劉以琳，田宏玉

(1.淮陰工學(xué)院 電子信息工程學(xué)院,江蘇 淮安,223001；2.鹽城工學(xué)院 數(shù)理學(xué)院,江蘇 鹽城,224051)

?

非對(duì)稱石墨烯p-n結(jié)中電子聚焦點(diǎn)移動(dòng)現(xiàn)象

劉以琳1，田宏玉2

(1.淮陰工學(xué)院 電子信息工程學(xué)院,江蘇 淮安,223001；2.鹽城工學(xué)院 數(shù)理學(xué)院,江蘇 鹽城,224051)

基于電子光學(xué)性質(zhì),文章研究了非對(duì)稱石墨烯p-n結(jié)中電子聚焦點(diǎn)的移動(dòng)現(xiàn)象.應(yīng)用非平衡格林函數(shù)方法展示了不同情況下聚焦點(diǎn)移動(dòng)規(guī)律,并根據(jù)電子在界面散射行為解釋了該現(xiàn)象.

電子光學(xué)性質(zhì);聚焦點(diǎn)移動(dòng)現(xiàn)象;界面散射

在對(duì)稱石墨烯p-n結(jié)中出現(xiàn)的電子束聚焦現(xiàn)象是電子學(xué)中一個(gè)很有趣的現(xiàn)象[1],而且最近實(shí)驗(yàn)上也對(duì)這一現(xiàn)象進(jìn)行了驗(yàn)證[2]。本文我們考慮了一個(gè)不對(duì)稱的石墨烯p-n結(jié)，研究電子經(jīng)過這種隧道結(jié)后的聚焦現(xiàn)象，這對(duì)于了解電子在不同的石墨烯p-n結(jié)界面散射行為有很重要的幫助。我們所采用的理論模型如圖1所示， 其中左邊半無限大石墨烯是電子型，右邊半無限大石墨烯是空穴型，界面位于x=0處。系統(tǒng)哈密頓如下：

(1)

圖1 (a)不對(duì)稱石墨烯p-n結(jié)，其中在兩個(gè)區(qū)域區(qū)域施加不相等的門壓(Vg1≠Vg2)或者將p區(qū)域放在h-BN襯底上。電子束由位于n區(qū)域的一個(gè)點(diǎn)電極注入，在p區(qū)域收集。(b)電子折射示意圖，其中θ1和θ2入射角和折射角，θc是發(fā)生反射的臨界角度。

Fig.1 (a)Plot of a schematic of the asymmetric graphene PN junction where the unequal gate voltate(Vg1≠Vg2) or deposited on the h-BN substrate.The electron flow is injected at the source point in thenregion and collected in thepregion with an extended drain.(b) Schematic diagram of the electron refraction process withθ1andθ2the incident and refraction angles,θcis the critical angle where the reflection occurs.

當(dāng)電子由n區(qū)域點(diǎn)電極注入以后，響應(yīng)信號(hào)會(huì)分布在整個(gè)區(qū)域。為了研究不對(duì)稱p-n結(jié)中的聚焦現(xiàn)象，我們考察散射區(qū)的局域電子密度δρ(i),

(2)

圖2 不對(duì)稱扶手型p-n結(jié)((a),(c),(d))和鋸齒型p-n結(jié)(d)電子密度δρ(i)圖。其中En=-0.05t,(a),(b)圖中Ep=0.05t,(c)圖Ep=0.02t,(d)圖Ep=0.06t。對(duì)于扶手型邊界石墨帶，源點(diǎn)電極在(-300×3a,0)，帶寬度為W=401×b。對(duì)于鋸齒型石墨帶，源電極在(-600b，0),帶寬為W=401×3a.Fig.2 Distribution of the local particle density variation δρ(i) in an armchair PN junction((a),(c),(d))and zigzag PN junction(b) with En=-0.05t,Ep=0.05t in (a),(b),Ep=0.02t in (c)and Ep=0.06t in(d).For the armchair ribbon,the source flow is injected from the honeycomb unit cell at (-300×3a,0)with ribbon width W=401×b.For the zigzag one,the source position is at (-600b,0) with ribbon width W=401×3a.

在圖2中，展現(xiàn)了不對(duì)稱石墨烯p-n結(jié)中電子聚焦點(diǎn)的偏移現(xiàn)象。對(duì)于對(duì)稱的p-n結(jié)，由(-L,0)點(diǎn)注入的電子總能在(L,0)點(diǎn)聚焦，如圖(a)和(b)所示，但是扶手型邊界石墨帶聚焦效果更好，更適合用來研究聚焦點(diǎn)偏移現(xiàn)象。在圖2(c)和(d)中，我們發(fā)現(xiàn)，當(dāng)Ep<-En的時(shí)候，聚焦點(diǎn)會(huì)向左側(cè)移動(dòng)，而Ep>-En時(shí)候，聚焦點(diǎn)會(huì)向右側(cè)移動(dòng)。這是因?yàn)楦鶕?jù)y方向動(dòng)量守恒knsinθ1=-kpsinθ2，當(dāng)Ep減小時(shí)，相應(yīng)費(fèi)米動(dòng)量Kp也會(huì)減小，從而散射角θ2增加，聚焦點(diǎn)向左側(cè)移動(dòng)。

圖3 扶手型石墨帶p-n結(jié)電子密度δρ(i)圖，(a)和(b)參數(shù)同圖2(2)和(c)相同。Fig.3 Distribution of the local particle density variation δρ(i) in an armchair PN junction near the interface.Parameters are the same as Fig.2(a) and(c).

