第二课：表面晶体结构

1、表表 面面 物物 理理 学学江 颖量子材料中心上一课提示：上一课提示：1. 表面物理研究什么？2. 研究表面物理有什么意义？3. 表面物理的发展历史。4. 表面物理的现状和发展趋势。 表面不仅仅是几何意义上它在物体的最外面，而是从物理上来讲，表面是一种物质的新相。1877年吉布斯(J.W. Gibbs)首先提出“表面相”的概念，指出在气固界面处存在一种二维的凝聚物质相，表面相不管是结构、还是物理性质，化学性质，都和固体体相有很大的差异。这是我们要研究表面的根本原因（基础研究基础研究）。 由于固体只有通过其表面才能与周围的环境发生相互作用，这种表层的存在将对固体的物理化学特性有很大影响。很多重要

2、的应用课题，如金属的腐蚀与回火变脆、多相催化、材料的外延生长和表面电子器件等都和固体表面的状态有密切的关系（应用研究应用研究）。 第二课第二课: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:本课内容提要：本课内容提要：表面的平移群和

4、点群二维晶格的分类二维空间群表面的弛豫和重构的基本概念几种典型的金属和共价键半导体的表面弛豫和重构表面重构的标记方法倒易空间和布里渊区低能电子衍射和实例 表面晶体学表面晶体学研究的是表面层上二维结构的周期性；原胞中原子的种类、数目与排列；表面晶格与衬底晶格的位置及取向的关系。 abR = m a + n b二维晶格点阵二维晶格点阵: 二维的周期结构可以抽象成二维点阵，点阵的每 个格点代表一个周期结构的单元。基矢和原胞的定义: 二维晶格点阵二维晶格点阵R = m a + n b二维晶格点阵二维晶格点阵R = m a + n bab二维晶格点阵二维晶格点阵Wigner-Seitz CellR =

5、m a + n b二维晶格点阵二维晶格点阵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二维点群二维点群: 旋转：二维的完整对称性，只允许有限的几种旋转。旋转2 /n角度(n = 1, 2, 3, 4, 6)。镜面反映。 将5个许可的旋转操作和镜象反映

9、组合起来就得到10个二维点群：1, 2, 1m, 2mm, 4, 4mm, 3, 3m, 6, 6mm。国际符号熊夫利符号二维点群二维平移群互相制约（二维空间群）五种二维布拉菲格子(Bravais Lattice)四种平面晶系10个空间点群The five two-dimensional Bravais lattices. Besides primitive unit cells (dashed lines) also a non-primitive cell (dotted lines) is shown.由于两维周期结构只存在有限由于两维周期结构只存在有限的的1010个点群，它们将限制可能

10、个点群，它们将限制可能出现的原始平移的种类。可以出现的原始平移的种类。可以证明这种相互限制的结果，使证明这种相互限制的结果，使得只可能有得只可能有5 5种二维布喇菲点阵种二维布喇菲点阵存在。存在。元格形状元格形状晶格符号晶格符号轴和夹角轴和夹角晶系名称晶系名称平行四边形平行四边形长方形长方形正方形正方形60o菱形菱形PP, CPPa b, 90oa b, =90oa=b, =90oa=b, =120o斜方斜方长方长方正方正方六角六角v 晶体表面总的对称性由布喇菲网格和结晶学点群结合起来描述。将点群操作应用于无限晶格，并考虑到可能有的平移对称性就得到空间群。v 10种结晶学点群和5种布喇菲网格以

11、及滑移线对称操作(g)共有17种可能的结合，即有17种不同的对称群，称为二维空间群。 - 13个是由点群与适当的布喇菲网格联合起来得到的； - 4个空间群是包括了滑移对称后产生的。 晶晶 系系点点 群群（10种）种）空间群符号空间群符号空间群编号空间群编号（17种）种）全全 名名简简 称称斜方斜方P12P1P211P1P212矩形矩形P,CmP1m1P1g1C1m1PmPgCm345正方正方P2mm 44mm P2mmP2mgP2ggC2mmP4P4mmP4gmPmmPmgPggCmmP4P4mP4g6789101112六角六角P33mm 66mmP3P3m1P31mP6P6mmP3P3m1P

