Electric Field--Definition. Brownian motion and drift velocity

Transcription

1 Electric Field--Definition Definition of electrostatic (electrical) potential, energy diagram and how to remember (visualize) relationships E x Electrons roll downhill (this is a definition ) Holes are like bubbles that float uphill (We didn t show E F relative to E fi so we don t know about external bias, internal doping etc.) 5 Brownian motion and drift velocity Particles (electrons and holes) move around at thermal velocity (v th ) and here over time have no NET motion they return to the same point (5.26) In the presence of electric field (E) the carriers (here holes) over time do have NET motion. This effective drift velocity (v d ) is much less than v th PS#2 Problem 6 3

2 Mobility--the linear part of v-e curve Ge V sat ~v th Si PS#2 Problem 7 Drift current--total charge x velocity (5.1) (5.4) (5.5) (5.8) e - h + + E - 8 4

3 Mobility (definition and physical parameters) Force equation leads to definition of drift velocity. Given constant electric field, then τ cp is the mean time between collisions (5.10) (5.11) (5.12a) 9 (5.12b) (5.13) (5.14) Based on linearity of problem, the average drift velocity is 1/2 the peak But (as per text) more careful analysis gives following definition of mobility in terms of fundamental quantities: Mean time between collisions Carrier effective mass τ/m * 10 5

4 Process/Ambient dependence of mobility Mobility depends on ambient conditions as well as NET IONIZED doping (N I ): (5.16) where Always REMEMBER This! (5.15) (5.18) We ll talk more about this summation rule or Mathieson s rule (Lect. 7) 11 Temperature Dependence of µ Region where µ I ~T +3/2 Region where µ L ~T -3/2 Data circa 1949 by Bardeen (two-time Nobel Laureate) and Pearson (Stanford Prof., advisor of Prof. Jim Harris) 12 6

5 Mobility vs. Doping Really Important Points: Mobility depends strongly on doping (=ionized impurity scattering centers) The number of scattering centers is determined (in any region) by the TOTAL number of dopants (N), not the effective number. Where T n =T/ Ohm s Law--Fundamental & Circuit-level Ohm s Law (Fundamental) (5.19) We ve already said this (but as a reminder ) (5.21a,b) ~1/doping (5.20) (5.22a) Ohm s Law (Circuit/Device) (5.22b) 14 7

6 Resistivity vs. Doping (Si, Ge ) Why n-type lower than p-type?? Silicon--holes (boron doping) and electrons (phosphorus doping) Compounds (GaAs & GaP) along with Germanium (Ge) 15 Resistor (our first device ) Example: L/W~3 W x J ρ A=WX J Useful DESIGN Formulation: ρ/x J has units Ω/ square L/W is a # of squares 16 8

7 Number of Squares x Sheet Resistance = Resistance R=3xR sh R=6xR sh R=14xR sh 17 An Example Create a resistor with value R=500Ω, using reasonable doping and geometry in a bulk p-type wafer (doping /cm 3). We will counter dope an n-type region let s assume that it is the n-well used in CMOS technology. If it is 20X higher doped than the substrate->2x10 16 /cm 3 (donors)--let s assume also that it s UNIFORMLY DOPED. If the junction depth (x J ) is 2µm, and the electron mobility is ~1100cm 2 /Vsec, then: Units!! µm=10-4 cm 18 9

8 Now, using the formula for R=L/W * sheet resistance, to achieve 500Ω, we will need L/W=500(Ω)/1420(Ω/sq) If we rounded off to tenth-of-microns, then L=5µm and W=14.2µm would work [Note: we are ignoring all end effects] (we could have used L=0.5µm and W=1.4µm but, given x J =2µm, variations might not give such good results ) L=0.5µm W=1.4µm x J =2µm 19 View of MOSFET as Resistor - V + I A n + n + PS#1 Problem 5 P-type Substrate L 1 L 2 L 3 n - region (induced by +V gate ) (J=σE) Ohm s Law--each R i 20 10

