Chap 6 Electron motion in the crystal

1. Chap 6 Electron motion in the crystal Objectives At the end of this Chapter, you should: 1.Be able to identify a wave packet in a crystal. 2. Be able to calculate the effective mass at different points in the band. 3. Understand the Electron quasi-momentum. 4. Understand the difference of electron motion in k space and real space. 5. Understand the concept of the hole, and understand its motion behavior.

2. When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in a metal. To my delight，by straight Fourier analysis, I found that the wave differed from the plane wave of free electrons only by a periodic modulation F. BLOCH When an external field (electric and magnetic fields, etc.) is applied to the crystal, the electrons in the crystal does not just feel the role of the external field, but also feel the role of the crystal in a periodic potential. Normally, the external field is much weaker than the crystal periodic potential. Because the periodic field intensity is generally 108 V/cm, very difficult for the external electric field to achieve such magnitude. Therefore, the movement of electrons in the external field in crystal must be discussed based on the eigenstate basis of the periodic field.

3. 6.1 The quasi-classical description of Bloch electrons： There are two methods used: Solving the single-electron wave equation in the presence of the external field. Under certain conditions, solving the electronic movement with external field as a quasi-classical particle. Wave equation with external field Typically, we can only approximately solve the wave equation in the presence of the external field. Very hard. Another method could be considered if：

4. Classical particle have defined energy and momentum, however, for microscopic particles governed by quantum mechanics, it is valid. Wave packet is often used for the description of quantum state classically, whichmeans that the particles is spatially distributed at r0 with uncertainty of △r. The momentum is with uncertainty of . r0 is treated as the centre of the particle, results from the uncertainty princible. The field is weak and constant. Neglect the electronic transitions between different energy bands. Neglect the electron diffraction, interference and collision. This method can be simply used and show a clear physical image. We are willing to adopt it.

5. Wave packet and electron velocity: In crystal, the quasi-classical movement of e- can be described by wave packet composed of Bloch functions. There are eigenstates with different energy in the wave packet. So,time factor should be considered for the Bloch functions. Firstly, consider the 1D case, assume the wave packet is composed of wave functions in the range of k centered at k0.k is small and we can think that: independent of k . For defined k , the Bloch function with time factor is, the neighboring k’ states will be mixed with quantum states k0:

6. Wavelet packet set

7. 0 For the analysis of wave packet movement, only need to analyse2 (that is, the probability distribution). set The wave function centered in the range of , The center of wave packet is : w＝0.

8. There is If the wave packet is a quasi-particle, the particle velocity is The width of Brillouin zone: 2/a ，assume that k is small, generally require that is For 3D case, the velocity of electron is:

9. For the dispersion of Bloch wave, a stable wave packet requires that the wave vector range △k should be very small. Considering the uncertainty relation suggesting that if the wave packet size is much larger than the cell size, the movement of electrons can be described in terms of wave packet. For transportation, only when the electron mean free path is much larger than the cell size, the electrons in the crystal can be solved as a quasi-classical particle. The motion speed of wave packet (group velocity) is equal to the average velocity of the particles in the center of wave packet .

10. important fact 1: The average velocity of electrons are fully determined by energy and wave vector, which is independent of time and space. Therefore, average velocity will keep unchanged without attenuation. This means: the electrons can not be scattered by static atoms. The resistance of a strictly periodic crystal is zero.This point is entirely different from the concept of free electron theory: ion (scattering centers) will affect the average velocity (drift) of the electrons. In other words: If the electron is in a certain state , as long as the lattice periodicity is kept unchanged, the electrons will be in constant motion throughout the crystal with the same velocity. Because the lattice influence on the propagation velocity, have been included in the definition of energy.

11. important fact 2: the direction of electron velocity in k space is the direction of the energy gradient, ie. perpendicular to the energy surface. Therefore, the direction of motion of electrons is determined by the shape of the energy surface, under normal circumstances, in k space, the surface is not spherical. therefore, generally the direction of v is not k direction. The following figure is a more accurate reflection of the Bloch electrons.

