第二課：表面晶體結(jié)構(gòu)

1、表表 面面 物物 理理 學(xué)學(xué)江 穎量子材料中心上一課提示：上一課提示：1. 表面物理研究什么？2. 研究表面物理有什么意義？3. 表面物理的發(fā)展歷史。4. 表面物理的現(xiàn)狀和發(fā)展趨勢(shì)。 表面不僅僅是幾何意義上它在物體的最外面，而是從物理上來(lái)講，表面是一種物質(zhì)的新相。1877年吉布斯(J.W. Gibbs)首先提出“表面相”的概念，指出在氣固界面處存在一種二維的凝聚物質(zhì)相，表面相不管是結(jié)構(gòu)、還是物理性質(zhì)，化學(xué)性質(zhì)，都和固體體相有很大的差異。這是我們要研究表面的根本原因（基礎(chǔ)研究基礎(chǔ)研究）。 由于固體只有通過(guò)其表面才能與周?chē)沫h(huán)境發(fā)生相互作用，這種表層的存在將對(duì)固體的物理化學(xué)特性有很大影響。很多重要

2、的應(yīng)用課題，如金屬的腐蝕與回火變脆、多相催化、材料的外延生長(zhǎng)和表面電子器件等都和固體表面的狀態(tài)有密切的關(guān)系（應(yīng)用研究應(yīng)用研究）。 第二課第二課:Surface phenomena: crystal growth, adsorption, oxidation, etching, catalysis; Bulk phenomena: transport, optical, magnetic, mechanical, thermal properties; (A silicon cube of 1 cm3 has 5 X 1022 bulk atoms and 4 X 1015 surface at

3、oms.) Effects determined by the interplay of bulk and surface, for example: Topological Insulator. Thermally activated adatom gas at high temperatureLow energy electron microscopy (LEEM) observationR. M. Tromp and M. Mankos, Phys. Rev. Lett. 81,1050 (1998).Clean Si(100) surface:本課內(nèi)容提要：本課內(nèi)容提要：表面的平移群和

4、點(diǎn)群二維晶格的分類(lèi)二維空間群表面的弛豫和重構(gòu)的基本概念幾種典型的金屬和共價(jià)鍵半導(dǎo)體的表面弛豫和重構(gòu)表面重構(gòu)的標(biāo)記方法倒易空間和布里淵區(qū)低能電子衍射和實(shí)例 表面晶體學(xué)表面晶體學(xué)研究的是表面層上二維結(jié)構(gòu)的周期性；原胞中原子的種類(lèi)、數(shù)目與排列；表面晶格與襯底晶格的位置及取向的關(guān)系。 abR = m a + n b二維晶格點(diǎn)陣二維晶格點(diǎn)陣: 二維的周期結(jié)構(gòu)可以抽象成二維點(diǎn)陣，點(diǎn)陣的每 個(gè)格點(diǎn)代表一個(gè)周期結(jié)構(gòu)的單元?；负驮亩x: 二維晶格點(diǎn)陣二維晶格點(diǎn)陣R = m a + n b二維晶格點(diǎn)陣二維晶格點(diǎn)陣R = m a + n bab二維晶格點(diǎn)陣二維晶格點(diǎn)陣Wigner-Seitz CellR =

5、m a + n b二維晶格點(diǎn)陣二維晶格點(diǎn)陣Wigner-Seitz CellR = m a + n bLattice planes( hkl ) Miller indices: h, k, l are the integer reciprocal axis intervals given by the intersections of the lattice planes with the three crystallographic axes; hkl The collection of such planes that are equivalent by symmetry;Lattice

6、directions hkl : Used to specify directions in the direct lattice; hkl : The collection of such directions that are equivalent by symmetry.Low-index surfaces of cubic latticeVertical and horizontal markings indicate the second and third layers, respectively.(010)Lattice planes(hkil ) Bravais indices

7、: In the case of trigonal and hexagonal lattices, four crystallographic axes are considered. h+k+i=0. l is perpendicular to the hexagonal basal plane.Characteristic planes in a hexagonal Bravais lattice. The vectors x1 (h), x2 (k), x3 (i), and c (l) can be identified with the primitive Bravais latti

8、ce vectors. 1 (C1): E 1m (C1V): E, h 2 (C2): E, C2 2mm (C2v): E, C2, 2v 3 (C3): E, 2C3 3m (C3v): E, 2C3, 3v 4 (C4): E, 2C4, C2 4mm (C4v): E, 2C4, C2 , 2v , 2d 6(C6): E, 2C6, 2C3, C26mm (C6v): E, 2C6, 2C3 , C2 , 3v , 3d二維點(diǎn)群二維點(diǎn)群: 旋轉(zhuǎn)：二維的完整對(duì)稱(chēng)性，只允許有限的幾種旋轉(zhuǎn)。旋轉(zhuǎn)2 /n角度(n = 1, 2, 3, 4, 6)。鏡面反映。 將5個(gè)許可的旋轉(zhuǎn)操作和鏡象反映

