Lecture3表面电子态.ppt

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1、本节课主要内容,凝胶模型近自由电子近似紧束缚近似镜像态实例1：贵金属表面态实例2：半导体表面态实例3：拓扑绝缘体表面态实例4：高温超导体表面态,表面电子态的分类,表面态的发现者,Nobel Prize for Physics in 1958,for the 1937 work unraveling the science behind the blue glow of radioactive material immersed in liquid,called the Cherenkov effect.In 1932,he predicted what are now called surfa

2、ce states or Tamm states.He is also famous for his work on the Soviet Unions hydrogen bomb project.,Igor Y.Tamm(1895-1971),1.凝胶模型,The jellium model,in which the valance electrons are in interaction with their own average charge and with an ionic charge uniformly spread in half the space,equilibratin

3、g the electronic density and,thus,are free.It applies to normal metals.,M.C.Desjonqeres and D.Spanjaard,Concepts in Surface Physics,Springer-Verlag,1996.,一维无限深势阱,z,Assuming the wave function in the well can be:,With the boundary conditions:,Then,the normalized wave function is:,where,p=1,2,3,The den

4、sity of k states is thus L/,无边界情况,For infinite one dimensional electron gas,the Born-Von Karman boundary condition is:,The corresponding wave functions are:,where n=0,1,2,3,The density of k states is thus L/2,长度为L的一维电子气（周期性边界条件）,可填充电子数 vs 费米波矢,无边界的一维电子气：,一维无限深势阱：,L,L,For a given kF,we loose one stat

5、e at the bottom of the band on making two surfaces.,有边界情况下的电子密度（一维无限深势阱）,At the continuum limit,N,L,but 2N/L remains constant and equal to the homogeneous bulk electron density 0-:,If we integrate-(z)-0-from z=0 to z=,we find:,We have:,Wave function：,For z L:,z,拓展到三维情况,Assuming the electrons are bou

6、nded in z-direction by impenetrable potential at z=0 and z=L,and free to move in xy-direction:,The corresponding normalized wave function is:,有表面存在情况下的动量空间,表面处的电子密度,体电子密度,Friedel oscillations,z-2,when z is large:,一维有限深势阱,z0,z0,Where:,Phase shift,当前模型的局限,没有考虑电子间的交换关联作用。忽略了原子核的周期性分布。非自洽的计算：势场应该从波函数得到。

7、,更精确的方法：DFT-LDA,Ves(r),Remarks,The jellium model description of a metal surface neglects the details of the electron-ion interaction and emphasizes the nature of the smooth surface barrier.The nearly free electron model emphasizes the lattice aspects of the problem and simplifies the form of the sur

8、face barrier.,2.近自由电子模型,The nearly free electron model,which is valid when the lattice potential is weak.Consequently,this potential is treated as a perturbation,the unperturbed states being free electron plane waves.This model can describe the electronic structure of normal metals and some narrow-g

9、ap semiconductors.,M.C.Desjonqeres and D.Spanjaard,Concepts in Surface Physics,Springer-Verlag,1996.,布洛赫定理,一维能带理论,设：e=m=1,a/2,0,在体内(za/2)，该方程可以利用简并微扰法求解:,将上述试解代入薛定谔方程可得:,能量本征值：,波函数：,where:,能量色散关系E(2),能隙,0,In the bulk,k has to be real due to the periodic boundary conditions.However,the termination of

10、 a crystal,i.e.the formation of a surface,obviously causes deviation from perfect periodicity.Therefore,k can be taken as complex number.,za/2,za/2,where:,当k为虚数时，薛定谔方程存在下列形式解：,根据表面处的波函数连续性，可以唯一确定k的取值。该k值对应的电子态能量位于体能隙之中，其波函数局域在表面附近，在表面外和体内都呈衰减行为。该电子态被称作表面态。,波函数在表面处的连续性,Vg0,Vg0,Shockley state,E,1,2,3,

11、The curvature of k can match the decaying vacuum solution only for Vg0,近自由电子模型的使用范围,Assuming that the valence electrons are completely delocalized when the solid is formed（metal）.The perturbation of the lattice potential is weak（narrow-band semiconductor）.,3.紧束缚近似模型,The tight-binding approximation u

12、sing wave functions written as linear combinations of atomic orbitals centered at each lattice site.This approximation applies to fairly localized electrons.It is successful in the treatment of transition metals and also for wide-bandgap semiconductors and insulators.,物理图像,This picture,it is easy to

13、 comprehend that the existence of a surface will give rise to surface states with energies different from the energies of the bulk states.Since the atoms residing in the topmost surface layer are missing their bonding partners on one side their orbitals have less overlap with the orbitals of neighbo

14、ring atoms.The splitting and shifting of energy levels of the atoms forming the crystal is therefore smaller at the surface than in the bulk.,一维线性链模型,体系的周期性势场为VL(r)为各格点原子势场Va(r-na)之和：,考虑由N个原子组成的线性链，原子间距为a：,其中孤立原子的薛定谔方程为：,原子能级,原子束缚态,则一维线性链体系的薛定谔方程为：,微扰,根据简并微扰方法，线性链共有化电子的波函数可以写为：,将上式代入体系的薛定谔方程，并只考虑最近邻

15、格点的交叠积分：,:on-site matrix element:nearest neighbor hopping matrix element,可以得到关于展开系数cn的齐次方程：,可以证明，该方程有下列简单形式的解：,A,B为任意常数,将cn的解代入上述cn的齐次方程可得：,在没有表面的情况下，根据周期性边界条件：,可以得知，k为简约波矢，在第一布里渊区内共有N个值，密度为Na/2,在有表面存在的情况下，展开系数cn在表面处应满足：,n=1,n=N,考虑到表面处的on-site matrix element可能与体内不同：,所以有如下关于cn的齐次方程组：,n=1,n=N,1nN,上述齐次

