Carrier transport: Drift and Diffusion

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1 . Carrier transport: Drift and INEL Solid State Devices - Spring 2012 Manuel Toledo April 10, 2012 Manuel Toledo Transport 1/ 32

2 Outline...1 Drift Drift current Mobility Resistivity Resistance Hall Effect...2 Haynes-Shockley Experiment Manuel Toledo Transport 2/ 32

3 Drift Drift: carrier motion due to applied electric field Manuel Toledo Transport 3/ 32

4 Drift current Drift current drift velocity: v d distance traveled by hole in time t: v d t holes crossing the plane in time t: pv d ta drift current for holes: i p,drift = qpv d A Manuel Toledo Transport 4/ 32

5 Drift current Drift current In vector notation: J p,drift = qpv d J n,drift = qnv d mobility: constant of proportionality between v d and E J p,drift = qpµ p E J n,drift = qnµ n E < µ >: cm 2 /V sec; For Si: µ n = 1360cm 2 /V sec and µ p = 460cm 2 /V sec at T = 300K and N D = /cm 3, N A = /cm 3, respectively. For GaAs: µ n = 8000cm 2 /V sec and µ p = 320cm 2 /V sec at T = 300K and N D, N A < /cm 3. Manuel Toledo Transport 5/ 32

6 Drift current Manuel Toledo Transport 6/ 32

7 Mobility Mobility is affected by scattering events with: i Phonons, ii Ionized impurity atoms, iii neutral impurity atoms and defects, iv other carriers, v internal electric field created by the piezoelectric effect. Most important are i and ii. Mobility if only i is present: µ L T 3/2 Mobility if only ii is present: µ I T 3/2 /N I Manuel Toledo Transport 7/ 32

8 Mobility ( ) 1 µ n = 1 µ Ln + 1 µ In + 1 µ p = 1 µ Lp + 1 µ Ip + Manuel Toledo Transport 8/ 32

9 Mobility Experimental fit µ = µ min + µ (N/N ref ) α N is either N D or N A. Other parameters exhibit a temperature dependence of the form A = A 0 (T/300) ν where A 0 is a constant, T is the absolute temperature, and ν is the temperature exponent. Manuel Toledo Transport 9/ 32

10 Mobility Manuel Toledo Transport 10/ 32

11 Mobility Small-device effects velocity saturation of electrons on Si: v dsat = v 0 dsat 1 + Ae T/T d where A = 0.8, v 0 dsat = cm/sec, T d = 600K, and T is temperature in degrees Kelvin. Inter-valley carrier transfer Ballistic transport Manuel Toledo Transport 11/ 32

12 Mobility Manuel Toledo Transport 12/ 32

13 Resistivity Resistivity Resistivity ρ = 1 q ( µ n n + µ p p ) = 1 σ For n-type let p 0 and set n = N D. For p-type let n 0 and set p = N A. Ohm s Law: J = σe = E ρ E = ρj Manuel Toledo Transport 13/ 32

14 Resistivity Manuel Toledo Transport 14/ 32

15 Resistance Manuel Toledo Transport 15/ 32

16 Resistance Ohm s Law: I = V/R Integrated Circuit Resistor (see previous slide) ρ is the material s resistivity conductivity = σ = 1 ρ Sheet resistance: R S = 1 σx j Sheet resistance is expressed in Ω/. R = ρ L x j W R = R S L W Example: For R = 3.5kΩ, and a sheet resistance of R S = 200Ω/ and a feature size W = 1µm, use L = 17.5µm. The smaller W the better. Manuel Toledo Transport 16/ 32

17 Resistance Integral = Average like above form of Ohm s Law Differential = local more accurate model. Conductivity varies with depth. See next slide. Average value of conductivity is used as an approximation. σ = 1 σ(x)dx x j Average conductivity no information about current distribution in resistor. 0 Manuel Toledo Transport 17/ 32

18 Resistance Typical conductivity profile. Manuel Toledo Transport 18/ 32

19 Resistance Using x 0 = 2, we can find the average conductivity: σ(x) = σ(0)e (x/x 0) 2 σ = 10 x j 0 e (x2 /2) dx 10(Ω cm) 1 = π/2µm 3µm = 4.18(Ω cm) 1 Integration was performed using what is known as Laplace integral. Manuel Toledo Transport 19/ 32

20 Resistance Using this device, to build a 100kΩ resistor R S = 1 σx j 1 = 4.18(Ω cm) 1 3µm = 797.4Ω/ L W = R R S = 100, 100Ω 797.4Ω = 125 For a feature size of W = 1µm, L = 125µm is required. Manuel Toledo Transport 20/ 32

