2 Numerical ample: Carrier Concentrations Donor concentration: N d = 10 15 cm -3 Thermal equilibrium electron concentration: n o N d = 10 15 cm 3 Thermal equilibrium hole concentration: 2 2 p o = n i no n i Nd = ( 10 10 cm 3 ) 2 10 15 cm 3 = 10 5 cm 3 Silicon doped with donors is called n-type and electrons are the majority carriers. Holes are the (nearly negligible) minority carriers.
Doping with Acceptors Acceptors (group III) accept an electron from the lattice to fill the incomplete fourth covalent bond and thereby create a mobile hole and become fied negative charges. ample: Boron. B + mobile hole and later trajectory immobile negatively ionized acceptor Acceptor concentration is N a (cm -3 ), we have N a >> n i typically and so: one hole is added per acceptor: p o = N a equilibrium electron concentration is:: n o = n i 2 / N a
Compensation ample shows N d > N a positively ionized donors As + As + B mobile electron and trajectory negatively ionized acceptor Applying charge neutrality with four types of charged species: ρ = qn o + qp o + qn d qn a = qp ( o n o + N d N a ) = 0 we can substitute from the mass-action law n o p o = n i 2 for either the electron concentration or for the hole concentration: which one is the majority carrier? answer (not surprising): N d > N a --> electrons N a > N d --> holes
Carrier Concentrations in Compensated Silicon For the case where N d > N a, the electron and hole concentrations are: 2 n i n o N d N a and p o ------------------ N d N a For the case where N a > N d, the hole and electron concentrations are: 2 n i p o N a N d and n o ------------------ N a N d Note that these approimations assume that N d - N a >> n i, which is nearly always true.
Carrier Transport: Drift If an electric field is applied to silicon, the holes and the electrons feel an electrostatic force F e = (+q or - q). Picture of effect of electric field on representative electrons: moving at the thermal velocity = 10 7 cm/s... very fast, but colliding every 0.1 ps = 10-13 s. Distance between collsions = 10 7 cm/s 10-13 cm = 0.01 µm (a) Thermal quilibrium, = 0 (b) lectric Field > 0 lectron # 1 lectron # 1 i f,1 i f,1 lectron # 2 lectron # 2 f,2 i f,2 i lectron # 3 lectron # 3 f,3 i f,3 i * i = initial position * f, n = final position of electron n after 7 collisions The average of the position changes for the case with > 0 is < 0
Drift Velocity and Mobility The drift velocity v dn of electrons is defined as: v dn = ----- t periment shows that the drift velocity is proportional to the electric field for electrons v dn = µ n, with the constant µ n defined as the electron mobility. Holes drift in the direction of the applied electric field, with the constant µ p defined as the hole mobility. v dp = µ p How do we know what s positive and what s negative? positive: negative:
lectron and Hole Mobilities mobilities vary with doping level -- plot is for 300 K = room temp. 1400 1200 mobility (cm 2 /Vs) 1000 800 600 400 holes electrons 200 0 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 N d + N a total dopant concentration (cm 3 ) typical values for bulk silicon - assuming around 5 10 16 cm -3 doping µ n = 1000 cm 2 /(Vs) µ p = 400 cm 2 /(Vs) at electric fields greater than around 10 4 V/cm, the drift velocities saturate --> ma. out at around 10 7 cm/s. Velocity saturation is very common in VLSI devices, due to sub-micron dimensions
Carrier Transport: Drift Current Density lectrons drifting opposite to the electric field are carrying negative charge; therefore, the drift current density is: J n dr = (-q) n v dn units: Ccm -2 s -1 = Acm -2 J n dr = (-q) n (- µ n ) = q n µ n Note that J n dr is in the same direction as the electric field. For holes, the mobility is µ p and the drift velocity is in the same direction as the electric field: v dp = µ p The hole drift current density is: J p dr = (+q) p v dp J p dr = q p µ p
Drift Current Directions and Signs For electrons, an electric field in the + direction will lead to a drift velocity in the - direction (v dn < 0) and a drift current density in the + direction (J n dr > 0). electron drift current density J n dr v dn For holes, an electric field in the + direction will lead to a drift velocity in the + direction (v dp >0) and a drift current density in the + direction (J n dr > 0). hole drift current density J p dr v dp