Lecture3 表面电子态

In the bulk, k has to be real
due to the periodic boundary conditions. However, the termination of a crystal, i.e. the formation of a surface, obviously causes deviation from perfect periodicity.
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：
N
(r) cn(r na) n1
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：
l | VL Va | m l,m l,m1
α: on-site matrix element β: nearest neighbor hopping matrix element
M. C. Desjonqùeres and D. Spanjaard, Concepts in Surface Physics, Springer-Verlag,
一维无限深势阱
Assuming the wave function in the well can be :
With the boundary conditions:
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）.
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
Igor Y. Tamm (1895-1971)
• 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.
The corresponding wave functions are:
长度为L的一维电子气 （周期性边界条件）
where n=0, 1, 2, 3……
The density of k states is thus L/2π
可填充电子数 vs 费米波矢
无边界的一维电子气： L
一维无限深势阱： L
For a given kF, we loose one state 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）
Wave function：
At the continuum limit, N, L, but 2N/L remains constant and equal to the homogeneous bulk electron density ρ0- :
We have:
For z << L:
z
If we integrate ρ-(z)- ρ0- from z=0 to z=, we find:
拓展到三维情况
有表面存在情况下的动量空间
Assuming the electrons are bounded in zdirection by impenetrable potential at z=0 and z=L, and free to move in xy-direction:
z
Then, the normalized wave function is:
where, p=1,2,3… The density of k states is thus L/π
无边界情况
For infinite one dimensional electron gas, the Born-Von Karman boundary condition is:
k (x, y, z) k (x L, y, z) k (x, y L, z) k (x, y,0) k (x, y, L) 0
The corresponding normalized wave function is:
E h2k2 2m
1
kF (2mEF ) 2 / h
表面处的电子密度
Friedel oscillations ~z-2
(z) k* k k kF
体电子密度
when
z
is
large: (z) ;
0
1
3
cos 2kF z (2kF z)2
0
k
3 F
/ 3 2
一维有限深势阱
Phase shift z>>0
z<0
Where: sin kz
k0
k02
2mW0 h2
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
• 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.
• 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 comprehend that the existence of a surface will give rise to surface states with energies different from the energies of the bulk states.
可以得到关于展开系数cn的齐次方程：
cn (E E0 ) (cn1 cn1) 0
可以证明，该方程有下列简单形式的解：
cn Aeinka Beinka A, B为任意常数
在没有表面的情况下，根据周期性边界条件：
(r) (r na)
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/2π 将cn的解代入上述cn的齐次方程可得：
Therefore, k can be taken as complex
number.
当k为虚数时，薛定谔方程存在下列形式解：
(z) ez cos( g z )
2
(z) eqz
z<a/2 z>a/2
where: q2 V0 E 0 Vg / g
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
3. 紧束缚近似模型
• 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.
a a a a ……..
12 3 5 6
N
体系的周期性势场为VL(r)为各格点原子势场Va(r-na)之和：
N
VL (r) Va (r na) n1
其中孤立原子的薛定谔方程为：
[ 2 Va (r) Ea ](r) 0
原子能级 原子束缚态
则一维线性链体系的薛定谔方程为：
{2 Va (r) [VL (r) Va (r)]} (r) E (r)
• 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
考虑由N个原子组成的线性链，原子间距为a：
M. C. Desjonqùeres and D. Spanjaard, Concepts in Surface Physics, Springer-Verlag,
uk (r Rn ) uk (r)
布洛赫定理
一维能带理论
0 a/2 设： e=ħ=m=1
在体内(z<a/2)，该方程可以利用简并微扰法求解:
• In 1932, he predicted what are now called surface states or Tamm states.
• He is also famous for his work on the Soviet
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, equilibrating the electronic density and, thus, are free. It applies to normal metals.
• The nearly free electron model emphasizes the lattice aspects of the problem and simplifies the form of the surface 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-gap semiconductors.
k (z) Aeikz Bei(k g )z g 2 / a
将上述试解代入薛定谔方程可得:
能量本征值： 波函数：
E
V0
(
g 2
)2
2
(g
2
2
Vg2
)1 2
where:
k
g 2
能量色散关系E(ĸ2)
能隙
0 Vg / g
0
2 0
2 0
E
V0
(
g 2
Байду номын сангаас
)2
2
(g
2
2
Vg2
)1 2
k
g 2
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
E 3 2
1
Vg<0 Shockley state
Vg>0
The curvature of Ψk can match the decaying vacuum solution only for Vg>0
• 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 neighboring atoms.