ECE 340 Lecture 6 : Intrinsic and Extrinsic Material I Class Outline:

Transcription

1 ECE 340 Lecture 6 : Intrinsic and Extrinsic Material I Class Outline: Effective Mass Intrinsic Material Extrinsic Material

2 Things you should know when you leave Key Questions What is the physical meaning of the effective mass What does a negative effective mass mean? What is intrinsic material? What is thermal equilibrium? What is extrinsic material? How does doping work?

3 Effective Mass At the end of lecture 5, we talked about effective mass Electric Field Electric Field In a vacuum, we can apply Newton s second law: F = qe = m0 In a semiconductor, we cannot. For overall motion NO! For motion in-between scattering NO! We defined a new effective mass which incorporated all of the complicated interactions. * dv F = qe = mn dt dv dt

4 Effective Mass We even defined the effective mass We can define the effective mass as: m * = d E / dk Nevertheless, two questions remain: 1. Where does this definition come from?. What does it mean physically?

5 Effective Mass Let s begin to think about where effective mass comes from Start with the energy-wavevector (dispersion) relation for free electrons: k E k = m Now look at the equation of motion for how electrons move in an energy band in an electric field. Suppose that the wavepacket is made of wavefunctions near a particular k. Ψ(x) k k E The wavepacket is moving with some group velocity, v g : v g = 1 de dk (5.1) (5.) All of the information of the effects of the crystal on the motion of the electron are in the dispersion relation.

6 Effective Mass What are the forces that the electron is experiencing? Ψ(x) E v g How much work is the field doing on the electron? δe = ee v δt (5.3) field g k k We observe that by using eq. 5. δe de dk δk = vgδk = (5.4) Combine eqns. 5.3 and 5.4 to arrive at an external force that is exerted on the electrons by the applied electric field. ee δk = where, dk dt = ee field field δt = F dv g dt d E 1 d E = = dkdt dk 1 d E = F dk dk dt Newton s nd law! 1 m *

7 Effective Mass Simple Example Consider a simple cosine approximation to the band: Sample parameters W (Band Width) ~ 5 ev a (lattice spacing) ~ 0.5 nm 5 0 E 1 ka ( k) = W ( 1 cos ka) = W sin E( k) ev What are the group velocity and the effective mass? Group velocity: ( k) v g ( k) 1 de dk aw = sin ( ka) = v g (k) π a π a

8 Effective Mass The group velocity goes to zero!! What about the effective mass? 0.3 Effective mass: π a -0.3 The effective mass becomes negative! States of positive mass occur near the bottom of the bands due to positive band curvature. States of negative mass occur at the top of bands. Physically, it means that on going from k to k+δk the momentum transfer to the lattice from the electron is larger than that of the momentum transfer from the applied force to the electron. As we approach Bragg reflection at the edge, when we increase the wavevector we can get an overall decrease in the forward momentum. π a m m a * ( k) = sec( ka) 0 W

9 Intrinsic Material Intrinsic Material is pure with no additional contaminants T = 0 K T = 300 K At T = 0 K, there is no energy in the system. All of the covalent bonds are satisfied. Valence band is full and conduction band is empty. At T > 0 K, thermal energy breaks bonds apart Crystal lattice begins to vibrate and exchange energy with carriers. Electrons leave the valence band to populate the conduction band.

10 Intrinsic Material But there are more processes at work Generation Rate: G = G + G + G cm th opt mech 3 s Generation Break up of a covalent bond to form an electron and a hole. Requires energy from thermal, optical, mechanical or other external sources. Supply of bonds to break is virtually inexhaustible. Atomic density >> # of electrons or # of holes.

11 Intrinsic Material Since we are in thermal equilibrium, there must be an opposite process Recombination Rate: 1 n p cm R 3 s N number of electrons P number of holes Recombination Formation of a bond by bringing together and electron and a hole. Releases energy in the form of thermal or optical energy. Recombination events require the presence of 1 electron and 1 hole. These events are most likely to occur at the surfaces of semiconductors where the crystal periodicity is broken.

12 Intrinsic Material In the steady state = The generation rate must be balanced by the recombination rate. 0 = R0 n0 p0 ni n 0 = p0 n0 = p0 = ni G = Important consequence is that for a given semiconductor the np product depends only on the temperature.

13 Intrinsic Material Putting numbers to the intrinsic concentrations Silicon n i ~ cm -3 Germanium n i ~ x cm -3 GaAs n i ~ x 10 6 cm -3 For silicon 5 x 10 3 atoms/cm 3 4 bonds per atom x 10 3 bonds/cm 3 n i (300 K) ~ cm -3 1 broken bond per bonds.

14 Extrinsic Semiconductors The great strength of semiconductors We can change their properties many orders of magnitude by introducing the proper impurity atoms. Which columns add Electrons? Holes? What about impurities?

15 Extrinsic Materials How does a donor work? Silicon (Si) 4 valence electrons Phosphorous (P) 5 valence electrons

16 Extrinsic Materials How does an acceptor work? Silicon (Si) 4 valence electrons Boron (B) 3 valence electrons Si B

17 Extrinsic Materials In general, we can modify the materials properties with the introduction of immobile impurity atoms We can Selectively create regions of n and p. Needed for CMOS. Modify the conductivity over several orders of magnitude. Manipulate the number of conduction electrons over 5 orders of magnitude.