Lecture 17: Semiconductors - continued (Kittel Ch. 8)

Transcription

1 Lecture 17: Semiconductors - continued (Kittel Ch. 8) Fermi nergy Conduction Band All bands have the form - const 2 near the band edge Valence Bands X = (2,,) π/a L = (1,1,1) π/a Physics 46 F 26 Lect 17 1

2 Outline lectrical carriers in Semiconductors Bands near maximum of filled bands, and minimum of empty bands quations of motion in electric and magnetic fields ffective mass lectrons and Holes Intrinsic concentrations in a pure material Law of mass action (Read Kittel Ch 8) Physics 46 F 26 Lect 17 2

3 Real Bands in a Semiconductor - Ge Fermi nergy 3,4 An accurate figure for Ge is given in Kittel Ch 8, Fig Lowest empty state indirect gap 3,4 Filled lower bands since there are 8 electrons per cell All bands have the form - const 2 near the band edge 1 X = (2,,) π/a 1 L = (1,1,1) π/a Physics 46 F 26 Lect 17 3

4 Bands in semiconductor near = Applies to direct gap semiconductors lie GaAs, InAs, Conduction Band gap All bands have the form - const 2 near the band edge Heavy hole Valence Bands Light hole Physics 46 F 26 Lect 17 4

5 Motion of carrier in field Consider one electron in an otherwise empty band (a similar analysis applies to a missing electron in an otherwise full band) dω Group velocity: v = = d 1 h d d If a force is applied the wor done on the electron is the change in energy d d/dt = F. v =. d /dt Using the above relations we find F = h d /dt just as in free case! - independent of the form of the bands! d Physics 46 F 26 Lect 17 5

6 ffective Mass Consider the acceleration of the electron in a band in the presence of a force (e.g. F = -e ) d 1 d d 1 d Acceleration: v = = 2 d 1 d 2 = F h dt d d 2 dt h 2 d 2 dt Thus the electron acts lie it has an effective mass m*, where 1 d 2 = d 2 m* 1 h 2 This is the same as for free electrons, but with an effective mass m* - the motion of the electrons is changed because the electron is in a periodic potential (remember - d /dt does not depend on the bands - but the relation of the velocity to does depend on the bands! Physics 46 F 26 Lect 17 6 h

7 The Simplest Case - added electrons in the conduction band with near Applies to direct gap semiconductors lie GaAs, InAs, State filled by One electron mpty States (Schematic) 2 Physics 46 F 26 Lect 17 7

8 Motion in a field (e.g., F = -e) Time increasing to the right in equal increments In this schematic picture, increases in increments of 4 steps each time unit Velocity increases as (1/m*) (d/dt) F F F State filled by One electron Physics 46 F 26 Lect 17 8

9 Violation of Newton s Laws? How can an electron (mass m e ) act lie it has mass m*? 1 That is: (dv/dt) = (1/m*) (d/dt) = (1/m*) F h The lattice provides the missing momentum! It is the lattice that causes the effect and it is properly included in m*. NOT a violation of Newton s laws! F F F Physics 46 F 26 Lect 17 9

10 What about the valence bands? Consider one empty state in an otherwise filled band. What is the momentum? Since the total for the filled band is, the momentum is the of the unbalanced electron -- The momentum is to the right! State with one missing electron Filled States (Schematic) unbalanced electron 2 Physics 46 F 26 Lect 17 1

11 Motion in a field (e.g., F = -e) Time increasing to the right in equal increments In this schematic picture, all the states move to the right in increments of 4 steps each time period Unbalanced State moves to left! F unbalanced state F F mpty State Physics 46 F 26 Lect 17 11

12 What is going on? There are two ey points: 1. The electrons actually accelerate to the left - opposite to the force - acts lie a hole that has positive charge and is moving to the right 2. The energy of the system is also opposite to energy plotted - the total energy increases as the hole moves downward F unbalanced state F F mpty State Physics 46 F 26 Lect 17 12

13 Conductivity Both electrons and holes contribute 1. An electron in the conduction bands has negative charge 2. A hole in the valence band has positive charge e F = - e J h F = e J Ohm s law results from scattering that limits the velocity Physics 46 F 26 Lect 17 13

14 Holes in semiconductors This can all be put together (see Kittel p ) by defining: 1. hole = - missing electron 2. hole = - missing electron 3. v hole = + v missing electron 4. m* hole = - m* missing electron > hole 5. q hole = - q missing electron = + e (positive!) hole mpty State Physics 46 F 26 Lect 17 14

15 quilibrium Concentration Details - See Kittel p Density of electrons = n = c Dc () f() d Parabolic Approx. for conduction band: n = 2(m c B T/ 2 π 2 ) 3/2 exp( -( c - µ)/ B T) Density of holes = p = v Dv () (1-f() )d Parabolic Approx. for valence band: p = 2(m v B T/ 2 π 2 ) 3/2 exp( -(µ - v )/ B T) Product: n p = 4 ( B T/ 2 π 2 ) 3 (m c m v ) 3/2 exp( -( c - v )/ B T) Physics 46 F 26 Lect 17 15

16 Law of Mass Action Product n p = 4 ( B T/ 2 π 2 ) 3 (m c m v ) 3/2 exp( -( c - v )/ B T) is independent of the Fermi energy ven though n and p vary by huge amounts, the product np is constant! Why? There is an equilibrium between electrons and holes! Lie a chemical reaction, the reaction rate for an electron to fill a hole is proportional to the product of their densities. If one creates more electrons by some process, they will tend to fill more of the holes leaving fewer holes, etc. Physics 46 F 26 Lect 17 16

17 Summary lectrical carriers in semiconductors involve bands near maximum of filled bands, minimum of empty bands quations of motion in electric and magnetic fields ffective mass Acts lie m*, with 1/m* = d 2 /d 2 lectrons and Holes A hole is the absence of electron in a filled band - Acts lie positive charge, with change of sign of and, positive m*, with 1/m* - d 2 /d 2 Intrinsic concentrations in a pure material Law of mass action n p = value that depends on material and T (Read Kittel Ch 8) Physics 46 F 26 Lect 17 17

18 Next time More on concentrations of electrons and holes in Semiconductors Control of conductivity by doping (impurities) Mobility Carriers in a magnetic field Cyclotron resonance Hall effect Thermoelectric effect (Read Kittel Ch 8) Physics 46 F 26 Lect 17 18