多空穴錯(cuò)位分布對(duì)石墨納米帶中熱輸運(yùn)的影響?

周欣 高仁斌 譚仕華 彭小芳?蔣湘濤包本剛

1)(中南林業(yè)科技大學(xué)理學(xué)院,長沙 410004)

2)(中南林業(yè)科技大學(xué)計(jì)算機(jī)與信息工程學(xué)院,長沙 410004)

3)(湖南科技學(xué)院教務(wù)處,永州 425100)

多空穴錯(cuò)位分布對(duì)石墨納米帶中熱輸運(yùn)的影響?

周欣1)高仁斌1)譚仕華1)彭小芳1)?蔣湘濤2)?包本剛3)

1)(中南林業(yè)科技大學(xué)理學(xué)院,長沙 410004)

2)(中南林業(yè)科技大學(xué)計(jì)算機(jī)與信息工程學(xué)院,長沙 410004)

3)(湖南科技學(xué)院教務(wù)處,永州 425100)

(2017年2月22日收到;2017年3月31日收到修改稿)

利用非平衡格林函數(shù)方法研究了石墨納米帶中三空穴錯(cuò)位分布對(duì)熱輸運(yùn)性質(zhì)的影響.研究結(jié)果發(fā)現(xiàn):三空穴豎直并排結(jié)構(gòu)對(duì)低頻聲子的散射較小,導(dǎo)致低溫區(qū)域三空穴豎直并排時(shí)熱導(dǎo)最大,而在高頻區(qū)域,三空穴豎直并排結(jié)構(gòu)對(duì)高頻聲子的散射較大,導(dǎo)致較高溫度區(qū)域三空穴豎直并排時(shí)熱導(dǎo)最小;三空穴的相對(duì)錯(cuò)位分布僅能較大幅度地調(diào)節(jié)面內(nèi)聲學(xué)模高頻聲子的透射概率,而三空穴的相對(duì)錯(cuò)位分布能較大幅度地調(diào)節(jié)垂直振動(dòng)膜高頻聲子和低頻聲子的透射概率,導(dǎo)致三空穴的相對(duì)錯(cuò)位分布不僅能大幅調(diào)節(jié)面內(nèi)聲學(xué)模和垂直振動(dòng)模的高溫?zé)釋?dǎo),也能大幅調(diào)節(jié)垂直振動(dòng)模的低溫?zé)釋?dǎo).研究結(jié)果闡明了空穴位置不同的石墨納米帶的熱導(dǎo)特性,為設(shè)計(jì)基于石墨納米帶的熱輸運(yùn)量子器件提供了有效的理論依據(jù).