9 Another Driving Force --Diffusion Flux Diffusion also moves carriers and gives rise to currents-- basically this compliments the drift current (Ohm s Law) Driving Force: gradients of charge n(x,0) n(x,t) x=0 The text (and next slide) formulates the problem in terms of positive gradient as shown next. 15 Formulation of Diffusion Equation Given this distribution of charge n(x) F n DEFINED in +x direction resulting flux F n e - J n dn/dx>0 (5.28) (5.29) (5.30) (5.31) 16 8

10 Diffusion Currents J n J p 17 Drift-Diffusion (DD) Equations We can use these equations individually (as shown in examples that follow) --or-- We can combine the equations to get the TOTAL current as shown next. 18 9

11 Combination of Ohm s Law and Diffusion Current contributions Called Drift-Diffusion (DD) Equations DIFFUSION (5.33) DRIFT (5.34) Ohm s Law Terms (signs same) Diffusion Terms (opposite signs) Comment: Typically we don t solve these equations with BOTH carriers at once, we find ways to simplify the problem by considering DRIFT for the majority carriers and DIFFUSION for the minority carrier species more to come 19 Electric Field--Definition Definition of electrostatic (electrical) potential, energy diagram and how to remember (visualize) relationships E x Electrons roll downhill (this is a definition ) Holes are like bubbles that float uphill (We didn t show E F relative to E fi so we don t know about external bias, internal doping etc.) 20 10

12 Electric Field--Definition Definition of electrostatic (electrical) potential, energy diagram and how to remember (visualize) relationships E x Electrons roll downhill (this is a definition ) Holes are like bubbles that float uphill (We didn t show E F relative to E fi so we don t know about external bias, internal doping etc.) 5 Electro-Static and Energy Perspectives +V E x Ground potential (0) +V (ref. Ground) 6 3

13 Reminders about E f and carriers E f E i E i E f 7 What happens if we have a NON-UNIFORMLY DOPED material? (motivation this is very much like a problem on PS#2) (Log) Concentration N d (x) N a Depth 8 4

14 Non-uniform Doping (and E x ) Suppose that doping is non-uniform, as shown below (Log) Concentration N d (x)~n d+ (x) E x n(x) given that E F is flat (in equilibrium ) (5.37) (5.38) Depth Note: N d and n(x) are not exactly equal this is required in order to have NET charge that supports the electric field 9 (5.39) Negative quantity (5.40) Example: If N(x) is an exponential, then the resulting E-field is CONSTANT (try an example with N(x)=No exp(-x/l) to see for your self) 10 5

15 Energy Band Diagram E C E F E Fi E V n-type Electrons roll down (due to E-field) p-type (Same as previous slide but now showing an acceptor doping*) (Log) Concentration N d (x)~n d+ (x) N a (constant) - n(x) Depth E F =E Fi *Without a p-type region E F DOES NOT CROSS E Fi n=p 11 Comment (and Preview of PN) N E f E i P E i E f If we stuck these two together E f MUST be flat (constant) why? ΔE E i E f Bands (E c, E v ) as well as E i will bend smoothly. They are FLAT at some point where there is NO electric field. E f FLAT everywhere 12 6

16 Another Example Another example, this time starting from diagram where bands are sloped but straight lines * Let s consider the problem in reverse what is the corresponding n(x)? and field is constant (bands constant slope) using this constant and integrating. *NOTE: bands are sloped due to DOPING (not E-field) also note that E F is FLAT!! 13 (Basically, this example is the inverse problem of what you ll do on the HW) 14 7

17 The relationship between Drift and Diffusion (5.41) (5.42) (given) (5.40) (required E x to make J n =0) (required relationship to satisfy the equation) Einstein Relationship (5.45) 15 Auxiliary Derivation Based on the equations from the previous page, we can generalize this relationship between electric field and electrostatic potential in terms of carrier concentrations as follows: Example: For PN device, without solving for details of how the field gets established 16 8

18 Physical (microscopic) Interpretation of Einstein Relationship Basic Definitions Using Einstein Relationship ( macroscopic result) (Recall this how we got v th ) Physical relationship ( microscopic interpretation) 17 9