氧空位遷移造成的氧化物介質(zhì)層時變擊穿的蒙特卡羅模擬

栗蘋 許玉堂

(北京理工大學機電學院,北京 100081)

氧空位遷移造成的氧化物介質(zhì)層時變擊穿的蒙特卡羅模擬

栗蘋 許玉堂?

(北京理工大學機電學院,北京 100081)

(2017年5月19日收到;2017年7月3日收到修改稿)

氧空位,蒙特卡羅,氧化物介質(zhì),模擬

1 引 言

半導體產(chǎn)業(yè)是近幾十年信息技術革命的基礎,其核心技術是互補金屬氧化物半導體晶體管組成的集成電路.根據(jù)摩爾定律,金屬氧化物半導體(MOS)晶體管尺寸不斷縮小,將在10年內(nèi)逼近其物理極限.其中柵介質(zhì)厚度的按比例縮小是其中最先達到物理極限的部分,為了進一步縮小晶體管,現(xiàn)在廣泛采用的方法是使用金屬氧化物高κ介質(zhì)層作為柵極絕緣層[1].但由于可應用于工業(yè)生產(chǎn)的金屬氧化物高κ介質(zhì)層的組分、形態(tài)及晶體結構等都十分復雜,其可靠性問題亦是實際應用的瓶頸[2].而在金屬氧化物柵介質(zhì)的可靠性問題中,其在電壓應力作用下的時變擊穿特性(time dependent dielectric breakdown)是其中的核心的問題之一[3,4].隨著MOS器件按比例縮小,其工作時施加在柵介質(zhì)層上的電場愈來愈強.當電場強到一定程度氧空位的形成與遷移將成為金屬氧化物介質(zhì)層時變擊穿的主導因素.氧空位隨著時間在介質(zhì)層內(nèi)部積累并演化形成導電通道引起介質(zhì)層損傷并失效,稱為氧空位遷移導致的金屬氧化物介質(zhì)時變擊穿[5].對于較強電場下的氧空位形成與遷移引起的金屬氧化物介質(zhì)擊穿的研究還僅僅處于起步階段.雖然氧空位形成與遷移的微觀機理已經(jīng)被初步研究[6,7],但其如何具體地影響介質(zhì)層擊穿尚未得到深入研究.

本文基于微觀機理,建立可靠的模擬工具來仿真氧空位形成與遷移引起的金屬氧化物介質(zhì)擊穿過程,為進一步研究介質(zhì)擊穿機理與評估高κ柵介質(zhì)的可靠性提供依據(jù).

2 金屬氧化物中氧空位形成與遷移的微觀機理

如圖1所示,在電場作用下金屬氧化物介質(zhì)內(nèi)部的晶格氧有一定概率被激發(fā)形成氧空位,但其遷移功函數(shù)非常高,只有極低的概率會形成氧空位.而若其周圍有氧空位存在,晶格氧不需完全克服共價鍵的束縛就可以遷移到近鄰氧空位的位置.因此可以等效地認為是氧空位在介質(zhì)層中遷移,且氧空位的存在會造成遷移勢壘(遷移功函數(shù))降低.此外,在介質(zhì)內(nèi)氧空位可以越過勢壘向任意近鄰的含有晶格氧的位置遷移.在半導體工藝制程中,金屬氧化物介質(zhì)層上下兩端界面一般是金屬性材料或溝道材料,而這些材料都會促進氧空位在絕緣界面處的形成,其遷移功函數(shù)與界面性質(zhì)相關.當施加電場時,沿電場方向的形成氧空位遷移功函數(shù)與遷移勢壘會顯著降低,氧空位會在電場較強處較快地形成與遷移,最終積累形成導電通道造成介質(zhì)擊穿.

氧空位在介質(zhì)界面處的形成概率可以用(1)式來描述:

其中Pi為氧空位形成概率,ν為氧空位形成與遷移的振動頻率,n為氧空位所帶電荷數(shù)目,q為單位電荷量,E為局域電場,k為玻爾茲曼常數(shù),T為環(huán)境溫度,Ea為形成氧空位遷移功函數(shù).