12. when the energy surface is a sphere, or in some special direction, v and k are in the same direction. For the 1D case, At the bottom and top of the band, E(k) reaches its extremity: So, v＝0. at a certain point of the band, v(k) E(k) The electron velocityismaximum, which completely different from the fact that the velocity of free electrons is always monotonically increase with the energy.

13. NEF model explanation: at the FBZcenter, the electron can be approximately regarded as the plane wave, v is proportional to the k. With the increase of k, it is around the border of the FBZ, and the influence of lattice scattering increase. At the Brillouin zone boundary, a strong Bragg reflection make the scattered wave and incident wave equal, so the wave velocity is zero. This result can be applied to all the radiated wave propagation in a periodic crystal.

14. 6.2. Electron quasi-momentum ： Set the external field force on electron is F. In dt time, the work of external field on the electrons isFv dt According to the principle of work and power In parallel direction of v, dk/dt and the component of F are equal; when F is perpendicular to velocity v, the principle of work and power can not be used to discuss the changes in the electron energy state, but we can still prove that in the direction perpendicular to the speed, dk/dt and the component of F is also equal.

15. k is the electron quasi-momentum,not the real momentum of Bloch electrons in a strict way.The strict meaning of the rate of momentum change is the force exerted on electrons, while the rate of change of quasi-momentum is only determined by the external field force, here,the potential field of lattice force is not included, which has been embodied in the quasi-momentum. For free electrons, k=p/ is the electron momentum. For Bloch electrons Bloch wave is not the eigen function of momentum.In crystal periodic field, k is the expanded concept of momentum, quasi-momentum or electron-lattice momentum.

16. 6.3 The accelerated velocity and effective mass the external field is dealt in a way of the combination of classic way and the band theory.The basic relationship of quasi-classical motion is: Newton law In addition, assume that the band index n is a constant, that is, ignoring the electron transitions between bands. Based on the above electron motion equation, we can with ease get the electron acceleration.

17. a）1D case the effective mass Due to the periodic field, the acceleration form is obviously not consistent with the electron inertial mass In periodic field, the function between E(k) and k is not a parabola, therefore, m* is not constant, dependent of E and k. m*>0； E(k), minimum At the band bottom, At the band top E(k), maximam m*<0 .

18. b) 3D case in component form, written as ＝1, 2, 3

19. The matrix form Compared with Newton's laws Here, we substitute with a second-order tensor

20. effective mass tensor Which is a symmetric tensor. At the same time, the point group symmetry of the crystal can reduce the tensor. For isotropic crystal, it degenerates into a scalar. The role of the effective mass is to embody the periodic field, so that we can determine the acceleration of electrons only by the external field force. taking kx、ky、kz as the major axis of the tensor, the matrix can be diagonalized as:

21. where The direction of the acceleration is not necessarily consistent with the direction of the external field force, which is determined by the inverted effective mass tensor. The electron effective mass commonly can be experimentally obtained by the electronic specific heat: 0 is the free electronic specific heat coefficient, expis experimental value.

22. Example: seeking electron effective mass of s states for simple cubic crystal .    , 1, 2, 3    kx , ky, kzis the direction of principle axis of tensor, so,

23. The principle component of effective mass are inversely proportional to J1. the bigger interatomic distance, the smaller the J1, thus, the bigger the effective mass. At the bottom point of energy band : k = (0, 0, 0)， The effective mass tensor degraded to be a scalar. At the band top R:

24. At the bottom and the top of the band, the electron effective mass is isotropic, and is a scalar, due to the cubic symmetry.At X point： Normally, effective mass is a tensor. In exceptional circumstances, it can degrade to be a scalar (positive or negative) Near the bottom of the band, effective mass is always positive. Near the top of the band, effective mass is always negative. At the band bottom and top, E (k) take the maximum and minimum values​​, respectively, with positive and negative divalent derivative.

25. Velocity Bloch wave Wave function uncertain simple form，