12、31mP6P6m1314151617二维格子的点群与空间群 1717个二维空间群所含有的对称性的图形表示个二维空间群所含有的对称性的图形表示 Real surface: Reconstruction and Relaxation (重构和弛豫重构和弛豫)Relaxation (表面晶格弛豫)： 不改变表面晶格周期性Relaxation (表面晶格弛豫)： 不改变表面晶格周期性Reconstruction (表面重构)：表面晶格周期性改变Relaxation (表面晶格弛豫)： 不改变表面晶格周期性Reconstruction (表面重构)：表面晶格周期性改变清洁表面的重构吸附表面的重构Cont

13、ractive relaxation of low index metal surfaces金属表面: 典型的弛豫机制金属: 电子公有化, jellium模型, 没有方向性.23x3 reconstruction on Au(111)0.5 nmHILO2.88 8.14 Au (110)-1x2 surface半导体表面:典型的重构机制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吸附表面的重构吸附表面的重构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 再构表面对应的平移群： Ts= mas1 + nas2其中， Ias1I=pIa1I, Ias2I=qIa2I p和q为整数，表示基矢倍数。 再构表面的表达方式为 E(khl) p X qE E为衬底元素符号，(hkl)为再构表面的晶面指数。 Wood 方法方法 （1963） 如果再构想对于衬底基失有转角 ，基失的关系

17、变为： as1=p1a1+q1a2, as2=p2a1+q2a2再构表面的表达方式为 E(khl) p X q - R 当有外来原子吸附D时，再构表面的表达方式为 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表面的晶体结构表面的晶体结构* 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面间距面间距

24、: d=1/ hb1+kb2+lb3指数小的晶面系，晶面有较大的间距。指数小的晶面系，晶面有较大的间距。Reciprocal Space (倒易空间倒易空间)* 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).在倒空间的点阵中，任一倒格点的位矢在倒空间的点阵中，任一倒格点的位矢 倒格矢，可以表示为倒格矢，可以表示为 ghk = hb1 + kb2.其中，其中，h和和k为相应正格子的晶列指数。在倒空间中的任何一个倒格点，为相应正格子的晶列指数。在倒空间中的任何一个倒格点，均可通过该式所决定的平移操作来得到。均可通过该式所决定的平移操作来得到。Reciprocal Space (倒易空间倒易空间)Fig. 17. Direct lattice (left) and corresponding reciprocal lattice (right) of five

26、2D Bravais lattices. 设设 正格矢：正格矢： Rhk= ha1 + ka2 倒格矢倒格矢: ghk = hb1 + kb2 则它们成为互为倒易关系的充要条件：则它们成为互为倒易关系的充要条件： Rhk ghk = 2n (a)n为任意整数。为任意整数。 设设 Ki和和Ks分别为入射和衍射波矢，有分别为入射和衍射波矢，有Ki=ci/, Ks=cs/. (b) 衍射与入射波光程差衍射与入射波光程差 CO+OD= - Rhk ci + Rhk cs = Rhk (cs-ci) 对于单色波，衍射加强条件是：光程差等于波长的整数倍，对于单色波，衍射加强条件是：光程差等于波长的整数倍，

27、则衍射方程可写为：则衍射方程可写为： Rhk (cs-ci)=n由由(b)得得 Rhk (Ks-Ki)=n, 此为劳厄衍射方程的波矢表达式。将此式与此为劳厄衍射方程的波矢表达式。将此式与(a)比较，有比较，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为一级衍射方程。为一级衍射方程。 CO DRhkKsKi 此式表示倒格矢等于反射波与入射波的波矢之差。它可以把衍射此式表示倒格矢等于反射波与入射波的波矢之差。它可以把衍射斑点同倒易点阵联系起来。斑点同倒易点阵联系起来。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二维正方晶格的布里

32、渊区二维长方晶格的布里渊区二维六方晶格的十个布里渊区 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渊区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