非平衡格林函數(shù),聲學(xué)聲子輸運(yùn),熱導(dǎo),量子體系

1 引 言

近年來,隨著微加工技術(shù)的不斷進(jìn)步,已能夠制造尺寸小于30 nm的低維尺度納米結(jié)構(gòu)[1,2],一些由硅、黑磷等單元素或硼氮、二硫化鉬等多元素構(gòu)成的二維六角晶體單原子層材料也在實(shí)驗(yàn)中被成功制造[3].石墨烯是新型單原子層二維碳納米材料,自從2004年首次發(fā)現(xiàn)以來,由于具有特異的物理和化學(xué)性質(zhì),例如低能無質(zhì)量Dirac電子的線性色散關(guān)系[4]、已知材料中最強(qiáng)的機(jī)械強(qiáng)度[5]、極高的電子遷移率(106cm2/(V·s))等[6]引起了人們的極大興趣.特別地,由于擁有創(chuàng)記錄的極高的熱導(dǎo)性質(zhì)(約3000 Wm/K)[7],石墨烯的熱輸運(yùn)性質(zhì)受到許多科研群體的廣泛關(guān)注[8?23],利用機(jī)械切割、電子束刻蝕等方法[24],石墨烯能夠被制備成各種寬度以及各種幾何邊界的準(zhǔn)一維條形量子結(jié)構(gòu),即石墨納米帶.在這些低維尺度的石墨納米帶中,熱導(dǎo)(包括聲子熱導(dǎo)、電子熱導(dǎo)以及光子熱導(dǎo))是量子化的,熱導(dǎo)的基本量子化單元是π2k2BT/(3h)(kB是玻爾茲曼常數(shù),h是布朗克常數(shù),T是溫度),獨(dú)立于材料以及載流子性質(zhì).研究表明,當(dāng)石墨納米帶帶寬低于10 nm時(shí)[1]其具有半導(dǎo)體性質(zhì),熱導(dǎo)的主要載體是聲學(xué)聲子,由于量子受限制約著聲子的色散關(guān)系和群速,導(dǎo)致其熱導(dǎo)性質(zhì)有別于體材料.一些科研群體深入研究了基于石墨烯構(gòu)建的量子結(jié)構(gòu)的熱輸運(yùn)性質(zhì)[15]并發(fā)現(xiàn)了許多新奇的熱輸運(yùn)現(xiàn)象,例如:缺陷導(dǎo)致的熱環(huán)流[25]、被突變結(jié)構(gòu)加強(qiáng)的低溫?zé)釋?dǎo)[26]、不對(duì)稱三端石墨納米帶結(jié)構(gòu)中的熱整流[27,28]、拓?fù)浣Y(jié)構(gòu)中反常的熱導(dǎo)性質(zhì)[29]等.研究表明,石墨納米帶中的熱輸運(yùn)性質(zhì)與石墨納米帶邊緣的粗糙程度[30]、各種缺陷[10?18]、彎曲程度[31]、超晶格[32]、同位素[33]、外力應(yīng)變以及界面接觸[34,35]、結(jié)構(gòu)尺寸[36]、異質(zhì)結(jié)[37]等密切相關(guān),且能被設(shè)計(jì)成具有優(yōu)良品質(zhì)的熱電材料[38,39]、高品質(zhì)的熱導(dǎo)調(diào)制器[31]、具有負(fù)微分效應(yīng)的熱導(dǎo)器件[40].這些研究中空穴對(duì)石墨納米帶熱輸運(yùn)的調(diào)制受到了極大的關(guān)注,例如空穴導(dǎo)致的透射深谷[41]、量子環(huán)加強(qiáng)的熱電性質(zhì)[42]等.最近,Peng和Chen[18]對(duì)比研究了石墨納米帶中空穴對(duì)面內(nèi)聲學(xué)模(IPMs)與垂直平面聲學(xué)模(FPMs)的聲學(xué)聲子輸運(yùn)和熱導(dǎo)的不同影響,主要闡述了空穴位于石墨納米帶中心位置時(shí)對(duì)熱導(dǎo)的影響.然而實(shí)際制備的石墨納米帶中空穴的分布應(yīng)該是更復(fù)雜隨機(jī)的錯(cuò)位分布情形,而多空穴的錯(cuò)位分布對(duì)熱導(dǎo)的調(diào)制很少見諸報(bào)道,有關(guān)多空穴的錯(cuò)位分布對(duì)熱輸運(yùn)的影響的系統(tǒng)研究至今仍然很缺乏.因此,有必要進(jìn)一步研究錯(cuò)位分布的多空穴結(jié)構(gòu)對(duì)熱輸運(yùn)的影響,這對(duì)于量子器件的設(shè)計(jì)具有非常重要的物理意義.

眾所周知,根據(jù)邊緣碳原子幾何排列的不同,石墨納米帶可以分為扶手椅型石墨納米帶和鋸齒型石墨納米帶[22],且在這兩種不同種類的石墨納米帶中熱輸運(yùn)性質(zhì)存在明顯的差異[43].本文主要關(guān)注鋸齒型石墨納米帶,研究結(jié)果發(fā)現(xiàn)三空穴在石墨納米帶中相對(duì)旋轉(zhuǎn)或相對(duì)平移能大幅調(diào)節(jié)石墨納米帶中的熱導(dǎo);在低溫區(qū)域,當(dāng)三空穴并列且豎直于石墨納米帶橫向方位排列時(shí)(即豎直并排結(jié)構(gòu),如圖1所示)熱導(dǎo)最大,而在高溫區(qū)域三空穴豎直并排結(jié)構(gòu)的熱導(dǎo)最小;三空穴的相對(duì)位置變化僅能大幅調(diào)節(jié)IPMs高頻聲子的透射概率和高溫?zé)釋?dǎo),而三空穴的相對(duì)位置變化不僅能大幅調(diào)節(jié)垂直振動(dòng)模高頻聲子的透射概率和高溫?zé)釋?dǎo),也能大幅調(diào)節(jié)垂直振動(dòng)模低頻聲子的透射概率和低溫?zé)釋?dǎo).上述結(jié)果為這個(gè)領(lǐng)域提供了新的內(nèi)容,有助于研究人員從理論和實(shí)驗(yàn)上更深入地理解相關(guān)結(jié)構(gòu)中的聲學(xué)聲子輸運(yùn)和熱導(dǎo)性質(zhì).