16、方程组有非零解的条件是cn的系数矩阵的行列式为0,由此可得到关于E的一个超越方程，共有N个解，大多数解对应的波函数在链中每个原子上有差不多的强度，,其中有两个解不同于体态值，其对应的波函数在表面上有很大的强度，而在体内原子上的强度很小，这便对应于表面态，一般称作Tamm state。,但是如果：,n=1,n=N,1nN,例子：三原子线性链,由于on-site matrix element 只引起能级的整体移动，所以我们这里忽略，并把E0作为能级原点，则有：,关于cn的齐次方程组可以简化为：,由cn的系数矩阵的行列式为0，可得：,从而得到能量本征值E：,对应的本征波函数为：,可见，2（r）只在表

17、面处原子上有分布，对应于表面态,Surface state,镜像态-强关联效应,前面的模型都假设电子-电子相互作用比较弱，或者不考虑电子-电子相互作用。如果电子-电子相互作用较强，将出现一种新的表面态：镜像态。不同于一般的表面态，镜像态往体内迅速衰减，而在表面外可以延伸很长的距离。,镜像势,Strongly correlated interaction,En=-0.85 eV/(n+a)2,n=1,2,.,类似于氢原子的能级,金属电子对表面外电荷的屏蔽效应,Cu(100)和Cu(111)表面的镜像态,Non-resonant,Resonant,金属/绝缘体界面的镜像态,Progress in

18、Surface Science 80(2005)4991,两个界面都有镜像态！,2PPE 测得的Cu(100)表面的镜像态,Progress in Surface Science 80(2005)4991,STM z(V)spectroscopy,利用STM探测表面的镜像态,休息15分钟,实例1：贵金属表面态,Ag(111)表面在费米面附近存在能隙，而其他表面不存在能隙,Ag各个表面的能带结构,各种贵金属的（111）面,Au(111)表面态的测量,ARPES,STM dI/dV spectroscopy,Constant density of states(2D),自旋轨道耦合,表面驻波,St

19、anding waves on Ag(111)surface,局域态密度：,表面态电子被台阶的散射,kx,ky,k1i,k1s,k2i,k2s,0,台阶,等能面E,M.F.Crommie et al.,Science 363,524(1993),局域态密度：,表面态的能量色散,Y.Hasegawa and Ph.Avouris,Phys.Rev.Lett.71,1071(1993),Quantum corrals,Fe adatoms on Cu(111),1993-IBMs Quantum Corral,Other arrangements of iron atoms on copper s

20、ubstrates.Note that only the perfect circle shows the standing wave of electrons.,表面态的应用1:原子超晶格,F.Silly et al.,Phys.Rev.Lett.92,016101(2004),表面态的应用2：Molecular graphene,人工石墨晶格：CO on Cu(111),CO,K.K.Gomes et al.,Nature 483,306(2012),表面态的应用2：molecular graphene,实例2：半导体表面态,There is a natural quasi-chemica

21、l view of the creation of a surface,as process accompanied by breaking of interatomic bonds,which can be discussed in the framework of the tight-binding approach.,半导体表面态可以由紧束缚近似很好的描述,Principle of semiconductor surface reconstruction,Basic principle:the surface structure observed will be the lowest f

22、ree-energy structure kinetically accessible under the preparation conditions.Principle 1:A surface tends to minimize the number of dangling bonds by formation of new bonds.The remaining dangling bonds tend to saturated.Principle 2:A surface tends to compensate charges.Principle 3:A semiconductor sur

23、face tends to be insulating(or semi-conducting).,sp3杂化,s,p 轨道杂化,表面的断键,(111),(100),Illustration of bond cutting during surface formation,Dangling bond,back bond,Si(111)和Si(100)的表面态,Si(100)的表面杂化,Si(100)表面的二聚体重构（2x1）,Four degenerated dangling bonds,*,Dimer-bond formation:and anti-levels,降低系统的总能量！,二聚体的b

24、uckling,*,DB down,DB up,Further separation of DB levels by dimer buckling,*,Splitting of dangling bond(DB)levels by-*interaction,*,Si(100)-c(4x2)STM形貌图,占据态,空态,STS隧道谱,T.Yokoyama et al.,Phys.Rev.B 61,R5078(2000),*state色散,实例3：拓扑绝缘体表面态,拓扑表面态的形成,T.Zhang et al.,PRL 103,266803(2009),实例4：高温超导体表面态,Bi2212的晶体结

25、构和电子结构,REVIEWS OF MODERN PHYSICS 79,353(2007),超导能隙,单晶,层状结构,动量空间中的准粒子弹性散射,超导能隙中的等能面,动量空间中的散射矢量分布,K.McElroy et al.,Nature 422,592(2003),弹性散射,解理后的BiO表面,STM 形貌图,局域态密度图（准粒子干涉条纹）,K.McElroy et al.,Nature 422,592(2003),局域态密度图的傅里叶变换,K.McElroy et al.,Nature 422,592(2003),能量色散和等能面,K.McElroy et al.,Nature 422,592(2003),色散关系,等能面,本节课小结,凝胶模型近自由电子近似紧束缚近似镜像态实例1：贵金属表面态实例2：半导体表面态实例3：拓扑绝缘体表面态实例4：高温超导体表面态,参考书：M.C.Desjonqeres and D.Spanjaard,Concepts in Surface Physics,Springer-Verlag,1996.（第5章）Zangwill,Physics at Surfaces,Cambridge,1998(第2章),