表面电子态的分类概述
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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
k
3 F
/ 3 2
一维有限深势阱
Phase shift z>>0
z<0
Where: sin kz
k0
k02
• 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.
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
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.
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:
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)百度文库
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.
• 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 /
表面处的电子密度
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:
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:
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.
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- :
(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
k
3 F
/ 3 2
一维有限深势阱
Phase shift z>>0
z<0
Where: sin kz
k0
k02
• 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.
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
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.
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:
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)百度文库
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.
• 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 /
表面处的电子密度
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:
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:
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.
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- :