9、組合起來(lái)就得到10個(gè)二維點(diǎn)群：1, 2, 1m, 2mm, 4, 4mm, 3, 3m, 6, 6mm。國(guó)際符號(hào)熊夫利符號(hào)二維點(diǎn)群二維平移群互相制約（二維空間群）五種二維布拉菲格子(Bravais Lattice)四種平面晶系10個(gè)空間點(diǎn)群The five two-dimensional Bravais lattices. Besides primitive unit cells (dashed lines) also a non-primitive cell (dotted lines) is shown.由于兩維周期結(jié)構(gòu)只存在有限由于兩維周期結(jié)構(gòu)只存在有限的的1010個(gè)點(diǎn)群，它們將限制可能

10、個(gè)點(diǎn)群，它們將限制可能出現(xiàn)的原始平移的種類(lèi)?？梢猿霈F(xiàn)的原始平移的種類(lèi)?？梢宰C明這種相互限制的結(jié)果，使證明這種相互限制的結(jié)果，使得只可能有得只可能有5 5種二維布喇菲點(diǎn)陣種二維布喇菲點(diǎn)陣存在。存在。元格形狀元格形狀晶格符號(hào)晶格符號(hào)軸和夾角軸和夾角晶系名稱(chēng)晶系名稱(chēng)平行四邊形平行四邊形長(zhǎng)方形長(zhǎng)方形正方形正方形60o菱形菱形PP, CPPa b, 90oa b, =90oa=b, =90oa=b, =120o斜方斜方長(zhǎng)方長(zhǎng)方正方正方六角六角v 晶體表面總的對(duì)稱(chēng)性由布喇菲網(wǎng)格和結(jié)晶學(xué)點(diǎn)群結(jié)合起來(lái)描述。將點(diǎn)群操作應(yīng)用于無(wú)限晶格，并考慮到可能有的平移對(duì)稱(chēng)性就得到空間群。v 10種結(jié)晶學(xué)點(diǎn)群和5種布喇菲網(wǎng)格以

11、及滑移線對(duì)稱(chēng)操作(g)共有17種可能的結(jié)合，即有17種不同的對(duì)稱(chēng)群，稱(chēng)為二維空間群。 - 13個(gè)是由點(diǎn)群與適當(dāng)?shù)牟祭凭W(wǎng)格聯(lián)合起來(lái)得到的； - 4個(gè)空間群是包括了滑移對(duì)稱(chēng)后產(chǎn)生的。 晶晶 系系點(diǎn)點(diǎn) 群群（10種）種）空間群符號(hào)空間群符號(hào)空間群編號(hào)空間群編號(hào)（17種）種）全全 名名簡(jiǎn)簡(jiǎn) 稱(chēng)稱(chēng)斜方斜方P12P1P211P1P212矩形矩形P,CmP1m1P1g1C1m1PmPgCm345正方正方P2mm 44mm P2mmP2mgP2ggC2mmP4P4mmP4gmPmmPmgPggCmmP4P4mP4g6789101112六角六角P33mm 66mmP3P3m1P31mP6P6mmP3P3m1P

12、31mP6P6m1314151617二維格子的點(diǎn)群與空間群 1717個(gè)二維空間群所含有的對(duì)稱(chēng)性的圖形表示個(gè)二維空間群所含有的對(duì)稱(chēng)性的圖形表示 Real surface: Reconstruction and Relaxation (重構(gòu)和弛豫重構(gòu)和弛豫)Relaxation (表面晶格弛豫)： 不改變表面晶格周期性Relaxation (表面晶格弛豫)： 不改變表面晶格周期性Reconstruction (表面重構(gòu))：表面晶格周期性改變Relaxation (表面晶格弛豫)： 不改變表面晶格周期性Reconstruction (表面重構(gòu))：表面晶格周期性改變清潔表面的重構(gòu)吸附表面的重構(gòu)Cont

13、ractive relaxation of low index metal surfaces金屬表面: 典型的弛豫機(jī)制金屬: 電子公有化, jellium模型, 沒(méi)有方向性.23x3 reconstruction on Au(111)0.5 nmHILO2.88 8.14 Au (110)-1x2 surface半導(dǎo)體表面:典型的重構(gòu)機(jī)制Surface Reconstruction: Destroy the translation symmetry of the ideal surface. In tetrahedrally bonded semiconductors, systems wit

