纖鋅礦氮化鎵基階梯量子阱中界面聲子的色散譜與散射率

張立

(廣州番禺職業(yè)技術學院 電子技術與材料開發(fā)研究所,廣州　511483)

纖鋅礦氮化鎵基階梯量子阱中界面聲子的色散譜與散射率

張立

(廣州番禺職業(yè)技術學院 電子技術與材料開發(fā)研究所,廣州511483)

摘要:本文理論分析了纖鋅礦GaN-基階梯量子阱中的電子界面光學聲子散射性質(zhì)。階梯量子阱中的解析的界面聲子態(tài)及Fr?hlich電子聲子相互作用哈密頓被導出了。在考慮強內(nèi)建電場效應及能帶的非拋物性特性的情況下,階梯量子阱結(jié)構(gòu)精確解析的電子本征態(tài)也被給出了。以一個四層纖鋅礦AlN-基階梯量子阱為例進行了數(shù)值計算。結(jié)果發(fā)現(xiàn),系統(tǒng)中存在四支界面光學聲子模,這一觀察明顯不同于對稱的GaN/AlN單量子阱與雙量子阱的情況。這一差異被歸結(jié)為階梯量子結(jié)構(gòu)的非對稱性。GaN-基階梯量子阱中的子帶內(nèi)散射率與子帶間散射率比GaAs-基階梯量子阱的結(jié)果大一個數(shù)量級,這被歸因于GaN-基晶體大的電子聲子耦合常數(shù)。GaN-基階梯量子阱的子帶內(nèi)散射率表現(xiàn)出與GaAs-基體系類似的結(jié)構(gòu)參數(shù)依賴關系,但兩類體系的子帶間散射率對階梯量子阱結(jié)構(gòu)參數(shù)依賴則明顯不同,這被歸結(jié)為GaN-基階梯量子阱結(jié)構(gòu)中強的內(nèi)建電場效應及帶的非拋物性。結(jié)果還表明,高頻界面聲子模相對于低頻界面聲子模,對散射率的貢獻更大。

關鍵詞:電子聲子散射率;界面聲子模;纖鋅礦階梯量子阱;內(nèi)建電場;帶非拋物性

1引言

自從上世紀九十年代以來,隨著先進的晶體生長技術(如金屬有機物化學氣相沉積、分子束外延等)巨大進步及一系列優(yōu)異光電性能的發(fā)現(xiàn),如寬的可調(diào)帶隙、高的電子遷移率、優(yōu)良的熱穩(wěn)定性及高光學效率等[14],包括GaN、AlN、InN及其三元混晶AlGaN、InGaN等短波長氮化物半導體,吸引的研究人員的持續(xù)興趣。自然,極化光學聲子模研究及聲子參與躍遷的研究工作也從窄帶GaAs-基半導體異質(zhì)結(jié)體系擴展到了寬帶GaN-基半導體納米異質(zhì)結(jié)體系[15-21]。

Fig.1Schematic diagram of confined potential profile inz-direction of the wurtzite GaN-based step QW

讓我們考慮一個纖鋅礦四層階梯量子阱,它的阱寬與壘寬分別設為Lw與Lb(Lw=z1,Lb=z2-z1)。我們?nèi)±w鋅礦晶體的c-軸方向作為坐標系的z-方向,并用字母t(z)表示徑向(軸向)。三個異質(zhì)結(jié)界面分別位于z=z0,z1,z2處(詳細情況參考圖1)。

在宏觀的介電連續(xù)及London的單軸晶體模型框架下,考慮到界面聲子勢的特征,即從界面處向兩邊衰減,圖1所示的纖鋅礦階梯量子阱中的界面聲子勢函數(shù)可寫為

(1)

其中kt是xy-平面內(nèi)的聲子波數(shù),ai和bi是界面聲子模的耦合系數(shù),γi(i=1,2,3,4分別相應于左側(cè)AlN勢壘、GaN、AlxGa1-xN及右側(cè)AlN勢壘)定義如下

(2)

其中

(3)

在方程(3)中,ωz,L,ωz,T,ωt,L,ωt,T分別表示晶體材料的A1(LO),A1(TO),E1(LO)及E1(TO)對稱性的中心特征頻率。

采用標準的量子化步驟[15-21],界面聲子場由下式給出,即

(4)

與湮滅算符,他們滿足下面的玻色對易規(guī)則,

(5)

[bktexp(ikt·ρ)+H.c.]