氧空位在介質(zhì)內(nèi)部的遷移概率可以用(2)式來描述:

其中Ph為氧空位遷移概率,Eh為氧空位遷移勢壘.

圖1 (網(wǎng)刊彩色)氧空位在金屬氧化物中遷移的微觀機理(金屬氧化物介質(zhì)內(nèi)部氧空位的存在會造成勢壘降低,在介質(zhì)內(nèi)氧空位可以越過勢壘向任意近鄰含有晶格氧的位置遷移,電場能夠造成氧空位遷移勢壘的變化;介質(zhì)界面處會有氧空位形成,其遷移功函數(shù)受電場調(diào)控)Fig.1.(color online)The microscopic mechanism of the migration of oxygen vacancies in metal oxides.The presence of oxygen vacancies in the metal oxide dielectric can cause a reduction in barrier,the oxygen vacancies can migrate over the barrier to the position of any nearest neighbor containing lattice oxygen in the medium.The electric field can cause the change of oxygen vacancy migration barrier;the oxygen vacancies are formed at the interface of the dielectric,and the transfer work function is controlled by the electric field.

3 模擬方法

如圖2(a)所示,首先根據(jù)對稱性建立二維網(wǎng)格來表示三維的金屬氧化物介質(zhì)層,本模擬中氧化層厚度設置為9個格點,從圖1可得氧空位的最小遷移距離約為其晶格常數(shù)的一半,而金屬氧化物的晶格常數(shù)約為0.5—0.7 nm,在本模擬中氧空位的單位遷移距離為0.25 nm(非晶態(tài)情況下,一般的氧空位的最小遷移距離小于晶格常數(shù)的1/2).因此可得本模擬中的金屬氧化物介質(zhì)層厚度約為2.3 nm,符合14 nm以下技術節(jié)點的MOS晶體管柵介質(zhì)的物理厚度,且模擬的介質(zhì)層物理寬度為14 nm.由于在金屬氧化物介質(zhì)層內(nèi)氧空位可以越過勢壘向任意近鄰的含有晶格氧的位置遷移,所以如圖2(a)所示,在模擬中設置氧空位可以向鄰近的8個任意格點位置遷移.根據(jù)氧空位分布,可以計算介質(zhì)層內(nèi)部電場分布與電流,將得到的電場迭代到(1)和(2)式中計算氧空位形成與遷移概率.使用蒙特卡羅方法模擬氧空位的形成與遷移得到其新的分布[8].在本模擬中,計算電勢分布、局域電場及電流是通過構建以各個格點作為節(jié)點的無源電路網(wǎng)絡,各格點與相鄰的8個格點都形成電流支路,再利用基爾霍夫定理計算得到.各點間的電流由(3)—(5)式描述.其中當兩格點都被晶格氧占據(jù)時,電子有極低的概率在其間跳躍輸運,

其中Δφ為格點間電勢差;α,R2為表示電子跳躍輸運的特征參數(shù).

金屬電極與晶格氧占據(jù)的格點或單側(cè)被氧空位占據(jù)的格點間的輸運可由(4)式表示:

由于氧空位可供電子跳躍輸運的能級較多,所以比兩格點都被晶格氧占據(jù)時跳躍概率要高;β為引入的特征參數(shù).

當兩格點都被氧空位占據(jù)時,電子輸運為金屬性,可由(5)式描述,

其中R1為格點間電阻.

具體的模擬流程及使用的關鍵參數(shù)在圖2(b)和圖2(c)中給出.

