《固体物理学》课程教学课件（PPT讲稿）Chapter 6 Electron motion in the crystal 6（supplement - a more concise description of quasi-momentum）

6-1 supplement:a more concise description of quasi- momentum Quantum mechanics tells us that the average velocity of electrons in the state of in crystal is equivalent to the speed of the wave packet centered at ko in the classical approximation The propagation velocity of the wave packet group velocity is: 80(k) V8- ak Quantum mechanics of de Broglie:E=h -1E(k) So,the average velocity of electron is:v= h ok Considering the electrons in different bands,the general velocity of electrons in crystal can be written as, ,=VE,()

6-1 supplement: a more concise description of quasi￾momentum Quantum mechanics tells us that the average velocity of electrons in the state of in crystal is equivalent to the speed of the wave packet centered at k0 in the classical approximation. The propagation velocity of the wave packet group velocity is: Quantum mechanics of de Broglie: So, the average velocity of electron is: ( ) g k v k  =  E =  1 E k( ) v k  =  0  k Considering the electrons in different bands, the general velocity of electrons in crystal can be written as, ( ) 1 (k ) E k n k n     = 

6-2 supplement:the reunderstanding of effective mass The electron motion is influenced by the lattice force F and external field force F dv=L(F+F) dt m in practice,it is difficult to deal with F clearly,so it can be rewritten as: dv=F dt m We need to introdue the effective mass m*(replace the real mass m and take the unknown lattice force into account).After using the effective mass,the of-used Newton's laws could be applied for the crystal electronic behavior in the external field.But for the sake of the lattice force,m*is different from m,therefore,the motion of electrons in the crystal is a "quasi-particles",which is called the Bloch electrons

The electron motion is influenced by the lattice force Fl and external field force F, ( ) d 1 d l v F F t m = + in practice, it is difficult to deal with clearly, so it can be rewritten as: Fl * d 1 d v F t m = We need to introdue the effective mass m* (replace the real mass m and take the unknown lattice force into account). After using the effective mass, the of-used Newton's laws could be applied for the crystal electronic behavior in the external field. But for the sake of the lattice force, m* is different from m, therefore, the motion of electrons in the crystal is a "quasi-particles", which is called the Bloch electrons. 6-2 supplement: the reunderstanding of effective mass

rewrite as: Fdt Fdt Fdt m* m m Obviously,when what the electron momentum get from the field is greater than that the momentum transfer to the lattice,effective mass m*>0.On the contrary,what the electron momentum get from the field is less than that the momentum transfer to the lattice,m*<0.When the momentum of all electrons obtained from the external field is passed to the lattice,m*-o and the average acceleration of the electrons is zero.One can also see from the above equation that:the direction of electronic acceleration is the resultant force direction of the field force and the lattice force,not necessarily consistent with the external force direction

rewrite as： d d d * F t F t F t l m m m = + Obviously, when what the electron momentum get from the field is greater than that the momentum transfer to the lattice, effective mass m*>0. On the contrary, what the electron momentum get from the field is less than that the momentum transfer to the lattice, m*< 0. When the momentum of all electrons obtained from the external field is passed to the lattice, m* →∞ and the average acceleration of the electrons is zero. One can also see from the above equation that: the direction of electronic acceleration is the resultant force direction of the field force and the lattice force, not necessarily consistent with the external force direction