2 理論模型與公式

圖1為含三個(gè)空穴鋸齒型石墨納米帶的模型圖,分為3個(gè)區(qū)域,根據(jù)Landauer公式和線性響應(yīng)假設(shè),石墨納米帶中的熱流可以寫為

其中fL/R(ω)=1/[exp(hω/kBTL/R)? 1]是石墨納米帶中的聲子Bose-Einstein分布函數(shù),τi(ω)是第i支模的聲子從左至右穿越散射區(qū)域的透射系數(shù).圖1左邊紅色區(qū)域(L)是溫度為T1的能量輸入熱庫,中間是聲子散射區(qū)域(C),右邊藍(lán)色區(qū)域(R)是溫度為T2的能量輸出熱庫.假設(shè)溫度差δT(δT=T1?T2>0)趨近于0,則可直接采用平均溫度T[T=(T1+T2)/2]計(jì)算.由于常溫下石墨烯的聲子平均自由程遠(yuǎn)大于同等尺寸大小的含空穴的石墨烯納米帶GNRs的平均自由程,所以電-聲相互作用忽略不計(jì).在彈性散射近似下,計(jì)算熱導(dǎo)的公式可寫為

其中,β=1/(kBT).研究表明[18],石墨納米帶中存在兩種聲學(xué)模:IPMs和FPMs,且這兩種聲學(xué)模的耦合可以忽略.因此,其哈密頓方程可表示為

石墨納米帶中總的聲子透射概率和熱導(dǎo)可分解為IPMs和FPMs的聲子透射概率和熱導(dǎo),并可對(duì)其進(jìn)行獨(dú)立計(jì)算.

圖1 (網(wǎng)刊彩色)具有三個(gè)空穴的鋸齒型石墨納米帶的結(jié)構(gòu)示意圖Fig.1. (color online)Schematics zigzag graphene nanoribbons with three cavities.

在諧波近似的情況下,FPMs的哈密頓方程可表示為

IPMs的哈密頓方程表示為

由于Brenner’s經(jīng)驗(yàn)勢(shì),可以在β計(jì)算碳原子之間的力常數(shù),對(duì)于FPMs,完整線性系統(tǒng)的動(dòng)力矩陣可表示為

IPMs的動(dòng)力矩陣可表示為

通過非平衡格林函數(shù)方法,FPMs的格林函數(shù)可以寫成

IPMs的格林函數(shù)可以寫成

本文中計(jì)算聲子的透射概率τi是計(jì)算熱導(dǎo)的關(guān)鍵,FPMs的透射概率可以表示為

IPMs的透射概率表示為

其 中ΓL(R),z = ?2ImΣL(R),z和ΓL(R),x-y =?2ImΣL(R),x-y分別是在IPMs和FPMs中作為能量輸入端和能量輸出端的接觸擴(kuò)展函數(shù),Gr(a)z和Gr(a)x-y分別是IPMs和FPMs中間散射區(qū)域的推遲/超前格林函數(shù),分別表示為

其中,Gaz=(Grz)+和Gax-y=(Grx-y)+,通過上述方法可計(jì)算出該結(jié)構(gòu)中的聲子透射概率和熱導(dǎo).