圖2 (網(wǎng)刊彩色)(a)模擬方法示意圖(根據(jù)對稱性建立二維網(wǎng)格來表示金屬氧化物介質(zhì)層并通過蒙特卡羅方法計算氧空位在網(wǎng)格中的隨機產(chǎn)生與遷移;同時根據(jù)氧空位分布計算電勢分布與電流,進而來模擬介質(zhì)層的擊穿過程,其中氧空位可以向相鄰的8個格點遷移且沿電場方向的遷移概率較大;介質(zhì)層內(nèi)部電場分布與電流通過構建各個格點作為節(jié)點的電路網(wǎng)絡,然后利用基爾霍夫定理計算得到);(b)氧空位遷移導致的金屬氧化物介質(zhì)擊穿的蒙特卡羅模擬流程圖;(c)模擬過程使用的關鍵參數(shù)Fig.2.(color online)(a)Schematic diagram of simulation method:a two-dimensional grid is constructed to represent the metal oxide dielectric layer according to symmetry,and the Monte Carlo method is used to calculate the random generation and migration of oxygen vacancies in the grid;at the same time according to the oxygen vacancy,we first calculate the potential distribution and current,simulate the breakdown process of dielectric layer;the oxygen vacancies can migrate to eight adjacent grid sites,and the probability of migrate along the electric field direction is larger;we construct a circuit network in which each lattice is as a node,and then use Kirchho ff’s theorem to calculate the electric field distribution and current in the dielectric layer.(b)Flow chart of Monte Carlo simulation of metal oxide dielectric breakdown induced by oxygen vacancy migration.(c)Key parameters used in the simulation process.

4 模擬結果與討論

通過蒙特卡羅模擬可得到界面形成氧空位遷移功函數(shù)較低(Ea=1.15 eV)時金屬氧化物介質(zhì)層的擊穿過程.如圖3(a)所示,在電壓應力的作用下,通過介質(zhì)層的電流將逐漸增加,最后劇烈上升造成擊穿.圖3(b1)—(b4)為模擬得到的圖3(a)中相應時間(a,b,c,d)的氧空位在金屬氧化物介質(zhì)中的分布情況,首先氧空位在界面附近大量形成,然后向介質(zhì)內(nèi)部逐漸遷移,最后形成導電通道.圖3(c1)—(c4)為模擬得到的與圖3(a)和圖3(b)對應的氧化物介質(zhì)中的電勢分布,從模擬結果可見,氧空位積累較多的區(qū)域電場較強,形成正反饋促進了導電通道的形成.圖3(d1)—(d4)為模擬得到的與圖3(a)、圖3(b)和圖3(c)對應的流經(jīng)介質(zhì)層內(nèi)部(各格點)的電流,隨著氧空位的積累,介質(zhì)內(nèi)部電流越來越大且越來越集中,最后模擬結果顯示了清晰的導電通道.

圖4為模擬得到的界面形成氧空位遷移功函數(shù)較高(Ea=1.35 eV)時,氧空位形成與遷移造成的金屬氧化物介質(zhì)層的擊穿過程.如圖4(a)所示,開始時在電壓應力的作用下通過介質(zhì)層的電流變化較小,然后將逐漸增加,最后劇烈上升造成擊穿;圖4(b1)—(b4)為模擬得到的圖4(a)中相應時間(a,b,c,d)的氧空位在介質(zhì)層中的分布,首先少量氧空位在界面附近形成,然后向介質(zhì)內(nèi)部逐漸遷移,在另一側(cè)界面形成堆積,最后形成導電通道;圖4(c1)—(c4)為模擬得到的對應的氧化物介質(zhì)中的電勢分布,從模擬結果同樣可以發(fā)現(xiàn)氧空位積累較多的區(qū)域電場較強,形成正反饋促進了導電通道的形成,但其氧空位的遷移與其形成相比較快,存在顯著的再分布,所以電勢變化也較為隨機;圖4(d1)—(d4)為模擬得到的對應的流經(jīng)介質(zhì)層內(nèi)部(各格點)的電流,隨著氧空位的積累介質(zhì)內(nèi)部電流越來越大且分布越來越不均勻,最后形成導電通道.

