些與之有相似電子結構的材料，如LaCo2B2化合物［2］和BiS2基化合物［3］.在探索新的超導體系的過程中，最理想的是像鐵基超導體［1，4］那樣，具有自旋密度波（SDW）不穩(wěn)定性或者電荷密度波（CDW）不穩(wěn)定性（由費米面嵌套導致）的層狀結構材料.這是由于在結晶材料中超導態(tài)和自旋密度波態(tài)是兩個相互關聯(lián)又相互競爭的集合電子現(xiàn)象.自旋密度波態(tài)通常出現(xiàn)在二維體系中，該體系中存在費米面嵌套導致的自旋或電荷調制［5］現(xiàn)象.超導也可以被看做是另一種基于庫伯配對效應的費米面不穩(wěn)定性.超導態(tài)與自旋密度波態(tài)之間，通常是相互競爭又相互關聯(lián)的［6］.

近來Yajima等人［7］在BaTi2Bi2O材料中發(fā)現(xiàn)了超導（T c＝1.2 K），并且其在50 K時發(fā)生類似于自旋密度波的相變.Doan等人［8］進一步發(fā)現(xiàn)，通過用鈉原子替代Ba原子而進行空穴摻雜，其超導轉變溫度T c可以升高至5.5 K，同時自旋密度波轉變溫度降低至30 K左右.截至目前，理論與實驗結果均顯示Na2Ti2Sb （As）2O［9－11］、BaTi2As2O［12，16］、Ba2Ti2Fe2As4O［12］和 BaTi2Sb2O［13－14］中可能存在自旋密度波，通過觀察Na2Ti2Sb2O的光譜學結果［11］，以及BaTi2Sb2O的電聲耦合的結果［13］，人們已經(jīng)證實了電荷密度波相轉變的存在，以上這些發(fā)現(xiàn)都需要我們對SDW以及它與SC之間的關系作進一步的探究.本文用第一性原理研究Ba（1－x）NaxTi2Bi2O的電子結構、磁性和費米面嵌套情況.

1 研究方法

圖1 BaTi2Bi2O非磁性態(tài)的晶體結構Fig.1 Structure of BaTi2Bi2O

計算采用基于超軟贗勢和平面波方法的STATE（Simulational Tool for Atom TEchnology）軟件包［21］.選用的晶格參數(shù)和原子位置都來自于XRD數(shù)據(jù)［15］，晶體結構如圖1，其中晶格常數(shù)為a＝b＝4.123 2?，c ＝8.344 7?，原子內部坐標為Ba（1d）：0.5，0.5，0.5；Ti（2 f）：0.5，0，0；Bi（2g）：0，0，0.748 7；O（1c）：0.5，0.5，0.為了探索材料真正的磁性基態(tài)，我們討論六種不同的情況：非磁態(tài)（即無自旋極化態(tài)）、鐵磁態(tài)以及四種不同的反鐵磁態(tài).對于這種磁性態(tài)，我們沒有優(yōu)化其結構，而是用用實驗得到的非磁態(tài)結構進行的計算.第一種反鐵磁類型，其最近鄰的Ti原子自旋方向相反（如圖2（a）），我們稱為 checkerboard型反鐵磁（AFM1）；第二種反鐵磁類型中，由Ti原子構成的正方形點陣的對角線上兩原子自旋方向均相反（如圖2（b）），我們稱之為collinear型反鐵磁（AF2）；第三種反鐵磁沿著由Ti原子構成的正方形點陣的一條對角線方向是鐵磁態(tài)，而另一個對角線方向是反鐵磁態(tài)（如圖2（c）），稱為bi?collinear［17］型反鐵磁（AF3）；第四種反鐵磁如圖2（d），Ti2Bi2O層被淺色窄虛線分成2×2的正方形點陣，其內部磁性態(tài)一致，相鄰正方形內磁性態(tài)相反，并且其磁性單胞在ab面是一個2a×2a的正方形點陣，我們稱這種反鐵磁類型為blocked checkerboard型反鐵磁（AF4）.為了保證體系收斂，我們采用340 eV的截斷能和20×20×10的K點網(wǎng)格數(shù)，用以優(yōu)化交換關聯(lián)能的廣義梯度近似（GGA）［23］以PBE的形式應用于我們的計算中，Ti?3d態(tài)的電子關聯(lián)性采用GGA＋U［18］的方法處理.

圖2 Ti2Bi2O層俯視圖（a）AF1，（b）AF2，（c）AF3，（d）AF4（其中虛線框代表ab面上不同反鐵磁構型的磁單胞，不同原子種類和Ti原子磁性方向已在圖中標出.圖（a）中的J1和J2分別表示緊鄰和次近鄰交換常數(shù).）Fig.2 Schematic top view of Ti2Bi2O layer（a）AF1，（b）AF2，（c）AF3，（d）AF4 （The dashed lines correspond tomagnetic unit cells of the different antiferromagnetic configurations. Atoms and directions of Timagneticmoments are denoted in the figure.）

