ECE-305: Spring 015 Carrier Action: II Pierret, Semiconductor Device Fundamentals (SDF) pp. 89-104 Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu Lundstrom ECE 305 S15 /4/15 drift current moving carriers produce a current a electric field cause carries to move at a velocity that is +/- the mobility times the electric field. J n drift = n( q)υ dn = σ n E = +nqµ n E drift = σ p E = pqµ p E J tot drift = ( σ n + σ p )E = ( nqµ n + pqµ p )E
Question 1 1) What is the resistivity of an intrinsic semiconductor? a) b) c) d) e) ρ = 0 ρ = ρ = 1 ( n i qµ p + n i qµ n ) ρ = 1 ( n i qµ p n i qµ n ) ρ = 1 ( n i qµ n n i qµ p ) σ n = nqµ n σ p = pqµ p outline 1. Diffusion current. The Fermi level is constant 3. Energy band diagrams 4. The relation between D and µ Lundstrom ECE 305 S15 4
Fick s Law of diffusion p( ) F F = q D cm s F = D p = D dp d # cm -s 0 (Adolph Fick, 1855) 5 diffusion current p( ) + l is the mean-freepath for scattering = + p L p R 0 l 6 0 0 + l
diffusion current p( ) l is the mean-freepath for scattering + p L p R 0 l 0 0 + l 7 + = q p L υ T = q p R υ T = J + p J p = q υ T ( p p L R ) = q υ T l ( p L p R ) l = qd p dp d D p υ T l cm s diffusion currents for electrons and holes diff = qd p p D p υ l Tp p cm J n diff = qd n n D n υ l Tn n cm s s 8
diffusion current Question : What direction is the electric field? a) To the right b) To the left p( ) E 0 diff = qd p dp d Is there a current or flu ( /q) of holes? in equilibrium, NO There must be a drift current that eactly cancels the diffusion current. 9 drift- diffusion equation = J p drift + J p diff = pqµ p E qd p p current = drift current + diffusion current J n = J n drift + J n diff = nqµ n E + qd n n µ p = qτ m p * D p = υ Tp l p µ n = qτ m n * D n = υ Tn l n J = + J n mobility and diffusion coefficient 10
is there a relation between D and µ? L n-type semiconductor I υ d = µ n E V + the drift current is also diffusive 11 outline 1. Diffusion current. The Fermi level is constant 3. Energy band diagrams 4. The relation between D and µ Lundstrom ECE 305 S15 1
the Fermi level in equilibrium E E 1 E F E C E V ( ) ( ) E i ( ) f ( E) = 1 ( ) k B T 1+ e E 1 E F Lundstrom ECE 305 S15 13 the Fermi level in equilibrium Very important point: The Fermi level is constant in equilibrium. Lundstrom ECE 305 S15 14
outline 1. Diffusion current. The Fermi level is constant 3. Energy band diagrams 4. The relation between D and µ Lundstrom ECE 305 S15 15 band bending E? KE E C +V G E i V = 0 E F E V 16
Voltage and electron potential energy E = qv - +V 17 V ( ) band bending +V G V = 0 18
band bending E E C ( ) = E C + ( ) qv ( ) E C E i V = 0 E F E V ( ) de C d = q dv ( ) = qe d 19 reading an energy band diagram E E C V ( ) = 1 q E ( ) C E i E F ( ) E = 1 de C q d E V ( ) = n i e E F E i n ( ( )) k B T ( ) = n i e E i p ( ( ) E F ) k B T 0
practice E E C Sketch: Electrostatic potential E i E F E V Electric field Electron density Hole density 1 Kroemer s lemma of proven ignorance Whenever I teach my semiconductor device physics course, one of the central messages I try to get across early is the importance of energy band diagrams. I often put this in the form of Kroemer s lemma of proven ignorance : If, in discussing a semiconductor problem, you cannot draw an Energy Band Diagram, this shows that you don t know what you are talking about. coroallary: If you can draw one, but don t, then your audience won t know what you are talking about. (Nobel Lecture, 000)
draw the energy band diagram p( ) 1) Find the potential vs. position ) Find the electric field vs. position 3 outline 1. Diffusion current. The Fermi level is constant 3. Energy band diagrams 4. The relation between D and µ Lundstrom ECE 305 S15 4
Einstein Relation = pqµ p E qd p dp d = 0 ( p = n i e E i E F ) k B T (non-degenerate semiconductor) D p = k T B µ p q D n = k T B µ n q (Einstein, 1905) Fig. 3.5 of SDF gives both the mobility and diffusion coefficient. 5 mobility vs. doping D µ = k T B q from R.F. Pierret, Semiconductor Device Fundamentals, Fig. 3.5 (a) 6
drift- diffusion equation µ p = qτ m p * µ n = qτ m n * D p = υ Tp l p D n = υ Tn l n = J p drift + J p diff = pqµ p E qd p p current = drift current + diffusion current J n = J n drift + J n diff = nqµ n E + qd n n total current = electron current + hole current J = J p + J n D p µ p = D n µ n = k B T q 7 energy band diagrams Drawing them Reading them 8