3 數(shù)值結(jié)果和分析

圖2描述了寬度n=15且含三個(gè)空穴的石墨納米帶中總的約化熱導(dǎo)隨溫度的變化關(guān)系.從圖2(a)和圖2(c)可以看到,當(dāng)溫度趨近于0 K時(shí),約化熱導(dǎo)總是趨近于3,這與三個(gè)空穴在量子結(jié)構(gòu)中的分布位置無關(guān),是由于當(dāng)T→0時(shí),僅有三支截止頻率為零的長波長聲學(xué)模被激發(fā),即兩支截止頻率為零的IPMs和一支截止頻率為零的FPMs,這些長波長聲學(xué)聲子的波長遠(yuǎn)長于散射結(jié)構(gòu)的結(jié)構(gòu)尺寸,散射結(jié)構(gòu)對(duì)長波長聲學(xué)聲子的散射很小的緣故.隨著溫度的升高,頻率較大的IPMs和FPMs被激發(fā),入射聲子與被結(jié)構(gòu)反射回的聲子相互作用,導(dǎo)致透射譜呈現(xiàn)振蕩透射峰-谷結(jié)構(gòu);同時(shí),這些較大頻率的聲學(xué)聲子被散射結(jié)構(gòu)散射,使聲子透射概率降低,導(dǎo)致約化熱導(dǎo)隨著溫度的升高而降低.隨著溫度的繼續(xù)升高,截止頻率大于零的聲學(xué)模被激發(fā),總的聲子透射概率明顯增大,因此約化熱導(dǎo)也隨溫度的升高而增大.比較圖2(a)中的實(shí)線、劃線、點(diǎn)線、點(diǎn)劃線發(fā)現(xiàn),三空穴的不同位置分布對(duì)應(yīng)的熱導(dǎo)曲線存在明顯的區(qū)別:從內(nèi)插圖中可以看出,在極低的溫度區(qū)域(0—6 K),三空穴豎直并排時(shí)熱導(dǎo)最大,當(dāng)空穴1和空穴3以空穴2為中心旋轉(zhuǎn)時(shí),約化熱導(dǎo)隨之降低;而在較高的溫度區(qū)域,三空穴豎直并排時(shí)熱導(dǎo)最小,當(dāng)空穴1和空穴3以空穴2為中心旋轉(zhuǎn)時(shí),約化熱導(dǎo)明顯增大;而且隨著溫度的升高,它們的約化熱導(dǎo)差值也隨著擴(kuò)大.這些熱導(dǎo)性質(zhì)能從圖2(b)中相對(duì)應(yīng)的透射譜得到很好的解釋:在低頻區(qū)域,雖然散射結(jié)構(gòu)對(duì)低頻聲子的散射導(dǎo)致了這些聲子的透射都呈現(xiàn)不規(guī)則的峰-谷結(jié)構(gòu),但是在圖2(b)中可清楚地看到,當(dāng)三空穴豎直并排時(shí)對(duì)聲子的散射影響最小,因此聲子透射相對(duì)較大,從而導(dǎo)致約化熱導(dǎo)在低溫區(qū)域相對(duì)較高;而在高頻區(qū)域,三空穴豎直并排時(shí)對(duì)聲子的透射概率明顯低于旋轉(zhuǎn)后的結(jié)構(gòu)或者旋轉(zhuǎn)后并平移的結(jié)構(gòu)對(duì)聲子的透射概率,因而導(dǎo)致三空穴豎直并排時(shí)的高溫?zé)釋?dǎo)明顯低于旋轉(zhuǎn)后的結(jié)構(gòu)或者旋轉(zhuǎn)后并平移的結(jié)構(gòu)的熱導(dǎo).觀察圖2(a)和圖2(c)所示的熱導(dǎo)曲線,不僅發(fā)現(xiàn)三空穴豎直并排時(shí)的約化熱導(dǎo)總是在低溫區(qū)域最大,在高溫區(qū)域最小,還發(fā)現(xiàn)兩幅圖中三空穴豎直并排時(shí)的約化熱導(dǎo)相對(duì)較大的低溫度區(qū)域范圍卻不一致:從內(nèi)插圖中可以看出,圖2(a)的低溫度范圍為0—6 K,而圖2(c)的低溫度范圍為0—15 K.三空穴豎直并排結(jié)構(gòu)與旋轉(zhuǎn)空穴結(jié)構(gòu)的透射譜比較發(fā)現(xiàn),三空穴豎直并排結(jié)構(gòu)對(duì)應(yīng)的透射曲線相對(duì)較大區(qū)域的范圍為0—10 cm?1(圖2(b)),而三空穴豎直并排結(jié)構(gòu)與平移空穴結(jié)構(gòu)的透射譜比較發(fā)現(xiàn),三空穴豎直并排結(jié)構(gòu)對(duì)應(yīng)的透射曲線較大區(qū)域的范圍為0—30 cm?1(圖2(d)).這表明,空穴的平移對(duì)低頻聲子的散射比空穴的旋轉(zhuǎn)對(duì)低頻聲子的散射影響更大,因而導(dǎo)致較大溫度范圍內(nèi)更低的低溫?zé)釋?dǎo).這些研究表明,空穴結(jié)構(gòu)在量子線中的分布能明顯調(diào)節(jié)量子線中的熱導(dǎo).

