1、本節(jié)課主要內容,凝膠模型 近自由電子近似 緊束縛近似 鏡像態(tài) 實例1：貴金屬表面態(tài) 實例2：半導體表面態(tài) 實例3：拓撲絕緣體表面態(tài) 實例4：高溫超導體表面態(tài),表面電子態(tài)的分類,表面態(tài)的發(fā)現者,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 c

2、alled surface states or Tamm states. He is also famous for his work on the Soviet Unions hydrogen bomb project.,Igor Y. Tamm (1895-1971),1. 凝膠模型,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 sp

3、ace, equilibrating 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.,一維無限深勢阱,z,Assuming the wave function in the well can be :,With the boundary conditions:,Then, the normalized wave functi

4、on is:,where, p=1,2,3,The density of k states is thus L/,無邊界情況,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,長度為L的一維電子氣 （周期性邊界條件）,可填充電子數 vs 費米波矢,無邊界的一維電子氣：,一維無限深勢阱：,L,

5、L,For a given kF, we loose one state at the bottom of the band on making two surfaces.,有邊界情況下的電子密度（一維無限深勢阱）,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

6、L:,z,拓展到三維情況,Assuming the electrons are bounded 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:,有表面存在情況下的動量空間,表面處的電子密度,體電子密度,Friedel oscillations,z-2,when z is large:,一維有限深勢阱,z0,z0,Where:,Phase shift,當前模型的局限,沒有

7、考慮電子間的交換關聯作用。 忽略了原子核的周期性分布。 非自洽的計算：勢場應該從波函數得到。,更精確的方法： 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 o

8、f the problem and simplifies the form of the surface barrier.,2. 近自由電子模型,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 e

9、lectronic structure of normal metals and some narrow-gap semiconductors.,M. C. Desjonqeres and D. Spanjaard, Concepts in Surface Physics, Springer-Verlag, 1996.,布洛赫定理,一維能帶理論,設： e=m=1,a/2,0,在體內(za/2)，該方程可以利用簡并微擾法求解:,將上述試解代入薛定諤方程可得:,能量本征值：,波函數：,where:,能量色散關系E(2),能隙,0,In the bulk, k has to be real due

10、to the periodic boundary conditions. However, the termination of 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:,當k為虛數時，薛定諤方程存在下列形式解：,根據表面處的波函數連續(xù)性，可以唯一確定k的取值。該k值對應的電子態(tài)能量位于體能隙之中，其波函數局域在表面附近，

11、在表面外和體內都呈衰減行為。該電子態(tài)被稱作表面態(tài)。,波函數在表面處的連續(xù)性,Vg0,Vg0,Shockley state,E,1,2,3,The curvature of k can match the decaying vacuum solution only for Vg0,近自由電子模型的使用范圍,Assuming that the valence electrons are completely delocalized when the solid is formed（metal）. The perturbation of the lattice potential is weak（n

12、arrow-band semiconductor）.,3. 緊束縛近似模型,The tight-binding approximation using 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 wi

13、de-bandgap semiconductors and insulators.,物理圖像,This picture, it is easy to 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 partner

14、s on one side their orbitals have less overlap with the orbitals of neighboring atoms. The splitting and shifting of energy levels of the atoms forming the crystal is therefore smaller at the surface than in the bulk.,一維線性鏈模型,體系的周期性勢場為VL(r)為各格點原子勢場Va(r-na)之和：,考慮由N個原子組成的線性鏈，原子間距為a：,其中孤立原子的薛定諤方程為：,原子能

15、級,原子束縛態(tài),則一維線性鏈體系的薛定諤方程為：,微擾,根據簡并微擾方法，線性鏈共有化電子的波函數可以寫為：,將上式代入體系的薛定諤方程，并只考慮最近鄰格點的交疊積分：,: on-site matrix element : nearest neighbor hopping matrix element,可以得到關于展開系數cn的齊次方程：,可以證明，該方程有下列簡單形式的解：,A, B為任意常數,將cn的解代入上述cn的齊次方程可得：,在沒有表面的情況下，根據周期性邊界條件：,可以得知，k為簡約波矢，在第一布里淵區(qū)內共有N個值，密度為Na/2,在有表面存在的情況下，展開系數cn在表面處應滿足：