(6)

(7)

式子(7)中的Nkt表示為

(8)

2.2帶強內(nèi)建電場的階梯量子阱中的電子本征態(tài)

由于纖鋅礦氮化物異質(zhì)結(jié)附近存在強的壓電極化及氮化物晶體存在的自發(fā)極化[27-28],因而導致氮化物量子阱內(nèi)存在強的內(nèi)建電場,其量級約為MV/cm。為了簡單起見,我們采用跟Shi及Takeuchi[27,31]類似處理,即忽略最左側(cè)與最右側(cè)的厚壘層AlN材料中的復雜應變,僅考慮中間有限厚度材料層中的應變。在這種情況下,階梯量子阱中阱層GaN材料中的內(nèi)建電場強度可寫為[27-28]

(9)

同樣在階梯勢壘AlGaN層中的內(nèi)建電場強度可寫為

(10)

(11)

其中k‖與r‖分別是xy-平面上的電子波矢與坐標,Uc(r)是導帶k=0處Bloch函數(shù)的周期部分,Ai(x)與Bi(x)是兩個Airy函數(shù)。在方程(11)中的符號ki(i=1,4)代表在第i個區(qū)域[-∞,-z0]及[z0,+∞]中的電子波數(shù),它們由下面兩個式子給出

(12)

k4=

(13)

方程(11)中的另外兩個函數(shù)ξi(z)(i=2,3)定義為

(14)

(15)

(16)

(17)

基于通常的費米黃金規(guī)則,在僅考慮單聲子激發(fā)的情況下,纖鋅礦階梯量子阱中電子界面光學聲子散射率可寫為

(18)

其中Nph是界面聲子的占有數(shù),上面(下面)的符號表示聲子的吸收(發(fā)射),|ki〉是吸收或發(fā)射聲子前的電子狀態(tài),而|kf〉表示吸收或發(fā)射聲子之后的狀態(tài)。Fm(k)是交疊積分函數(shù),它被定義為

(19)

對第一子帶內(nèi)(即基態(tài)內(nèi))的散射過程,我們考慮聲子發(fā)射過程的散射率W(1→1)。在電子能量剛好足夠發(fā)射一個界面聲子的情況下,基于方程(18),我們獲得[3]

(20)

其中k11由子帶內(nèi)躍遷散射過程中的能量守恒定律來決定,即

(21)

對第一與第二子帶間的躍遷,我們也考慮剛好發(fā)射一個界面聲子情況的散射率W(2→1),這樣我們能得到子帶間散射率表達式為

(22)

其中k21滿足下面的關系

(23)

其中(E2-E1)是第一子帶與第二子帶間的能量間隔。

3數(shù)值結(jié)果與討論

我們以一個四層AlN/GaN/AlxGa1-xN/AlN階梯量子阱為例進行了數(shù)值計算。計算中所用到的材料參數(shù)列在了表1中。

表1　GaN,AlN及AlxGa1-xN的材料參數(shù)[18,27,34,35]

Fig.2Dispersive spectra ?ω of the IO modes as a function of the free wave-numberktinxy-plane

Fig.3Electrostatic potential distributions of the four branches of IO modes in confinedz-direction when wave-numberktis kept at 0.32 nm-1

Fig.4Intrasubband electron-phonon scattering ratesW(a) and intersubband scattering ratesW(b) as a function ofLwwhenLbis kept ataB(aBis the Bohr radius of the GaN material,aB2.5 nm).The solid(dash)lines denote the high frequency mode 4(the low frequency mode 1)