圖2 (網(wǎng)刊彩色)(a),(b)鋸齒型石墨納米帶中三空穴分布為圖1結(jié)構(gòu)(實(shí)線),以圖1結(jié)構(gòu)中的空穴2為中心,空穴1和3旋轉(zhuǎn)49?(短劃線),90?(點(diǎn)線),以及旋轉(zhuǎn)90?后三個(gè)空穴整體下移6α(點(diǎn)劃線)時(shí),總的約化熱導(dǎo)隨溫度的變化以及總的√透射概率隨頻率√的變化;(c),(d)鋸齒型石墨納√米帶中三空穴分布為圖1結(jié)構(gòu)(實(shí)線√),圖1結(jié)構(gòu)中固定空穴3,空穴2右移43α(短劃線),83α(點(diǎn)線),以及空穴2右移83α后再下移6α,且空穴1右移43α(點(diǎn)劃線)時(shí),總的約化熱導(dǎo)隨溫度的變化以及總的透射概率隨頻率的變化;α是原子間的鍵長;內(nèi)插圖表示相對(duì)應(yīng)結(jié)構(gòu)中低溫區(qū)域的約化熱導(dǎo)和低頻區(qū)域的透射概率Fig.2.(color online)(a),(c)The total reduced thermal conductance as a function of temperature;(b),(d)the total transmission rate as a function of frequency;in panels(a),(b),for the distributions of three cavities of zigzag graphene nanoribbons,solid curves correspond to the case shown in Fig.1,dashed and dotted curves respectively correspond to the cases that the cavities 1 and 3 rotate 49?and 90?about the cavity 2 fi xed as the centre in Fig.1,dash-dotted curves correspond to the case that the cavities 1 and 3 fi rst rotate 90?about the cavity 2,and then the three cavities move down 6α;in panels(c),(d),for the distributions of three cavities of zigzag graphene nanoribbons,solid curves correspond to the √case shown√ in Fig.1,dashed and dotted curves respectively correspond to the cases that the cavity 2 move right 43α and 83α w√hen the cavity 3 is fi xed in Fig.1,dash-dotted curves corres√pond to the case that the cavity 2 fi rst moves right 83α and then moves down 6α,and the cavity 1 moves right 43α.Here,α is the bond length between the atoms.The insets describe thetotalreducedthermalconductancein the low temperatureregion andthe total transmissionrate in the low frequency region,respectively.

為了研究空穴在石墨納米帶中不同位置分布對(duì)IPMs和FPMs熱輸運(yùn)性質(zhì)的影響,圖3給出了IPMs和FPMs的透射譜.從圖3可清楚地看到當(dāng)頻率趨近于0 K時(shí),IPMs與FPMs的透射概率分別趨近于2和1,這是由于分別有兩支截止頻率為零的長波長IPMs與一支截止頻率為零的FPMs被激發(fā)的緣故.隨著頻率的增加,截止頻率大于零的高頻聲學(xué)支被激發(fā),這些被激發(fā)的聲學(xué)模被散射結(jié)構(gòu)散射后與入射模相互作用,導(dǎo)致出現(xiàn)復(fù)雜的透射峰-谷結(jié)構(gòu).然而IPMs的透射譜的峰-谷結(jié)構(gòu)的振蕩劇烈程度明顯大于FPMs,這是由于在高頻區(qū)域,面內(nèi)的橫向聲學(xué)模與縱向聲學(xué)模都被激發(fā)且相互耦合,而面內(nèi)被激發(fā)的IPMs明顯多于FPMs,因此更多被激發(fā)的IPMs耦合導(dǎo)致其呈現(xiàn)振蕩更加劇烈的透射峰-谷結(jié)構(gòu).此外,研究發(fā)現(xiàn),雖然三空穴的平移與旋轉(zhuǎn)都能調(diào)節(jié)IPMs與FPMs的透射概率,然而對(duì)這兩類振動(dòng)模的調(diào)節(jié)幅度也存在明顯的差異:三空穴的相對(duì)位置改變對(duì)IPMs的低頻范圍的聲子透射調(diào)節(jié)不是很明顯,而對(duì)高頻范圍的聲子透射調(diào)節(jié)卻較大;而三空穴的相對(duì)位置改變對(duì)FPMs的低頻區(qū)域與高頻區(qū)域的聲學(xué)聲子的透射概率的調(diào)節(jié)都非常大.這些研究表明可通過調(diào)節(jié)空穴的不同位置分布來有區(qū)別地調(diào)制IPMs與FPMs的透射概率.

