SolidStatePhysicsPPT学习课件PPT课件

1、Chapter 6: Free Electron Fermi Gas第1页/共32页123返回退出自由电子气模型自由电子气模型一、价电子与传导电子一、价电子与传导电子自由Na原子的的电子组态电子组态16223221SPSS价电子轨道半径0.19nm价电子决定了元素的大多数化学性质第2页/共32页123返回退出自由电子气模型自由电子气模型一、价电子与传导电子一、价电子与传导电子 价电子 离子实间距0.37nm 传导电子传导电子决定了金属的大多数特性第3页/共32页123返回退出自由电子气模型自由电子气模型二、德鲁特模型二、德鲁特模型1900年 德鲁特自由电子气模型第4页/共32页123返回退出自

2、由电子气模型自由电子气模型二、德鲁特模型二、德鲁特模型自由电子气系统32910mMZNVNnmA理想气体系统 32510m浓度约第5页/共32页123返回退出自由电子气模型自由电子气模型二、德鲁特模型二、德鲁特模型自由电子气系统理想气体系统 浓度大 浓度小电子气带电理想气体分子电中性第6页/共32页123返回退出自由电子气模型自由电子气模型二、德鲁特模型二、德鲁特模型电子的无轨热运动和漂移运动的叠加mne2第7页/共32页123返回退出自由电子气模型自由电子气模型二、德鲁特模型二、德鲁特模型电子运动方程稳态解电流密度电导率 mne2VmEedtvdmEmeVdEmenVnejd2第8页/共32

3、页123返回退出自由电子气模型自由电子气模型三、成功与失败三、成功与失败电电子子浓浓度度 n: 32928/1010m室室温温下下 : m810求得求得 驰豫时间驰豫时间 : s14151010 电子平均速度电子平均速度 v: sm/105求得求得 平均自由程平均自由程 v: nm101 . 0 似乎自恰性相当好似乎自恰性相当好实实 验验 结结 果果 ： : 100 nm 以以 上上 此此 外外 ： 电电 子子 比比 热热 、 磁磁 化化 率率 等等 也也 遇遇 到到 了了 困困 难难 。第9页/共32页123返回退出自由电子气模型自由电子气模型三、成功与失败三、成功与失败 电子平均自由程：电

4、阻率： 电子对比热贡献： 自由电子气 模型计算实验结果100nm0.1-1.0nm较小仅为计算值的1/200较大怎么办？TT第10页/共32页Chapter 6: Free Electron Fermi Gas The alkali metals: Li, Na, K, Cs, Rb. Na, 3s conduction band. A monovalent crystal which contains N atoms will have N conduction electrons and N positive ion cores. The ion cores fill only about

5、 15 percent of the volume of a sodium crystal ( Na+, R=0.98A)第11页/共32页Chapter 6: Free Electron Fermi Gas The classical theory of the free electron model: Ohms law, the relation between the electrical and thermal conductivity; the heat capacity and the magnetic susceptibility (Maxwell distribution fu

6、nction). Experiments: a conduction electron in a metal can move freely in a straight path over many atomic distances.第12页/共32页Chapter 6: Free Electron Fermi Gas Why is condensed matter so transparent to conduction electrons? (1) A conduction electron is not deflected by ion cores arranged on a perio

7、dic lattice; (2) A conduction electron is only infrequently by other conduction electrons. (Pauli principle). A Free electron Fermi gas: a gas of free electrons subject to the Pauli principle)第13页/共32页1. Energy Levels in One Dimension Consider a free electron gas in one dimension, taking account of

8、quantum theory and of the Pauli principle. An electron of mass m is confined to a length L by infinite barriers. The Schrodinger equation (-2/2m)d2n/dx2=n n , where n is the energy of the electron in the orbital.第14页/共32页1. Energy Levels in One Dimension The Schrodinger equation (-2/2m)d2n/dx2=n n ,

9、 where n is the energy of the electron in the orbital. The boundary conditions are n (0)=0; n (L)=0. Then n (x)=Asin(2 x/ n); n n/2=L,where A is a constant. 第15页/共32页1. Energy Levels in One Dimension第16页/共32页1. Energy Levels in One Dimension The energy n of the electron is given by n = (2/2m)(n /L)2

10、 . In a linear solid the quantum numbers of a conduction electron orbital are n and ms(=1/2). The number of orbitals with the same energy is called the degeneracy. Let nF is the topmost filled energy level, the condition 2 nF=N determines nF.第17页/共32页1. Energy Levels in One Dimension第18页/共32页1. Ener

11、gy Levels in One Dimension The Fermi energy is defined as the energy of the topmost filled level in the ground state of the N electron system. ThusF = (2/2m)(nF /L)2 = (2/2m)(N /2L)2 .第19页/共32页2. Effect of Temperature on the Fermi-Dirac Distribution The ground state is the state of the N electron sy

12、stem at absolute zero. The Fermi-Dirac distribution gives the probability that an orbital at energy will be occupied in an ideal electron gas in thermal equilibrium:第20页/共32页2. Effect of Temperature on the Fermi-Dirac Distribution第21页/共32页2. Effect of Temperature on the Fermi-Dirac Distribution The

13、quantity is a function of the temperature and is the chemical potential. At absolute zero =F (the energy of the topmost filled orbital at T=0). At all temperatures f() is equal to 1/2 when =; When -kBT, f() =exp(- )/ kBT (Boltzmann or Maxwell distribution)第22页/共32页3. Free Electron Gas in 3D The free

14、-particle Schrodinger equation in 3D is第23页/共32页3. Free Electron Gas in 3D The periodic boundary conditions: (x+L,y,z)= (x,y,z), (x,y+L,z)= (x,y,z), (x,y,z+L)= (x,y,z). Wavefunctions satisfying the free-particle Schro-dinger equation and the periodicity condition are of the form of a travelling plan

15、e wave: k(r)=exp(ikr),第24页/共32页3. Free Electron Gas in 3D The wavevector k satisfy kx=0; 2/L; 4/L; , and similarly for kx and kz . Any component of k is of the form 2n /L, where n is an integer. The energy k of the orbital with wavevector k: k=2k2/2m= (2/2m)(k x2 + k y2 + k z2 ).第25页/共32页3. Free Ele

16、ctron Gas in 3D p k(r) =-i k(r) = k k(r), so that the plane wave k(r) is an eigenfunction of the linear momentum with the eigenvalue k; The particle velocity in the orbital k is given by v=k/m. In the ground state of a system of N free electrons the occupied orbitals may be represented as points ins

17、ide a sphere in k space.第26页/共32页3. Free Electron Gas in 3D 第27页/共32页3. Free Electron Gas in 3D The energy at the surface of the sphere is the Fermi energy. The wavevectors at the Fermi surface have a magnitude kF such that: k=2kF2/2m. In the sphere of volume 4kF3/3 the total number of orbitals is 2

18、(L/2 )3(4kF3/3 )=N,第28页/共32页3. Free Electron Gas in 3D 2(L/2 )3(4kF3/3 )=N, Then kF=(32N/V)1/3, which depends only on the particle concentration. The Fermi energy is F=(2/2m) (32N/V)2/3. The Fermi velocity is vF=(kF/m) =(/m) (32N/V)1/3. Fermi temperature TF= F/ kB第29页/共32页3. Free Electron Gas in 3D 第30页/共32页3. Free Electron Gas in 3D The total number of orbitals of energy : N=(V/32)(2m/ 2)3/2, so that the density of states