除了聚焦點(diǎn)的移動(dòng)現(xiàn)象，由圖2(c)我們還發(fā)現(xiàn)，當(dāng)Ep<-En時(shí)聚焦點(diǎn)的強(qiáng)度會(huì)變?nèi)酢榱私沂具@一現(xiàn)象的內(nèi)在性質(zhì)，我們畫出了圖2(a)和(c)在界面(x=0)附近的密度，如圖3(a)和(b)所示。發(fā)現(xiàn)在x<0端，(b)圖的密度要比(a)圖密度大，而在x>0端它的密度比(a)圖小，說明電子在界面上發(fā)生了反射，正是這種反射造成圖2(c)聚焦點(diǎn)強(qiáng)度變?nèi)酢?/p>

圖4 不同的交錯(cuò)勢(shì)Δ下扶手型石墨烯PN junction電子密度δρ(i)分布圖，(a)-(d)中Δ=0,0.01t,0.02t,0.03t，其他參數(shù)同圖2(a)相同。Fig.4 Distribution of the local particle density variation δρ(i) in an armchair PN junction for different sublattice potentials Δ,Δ=0,0.01t,0.02t,0.03t in(a)-(d).Other parameters are the same as that in Fig.2(a).

如果將p區(qū)域石墨烯放在h-BN基底上，聚焦點(diǎn)也會(huì)發(fā)生移動(dòng)，因?yàn)槭┖鸵r底之間相互作用會(huì)影響線性色散關(guān)系，結(jié)果如圖4所示。我們發(fā)現(xiàn)隨著Δ增加，聚焦點(diǎn)向左移動(dòng)并且其強(qiáng)度逐漸減弱，這是由于隨著Δ增加，p區(qū)域電子動(dòng)量逐漸減小，從而在界面上的反射逐漸增強(qiáng)導(dǎo)致的。

總之，我們探討了不對(duì)稱石墨烯p-n結(jié)中聚焦點(diǎn)的移動(dòng)現(xiàn)象，發(fā)現(xiàn)隨著p區(qū)域電子濃度改變，聚焦點(diǎn)會(huì)發(fā)生移動(dòng)，并且其強(qiáng)度也隨之發(fā)生變化。

[1]WILLIAMS J R,LOW T,LUNDSTROM M S,et al.Gate-controlled guiding of electrons in grapheme[J].Nature Nanotechnology,2011,6(4):222-225.

[2]CHEIANOV VV ,FAL’KO1 V,ALTSHULER BL.The focusing of electron flow and a veselago lens in graphene p-n junctions[J].Science,2007,315(5816):1252-1255.

[3]LEE GH,PARK GH,LEE HJ.Observation of negative refraction of Dirac fermions in graphene[J].Journal of Applied Physics,2015,113(15):666.

[4]CHEN S，HAN Z，ELAHI MM,et al,Electron optics with p-n junctions in ballistic graphene[J].Science,2016,353(6307):1522-1525.

[5]XING Y,WANG J,SUN Q.The focusing of electron flow in a bipolar graphene ribbon with different chiralities[J].Phys.Rev.B,2010,81(16):165425-165435.

[6]TIAN HY,CHAN KS,WANG J.Efficient spin injection in graphene using electron optics[J].Physical Review B Condensed Matter,2012,86(24):59-73.

[7]YOUNG AF，SANCHEZ-YAMAGISHI JD ，HUNT B，et al.Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state[J].Nature,2014,505(7484):528-532.

[8]XIAO D,YAO W,NIU Q.Valley-Contrasting Physics in Graphene:Magnetic Moment and Topological Transport[J].Phys.Rev.Lett.,2007,99(23):236809-236812.

The shift of the focal point in the asymmetric graphene p-n junction

LIU Yilin1,TIAN Hongyu2

(1.Faculty electric information engineering,Huaiyin Institute of Technology,Huaian 223001,China;2.Department of Physics,Yancheng Institute of Technology,Yancheng 224051,China )

We theoretically investigated the shift of the focal point in the asymmetric graphene p-n junction(PNJ) based on electron optics.In terms of the nonequilibrium Green’s function technique,we numerically displayed the shift rule of the focusing point,and explained the phenomenon according to the scattering properties of electrons at the interface.

electron optics；the shift of the focal point；interface scatter

1672-7010(2017)03-0085-04

2017-01-08

國(guó)家自然科學(xué)基金資助項(xiàng)目(11447218)

田宏玉(1981-)，男，江蘇鹽城人。講師，博士，從事電子輸運(yùn)輸運(yùn)性質(zhì)研究。

O472+.4

A