圖3 (網(wǎng)刊彩色)(a),(c)鋸齒型石墨納米帶中三空穴分布為圖1結(jié)構(gòu)(實(shí)線),以圖1結(jié)構(gòu)中的空穴2為中心,空穴1和3旋轉(zhuǎn)49?(短劃線),90?(點(diǎn)線),以及旋轉(zhuǎn)90?后三個(gè)空穴整體下移6α(點(diǎn)劃線)時(shí),IPM√s和FPMs的透√射概率;(b),(d)鋸齒型石墨納√米帶中三空穴分布為圖1結(jié)構(gòu)(實(shí)√線),圖1結(jié)構(gòu)中固定空穴3,空穴2右移43α(短劃線),83α(點(diǎn)線),以及空穴2右移83α后再下移6α,且空穴1右移43α(點(diǎn)劃線)時(shí),IPMs和FPMs的透射概率Fig.3.(color online)The transmission rates of IPMs(panels(a),(b))and FPMs(panels(c),(d));in panels(a),(c),for the distributions of three cavities of zigzag graphene nanoribbons,solid curves correspond to the case shown in Fig.1,dashed and dotted curves respectively correspond to the cases that the cavities 1 and 3 rotate 49?and 90?about the cavity 2 fi xed as the centre in Fig.1,dash-dotted curves correspond to the case that the cavities 1 and 3 fi rst rotate 90?about the cavity 2,and then the three cavities move down 6α;in panels(b),(d),for the distributions of three cavities of zigzag graphene nanoribbons,solid curves correspond to the√ case shown√ in Fig.1,dashed and dotted curves respectively correspond to the cases that the cavity 2 move right 43α and 83α w√hen the cavity 3 is fi xed in Fig.1,dash-dotted curves corresp√ond to the case that the cavity 2 fi rst moves right 83α and then moves down 6α,and the cavity 1 moves right 43α.

圖4 (網(wǎng)刊彩色)(a),(c)鋸齒型石墨納米帶中三空穴分布為圖1結(jié)構(gòu)(實(shí)線),以圖1結(jié)構(gòu)中的空穴2為中心,空穴1和3旋轉(zhuǎn)49?(短劃線),90?(點(diǎn)線),以及旋轉(zhuǎn)90?后三個(gè)空穴整體下移6α(點(diǎn)劃線)時(shí),IPM√s和FPMs的約√化熱導(dǎo);(b),(d)鋸齒型石墨納√米帶中三空穴分布為圖1結(jié)構(gòu)(實(shí)線√),圖1結(jié)構(gòu)中固定空穴3,空穴2右移43α(短劃線),83α(點(diǎn)線),以及空穴2右移83α后再下移6α,且空穴1右移43α(點(diǎn)劃線)時(shí),IPMs和FPMs的約化熱導(dǎo)Fig.4.(color online)The reduced thermal conductance of IPMs(panels(a),(b))and FPMs(panels(c),(d));in panels(a),(c),for the distributions of three cavities of zigzag graphene nanoribbons,solid curves correspond to the case shown in Fig.1,dashed and dotted curves respectively correspond to the cases that the cavities 1 and 3 rotate 49?and 90?about the cavity 2 fi xed as the centre in Fig.1,dash-dotted curves correspond to the case that the cavities 1 and 3 fi rst rotate 90?about the cavity 2,and then the three cavities move down 6α;in panels(b),(d),for the distributions of three cavities of zigzag graphene nanoribbons,solid curves correspond to the√ case show√nin Fig.1,dashed and dotted curves respectively correspond to the cases that the cavity 2 move right 43α and 83α w√hen the cavity 3 is fi xed in Fig.1,dash-dotted curves corresp√ond to the case that the cavity 2 fi rst moves right 83α and then moves down 6α,and the cavity 1 moves right 43α.