2 結果與討論

圖3描繪了由GGA方法計算所得的BaTi2Bi2O非磁態(tài)的總的態(tài)密度（DOS）和分態(tài)密度，從圖3（a）中我們可以看出每個化學單胞的費米能級處的態(tài)密度為N（EF）＝3.42 state·eV－1，通過費米能級處的態(tài)密度可以計算得出，相應磁化率為：χ0（q）＝1.1×10－4emu·mol－1，比熱系數(shù)為：γ0＝8.2 mJ·（K2·mol）－1，這與Suetin［16］的研究結果一致（χ0（q）＝1.1×10－4emu·mol－1，γ0＝8.06 mJ·（K2·mol）－1）.另外由圖可以看出，O?2p軌道的電子主要分布在從－7.4 eV到－4.7 eV的能量區(qū)域，從－5 eV到－1 eV的能量范圍內，Bi?6p軌道的電子對態(tài)密度的貢獻最大，Ti?3d軌道的電子形成的能帶主要位于－4 eV～－1 eV之間，在費米能級處（由－1 eV到1 eV）主要分布著Ti?3d軌道的電子和與之有著強烈雜化的Bi?6p態(tài)電子，它們對于費米面的貢獻分別是83%和17%.為了研究軌道特征，圖3（b）給出了Ti原子3d態(tài)的5個分態(tài)密度，該圖顯示費米能級處的態(tài)密度主要是由Ti原子3d態(tài)的dz2，dx2－y2和dxy軌道電子貢獻的，這一點與LaOFeAs［19］材料不同，后者費米能級處主要分布著dxy，dyz和dzx軌道的電子.它與Fe?As基超導體［24］不同的軌道特征，預示著兩者的超導電子配對也是不同的.

圖3 （a）BaTi2Bi2O的態(tài)密度圖及（b）Ti?3d態(tài)的投影態(tài)密度圖Fig.3 （a）DOS of BaTi2Bi2O and（b）projected density of state of Ti?3d

從投影能帶結構圖（圖4）可以看出，有兩條能帶穿越費米面，與BaTi2Sb2O［14］的軌道特征很相似，該圖中穿越費米面的這兩條能帶是由Ti原子的dz2，dx2－y2和dxy態(tài)電子構成的，這點與Ti的投影態(tài)密度圖（圖3）吻合.圖5描繪了三維費米面及在KZ＝0，π/2，π/3處截取的費米面橫截面.費米面有三部分構成，第一部分是沿著M?A連線（位于布里淵區(qū)的四個角）.這一部分費米面顯示出良好的二維特性，而圖4（a）?（c）的投影能帶結構圖顯示，這一部分的費米面主要是由Ti原子3d態(tài)中dz2，dx2－y2和dxy軌道的電子構成.第二部分位于布里淵區(qū)的中心位置（沿著Γ?Z連線），這部分費米面擁有復雜的三維形狀，結合圖4（a）可知其有明顯的dz2軌道特征.此外費米面的第一部分與第二部分來自于同一能帶.在費米面包括的區(qū)域內，載流子數(shù)量分別為0.248 electrons·cell－1和0.248 holes·cell－1，電子與空穴的濃度均為1.70×1021·cm－3.圍繞著X點的第三部分屬空穴型費米面，由于載流子中電子空穴各占一半，這部分費米面正好補償了電子型費米面部分.與電子型費米面（第一部分費米面）相同，這部分費米面擁有dz2，dx2－y2和dxy的混合特征，其中dz2軌道特性尤為顯著.

為了觀察費米面的分布及其嵌套情況，我們分別截取了KZ＝0、π/2、π/3時的費米面（圖5）.我們將費米面沿著矢量q1＝（π/a，0，0）或者q2＝（0，π/a，0）平移后，位于N點（0，π/a，0）的第三部分費米面（空穴型）與位于M點（π/a，π/a，0）的第一部分費米面（電子型）發(fā)生強烈嵌套（圖5（c））.如果費米面在某個晶格震動矢量q的平移下（這不是一個物理過程，是一個理論上假想的平移）能夠與原先的費米面基本重合，則這兩個電子態(tài)就會受到該晶格震動模式的散射，發(fā)生簡并微擾，同時打開能隙.而一旦能隙因此打開，能隙底部的態(tài)往往都是駐波態(tài)，并且具有相同的波動周期（因為它們是兩個被晶格動量向連接的態(tài)的線性組合，這種組合的波函數(shù)在振幅上的波動必然與晶格的波動吻合）.由于費米面上的態(tài)都是能級簡并的，大量這樣的電子態(tài)被費米子占據(jù)以后整個費米液體就會體現(xiàn)出密度波動，這就強化最初造成密度波動的漲落.這種正反饋就是費米面失穩(wěn)的根本機制.先前對Fe基超導體的研究已經(jīng)證實了：費米面的嵌套是導致SDW態(tài)主要原因［4，20］.

圖4 BaTi2Bi2O的投影能帶圖（圖中符號大小對應于布洛赫態(tài)在dz2、dx2－y2、dxy、dyz和dzx軌道上的投影比重.）Fig.4 Projected bands of BaTi2Bi2O（Symbol sizes correspond to projected weights of Bloch states onto dz2，dx2－y2，dxy，dyzand dzxorbitals.）

費米面嵌套效應可以由林哈德效應函數(shù)量化表示出來.對于無摻雜化合物，χ0（q）的峰值出現(xiàn)在X點（如圖6（a）），強嵌套效應的存在暗示某種有序（比如SDW或者CDW［4，20］）可能在低溫無摻雜化合物（如LaOFeAs［4，20］）中形成.我們通過將費米面上移0.2 eV來研究0.3個電子摻雜型的化合物的磁化率，同理，通過下移0.1 eV來研究0.3個空穴摻雜型化合物的χ0（q）.對化合物進行0.3個空穴摻雜時（Ba0.7Na0.3Ti2Bi2O，圖6（b）），χ0（q）的峰值降低，并且位置由X點向Γ點方向移動.這種現(xiàn)象曾在Fe?As基超導材料［24］中報導，其磁化率峰值（在M點）隨著空穴摻雜明顯降低并且變得有些不對稱.然而對化合物進行0.3個電子摻雜時，χ0（q）卻升高了，這與Fe?As基超導材料完全不同：首先，后者的磁化率峰值在非摻雜情況下出現(xiàn)在M點（并非X點），隨著電子或空穴的摻雜明顯降低并且略有不對稱［24］.我們的結果暗示，只有空穴摻雜才能將那些可以誘導SDW的費米面嵌套效應顯著的壓制.當SDW被完全壓制時，超導便出現(xiàn)了.這些正好可以解釋，為什么超導只能出現(xiàn)在空穴摻雜型化合物里而非電子摻雜型中.