在圖5中,我們描繪了子帶內(nèi)散射率(圖5a)與子帶間散射率(圖5b)隨階梯量子阱結(jié)構(gòu)中的階梯勢壘寬度Lb變化的函數(shù)曲線。在繪圖過程中,GaN阱寬保持在一個aB,階梯壘的摻雜深度x固定在0.5。與圖4的情況類似,圖5中的實線與虛線分別相應于高頻界面模與低頻界面模。從圖中可以看到,子帶間散射率W是Lb的單調(diào)遞減函數(shù)(圖5a),而子帶內(nèi)散射率W(21)則在某到特定的壘寬Lb處取最大值(圖5b)。對比GaAs-基階梯量子阱[3],再次發(fā)現(xiàn)此處的子帶內(nèi)與子帶間的散射率比GaAs-基階梯量子阱的要大一個數(shù)量級。兩類體系中子帶間散射率對Lb的依賴關系是一致的,即W隨著Lb的增加而減小。但兩類體系中的子帶內(nèi)的散射率W對Lb依賴特性明顯不同,這也被歸結(jié)為GaN-基量子結(jié)構(gòu)的非對稱性及結(jié)構(gòu)中存在的強內(nèi)電場效應。

圖6給出了GaN-基階梯量子結(jié)構(gòu)中子帶內(nèi)及子帶間散射率作為階梯壘的摻雜深度x的變化關系曲線。其中結(jié)構(gòu)中的GaN阱寬及AlxGa1-xN階梯壘寬均被固定在一個玻爾半徑aB。可以看到子帶內(nèi)散射率W是x的單調(diào)遞增函數(shù),而W則是x的單調(diào)遞減函數(shù)。GaN-基階梯量子阱中的子帶內(nèi)與子帶間的散射率明顯區(qū)別于GaAs-基階梯量子阱的情況。在Al0.35Ga0.65As/GaAs/AlxGa1-xN/ Al0.4Ga0.6As階梯量子阱中,子帶間散射率是x的敏感函數(shù),而子帶內(nèi)散射率則是x的復雜函數(shù)[3]。

Fig.5Intrasubband scattering rate(a)and the intersubband scattering rate(b)as a function ofLbwhenLwis kept ataB,andxof step AlxGa1-xN is kept atx= 0.5

Fig.6Intrasubband scattering rates(a)and intersubband rates(b)as a function of the Al doped concentrationxof step AlxGa1-xN barrier whenLwandLbare kept ataB

4結(jié)論

參考文獻

[1]Stroscio M A.Interface-phonon-assisted transitions in quantum-well lasers[J].J Appl Phys,1996,80:6864-6867.

[2]Shi J J,Liu Z,Pan S H.Electron-phonon scattering in asymmetric semiconductor quantum well structures [J].Superlatt Microstru,1995,17:329-334.

[3]Shi J,Pan S H.Electron-interface-phonon interaction and scattering in asymmetric semiconductor quantum-well structures [J].Phys Rev B,1995,51:17681-17688 .

[4]Shi J,Pan S H,Liu Z X.Interaction between an electron and optical phonons in polar semiconductor heterostructures [J].Z Phys B,1996,100:353-364.

[5]Kinsler P,Kelsall R W,Harrison P.Interface and confined phonons in stepped quantum wells [J].Physica B,1999,263-264:507-509.

[6]Duan W H,Zhu J L,Gu B L.Electron-interface-phonon scattering in graded quantum wells of Ga1-xAlxAs [J].Phys Rev B,1994,49:14403-14408.

[7]Teng H B,Sun J P,Haddad G I,etal.Phonon assisted intersubband transitions in step quantum well structures [J].J Appl Phys,1998,84:2155-2164.

[8]Shi J,Pan S H.Polar optical oscillations in coupled quantum wells:The electron-phonon interaction and scattering [J].J Appl Phys,1996:80:3863-3875.

[9]Gao X,Botex D,Knezevic I.Phonon confinement and electron transport in GaAs-based quantum cascade structures[J].J Appl Phys,2008,103:073101.