圖4給出了IPMs與FPMs的約化熱導(dǎo)隨溫度的變化.從圖4可以看出,當(dāng)溫度趨近于0 K時(shí),IPMs與FPMs的約化熱導(dǎo)分別趨近于2和1且不受散射區(qū)域的影響,這是由于溫度趨近于0 K時(shí)只有兩支起振頻率為零的面內(nèi)長波長聲學(xué)模和一支起振頻率為零的垂直平面長波長振動(dòng)聲學(xué)模被激發(fā)的緣故.在極低溫度范圍(0—10 K)內(nèi)約化熱導(dǎo)隨溫度的升高呈現(xiàn)下降的趨勢(shì),這是由于隨著溫度的升高,一些高頻聲子被激發(fā),被激發(fā)的高頻率聲子被散射,結(jié)構(gòu)散射導(dǎo)致了約化熱導(dǎo)的下降,隨著溫度的進(jìn)一步上升,起振頻率大于零的高頻聲學(xué)支被激發(fā),約化熱導(dǎo)迅速上升.對(duì)于IPMs,不管是兩空穴繞中間空穴的相對(duì)旋轉(zhuǎn),還是空穴間的相對(duì)平移,在約0—150 K的溫度范圍內(nèi),約化熱導(dǎo)幾乎不受影響.這是因?yàn)榈蜏胤秶臒釋?dǎo)主要由低頻聲子的輸運(yùn)貢獻(xiàn).從圖3(a)和圖3(b)可以看到,IPMs模的低頻聲子的輸運(yùn)幾乎不受空穴在量子線中的相對(duì)錯(cuò)位的影響,因此熱導(dǎo)也幾乎不受空穴相對(duì)錯(cuò)位的影響,包括空穴間的相對(duì)平移和相對(duì)旋轉(zhuǎn).而在溫度大于150 K的相對(duì)高溫區(qū)域,大量高頻聲子被激發(fā),熱導(dǎo)主要是由高頻聲子的輸運(yùn)而貢獻(xiàn)的.從圖3(a)和圖3(b)所示透射譜可清晰地看到,高頻聲子的透射明顯依賴于空穴在量子線中的位置分布.因此,IPMs貢獻(xiàn)的高溫?zé)釋?dǎo)也依賴于空穴在量子線中的位置分布.而對(duì)于FPMs貢獻(xiàn)的熱導(dǎo)在低溫到高溫范圍都高度依賴于三空穴的相對(duì)旋轉(zhuǎn)以及相對(duì)平移,這是由于FPMs的透射概率在低頻范圍和高頻范圍都高度依賴于這些空穴的相對(duì)平移和旋轉(zhuǎn),這一點(diǎn)可清楚地從圖3(c)和圖3(d)中看到.比較圖4(c)和圖4(d),可看到三空穴的相對(duì)旋轉(zhuǎn)對(duì)FPMs約化熱導(dǎo)的調(diào)節(jié)效果比三空穴的相對(duì)平移對(duì)FPMs約化熱導(dǎo)的調(diào)節(jié)效果明顯.這是由于三空穴的相對(duì)旋轉(zhuǎn)對(duì)FPMs高頻聲子的透射概率的影響大于三空穴的相對(duì)平移對(duì)FPMs高頻聲子的透射概率的影響.這些研究表明,可通過空穴平移或旋轉(zhuǎn)來分別調(diào)制IPMs與FPMs的約化熱導(dǎo).

4 結(jié) 論

采用非平衡格林函數(shù)方法,在保持石墨納米帶帶寬不變的情況下,研究了存在三個(gè)空穴的石墨納米帶中彈性聲學(xué)聲子輸運(yùn)和熱導(dǎo)特性,得到了一些有趣的物理結(jié)果:1)三空穴豎直并排結(jié)構(gòu)對(duì)低頻聲子的散射較小,導(dǎo)致低溫區(qū)域三空穴豎直并排時(shí)熱導(dǎo)最大;2)三空穴豎直并排結(jié)構(gòu)對(duì)高頻聲子的散射較大,導(dǎo)致在較高溫度區(qū)域三空穴豎直并排時(shí)熱導(dǎo)最小;3)三空穴的相對(duì)錯(cuò)位分布僅能較大幅度地調(diào)節(jié)IPMs高頻聲子的透射概率,導(dǎo)致三空穴的相對(duì)平移或旋轉(zhuǎn)僅能大幅調(diào)節(jié)IPMs的高溫?zé)釋?dǎo);4)三空穴的相對(duì)錯(cuò)位分布能較大幅度地調(diào)節(jié)FPMs的高頻聲子和低頻聲子的透射概率,導(dǎo)致三空穴的相對(duì)平移或旋轉(zhuǎn)不僅能大幅調(diào)節(jié)FPMs的高溫?zé)釋?dǎo),也能大幅調(diào)節(jié)FPMs的低溫?zé)釋?dǎo).此研究結(jié)果可為設(shè)計(jì)基于石墨納米帶的熱量子器件提供有價(jià)值的物理模型和理論參考.

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PACS:63.22.Rc,73.23.Ad,44.10.+iDOI:10.7498/aps.66.126302

In fl uence of multi-cavity dislocation distribution on thermal conductance in graphene nanoribbons?

