Lecture3 表面电子态

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,NobelPrizeforPhysicsin1958,forthe1937workunravelingthesciencebehindtheblueglowofradioactivematerialimmersedinliquid,calledtheCherenkoveffect.In1932,hepredictedwhatarenowcalledsurfacestatesorTammstates.HeisalsofamousforhisworkontheSovietUnionshydrogenbombproject.,IgorY.Tamm(1895-1971),1.凝胶模型,Thejelliummodel,inwhichthevalanceelectronsareininteractionwiththeirownaveragechargeandwithanionicchargeuniformlyspreadinhalfthespace,equilibratingtheelectronicdensityand,thus,arefree.Itappliestonormalmetals.,M.C.DesjonqeresandD.Spanjaard,ConceptsinSurfacePhysics,Springer-Verlag,1996.,一维无限深势阱,z,Assumingthewavefunctioninthewellcanbe:,Withtheboundaryconditions:,Then,thenormalizedwavefunctionis:,where,p=1,2,3,ThedensityofkstatesisthusL/,无边界情况,Forinfiniteonedimensionalelectrongas,theBorn-VonKarmanboundaryconditionis:,Thecorrespondingwavefunctionsare:,wheren=0,1,2,3,ThedensityofkstatesisthusL/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,ForagivenkF,welooseonestateatthebottomofthebandonmakingtwosurfaces.,有边界情况下的电子密度（一维无限深势阱）,Atthecontinuumlimit,N,L,but2N/Lremainsconstantandequaltothehomogeneousbulkelectrondensity0-:,Ifweintegrate-(z)-0-fromz=0toz=,wefind:,Wehave:,Wavefunction：,Forz0,z0,Where:,Phaseshift,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。,更精确的方法：DFT-LDA,Ves(r),Remarks,Thejelliummodeldescriptionofametalsurfaceneglectsthedetailsoftheelectron-ioninteractionandemphasizesthenatureofthesmoothsurfacebarrier.Thenearlyfreeelectronmodelemphasizesthelatticeaspectsoftheproblemandsimplifiestheformofthesurfacebarrier.,2.近自由电子模型,Thenearlyfreeelectronmodel,whichisvalidwhenthelatticepotentialisweak.Consequently,thispotentialistreatedasaperturbation,theunperturbedstatesbeingfreeelectronplanewaves.Thismodelcandescribetheelectronicstructureofnormalmetalsandsomenarrow-gapsemiconductors.,M.C.DesjonqeresandD.Spanjaard,ConceptsinSurfacePhysics,Springer-Verlag,1996.,布洛赫定理,一维能带理论,设：e=m=1,a/2,0,在体内(z0,近自由电子模型的使用范围,Assumingthatthevalenceelectronsarecompletelydelocalizedwhenthesolidisformed（metal）.Theperturbationofthelatticepotentialisweak（narrow-bandsemiconductor）.,3.紧束缚近似模型,Thetight-bindingapproximationusingwavefunctionswrittenaslinearcombinationsofatomicorbitalscenteredateachlatticesite.Thisapproximationappliestofairlylocalizedelectrons.Itissuccessfulinthetreatmentoftransitionmetalsandalsoforwide-bandgapsemiconductorsandinsulators.,物理图像,Thispicture,itiseasytocomprehendthattheexistenceofasurfacewillgiverisetosurfacestateswithenergiesdifferentfromtheenergiesofthebulkstates.Sincetheatomsresidinginthetopmostsurfacelayeraremissingtheirbondingpartnersononesidetheirorbitalshavelessoverlapwiththeorbitalsofneighboringatoms.Thesplittingandshiftingofenergylevelsoftheatomsformingthecrystalisthereforesmalleratthesurfacethaninthebulk.,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，并只考虑最近邻格点的交叠积分：,:on-sitematrixelement:nearestneighborhoppingmatrixelement,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-sitematrixelement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Tammstate。,但是如果：,n=1,n=N,1nN,例子：三原子线性链,由于on-sitematrixelement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）只在表面处原子上有分布，对应于表面态,Surfacestate,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,Stronglycorrelatedinteraction,En=-0.85eV/(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,ProgressinSurfaceScience80(2005)4991,两个界面都有镜像态！,2PPE测得的Cu(100)表面的镜像态,ProgressinSurfaceScience80(2005)4991,STMz(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,STMdI/dVspectroscopy,Constantdensityofstates(2D),自旋轨道耦合,表面驻波,StandingwavesonAg(111)surface,http:/www.omicron.de,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.Crommieetal.,Science363,524(1993),局域态密度：,表面态的能量色散,Y.HasegawaandPh.Avouris,Phys.Rev.Lett.71,1071(1993),Quantumcorrals,FeadatomsonCu(111),1993-IBMsQuantumCorral,Otherarrangementsofironatomsoncoppersubstrates.Notethatonlytheperfectcircleshowsthestandingwaveofelectrons.,表面态的应用1:原子超晶格,F.Sillyetal.,Phys.Rev.Lett.92,016101(2004),表面态的应用2：Moleculargraphene,人工石墨晶格：COonCu(111),CO,K.K.Gomesetal.,Nature483,306(2012),表面态的应用2：moleculargraphene,实例2：半导体表面态,Thereisanaturalquasi-chemicalviewofthecreationofasurface,asprocessaccompaniedbybreakingofinteratomicbonds,whichcanbediscussedintheframeworkofthetight-bindingapproach.,半导体表面态可以由紧束缚近似很好的描述,Principleofsemiconductorsurfacereconstruction,Basicprinciple:thesurfacestructureobservedwillbethelowestfree-energystructurekineticallyaccessibleunderthepreparationconditions.Principle1:Asurfacetendstominimizethenumberofdanglingbondsbyformationofnewbonds.Theremainingdanglingbondstendtosaturated.Principle2:Asurfacetendstocompensatecharges.Principle3:Asemiconductorsurfacetendstobeinsulating(orsemi-conducting).,sp3杂化,s,p轨道杂化,表面的断键,(111),(100),Illustrationofbondcuttingduringsurfaceformation,Danglingbond,backbond,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）,Fourdegenerateddanglingbonds,*,Dimer-bondformation:andanti-levels,降低系统的总能量！,二聚体的buckling,*,DBdown,DBup,FurtherseparationofDBlevelsbydimerbuckling,*,Splittingofdanglingbond(DB)levelsby-*interaction,*,Si(100)-c(4x2)STM形貌图,占据态,空态,STS隧