Lecture3-表面电子态分析

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
z<0
Where: sin kz
k0
k02

2mW0
2
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.
波函数在表面处的连续性
E 3 2
1
Shockley state
Vg<0 Vg>0
The curvature of Ψk can match the decaying vacuum solution only for Vg>0
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）.
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
ห้องสมุดไป่ตู้

(g
2
2
Vg2
)1 2
where:
k g2
能量色散关系E(ĸ2)
能隙
0 Vg / g
0
2 0
2 0
E

V0

(
g 2
)2


2

(g
2
2
Vg2
)1 2
k

g 2

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. Therefore, k can be taken as complex number.
{2 Va (r) [VL (r) Va (r)]} (r) E (r)
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
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.
• In 1932, he predicted what are now called surface states or Tamm states.
• He is also famous for his work on the Soviet Union's hydrogen bomb project.
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 2k2 2m
1
kF (2mEF ) 2 /
M. C. Desjonqùeres and D. Spanjaard, Concepts in Surface Physics, Springer-Verlag, 1996.
一维无限深势阱
Assuming the wave function in the well can be :
With the boundary conditions:
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:
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：
• 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：
• 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.
• 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.
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 z-direction by impenetrable potential at z=0 and z=L, and free to move in xy-direction:
表面处的电子密度
Friedel oscillations ~z-2
(z) k* k k kF
when z is large:
体电子密度
(z)
0
1
3
cos 2kF z (2kF z)2

0 kF3 / 3 2
一维有限深势阱
Phase shift z>>0
• This approximation applies to fairly localized electrons.
• It is successful in the treatment of transition metals and also for wide-bandgap semiconductors and insulators.
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.