圖5 （a）三維費米面圖及費米面在（b）KZ＝0、（c）π/2、（d）π/3處的橫截面（實線是最初計算所得的費米面，虛線表示沿著矢量q＝（π，0，0）平移之后的費米面.）Fig.5 （a）3D Fermi surfaces and cross?sections of Fermi surfaces（2D）at（b）KZ＝0，（c）KZ＝π/2，and（d）KZ＝π/3 （Black lines are calculated Fermi surfaces，while greg lines indicate Fermi surfaces shifted by q＝（π，0，0）.）

強烈的嵌套效應會誘導產(chǎn)生自旋密度波，所以我們分別研究了材料的鐵磁態(tài)和四種不同的反鐵磁構型.早前的研究顯示，Ti氧化物的電子關聯(lián)性很重要［21］，所以也考慮了 Ti?3d態(tài)的電子強關聯(lián)性（GGA＋U［22－23］）.加U值從0到3 eV（如圖7（b）），材料鐵磁態(tài)的Ti原子磁矩均為0，由此知其與NM態(tài)簡并.不加U時，AF1態(tài)、AF2態(tài)都和FM態(tài)（或者NM態(tài)）有相同的總能量（如圖7（a））.同時以上3個態(tài)的Ti原子的磁矩均為0（如圖7（b））.這就意味著在不考慮Ti原子電子關聯(lián)性的情況下，F(xiàn)M態(tài)、AF1態(tài)和AF2態(tài)都與非自旋極化態(tài)簡并.這個結果與Sing等人［20］關于BaTi2Sb2O的研究結果相同，后者發(fā)現(xiàn)以上各態(tài)都是不穩(wěn)定的.另一方面，我們的研究結果與Lu等人［10］關于NaTi2Sb2O的結論（以上三個態(tài)均與NM態(tài)簡并）也一致.對于AF3態(tài)和AF4態(tài)來說，與Fe?As基超導材料［1，20］的2μB相比，前者Ti原子磁矩小至0.13μB（Ueff＝0 E）. AF1態(tài)、AF3態(tài)和AF4態(tài)的能量隨著加U而升高，其中AF3態(tài)和AF4態(tài)的Ti原子的磁矩由0.13μB（Ueff＝0 eV）升高至0.82μB（Ueff＝3 eV）.正如上面所提到的，體系存在兩個相互獨立且相互正交的等效嵌套矢量q1＝（π/a，0，0）和q2＝（0，π/a，0）.可能存在的自旋密度波將會在Ti?Ti正方形點陣內以M cos（q·R）的形式振蕩，其中，R和M分別代表格點矢量和磁矢量.對應波矢量q＝q1＋q2和q1（q2），我們最終發(fā)現(xiàn)兩個穩(wěn)定的磁有序態(tài)即blocked checkerboard antiferromagnetic有序和bi?collinear antiferromagnetic有序.根據(jù)海森堡模型我們可以由這種磁性態(tài)的能量求得近鄰和次近鄰交換常數(shù)（如圖7（c））.由交換常數(shù)均為負數(shù)，可以說明體系為什么更傾向于呈反鐵磁態(tài).

圖6 計算所得的各個磁化系數(shù)χ0（q）圖：（a）無摻雜情況、（b）0.3空穴摻雜情況、（c）0.3電子摻雜情況以及（d）三種摻雜情況下沿Γ－X線的情形Fig.6 Calculated bare susceptibility（Lindhard response function）χ0（q）at KZ＝0，（a）without doping，（b）with 0.3 hole?doping，（c）with 0.3 electron?doping and（d）they alongΓ－X line

圖7 （a）四個反鐵磁態(tài)（AFM1、AFM2、AFM3、AFM4）相對于FM態(tài)的總能量，（b）隨著加U增加，五種構型中Ti原子磁矩的變化；（c）近鄰（J1）和次近鄰（J2）交換常數(shù)隨U變化Fig.7 （a）Total energies（with respect to FM state）of the checkerboard AF（AF1），collinear AF（AF2），bi?collinear AF（AF3），and blocked checkerboard AF（AF4），（b）Moments of Ti atoms in magnetic configurations as functions of electron correlation U，（c）First and second nearest neighbor exchange coupling constants as functions of U

3 結論

通過第一性原理計算研究了BaTi2Bi2O材料的電子結構和磁性.在非自旋極化態(tài)構型中，費米能級處的態(tài)密度主要由dz2，dx2－y2和dxy軌道的電子構成，這一點與Fe?As基超導材料不同.計算所得磁化率χ0（q）的峰值出現(xiàn)在X點，不像Fe?As基超導材料那樣出現(xiàn)在M點.空穴摻雜使得χ0（q）的峰值降低并且略微有些不對稱，而電子摻雜則使得該峰值升高.當SDW被完全壓制時，超導便出現(xiàn)了.這些正好可以解釋，為什么超導只能出現(xiàn)在空穴摻雜型化合物里而非電子摻雜型中.上述磁化率峰值誘導產(chǎn)生自旋密度波，使得BaTi2Bi2O材料的磁性基態(tài)是bi?collinear antiferromagnetism（AF3）與blocked checkerboard antiferromagnetism（AF4）的二度簡并態(tài).