圖3 (網(wǎng)刊彩色)(a)模擬得到的金屬氧化物介質(zhì)層的擊穿過程(Ea=1.15 eV),當施加電壓應力時通過介質(zhì)層的電流逐漸增大;(b1)—(b4)為模擬得到的圖(a)中相應時間(a,b,c,d)的氧空位分布,白色圓點為氧空位;(c)模擬得到的介質(zhì)層內(nèi)部的電場分布;(d)模擬得到的流過介質(zhì)層內(nèi)部(各格點)的電流Fig.3.(color online)(a)The simulated breakdown process of the metal oxide dielectric layer is(Ea=1.15 eV),and when the voltage stress is applied,the current passing through the dielectric layer gradually increases;(b1)–(b4)the oxygen vacancy distribution of the corresponding time(a,b,c,d)in panel(a),and the white dots are oxygen vacancies;(c)the obtained electric field distribution inside the dielectric layer by simulation;(d)the obtained current of flow through the dielectric layer(each lattice)by simulation.

圖4 (網(wǎng)刊彩色)(a)模擬得到的金屬氧化物介質(zhì)層的擊穿過程(Ea=1.35 eV),當施加電壓應力時,通過介質(zhì)層的電流逐漸增大;(b1)—(b4)為模擬得到的圖(a)中相應時間(a,b,c,d)的氧空位分布,白色圓點為氧空位;(c)模擬得到的介質(zhì)層內(nèi)部的電場分布;(d)模擬得到的流過介質(zhì)層內(nèi)部(各格點)的電流Fig.4.(color online)(a)The simulated breakdown process of the metal oxide dielectric layer(Ea=1.35 eV),and when the voltage stress is applied,the current passing through the dielectric layer gradually increases;(b1)–(b4)the oxygen vacancy distribution of the corresponding time(a,b,c,d)in panel(a),and the white dots are oxygen vacancies;(c)the obtained electric field distribution inside the dielectric layer by simulation;(d)the obtained current of flow through the dielectric layer(each lattice)by simulation.

圖5 (網(wǎng)刊彩色)氧空位遷移導致的金屬氧化物介質(zhì)層的擊穿過程的示意圖 (a)界面氧空位遷移功函數(shù)小于其遷移勢壘;(b)界面形成氧空位遷移功函數(shù)大于其遷移勢壘Fig.5.(color online)Schematic diagram of the breakdown process of metal oxide dielectric layer induced by oxygen vacancy migration:(a)The migration function of the oxygen vacancy at interface is smaller than its migration barrier;(b)the migration function of the oxygen vacancy at interface is larger than its migration barrier.

圖6 (網(wǎng)刊彩色)蒙特卡羅模擬得到的氧空位遷移導致的金屬氧化物介質(zhì)層的時變擊穿過程的韋伯分布 (a)界面形成氧空位遷移功函數(shù)為1.15 eV;(b)界面形成氧空位遷移功函數(shù)為1.35 eVFig.6.(color online)The Weber distribution of time dependent dielectric breakdown process of the metal oxide dielectric layer caused by the oxygen vacancy migration,which is obtained by Monte Carlo simulation:(a)The migration work function of formed oxygen of the interface is 1.15 eV;(b)the migration work function of formed oxygen of the interface is 1.35 eV.

通過圖3與圖4的對比可以發(fā)現(xiàn),在界面形成氧空位遷移功函數(shù)不同時,金屬氧化物介質(zhì)層擊穿過程不同(導電通道演化過程不同).圖5顯示了界面形成氧空位遷移功函數(shù)分別大于和小于其遷移勢壘時的介質(zhì)擊穿過程.當遷移功函數(shù)較小時,氧空位在其形成界面大量堆積,并從界面向介質(zhì)內(nèi)部遷移形成導電通道,擊穿時間由氧空位在介質(zhì)中的遷移勢壘決定.而當氧空位在界面遷移功函數(shù)較大時,形成的氧空位快速遷移到另一側(cè)界面,導電通道反向生長造成介質(zhì)擊穿,因此當氧空位在界面遷移功函數(shù)較大時,優(yōu)化界面可以有效提高其可靠性.此外介質(zhì)內(nèi)部的原生缺陷對氧空位的遷移有重要影響,擊穿往往在原生缺陷較多的位置發(fā)生.

