Numerical Example: Carrier Concentrations
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1 2 Numerical ample: Carrier Concentrations Donor concentration: N d = cm -3 Thermal equilibrium electron concentration: n o N d = cm 3 Thermal equilibrium hole concentration: 2 2 p o = n i no n i Nd = ( cm 3 ) 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.
2 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
3 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
4 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.
5 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 = s. Distance between collsions = 10 7 cm/s 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
6 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:
7 lectron and Hole Mobilities mobilities vary with doping level -- plot is for 300 K = room temp mobility (cm 2 /Vs) holes electrons N d + N a total dopant concentration (cm 3 ) typical values for bulk silicon - assuming around 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
8 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
9 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
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