［1］ Kamihara Y，Watanabe T，Hirano M，Hosono H.Iron?based layered superconductor La［O1－xFx］FeAs（x＝0.05－0.12）with Tc＝26 K［J］.Am Chem Soc，2008，130：3296－3297.

［2］ Mizoguchi H，Kuroda T，Kamiya T，Hosono H.LaCo2B2：A Co?based layered superconductor with a ThCr2Si2?type structure ［J］.Phys Rev Lett，2011，106：237001－237004.

［3］ Mizuguchi Y，Demura S，Deguchi K，Takano Y，F(xiàn)ujihisa H，Gotoh Y，Izawa H，Miura O.Superconductivity in novel BiS2?based layered superconductor LaO1－xFxBiS2［J］.Phys Soc Jpn，2012，81：114725－114729.

［4］ Dong J，Zhang H J，Xu G，Li Z，LiG，Hu W Z，Wu D，Chen G F，Dai X，Luo JL，F(xiàn)ang Z，Wang N L.Competing orders and spin?density?wave instability in La（O1－xFx）FeAs［J］.Europhysics Letters，2008，83：27006－27009.

［5］ Zhai H F，et al.Growth and characterizations of Ba2Ti2Fe2As4O single crystals［J］.Phys Rev B，2013，87：100502－100506.

［6］ Chen H，et al.Coexistence of the spin?density wave and superconductivity in Ba1－xKxFe2As2［J］.Europhysics Lett，2009，85：17006－17018.

［7］ Yajima T，Nakano K，Takeiri F，Ono T，Hosokoshi Y，Matsushita Y，Hester J，Kobayashi Y，Kageyama H. Superconductivity in Ba1－xKxTi2Sb2O（0≤x≤1）controlled by charge doping［J］.JPhys Soc Jpn，2012，81：103706－103709.

［8］ Doan P，et al.Growth and characterizations of Ba2Ti2Fe2As4O single crystals［J］.Am Chem Soc，2012，134：16520－16523.

［9］ Liu R H，et al.Physical properties of the layered pnictide oxides Na2Ti2P2O（P＝As，Sb）［J］.Phys Rev B，2009，80：144516－144520.

［10］ Yan XW，Lu Z Y，Layered pnictide?oxide Na2Ti2Pn2O（Pn＝As，Sb）：A candidate for spin density waves［J］.J Phys Condens Matter，2013，25：365501－365509.

［11］ Huang Y，Wang H P，WangW D，Shi Y G，Wang N L.Formation of the density wave energy gap in Na2Ti2Sb2O：An optical spectroscopy study［J］.Phys Rev B，2013，87：100507－100510.

［12］ Sun Y L，et al.Growth and characterizations of Ba2Ti2Fe2As4O single crystals［J］.Am Chem Soc，2012，134：12893－12897.

［13］ Subedi A.Electron?phonon superconductivity and charge density wave instability in the layered titanium?based pnictide BaTi2Sb2O［J］.Phys Rev B，2013，87：054506－054511.

［14］ Wang G T，Zhang H P，Zhang L，Liu C.The electronic structure and magnetism of BaTi2Sb2O［J］.JAppl Phys，2013，113：243904－243908.

［15］ Yajima T，Nakano K，Takeiri F et al.Tcenhancement by aliovalent anionic substitution in superconducting BaTi2（Sb1－xSnx）2O［J］.JPhys Soc Jpn，2013，82：074707－074711.

［16］ Suetin D V，Ivanovskiii A L.Electronic properties and fermi surface for new Fe?free layered pnictide?oxide superconductor BaTi2Bi2O from first principles［J］.JETP letters，2013，97：220－225.

［17］ Ma F J，JiW，Hu JP，Lu ZY，Xiang T.First?principles calculations of the electronic structure of tetragonal alpha?FeTe and alpha?FeSe crystals：Evidence for a bicollinear antiferromagnetic order［J］.Phys Rev Lett，2009，102：177003－177007.

［18］ Anisimov V I，Zaanen J，Anderson O K.Strong Coulomb correlations in electronic structure calculations［J］.Phys Rev B，1991，44：943－954.

［19］ Mazin I I，Johannes M D，Boeri L，Koepernik K，Singh D J.Problemswith reconciling density functional theory calculations with experiment in ferropnictides［J］.Phys Rev B，2008，78：085104－085110.

［20］ Xu G，Ming W，Yao Y，Dai X，Zhang S C，F(xiàn)ang Z.Doping?dependent phase diagram of LaO M As（M＝V－Cu）and electron?type superconductivity near ferromagnetic instability［J］.Europhysics Lett，2008，82：67002－67006.

［21］ Solovyev I，Hamada N，Terakura K.Crucial role of the lattice distortion in the magnetism of LaMnO3［J］.Phys Rev Lett，1996，76：4825－4828.

［22］ Fang Z，Terakura K.Structural distortion andmagnetism in transitionmetal oxides：Crucial roles of orbital degrees of freedom ［J］.JPhys Condens Matter，2002，14：3001－3014.

［23］ Perdew JP，Wang Y.Accurate and simple analytic representation of the electron?gas correlation energy［J］.Phys Rev B，1992，45：13244－13249.

［24］ Chen W Q，Yang K Y，Zhou Y，Zhang FC.Strong coupling theory for superconducting iron pnictides［J］.Phys Rev Lett，2009，102：047006－047009.