Zhou Xin1)Gao Ren-Bin1)Tan Shi-Hua1)Peng Xiao-Fang1)?Jiang Xiang-Tao2)?Bao Ben-Gang3)

1)(Institute of Mathematics and Physics,Central South University of Forestry and Technology,Changsha 410004,China)
2)(Institute of Computer and Information Engineering,Central South University of Forestry and Technology,Changsha 410004,China)
3)(Office of Academic A ff airs,Hunan University of Science and Engineering,Yongzhou 425100,China)

22 February 2017;revised manuscript

31 March 2017)

Using non-equilibrium Green’s function method and keeping the zigzag carbon chains unchanged,we investigate the transmission rate of acoustic phonon and the reduced thermal conductance in the graphene nanoribbons with three cavities.The results show that the reduced thermal conductance approaches to 3π2k2BT/(3h)in the limit T → 0 K.Due to the fact that only long wavelength acoustic phonons with zero cuto fffrequency are excited at such low temperatures,the scattering in fl uence on the long wavelength acoustic phonons by the dislocation distribution of three cavities in the graphene nanoribbons can be ignored and these phonons can go through the scattering region perfectly.As the temperature goes up,the reduced thermal conductance decreases.This is because the high-frequency phonons are excited and these high-frequency phonons are scattered easily by the scattering structures.With the further rise of temperature,acoustic phonon modes with the cuto fffrequency greater than zero are excited,which leads to a rapid increase of the reduced thermal conductance.This study shows that in higher frequency region,the transmission spectra display complex peak-dip structures,which results from the fact that in higher frequency region,more phonon modes are excited and scattered in the middle scattering region with three cavities,and the scattering phonons are coupled with the incident phonons.When the three cavities are aligned perpendicularly to the edge of the graphene nanoribbons,the scattering from low-frequency phonons by the scattering structures is smallest,which leads to the fact that the reduced thermal conductance is largest at low temperatures;however,at high temperatures,the reduced thermal conductance is smallest when the three cavities is aligned perpendicularly to the edge of the graphene nanoribbons.This is because the scattering from high-frequency phonons by the scattering structures is biggest.These results show that the acousticphonon transport and the reduced thermal conductance are dependent on the relative position of the three cavities.In addition,the dislocation distribution of the three cavities can only modulate obviously the high-temperature thermal conductance of the in-plane modes(IPMs).This is because the change of the relative position of the quantum dots can only modulate greatly the high-frequency phonon transmission rate and less modulate the low-frequency phonon transmission rate of the IPMs.However,the dislocation distribution of the three cavities can adjust obviously not only the high-temperature thermal conductance of the fl exural phonon modes(FPMs),but also the low-temperature thermal conductance of the FPMs.This is because the change of the relative position of the three cavities can modulate greatly phonon transmission rates of fl exural phonon modes in the low-frequency and high-frequency regions.These results provide an e ff ective theoretical basis for designing the thermal transport quantum devices based on graphene nanoribbons.

nonequilibrium Green’s functions,acoustic phonon transport,thermal conductance,quantum system

10.7498/aps.66.126302

?國家自然科學(xué)基金(批準(zhǔn)號(hào):11247030,61272147,61602529)、湖南省自然科學(xué)基金(批準(zhǔn)號(hào):14JJ4054)、湖南省教育廳基金(批準(zhǔn)號(hào):12B136,12C0446)、中南林業(yè)科技大學(xué)人才引進(jìn)計(jì)劃(批準(zhǔn)號(hào):104-0160)和中南林業(yè)科技大學(xué)研究生科技創(chuàng)新基金(批準(zhǔn)號(hào):CX2016B26)資助的課題.

?通信作者.E-mail:xiaofangpeng11@163.com

?通信作者.E-mail:xtjiang@csuft.edu.cn

?2017中國物理學(xué)會(huì)Chinese Physical Society

http://wulixb.iphy.ac.cn

*Project supported by the National Natural Science Foundation of China(Grant Nos.11247030,61272147,61602529),the Hunan Provincial Natural Science Foundation,China(Grant No.14JJ4054),the Research Foundation of Hunan Provincial Education Department,China(Grant Nos.12B136,12C0446),the Talent Introducing Foundation of Central South University of Forestry and Technology,China(Grant No.104-0160),and the Scienti fi c Innovation Fund for Graduate of Central South University of Forestry and Technology,China(Grant No.CX2016B26).

?Corresponding author.E-mail:xiaofangpeng11@163.com

?Corresponding author.E-mail:xtjiang@csuft.edu.cn