通過蒙特卡羅方法可以模擬得到服從韋伯分布(Weibull distribution)的金屬氧化物介質(zhì)擊穿時間統(tǒng)計,如圖6所示.在準確提取材料參數(shù)的情況下,建立的模擬工具可以有效地幫助評估金屬氧化物介質(zhì)的可靠性.此外,模擬顯示優(yōu)化界面提高形成氧空位遷移功函數(shù),可以有效增加擊穿時間,提高柵介質(zhì)可靠性.

5 結 論

通過施加電場能夠降低金屬氧化物內(nèi)部沿電場方向的氧空位遷移勢壘和遷移功函數(shù),利用蒙特卡羅方法計算氧空位的隨機形成與遷移可模擬介質(zhì)時變擊穿.利用建立的模擬工具發(fā)現(xiàn)界面形成氧空位遷移功函數(shù)對介質(zhì)層擊穿行為有重要影響,優(yōu)化界面提高遷移功函數(shù)可顯著提高介質(zhì)可靠性.此模擬工具可應用于MOS晶體管柵介質(zhì)擊穿研究并準確評估其可靠性.

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PACS:77.84.Bw,77.22.–d,85.30.–z DOI:10.7498/aps.66.217701

?Corresponding author.E-mail:xytang@foxmail.com

Monte Carlo simulation of time-dependent dielectric breakdown of oxide caused by migration of oxygen vacancies

Li Ping Xu Yu-Tang?

(Mechanical and Electrical College,Beijing Institute of Technology,Beijing 100081,China)

d 19 May 2017;revised manuscript

3 July 2017)

In this article,the Monte Carlo method is used to study the formation and migration of oxygen vacancies in metal oxide dielectric.The time-dependent breakdown of the dielectric is simulated.In the direction of the electric field across the metal oxide,the migration barrier and migration work function of oxygen vacancies are found to be reduced by the applied electric field.This finding provides a good foundation for further studying the breakdown mechanism and evaluating the reliability of highκgate dielectric.The Monte Carlo process is described as follows.Firstly,a three-dimensional metal oxide dielectric layer is built with two-dimensional symmetrical grid,where the thickness of the oxide layer is set to be 9 lattice points and the oxygen vacancies can migrate to the adjacent 8 arbitrary lattice positions in this simulation.Secondly,the possibilities of formation and migration of oxygen vacancies are calculated according to the distribution of oxygen vacancies.Finally,the Monte Carlo method is used to simulate the new distribution of oxygen vacancies.Therefore,we simulate the breakdown process of the metal oxide dielectric layer with different oxygen vacancy migration functions(Ea=1.15,1.35 eV)at the interface.And we obtain the results as follows.1)When the migration function is small,many oxygen vacancies accumulate largely at the forming interface.And the vacancies would migrate from the interface to the dielectric,forming a conductive channel.The breakdown time is determined by the migration barrier of oxygen vacancies in the dielectric.2)When the migration function of the oxygen vacancies at the interface becomes larger,the formed oxygen vacancies will rapidly migrate to the other interface,and the reverse propagation of the conductive channel causes the dielectric breakdown.Therefore,larger migration function of the oxygen vacancies at the interface can effectively improve its reliability.3)The original defects within the dielectric will seriously in fluence the migration of oxygen vacancies,and the breakdown is easier to occur with more primary defects.4)The simulation shows that the oxygen vacancy migration function can be improved by optimizing the interface formation process.And the breakdown time could also be prolonged.Therefore,this simulation tool can be applied to the research of metal-oxide-semiconductor transistor gate dielectric breakdown and the assessment of its reliability accurately.

oxygen vacancy,Monte Carlo,oxide,simulation

基于氧空位在金屬氧化物內(nèi)部遷移的微觀機理,利用蒙特卡羅方法建立了一種新型的可模擬金屬氧化物介質(zhì)時變擊穿的模擬工具.利用建立的模擬工具研究了界面形成氧空位遷移功函數(shù)對介質(zhì)層擊穿行為的影響.該工具可應用于金屬氧化物半導體晶體管柵介質(zhì)擊穿研究并準確評估其可靠性.

10.7498/aps.66.217701

?通信作者.E-mail:xytang@foxmail.com

?2017中國物理學會Chinese Physical Society