14、h dangling bonds are unstable, since rebonding usually lowers the total energy of the halfspace. This process is accompanied by bringing surface atoms closer together (Hahn-Teller displacement). (Argument: saturation of dangling bonds)Fig. 9 (a) Pairing reconstruction; (b) Missing row reconstruction

15、; (c) Relaxation of the uppermost atomic layer.Si(100)-2x1 SurfaceStructure model of the Si(100)-2x1Four degenerated dangling bonds*Dimer-bond formation: and anti- levels*Splitting of dangling bond(DB) levels by - * interaction*DB downDB upFurther separation of DB levels by dimer bucklingBonding con

16、figuration diagram TOP VIEWSIDE VIEW吸附表面的重構(gòu)吸附表面的重構(gòu)Atomically resolved O2 lattice: Au(110)-3x4-O234Wood 方法方法 （1963） 理想表面已知其平移群： T= ma1 + na2 再構(gòu)表面對(duì)應(yīng)的平移群： Ts= mas1 + nas2其中， Ias1I=pIa1I, Ias2I=qIa2I p和q為整數(shù)，表示基矢倍數(shù)。 再構(gòu)表面的表達(dá)方式為 E(khl) p X qE E為襯底元素符號(hào)，(hkl)為再構(gòu)表面的晶面指數(shù)。 Wood 方法方法 （1963） 如果再構(gòu)想對(duì)于襯底基失有轉(zhuǎn)角 ，基失的關(guān)系

17、變?yōu)椋?as1=p1a1+q1a2, as2=p2a1+q2a2再構(gòu)表面的表達(dá)方式為 E(khl) p X q - R 當(dāng)有外來(lái)原子吸附D時(shí)，再構(gòu)表面的表達(dá)方式為 E(khl) p X q R - D Fig. 13. A surface superstructure with the possible denotations c(2X2) and (2X2)R45.Another problem is related to the fact that one and the same reconstruction may be defined in different ways. Super

18、lattice at surface A periodicity with 2D primitive basis vectors a1 and a2 at topmost layer. The translational group T of the whole crystal with surface is thus the intersection T = Ts TbT is the largest common subgroup of both groups Ts and Tb.Ts characterizes the translation symmetry of the topmos

19、t layer by a1 and a2 . Tb characterizes the translation symmetry of the bulk by a1 and a2.Matrix notation i) When all matrix elements mij are integers, the surface is called a simple superlattice;ii) When all matrix elements mij are rational number, the surface is said to have a coincidence structur

20、e and the superlattice is referred to as commensurate;iii) When at least one matrix element mij is an irrational number, the superlattice is termed incoherent or incommensurate. When a surface superlattice is superimposed on the substrate lattice which exhibits the basic periodicity. The surface net

21、 of the topmost atomic layer may be determined in terms of the substrate net by: Fig.12. Three different types of surface reconstructions. (a) 1X2, (b) (3X3)R30, , and (c) general case. The Wood notation doesnt apply in this case, however, the matrix notation does with m11=5, m12=-1, m21=2, m22=2. 2

22、001111222151X2(3X3)R30a1a2b1b2a1a2a1a2b1b2b1b2Monolayer Al(111) on Si(111) surface4 aAl = 3 aSiaAl=3/4 aSiaAl = 2.86 aSi = 3.84 Si(111)-1x1 + Al(111)-1x1SiSiAlAlaaaa2121430043Si(111)-2x1 SurfaceSTM topography of Si(111)-2x1 SurfaceClean Si(111)-7x7 surface, filled state STM image, sample bias -1.2V,

23、 20pAFHFHUHUH27 Si(111)-7x7 Surface表面的晶體結(jié)構(gòu)表面的晶體結(jié)構(gòu)* 3D Reciprocal lattice（倒格矢）（倒格矢）The primitive basis vectors b1, b2, b3 of the 3D reciprocal lattice b1 = 2(a2 X a3)/(a1a2Xa3) b2 = 2(a3 X a1)/(a2a3Xa1) b3 = 2(a1 X a2)/(a3a1Xa2)Here ai bj = 2ij, where ij=1, if i=j (i, j = 1,2,3) ij=0, if ij hkl面間距面間距

24、: d=1/ hb1+kb2+lb3指數(shù)小的晶面系，晶面有較大的間距。指數(shù)小的晶面系，晶面有較大的間距。Reciprocal Space (倒易空間倒易空間)* 2D Reciprocal lattice（倒格矢）（倒格矢）The primitive basis vectors b1 and b2 of the 2D reciprocal lattice b1 = 2(a2 X n)/(|a1 X a2|) b2 = 2(n X a1)/(|a1 X a2|)Where ai bj = 2ij (i, j = 1,2). The length of these vectors are | bi