16、,n=1,n=N,考慮到表面處的on-site matrix element可能與體內不同：,所以有如下關于cn的齊次方程組：,n=1,n=N,1nN,上述齊次方程組有非零解的條件是cn的系數矩陣的行列式為0,由此可得到關于E的一個超越方程，共有N個解，大多數解對應的波函數在鏈中每個原子上有差不多的強度，,其中有兩個解不同于體態(tài)值，其對應的波函數在表面上有很大的強度，而在體內原子上的強度很小，這便對應于表面態(tài)，一般稱作Tamm state。,但是如果：,n=1,n=N,1nN,例子：三原子線性鏈,由于on-site matrix element 只引起能級的整體移動，所以我們這里忽略，并把E0

17、作為能級原點，則有：,關于cn的齊次方程組可以簡化為：,由cn的系數矩陣的行列式為0，可得：,從而得到能量本征值E：,對應的本征波函數為：,可見， 2（r）只在表面處原子上有分布，對應于表面態(tài),Surface state,鏡像態(tài)-強關聯效應,前面的模型都假設電子-電子相互作用比較弱，或者不考慮電子-電子相互作用。 如果電子-電子相互作用較強，將出現一種新的表面態(tài)：鏡像態(tài)。 不同于一般的表面態(tài)，鏡像態(tài)往體內迅速衰減，而在表面外可以延伸很長的距離。,鏡像勢,Strongly correlated interaction,En = -0.85 eV/(n+a)2, n = 1,2,.,類似于氫原子的

18、能級,金屬電子對表面外電荷的屏蔽效應,Cu(100)和Cu(111)表面的鏡像態(tài),Non-resonant,Resonant,金屬/絕緣體界面的鏡像態(tài),Progress in Surface Science 80 (2005) 4991,兩個界面都有鏡像態(tài)！,2PPE 測得的Cu(100)表面的鏡像態(tài),Progress in Surface Science 80 (2005) 4991,STM z(V) spectroscopy,利用STM探測表面的鏡像態(tài),休息15分鐘,實例1：貴金屬表面態(tài),Ag(111)表面在費米面附近存在能隙，而其他表面不存在能隙,Ag各個表面的能帶結構,各種貴金屬的（1

19、11）面,Au(111) 表面態(tài)的測量,ARPES,STM dI/dV spectroscopy,Constant density of states (2D),自旋軌道耦合,表面駐波,Standing waves on Ag(111) surface,http:/www.omicron.de,局域態(tài)密度：,表面態(tài)電子被臺階的散射,kx,ky,k1i,k1s,k2i,k2s,0,臺階,等能面E,M. F. Crommie et al., Science 363, 524 (1993),局域態(tài)密度：,表面態(tài)的能量色散,Y. Hasegawa and Ph. Avouris, Phys. Rev.

20、 Lett. 71, 1071 (1993),Quantum corrals,Fe adatoms on Cu(111),1993 - IBMs Quantum Corral,Other arrangements of iron atoms on copper substrates. Note that only the perfect circle shows the standing wave of electrons.,表面態(tài)的應用1: 原子超晶格,F. Silly et al., Phys. Rev. Lett. 92, 016101 (2004),表面態(tài)的應用2：Molecular

21、graphene,人工石墨晶格：CO on Cu (111),CO,K. K. Gomes et al., Nature 483, 306 (2012),表面態(tài)的應用2：molecular graphene,實例2：半導體表面態(tài),There is a natural quasi-chemical 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-bindin

22、g approach.,半導體表面態(tài)可以由緊束縛近似很好的描述,Principle of semiconductor surface reconstruction,Basic principle: the surface structure observed will be the lowest free-energy structure kinetically accessible under the preparation conditions. Principle 1: A surface tends to minimize the number of dangling bonds by

23、 formation of new bonds. The remaining dangling bonds tend to saturated. Principle 2: A surface tends to compensate charges. Principle 3: A semiconductor surface tends to be insulating (or semi-conducting).,sp3雜化,s,p 軌道雜化,表面的斷鍵,(111),(100),Illustration of bond cutting during surface formation,Dangling bond,back bond,Si(111)和Si(100) 的表面態(tài),Si(100)的表面雜化,Si(100)表面的二聚體重構（2x1）,Four degenerated dangling bonds,*,Dimer-bond formation: and anti- levels,降低系統(tǒng)的總能量！,二聚體的buckling,*,DB down,DB up,Further separation of DB levels by dimer buckling,*,Splitting