0 Introduction

Recently，graphite?related materials such as fullerenes， carbon nanotubes， especially graphene，have been gradually attracted broad attention due to their unique electronic and excellent physical properties.It has been demonstrated that graphene，amonolayer of carbon atoms arranged laterally in a honeycomb lattice，as the most promising candidate material for nanoelectronic devices，owing to its remarkable high electron and hole mobility，combined with high mechanical and thermal stability［1－3］.

Although two?dimensional graphene is a zero?gap material and makes it not suitable for transistor applications，it has been studied by researches that this problem can be circumvented by means of graphene nanoribbons（GNRs）due to quantum confinement in the width direction［4－6］.The electronic properties of GNRs depend sensitively upon their widths and shape of edges.Semiconducting GNRs can be used as channel material［7－8］of field effect transistors（FETs）which have been explored as potential alternatives to CMOS devices.Compared with siliconmaterials，due to an exceptionally high mobility，and near ballistic transport in GNR，GNRFETs can obtain a higher driving current，faster operation speed and a significant reduction in power consumption［9－11］.

In the recent past，gate underlap structures have been reported to reduce electric field in the drain of FETs and thus reduce interband tunneling in the drain which is the cause of ambipolar leakage current［12－14］.In addition，linear doping in the underlap region is generally used to suppress band to band tunneling（BTBT）and ambipolar conduction of MOSFETs.In this structure，the n?type impurity is at maximum level at source/drain side and reduces linearly toward theintrinsic channelwhich becomes zero at end of channel at source/drain side.

On the other hand，to enhance the immunity against short?channel effects（SCE），triple?material?gate（TMG）structures have been proposed to improve the performances of GNRFETs［15］. Compared to a single?material?gate?GNRFET（SMG?GNRFET），TMG?GNRFET has three materials with differentworkfunctions laterallymerged together in gate.Due to the field discontinuity along the channel，this structure leads to potential steps along the channel at the interfaces of different gate metals.As the drain potential changes，the step in potential increases，which provides a better shielding of the channel from the drain variation，leading to a considerable increase in average carrier velocity in the channel.

In order to possess the advantages of both triple?material?gate and linear doping profile，we propose a compound structure called linear doping triple?material?gate GNRFET（TL?GNRFET）.In this paper，based on a previous work［16］，we investigated the characteristics of GNRFETs，using self?consistentmethod solving the non?equilibrium Green's function（NEGF）coupled with a three?dimensional Poisson equation under the ballistic limits in the mode space.By comparing the conventional single?material?gate GNRFET with conventional doping GNRFET（SC?GNRFET），single?material?gate GNRFET with linear doping（SL?GNRFET），triple?material?gate GNRFET with conventional doping（TC?GNRFET）and triple?material?gate with linear doping（TL?GNRFET），it indicates that TL?GNRFET structure simultaneously reduces the off?state current and has better switching performance.In addition，asymmetric gate underlap and its effects have also been discussed and it is revealed that a structure in which top and bottom gates are both shifted towards the source has an improved subthreshold performance.

1 Model and Methods

In order to simulate the device characteristics，we perform a self?consistent calculation of potential and charge density in GNRFET.To obtain the self?consistentpotential，a nonlinear Poisson equation is solved for double gated GNRFETs by setting Dirichlet boundary conditions on the gate surface and Von Neumann boundary conditions along the exposed surface of the dielectric.To describe characteristics of the GNRFET，we used a tight?binding Hamiltonian with the atomistic nearest neighbor pz?orbital tight binding approximation.The recursive Green function algorithm is used to solve the NEGF equations for the density of states，charge density，and current.The self?energy generated by device's source and drain electrode can be solved by calculating the surface Green function using iterative approach［16－18］.

The calculation of charge density is using the NEGF［17－18］.Retarded Green's function of the device is

whereη＋is a positive infinitesimal，E is energy，HDis the Hamiltonian when electrons in GNR is under the best adjacent approximation，ΣSandΣDare self?energies generated by device's source and drain electrodes，which can be solved by calculating the surface Green function using iterative approach.Once we get the Green function，the density of electron and hole in any position in the device can be given by the following equations

where EIis partial Fermi level in GNR，EFD（S）is the Fermi level in drain（source）.The electrostatic potential profile can be obtained by solving the 3D?Poisson equation when the electron and holes density have been computed and Eq.（2）and Eq.（3）.Therefore，3D?Poisson equation is expressed as

where U is electrostatic potential，εis dielectric constant，ρis the distribution of net charge.To calculate the channel current，we can use the Landauer?Büttiker formula

where q is the electron charge，h is Planck's constant and T［E］＝Trace［ΓGΓG＋］，which is the transmission coefficient of electron tunneling through the channel，EFD（S）is Fermi level of the drain （source）.

In summary，the Poisson?Schr?dinger self?consistent algorithm in this paper can be simplified as follows（as shown in Fig.1）：

（a）Give the initial value of electrostatic potential of the device；

（b）Solve the NEGF equations for device specified to obtain the charge density Eqs.（2）－（3）；

（c）Put Eqs.（2）－（3）into Poisson Eq.（4）to obtain a new electrostatic potential；

（d）Repeat step（b）and（c）until the convergence of electrostatic potential and charge density is achieved；

（e）Put the convergence solution into the NEGF to obtain channel current（5）.