21 Hall Effect Hall Effect Hall effect setup (from Manuel Toledo Transport 21/ 32

22 Hall Effect Hall Effect Lorentz Force (due to magnetic field): F m = qvb For n-type, drift velocity = v x = µ n E x. Force due to Hall Effect s electric field: F H = qe H Equilibrium: F H = F m and (ignoring sign) qvb = qµe x B = qe H E H = µe x B V x = logitudinal voltage = E x L V H = Hall s voltage = E H W = µe x B W V H = µ ( ) W V x B L Manuel Toledo Transport 22/ 32

23 Hall Effect Hall Effect A Silicon sample contains phosphorous atoms per cc. A 1mA current is flowing through the sample, which is 1cm long. A transversal magnetic field B z = 10 4 Wb/cm (1Wb = 1V s = 1T m 2 = 10 8 G cm 2 ) is applied. Find the sample s dimensions if a Hall voltage V H = 20mV is measured. Manuel Toledo Transport 23/ 32

24 Manuel Toledo Transport 24/ 32

25 Manuel Toledo Transport 25/ 32

26 F 1 = 1 2 n( l) l τ c = 1 2 n( l) v th F 2 = 1 2 n(l) v th Net flow from left to right: F = F 1 F 2 = 1 2 v th (n( l) n(l)) 1 (( 2 v th n 0 l dn ) ( n 0 + l dn )) dx dx = v th l dn dx D dn n dx D n = diffusion coefficient = diffusivity. The diffusion current due to a gradient in electron concentration is: J n = qf = qd n dn dx Manuel Toledo Transport 26/ 32

27 Example Assume that, in an n-type semiconductor at T = 300K, the electron concentration varies linerly from cm 3 to cm 3 over a distance of 0.1 cm. Find the diffusion current if the electron diffusion coefficient is D n = 22.5cm 2 /s. Manuel Toledo Transport 27/ 32

28 Example Assume that, in an n-type semiconductor at T = 300K, the electron concentration varies linerly from cm 3 to cm 3 over a distance of 0.1 cm. Find the diffusion current if the electron diffusion coefficient is D n = 22.5cm 2 /s. J n,diff ( ) ( 22.5cm 2 /s ) ( ) = 10.8 A/cm Manuel Toledo Transport 27/ 32

29 Einstein Relationship Under equilibrium conditions, the Fermi level inside a material does not change with position. A nonzero electric field is established inside a non-uniformly doped semiconductor under equilibrium conditions. Manuel Toledo Transport 28/ 32

30 J n,drift + J n,diff = qµ n ne + qd n dn dx = 0 n = N C F 1/2 (η c ) where η c = (E F E c )/kt, under equilibrium de F /dx = 0, E = 1 de c q dx dn dx = 1 dn de c kt dη c dx = q dn E kt dη c ( qe µ n n q ) kt D dn n = 0 µ n = q dη c kt D 1 dn n n dη c non-degenerate limit: n N C exp η c, dn dη c n and µ n = q kt D n D n = kt q µ n Manuel Toledo Transport 29/ 32

31 Example Minority carriers (holes) are injected into a homogeneous n-type semiconductor sample at one point. An electric field of 50 V/cm is applied across the sample, and the field moves these minority carriers a distance of 1 cm in 100 µs Find the drift velocity and the diffusivity of the minority carriers. Manuel Toledo Transport 30/ 32

32 Example Minority carriers (holes) are injected into a homogeneous n-type semiconductor sample at one point. An electric field of 50 V/cm is applied across the sample, and the field moves these minority carriers a distance of 1 cm in 100 µs Find the drift velocity and the diffusivity of the minority carriers. v p = 1 cm 100 µs = 104 cm/s µ p = v p E = = 200 cm2 /V s D p = kt q µ p = = 5.18 cm 2 /s Manuel Toledo Transport 30/ 32

33 Haynes-Shockley Experiment Haynes-Shockley Experiment Light pulse creates excess minority carriers - in this case, excess holes, above thermal equilibrium. Excess carriers drift due to the electric field. Excess carriers drift due to the electric field. A second kind of current due to diffusion takes place: j diff = qd p p x Excess carriers are collected at the terminals and a voltage pulse is observed. Manuel Toledo Transport 31/ 32

34 Haynes-Shockley Experiment Haynes-Shockley Experiment Measure the time it takes the pulse to arrive to figure out carrier drift velocity and mobility: v d = L/t max µ p = v d E = L/t max L 2 = V/L V t max Due to diffusion, the width of the pulse widens with time. The hole distribution can be represented by a Gaussian: p = p max e (x x 2 max) 4Dpt For (x x max ) 2 = 4D p t, p = p max e. Identify this level in the measured pulse to determine x = p q 4D p t = 4D p (t max + t) Using v d = x t, v d = L, and solving for D t max p: D p = (( t L)/tmax)2 4(t max + t) Manuel Toledo Transport 32/ 32

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