6.4 The electron motion in a constant electric field Take the 1D tight-binding approximation as an example E'(k)=E;-Jo-2J coska s is the atomic energy level.SetJ>0,thus,k=0 is the band bottom;k=±π/a is the band top. ()=I ds2asinka h2 h2 ,m*= h dk h d2E dk2 2a2J coska At k=0 and k=tn/a,the velocity of electron is v(k)=0;but at k=+n/2a,v(k)is maximum or minimum values,respectively
Take the 1D tight-binding approximation as an example ( ) 0 1 2 cos i E k J J ka i = − − I is the atomic energy level. Set J1 >0, thus, k＝0 is the band bottom; k＝/a is the band top. ( ) 1 1 d 2 sin d E aJ v k ka k = = 2 2 2 1 2 2 d d 2 cos E k m a J ka = = At k = 0 and k= /a, the velocity of electron is v(k)=0; but at k= /2a, v(k) is maximum or minimum values, respectively. 6.4 The electron motion in a constant electric field
E(k) m (k) -h-2a -元归 a 0 2a xa -元da 0 π/a Clearly,at the top and bottom of energy band,the velocity is zero. There are maximum and minimum value between them.at the bottom:m*>0,at top:m*<0,in the middle:m*co. We will discuss the movement of electrons in constant electric field starting based on the images
Clearly, at the top and bottom of energy band, the velocity is zero. There are maximum and minimum value between them. at the bottom: m*>0，at top: m*<0, in the middle: m*→±∞. We will discuss the movement of electrons in constant electric field starting based on the images
a)The motion vision in the k space Apply a constant electric field along the -x direction,the force on the electrons equals:Fe along the +x direction. k dk es There is F=h =eg we can obtained: =const dt dt h which means that the speed of electrons is a constant in the k space. In the quasi-classical motion,electrons travel in the same band. Therefore,the motion of electrons with constant velocity in k space means that the curve of energy eigenvalues of electrons are periodically changed as a function of E(k),that is,electrons are doing circular movement in k space,which can be understood from the Brillouin zone map in the next page
In the quasi-classical motion, electrons travel in the same band. Therefore，the motion of electrons with constant velocity in k space means that the curve of energy eigenvalues of electrons are periodically changed as a function of E(k), that is, electrons are doing circular movement in k space, which can be understood from the Brillouin zone map in the next page. a) The motion vision in the k space Apply a constant electric field along the –x direction, the force on the electrons equals: F=e along the +x direction. There is , we can obtained: which means that the speed of electrons is a constant in the k space. d d k F e t = = d const d k e t = =
When electrons meet the Boundary of Brillouin Zone k=,the following situation happen: The difference betweenand reiprocal lattice oractually represent the same state.The electron shif from=and shift in from=to form a circular motion. eE k The electron motion in a constant electric field
When electrons meet the Boundary of Brillouin Zone , the following situation happen: The difference between and is a reciprocal lattice vector , actually represent the same state. The electron shift out from and shift in from to form a circular motion. k a = k a = k a = − k a k = a = − 2 a
when t=0,electron is located at the band bottom,k=0,m*>0. The velocity of electrons v can be accelerated by external force. When arrival at k=z/2a m*->o,v reach maximum.When k go over this point,m*0,so that the reverse speed decreases until k-0 and v-0.This is the speed of oscillation in a constant external field.(See above diagram
when t=0, electron is located at the band bottom, k=0, m*>0. The velocity of electrons v can be accelerated by external force. When arrival at , m*→∞, v reach maximum. When k go over this point, m*0, so that the reverse speed decreases until k=0 and v=0. This is the speed of oscillation in a constant external field. (See above diagram k a = 2 k a = k a = − 2
b)The motion vision in the real space The oscillation of the electron velocity means the electron oscillation in real space (coordinate space).Function E(k)represents the electron energy eigenvalues in the periodic field.When external electric field is applied,an electrostatic potential es will be attached, which makes the energy band inclined as shown below. band inclined by the electric field
b) The motion vision in the real space E x =0 E x band inclined by the electric field The oscillation of the electron velocity means the electron oscillation in real space (coordinate space). Function E(k) represents the electron energy eigenvalues in the periodic field. When external electric field is applied, an electrostatic potential will be attached, which makes the energy band inclined as shown below. e
The periodic oscillations of the electron velocity is in real space When t=0,electron is at point A,the band bottom of lower band. In the presence of external field force,electrons move along A->B >C,corresponding from k=0 to k=".The electron meet energy gap (equal to point A)at C point.In the classical movement, electrons are constrained in one energy band,so it will be reflected back.The electron transmit as C->B->A,that is,a oscillation process is completed during the movement from k=-z/a to k=0
A B C When t = 0, electron is at point A, the band bottom of lower band. In the presence of external field force, electrons move along A→B →C, corresponding from k = 0 to . The electron meet energy gap (equal to point A) at C point. In the classical movement, electrons are constrained in one energy band, so it will be reflected back. The electron transmit as C→B→A, that is, a oscillation process is completed during the movement from k= –π/a to k = 0. k a = The periodic oscillations of the electron velocity is in real space
Two points worthy of notification Point1 It is difficult to observe oscillation phenomenon.Electrons are scattered by phonons,impurity and defects in the traveling process. If the average time interval between neighbor two scatteringtis quite small,electrons have been scattered before oscillation.The time for a complete oscillation is: T- 2π/a2 eslh esa to observe the oscillation process,there aretT. In the crystal ~10-14s,a3X10-10m,so,s~2X 105 V/cm is needed. Generally,T~10-5s,t10-14s,109 times collision will occurs in a period?
Point 1 It is difficult to observe oscillation phenomenon. Electrons are scattered by phonons, impurity and defects in the traveling process. If the average time interval between neighbor two scatteringτis quite small, electrons have been scattered before oscillation. The time for a complete oscillation is: to observe the oscillation process, there are T. In the crystal 10－14 s, a 3×10－10 m, so, 2×105 V/cm is needed. Generally, T ~ 10-5 s,τ～10-14s, 109 times collision will occurs in a period？ Two points worthy of notification
Point 2 all electrons will return when arriving at the energy gap. According to quantum mechanics,some transmission probabilities exist,and only part of electrons will be reflected back π2 transmission probability sexp For insulator or semiconductor,a strong electric field will be established inside the materials,leading to electron interband tunneling,known as the electrical breakdown or Zener effect Similiar to the electrical breakdown,the strong magnetic field will cause the magnetic breakdown
Point 2 all electrons will return when arriving at the energy gap. According to quantum mechanics, some transmission probabilities exist, and only part of electrons will be reflected back. For insulator or semiconductor, a strong electric field will be established inside the materials, leading to electron interband tunneling, known as the electrical breakdown or Zener effect. Similiar to the electrical breakdown, the strong magnetic field will cause the magnetic breakdown. transmission probability
In the presence of the electrostatic field,Bloch electrons periodic oscillation in real space,completely different from free electrons.A brief description of the two-dimensional case are given below: Electron motion in the 2D lattice in the presence of the electric field (a)reduced BZ (b)expanded BZ In the presence of the electric field,electrons start linear motion from an arbitrary point P in k space.When encountering the border, electrons are rebounded to the symmetry point,and start again
In the presence of the electrostatic field, Bloch electrons periodic oscillation in real space, completely different from free electrons. A brief description of the two-dimensional case are given below: In the presence of the electric field, electrons start linear motion from an arbitrary point P in k space. When encountering the border, electrons are rebounded to the symmetry point, and start again