Fig.1 Poisson?Schr?dinger self?consistent algorithm

2 Results and discussions

Simulation results are presented for double?gate GNRFET structure made from armchair GNR with N＝13（width of 1.47 nm and bandgap of Eg≈0.72 eV）.Double?gate geometry is implemented using SiO2as the gate insulator.Unless otherwisementioned，the basic parameters of SMG?GNRFET are thickness of gate oxide Tox＝2 nm，n?doped GNR extensions length at source/drain ends Lsd＝20 nm，dielectric constantof gate oxideε＝3.9，the fraction of doping atoms in then＋source/drain 5×10－3，length of gate Lg＝10 nm，the channel GNR undoped.A schematic longitudinal cross?sectional view of TMG?GNRFET is shown in Fig.2.The gate of TMG?GNRFET consists of three laterally contacting metals M1，M2 and M3 with length LS，LMand LD，respectively.The work function of the main gate metal M2，unless otherwise mentioned，is larger than that of the two side gates M1 and M3 with the same metals.The rest parameters remain the same as the SMG?GNRFET.Transversal cross?section of the simulated GNRFET structure is shown in Fig.2（b），and the oxide layer thickness of the width direction S1＝S2＝1 nm.Different doping strategies are shown in Fig.2（a）.Unless otherwisementioned，the length of underlap region Lu1＝Lu2＝10 nm.

Fig.2 Simulated TMG?GNRFETs（A 50 nm?long armchair?edge GNR is used as channelmaterial.）（a）Longitudinal cross?section and doping profile，（b）Transversal cross?section

2.1 Comparison w ith experiment

Figure 3 shows a comparison of GNRFETs output characteristics ofmeasured and modeled［19］. The channel of the p?type GNRFET with back?gate topology is Lch＝110 nm，the width of GNR is w ≈2.5 nm corresponding to 21?AGNR，and the SiO2，tox＝10 nm.It is observed that our results arein good agreement with simulated results given in Ref. ［20］.However，due to the presence of elastic/inelastic scattering such as edge roughness scattering， optical phonon scattering and defect scattering which could limit the carrier transport of GNR channel，themeasured current is lower than the ballistic current of simulations for same device structure（A?GNR of 2.5 nm width and 110 nm channel length），as shown in Fig.3.As can be seen，the experimental GNRFET delivers about 20%of the ballistic current at low｜Vds｜and about40%of the ballistic current at high｜Vds｜.

Fig.3 Output characteristics of GNRFET in our result，simulated ballistic result in Ref.［20］and experimental data adopted from Ref.［19］

2.2 Compound effects of linear doping and triple?material?gate structures for GNRFET

Here，the effects of gate length on the static performances of GNRFET structure are studied. Figure 4 shows transfer characteristics of SC?GNRFET with different length.It is clear that the SC?GNRFET with longer gate has better sub?threshold characteristic which indicates lower power consumption.This is because the effect of DIBL which is becomingmore significantwith decreasing gate length.

In order to study the impact of linear doping and hetero?material gate on the performance of GNRFETs，we compare the transfer characteristics of SMG?GNRFET and TMG?GNRFET with and without linear doping in Fig.5.The following observations are indicated from the figure：（1）The current characteristics of TC?GNRFET is similar with SC?GNRFET at the same drive current，while the off?state current and subthreshold slope of TC?GNRFET is found lower than SC?GNRFET.（2）Linear doping has improvement in the subthreshold characteristics of both SMG?GNRFETs and TMG?GNRFETs.（3）Compared to other structures，TL?GNRFET has a slightly lower on?state current and the lowest off?state current，that is to say，the Ion/Ioffof TL?GNRFET is the best.

Fig.4 Transfer characteristics of SC?GNRFET，the gate length is 6，10，16 and 15 nm

Fig.5 Transfer characteristics of GNRFETs with different structures

The subthreshold swing defined by S＝d Vgs/d（ln I）is a key parameter of transistorminiaturization.A small subthreshold swing（S）is desired for low threshold voltage and low power operation for FETs scaled down to small sizes.Subthreshold swing of SC?GNRFET and SL?GNRFET and TL?GNRFET are compared in Fig.6.It is evident from the figure that，SC?GNRFET has the largest subthreshold swing，while TL?GNRFET has the smallest subthreshold swing.In reflects the TL?GNRFET has a better control ability for suppressing the SCE effect.

Fig.6 Threshold voltage swing of different GNRFETs at Vds＝0.7 V

In addition，DIBL is an importantmanifestation of SCE effect.To study the DIBL in TL?GNRFET，F(xiàn)ig.7 illustrates the comparison of threshold voltage shift betweens C?GNRFET and TL?GNRFET as the drain voltage increases from 0.01 V to 0.7 V.Here，we define the threshold voltage corresponding to the gate voltage of 10 nA current of channel.As shown in Fig.7，threshold voltage shift of SC?GNRFET is 0.11 V while that of TL?GNRFET is 0.05 V，which indicates that TL?GNRFET has better ability in suppressing DIBL effect compared to convention GNRFETs.It can be explained by electronic potential along channelwith different drain voltage of TL?GNRFET.It is clearly from Fig.8 that the increase of additional drain potential is absorbed under the gate near drain，while the gate near the source region is screened from drain potential variations.As a result，Vdshas only a small influence on drain current after saturation.Thus，the gate has better controlling over the channel and suppresses the DIBL effectively.

Fig.7 Shift in threshold voltage of SC?GNRFET and TL?GNRFET at Vds＝0.01 V and Vds＝0.7 V

Fig.8 Potential profile along channel of TL?GNRFET at different drain biases

To probe physical mechanisms responsible for the improved performance of TL?GNRFET，lateral electric field profiles along channel of different GNRFET at Vgs＝Vds＝0.3 V are shown in Fig.9.Comparing Fig.9（a）and Fig.9（b），we found that the implantation of linear doping has a prominent effect on reducing electric field in the drain region，so the linear doping has a great contribution to preventing the hot carrier effects.Comparing Fig.9（a）and Fig.9（c），we found that as triple?material?gate is applied，the changes aremainly in the gate region.The triple?material?gate promotes the average electric field in the channel，and thus enhances the electron transportefficiency.In addition，the linear doping reduces the electric field in the drain aswell as that in the source，while the triple?material?gate enlarges the source region electric field.So the triple?material?gate can be an compensation in structures with linear doping as shown in Fig.9（d），enhancing the electron transport efficiency，keeping the source electric field relatively high，and less affecting electric field in the drain.The curves in Fig.9 proved the explanation.The structure with linear doping and triple?material?gate has the best subthreshold characteristics and on?off current ratio.