[10]Michael A,Mitra D.Phonons in Nanostructures [M].Cambridge University Press,2001.

[11]Jeganathan K,Debnath R K,Meijers R,etal.,Raman scattering of phonon-plasmon coupled modes in self-assembled GaN nanowires [J].J Appl Phys,2009,105:123707(1-5).

[12]Zhang L,Shi J J,Shao P M.Surface polaronic effect on donor-impurity states of a wurtzite nitride nanowire:two-parameter variational approach [J].J Appl Phys,2011,110:013712(1-9).

[13]Zhang L,Shi J J.Polar optical phonon states and their degenerative behaviors of wurtzite ZnO/MgZnO coupling quantum dots [J].Int J Mod Phys B,2014,28:1430005(1-32).

[14]Nakamura S,Chichibu S F.Introduction to nitride semiconductor bule lasers and light emitting diodes [M].Taylor and Francis,London,2000.

[15]Zhang L,Shi J J,Gao S.Polaronic self-trapped energy and effective mass of a cylindrical wurtzite GaN nanowire [J].Semicond Sci Technol,2008,23:045014(1-10).

[16]Komirenko S M,Kim K W,Stroscio M A,etal.Dispersion of polar optical phonons in wurtzite quantum wells [J].Phys Rev B,59(1999):5013-5020.

[17]Lee B C,Kim K W,Dutta M,etal.Electron-optical-phonon scattering in wurtzite crystals [J].Phys.Rev.B 1997,56:997-1000; Optical-phonon confinement and scattering in wurtzite heterostructures [J].Phys Rev B,1998,58:4860-4865.

[18]Shi J J.Interface optical-phonon modes and electron-interface-phonon interactions in wurtzite GaN-AlN quantum wells [J].Phys Rev B,2003,68:165335(1-11).

[19]Shi J J,Chu X L,Goldys E M.Propagating optical-phonon modes and their electron-phonon interactions in wurtzite GaN∕AlxGa1-xN quantum wells [J].Phys Rev B,2004,70:115318(1-8).

[20]Zhang L,Shi J.Vibration spectra of quasi-confined optical phonon modes in an asymmetric wurtzite AlxGa1-xN/GaN/AlyGa1-yN quantum well [J].Commun Theor Phys,2007,47:349-354.

[21]Zhang L,Shi J.Polar vibration spectra of interface and surface optical phonons and their Frohlich electron-phonon interactions in freestanding symmetrical and asymmetrical wurtzite GaN/Ga1-xAlxN multi-layer heterostructures [J].Int J Mod Phys B,2006,20:559-578.

[22]Lv J T,Cao J C,Confined optical phonon modes and electron-phonon interactions in wurtzite GaN/ZnO quantum wells [J].Phys Rev B,2005,71:155304(1-6).

[23]Pokatilov E P,Nika D L,Balandin A A.Confined electron-confined phonon scattering rates in wurtzite AlN/GaN/AlN heterostructures [J].J Appl Phys,2004,95:5626-5632.

[24]Zhu J,Ban S L,Ha S H.Phonon-assisted intersubband transitions in wurtzite GaN/InxGa1-xN quantum wells [J].Chin Phys B,21,2012:097301(1-6).

[25]Wu X,Peeters F M.Cyclotron-resonance mass of two-dimensional electrons in GaN/AlxGa1-xN heterostructures [J].Phys Rev B,1997,55:15438-15440.

[26]Buonocore F,Iadonisi G,Ninno D.Polaron in quantum well wires [J].Phys Rev B,2002,65:205415(1-7).

[27]Shi J J,Xia C X,Wei SY,etal.Exciton states in wurtzite InGaN strained coupled quantum dots:Effects of piezoelectricity and spontaneous polarization [J].J Appl Phys,97(2005)083705(1-8).

[28]Xia C X,Wei SY.Built-in electric field effect in wurtzite InGaN/GaN coupled quantum dots [J].Phys Lett A,2005,346:227-231.