25、 | = 2/ai sin(a1,a2).在倒空間的點(diǎn)陣中，任一倒格點(diǎn)的位矢在倒空間的點(diǎn)陣中，任一倒格點(diǎn)的位矢 倒格矢，可以表示為倒格矢，可以表示為 ghk = hb1 + kb2.其中，其中，h和和k為相應(yīng)正格子的晶列指數(shù)。在倒空間中的任何一個(gè)倒格點(diǎn)，為相應(yīng)正格子的晶列指數(shù)。在倒空間中的任何一個(gè)倒格點(diǎn)，均可通過(guò)該式所決定的平移操作來(lái)得到。均可通過(guò)該式所決定的平移操作來(lái)得到。Reciprocal Space (倒易空間倒易空間)Fig. 17. Direct lattice (left) and corresponding reciprocal lattice (right) of five

26、2D Bravais lattices. 設(shè)設(shè) 正格矢：正格矢： Rhk= ha1 + ka2 倒格矢倒格矢: ghk = hb1 + kb2 則它們成為互為倒易關(guān)系的充要條件：則它們成為互為倒易關(guān)系的充要條件： Rhk ghk = 2n (a)n為任意整數(shù)。為任意整數(shù)。 設(shè)設(shè) Ki和和Ks分別為入射和衍射波矢，有分別為入射和衍射波矢，有Ki=ci/, Ks=cs/. (b) 衍射與入射波光程差衍射與入射波光程差 CO+OD= - Rhk ci + Rhk cs = Rhk (cs-ci) 對(duì)于單色波，衍射加強(qiáng)條件是：光程差等于波長(zhǎng)的整數(shù)倍，對(duì)于單色波，衍射加強(qiáng)條件是：光程差等于波長(zhǎng)的整數(shù)倍，

27、則衍射方程可寫(xiě)為：則衍射方程可寫(xiě)為： Rhk (cs-ci)=n由由(b)得得 Rhk (Ks-Ki)=n, 此為勞厄衍射方程的波矢表達(dá)式。將此式與此為勞厄衍射方程的波矢表達(dá)式。將此式與(a)比較，有比較，有 ghk=Ks-Ki (c)更普遍的形式：更普遍的形式： mghk=Ks-Ki ， 其中其中m=1為一級(jí)衍射方程。為一級(jí)衍射方程。 CO DRhkKsKi 此式表示倒格矢等于反射波與入射波的波矢之差。它可以把衍射此式表示倒格矢等于反射波與入射波的波矢之差。它可以把衍射斑點(diǎn)同倒易點(diǎn)陣聯(lián)系起來(lái)。斑點(diǎn)同倒易點(diǎn)陣聯(lián)系起來(lái)。Diffraction of an incident plane wave

28、with vector ki. The surface is represented by the corresponding 2D Bravais lattice. Parallel momentum conservation with any reciprocal lattice vector ghk creates well-defined diffracted beams (hk).The reciprocal lattice vectors have a direct physical meaning. In a diffraction experiment, e.g., LEED,

29、 each diffraction beam corresponds to a reciprocal lattice vector ghk and, in fact, each such beam can be labeled by the values h and k as the beam (hk).KiKsG1窗口LEED experimentLEED pattern of Si(111)-7x71x17x7Fig. 21. Sequence of LEED patterns (with same electron energy of 130 eV) for the Si-termina

30、ted surface of 6H-SiC(0001). The 1X1 bulk-terminated phase is stabilized by OH adsorption, whereby the following reconstruction by 800C, 1000C and 1100C annealing. 1 x 1Brillouin zonesThe surface BZ is defined as the smallest polygon in the 2D reciprocal space situated symmetrically with respect to

31、a given lattice point (used as coordinate zero) and bounded by points k satisfying the equation k g = |g|2/2Where k is restricted to 2D. The set of points defined by above equation gives a straight line at a distance |g|/2 from the zero point. Surface Brillouin zones of a 2D cubic latticekg二維正方晶格的布里

32、淵區(qū)二維長(zhǎng)方晶格的布里淵區(qū)二維六方晶格的十個(gè)布里淵區(qū) Fig. 22. BZ of five plane lattices: (a) oblique, (b) p-rectangular, (c) c-rectangular, (d) square, and (e) hexagonal. Symmetry lines and points are also shown, and their notations are introduced.面心立方晶格的第一布里淵區(qū)Projection of 3D onto 2D BZ Within an explicit procedure certain bulk directions and points of high symmetry in the 3D BZ are projected onto the 2D surface