Fig.9 Electric field profiles along channel of different GNRFET at Vgs＝Vds＝0.3 V （a）SC?GNRFET（b）SL?GNRFET（c）TC?GNRFET（d）TL?GNRFET

2.3 Effects of asymmetric gate underlap

In this section，we investigate effects of source/drain asymmetric gate underlap and developed new GNRFET architectures aiming the optimization of transistor ambipolarity and on/off ratio Ion/Ioff.In traditional GNRFETs，the gate is

located in the middle of the device.We simulated situations that the top and bottom gates are both shifted towards the source（Asy.S）or the drain （Asy.D）.And the linear doping single?material?gate（SL）and linear doping triple?material?gate （TL）structures are under investigation respectively，as shown in Fig.10.

Fig.10 Cross?sectional view of asymmetric gate GNRFET with different structures（a）Asy.S（b）Asy.D

To study the performance of asymmetric gate，comparisons are made between conventional doping and linear doping for SMG?GNRFET and TMG?GNRFET.The drain current versus gate voltage of SC?，SL?，T?，and TL?GNRFETwith Asy.Sstructure and Asy.D structure are plotted in Fig.11，respectively.It is obvious from Fig.11（a）that，as the top/bottom gatemoving towards the source，the four devices will have better subthreshold characteristics and smaller off?state current compared to the symmetrical gate structures（as shown in Fig.5）.On the other hand，from Fig.11 （b），it is noticed that，the four devices have worse subthreshold characteristics and larger off?state current compared to the symmetrical gate structures（as shown in Fig.5），as the top/bottom gate moving towards the drain.

Fig.11 Transfer characteristics of different GNRFETs with misalignment toward source 5 nm

To see clearly the variation of Ionand Ioffof four deviceswith asymmetrical gate structures，three tables are shown.Comparing Tab.1 with Tab.2，we found that Ion/Ioffof four structures become larger as the top/bottom gatemoving towards the source.On the contrary，comparing Tab.1 with Tab.3，Ion/Ioffof four structures become smaller as the top/bottom gatemoving towards the drain.

The effects of the source/drain asymmetric gate underlap can be found in electron potential profiles shown in Fig.12.Comparing Fig.12（a）with Fig.12（c），as gates shift a certain part to drain region，the electron potential distributions between the channel and source becomesmore gentlethan that as gates shift a certain part to source region，which makes the off?state current of Asy.S structure greater than that of Asy.D.

Table 1 On/off state current of four structures of GNRFETs w ith gate located in them iddle

Table 2 On/off state current of four structures of GNRFETs w ith both gatemoved together towards source（Asy.S）

Table 3 On/off state current of four structures of GNRFETs w ith both gatem oved together towardsdrain（Asy.D）

Fig.12 Electronic potential profiles along channel of different of GNRFET at VgsVds＝0.3 V

It can be explained in other ways as well. Figure 13 shows lateral electric field distributions of the source/drain asymmetric gate underlap of GNRFETs.As the gate ismoving towards the drain，the electric field in the source region decreases，whilethat in the drain region has a great promotion，which is not ideal for nanoscale electronic devices. However，as the gate is located near the source，the source electric field is enhanced and there is a relatively wealer electric field in the drain region，and thus efficiently suppress the hot carrier effects.What ismore，in the channel region，the average electric field is the greatest as the gate is more near the source than the drain，which means in that situation the device has higher electron transport efficiency.In summary，as the gate locates near the source，the device has better performance.

Fig.13 Electric field profiles along channel of different GNRFET with source/drain asymmetric gate underlap at Vgs＝Vds＝0.3 V，（a）and（b）Asy.S，（c）and（d）Asy.D

3 Conclusions

Based on a previous work，we investigated characteristics of GNRFETs，using self?consistent method solving the non?equilibrium Green's function（NEGF）coupled with a three?dimensional Poisson equation under the ballistic limits in themode space.By comparing TL?GNRFET with SC?GNRFETs，SL?GNRFETs and TC?GNRFETs，we find that the TL?GNRFET structure simultaneously reduces the off?state current and has better switching performance.In addition，the asymmetric gate underlap and its effects are discussed and it is revealed that the asymmetric structure in which the gate is located near the source has an improved on/off state current performance.

［1］ Neto A H C，Guinea F，Peres N M R，Novoselov K S，Geim A K.The electronic properties of graphene ［J］.Rev Mod Phys，2009，81：109－162.

［2］ Lemme M C，Echtermeyer T J，Baus M，Kurz H.A graphene field?effect device［J］.IEEE Electron Device Lett，2007，28（4）：282－284.

［3］ George S.Gate capacitancemodeling and width?dependent performance of graphene nanoribbon transistors ［J］.Microelectron Eng，2013，112：220－226.

［4］ Noei M，Moradinasab M，F(xiàn)athipour M.A computational study of ballistic graphene nanoribbon field effect transistors［J］.Physica E，2012，44（7－8）：1780－1786.

［5］ Li X L，Wang X R，Zhang L，Lee SW，Dai H J.Chemically derived，ultra smooth graphene nanoribbon semiconductors［J］.Science，2008，319：1229－1232.