[29]Tchernycheva M,Nevou L,Doyennette L,etal.Optical and theoretical study of strong electron coupling in double GaN/AlN quantum wells [J].Phys Stat Sol B,2006,243:1630-1633.

[30]Kinsler P.Tech Report,Imperial College,United Kingdom,2002.

[31]Takeuchi T,Sota S,Katsuragawa M.etal.Quantum-confined stark effect due to piezoelectric fields in GaInN strained quantum wells [J]. Jpn J Appl Phys,Part 2,36(1997)L382.

[32]Hrivnak L.Simple calculations of energy levels in quantum wells of lattice-matched semiconductors with nonparabolic bands [J].J Appl Phys,1992,71:4370-4376.

[33]Elabsy A M,Csabinszky P.Combined effect of the screening of a donor ion and the conduction band nonparabolicity on the binding energy of a donor at the center of a spherical quantum dot [J] .Int J Quantum Chem,1996,30:1719-1722.

[34]Harima H,Properties of GaN and related compounds studied by means of Raman scattering [J].J Phys:Condens Matter,2002,14:R967-R993.

[35]Vurgaftman I,Meyer J R,Band parameters for nitrogen-containing semiconductors [J].J Appl Phys,2003,94:3675-3696.

Dispersive Spectra and Scattering Rates of Interface Optical Phonon Modes in a Wurtzite GaN-Based Step Quantum Well with Strong Built-in Electric Field

ZHANG Li

(InstituteofElectronicTechniqueandMaterialDevelopment,GuangzhouPanyuPolytechnic,Guangzhou511483,China)

Abstract:Electron-interface optical(IO)phonon scattering rates in a wurtzite GaN-based asymmetrical step quantum well(QW)are theoretically analyzed by using the usual Fermi golden rule.Based on the dielectric continuum model and Loudon's uniaxial crystal model,the analytical IO phonon states and their Fr?hlich electron-phonon interaction Hamiltonian are derived.Taking into consideration the effects of strong built-in electric field(BEF)and the band nonparabolicity,the exact electronic eigen-states in the step QW are also obtained with the aid of two Airy functions.Numerical calculations on a wurtzite AlN/GaN/AlxGa1-xN/AlN step QW are performed.It is found that there are four branches of IO phonon modes in the asymmetric nitride step QWs,which is obviously different from the situation of symmetrical GaN/AlN single and coupling QWs.This is mainly ascribed to the asymmetry of the step QW structures studied here.The calculated results show that the intrasubband and intersubband scattering rates in wurtzite step QWs are one order of magnitude larger than those in GaAs-based step QWs,which is attributed to the larger electron-phonon coupling constants of GaN-based materials.The intrasubband scattering rates in wurtzite step QWs behave analogous dependent relation as those in cubic GaAs-based step QWs on the structural parameters,but the intersubband scattering rates here show obviously different dependent behavior on the structural parameters.This is ascribed to the effects of the strong BEF and the band nonparabolicity.Moreover,the high-frequency IO modes play more important role to the total scattering rates than the low-frequency ones.

Key words:electron-phonon scattering;interface phonon modes;wurtzite step quantum well;built-in electric field;band nonparabolicity

收稿日期:2015-09-21; 修改稿日期:2015-11-10

基金項目:國家自然科學基金項目(60906042,61178003及61475039)、廣州市創(chuàng)新學術團隊項目(13C17)、廣州市“羊城學者”學術帶頭人項目(10B010D)、廣州市教育局科技項目(10B001)、廣州市番禺區(qū)產(chǎn)學項目(2010-D-09-1103004)聯(lián)合資助

作者簡介:張立(1976-),男,教授,主要從事短波長納米半導體材料光電子特性研究工作及電子電氣專業(yè)教學工作。Email:zhangli-gz@263.net

文章編號:1004-5929(2016)02-0131-09

中圖分類號:O469,O482.3

文獻標志碼:A

doi:10.13883/j.issn1004-5929.201602007