［6］ Han M Y，Ozyilmaz B，Zhang Y B，Kim P.Energy band?gap engineering of graphene nanoribbons［J］. Phys Rev Lett，2007，98：206805－206808.

［7］ Obradovic B，Kotlyar R，Heinz F，Matagne P，Rakshit T，GilesM D，Stettler M A，Nikonov D E Analysis of graphene nanoribbons as a channelmaterial for field?effect transistors［J］.Appl Phys Lett，2006，88 （14）：142102（3）.

［8］ Chen Z，Lin Y M，Rooks M J，Avouris P.Graphene nanoribbon electronics［J］.Phys E，2007，40（2）：228－232.

［9］ Lin Y M，Dimitrakopoulos C，Jenkins K A，F(xiàn)armer D B，Chiu H Y，Grill A，Avouris Ph.100 GHz transistors from wafer?scale epitaxial graphene［J］.Science，2010，327：662－662.

［10］ Liao L，Bai J，Cheng R，Lin Y C，Jiang S，Qu Y，Huang Y，Duan X.Sub?100 nm channel length graphene transistors［J］.Nano Letters，2010，10（10）：3952－3956.

［11］ Liao L，Lin Y C，Bao M Q，Cheng R，Bai J，Liu Yuan，Qu Y，Wang K，Huang Y，Duan X.High?speed graphene transistorswith a self?aligned nanowire gate［J］.Nature，2010，467：305－308.

［12］ Alam K，Lake R.Role of doping in carbon nanotube transistors with source/drain underlaps［J］.IEEE Trans Nanotechnol，2007，6（6）：652－658.

［13］ Kordrostami Z，Sheikhi M H，Zarifkar A.Influence of channel and underlap engineering on the high?frequency and switching performance of CNTFETs［J］.IEEE Trans Nanotechnol，2012，11（3）：526－533.

［14］ Sarvari H，Ghayour R.Design of GNRFET using different doping profilesnear the source and drain contacts ［J］.International Journal of Electronics，2012，99（5）：673－682.

［15］ Orouji A A，Arefinia Z.Detailed simulation study of a dual material gate carbon nanotube field?effect transistor［J］.Phys E，2009，41（10）：552－557.

［16］ Wang W，Yang X，Li N，Zhang L，Zhang T，Yue G.Numerical study on the performance metrics of lightly doped drain and source graphene nanoribbon field effect transistors with double?material?gate［J］. Superlattices and Microstructures，2013，64（9）：227－236.

［17］ Mintmire JW，White C T.Universal density of states for carbon nanotubes［J］.Physics Review Letters，1998，81（12）：2506－2509.

［18］ Fiori G，Iannaccone G.Simulation of graphene nanoribbon field?effect transistors［J］.IEEE Electron Device Letters，2007，28（8）：760－762.

［19］ Wang X，Ouyang Y，Li X，Wang H，Guo J，Dai H.Room?temperature all?semiconducting sub?10 nm graphene nanoribbon field?effect transistors［J］.Phys Rev Lett，2008，100（20）：206803－206807.

［20］ Guo J.Modeling of graphene nanoribbon devices［J］.Nanoscale，2012，4：5538－5548.

Electronic Structure and M agnetism of BaTi2Bi2O

WANG Guangtao，ZHANG Lin，ZHANG Huiping，LIU Chang
（College of Physics and Information Engineering，Henan Normal University，Xinxiang 453007，China）

Electronic structure and magnetic structures of BaTi2Bi2O are studied with first?principles calculations.In nonmagnetic state，density of states at Fermi level aremostly derived from dz2，dx2－y2and dxyorbitals.Fermisurface（FS）consists of three sheets. The third FS sheet（along X?R line）nestswith the first FS sheet（along M?A line）by q?vector q1＝（π/a，0，0）or q2＝（0，π/a，0）shift.Calculated bare susceptibilityχ0（q）peaked at X?point，rather than at M?point in FeAs?based superconductors.Such peaked susceptibility induces spin density wave（SDW）.Magnetic ground state is nearly two degenerate antiferromagnetism of bi?collinear antiferromagnetism（AF3）and blocked checkerboard antiferromagnetism（AF4）.Peak of susceptibilityχ0（q）is obviously suppressed and becomes slightly in?commensurate with hole doping，but increased with electron doping.As spin fluctuation is suppressed superconductivity appears.It explains that superconductivity appears only in hole?doped compounds，and not in electron?doped ones. Key words： first?principles calculations；Fermi surface nesting；superconductivity

Performance of Asymmetric Linear Doping Triple?material?gate GNRFETsWANGWei，GAO Jian，ZHANG Ting，ZHANG Lu，LINa，YANG Xiao，YUE Gongshu（College ofElectronic Science Engineering，Nanjing University of Post and Telecommunications，Nanjing，Jiangsu 210023，China）

GNRFET；NEGF；triple?gate?material；linear doping

O484.3 Document code：A

O469

A

2013－09－22；

2014－07－30

國家自然基金（11274095）和河南省高校創(chuàng)新人才（2012HASTIT009）資助項目

王廣濤（1976－），男，博士，教授，從事第一性原理計算，E?mail：wangtao＠htu.cn

Received date： 2013－09－22；Revised date： 2014－07－30

Received date：2014－01－25；Revised date：2014－06－26

Foundation item s：Supported by Natural Science Fouudation of Higher Eduction in Jiangsu Province（10KJD510006）

Biography：WangWei（1964－），male，PhD，associate professor，major in nanoelectronics，E?mail：wangwej＠njupt.edu.cn