Presentation is loading. Please wait.

Presentation is loading. Please wait.

ECE 663 The first few chapters showed us how to calculate the equilibrium distribution of charges in a semiconductor np = n i 2, n ~ N D for n-type The.

Similar presentations


Presentation on theme: "ECE 663 The first few chapters showed us how to calculate the equilibrium distribution of charges in a semiconductor np = n i 2, n ~ N D for n-type The."— Presentation transcript:

1 ECE 663 The first few chapters showed us how to calculate the equilibrium distribution of charges in a semiconductor np = n i 2, n ~ N D for n-type The last chapter showed how the system tries to restore itself back to equilibrium when perturbed, through RG processes R = (np - n i 2 )/[  p (n+n 1 ) +  n (p+p 1 )] In this chapter we will explore the processes that drive the system away from equilibrium. Electric forces will cause drift, while thermal forces (collisions) will cause diffusion. The story so far..

2 Drift: Driven by Electric Field v d =  E Velocity (cm/s) Mobility (cm 2 /Vs) Electric field (V/cm) E Which has higher drift? x

3 ECE 663 DRIFT

4 Why does a field create a velocity rather than an acceleration? Terminal velocity Gravity Drag

5 Why does a field create a velocity rather than an acceleration? Random scattering events (R-G centers) The field gives a net drift superposed on top

6 Why does a field create a velocity rather than an acceleration? m n *(dv/dt + v/  n ) = -q E  n = q  n /m n *  p = q  p /m p *

7 ECE 663 From accelerating charges to drift

8 From mobility to drift current  n = q  n /m n *  p = q  p /m p * J n = qnv = qn  n E drift J p = qpv = qp  p E drift (A/cm 2 )

9 Resistivity, Conductivity  n = nq  n = nq 2  n /m n * J n =  n E drift J p =  p E drift  p = pq  p = pq 2  p /m p *  = 1/   =  n +  p

10 Ohm’s Law J n = E/  n drift J p = E/  p drift L A E = V/L I = JA = V/R R =  L/A (Ohms) V What’s the unit of  ?

11 So mobility and resistivity depend on material properties (e.g. m*) and sample properties (e.g. N T, which determines  ) Recall 1/  =  v th N T

12 Can we engineer these properties? What changes at the nanoscale?

13 ECE 663 What causes scattering? Phonon Scattering Ionized Impurity Scattering Neutral Atom/Defect Scattering Carrier-Carrier Scattering Piezoelectric Scattering

14 ECE 663 Some typical expressions Phonon Scattering Ionized Impurity Scattering

15 ECE 663 Combining the mobilities Matthiessen’s Rule Caughey-Thomas Model

16 ECE 663 Doping dependence of mobility

17 ECE 663 Doping dependence of resistivity  N = 1/qN D  n  P = 1/qN A  p  depends on N too, but weaker..

18 ECE 663 Phonon Scattering ~T -3/2 Ionized Imp ~T 3/2 Piezo scattering Temperature Dependence

19 ECE 663 Bailon et al Tsui-Stormer-Gossard Pfeiffer-Dingle-West.. Reduce Ionized Imp scattering (Modulation Doping)

20 ECE 663 Field Dependence of velocity Velocity saturation ~ 10 7 cm/s for n-Si (hot electrons) Velocity reduction in GaAs

21 ECE 663 Gunn Diode Can operate around NDR point to get an oscillator

22 ECE 663 GaAs bandstructure

23 ECE 663 Transferred Electron Devices (Gunn Diode)  E(GaAs)=0.31 eV Increases mass upon transfer under bias

24 ECE 663 Negative Differential Resistance

25 ECE 663 DIFFUSION

26 ECE 663 J 1 = qn(x)v J 2 = -qn(x+l)v l = v  J n = q(l 2 /  )dn/dx = qD N dn/dx DIFFUSION diff

27 ECE 663 Drift vs Diffusion t x t x ~  E t ~ Dt E1E1 E 2 > E 1

28 SIGNS ECEC E J n = qn  n E drift J p = qp  p E drift v n =  n E v p =  p E Opposite velocities Parallel currents

29 SIGNS J n = qD n dn/dx diff J p = -qD p dp/dx diff dn/dx > 0dp/dx > 0 Parallel velocities Opposite currents

30 ECE 663 In Equilibrium, Fermi Level is Invariant e.g. non-uniform doping

31 ECE 663 Einstein Relationship  and D are connected !! J n + J n = qn  n E + qD n dn/dx = 0 diff drift n(x)= N c e -[E C (x) - E F ]/kT = N c e -[E C -E F - qV(x)]/kT dn/dx = -(q E /kT)n qn  n E - qD n (q E /kT)n = 0 D n /  n = kT/q

32 ECE 663 Einstein Relationship  n = q  n /m n * D n = kT  n /m n * ½ m*v 2 = ½ kT D n = v 2  n = l 2 /  n

33 ECE 663 We know how to calculate fields from charges (Poisson) We know how to calculate moving charges (currents) from fields (Drift-Diffusion) We know how to calculate charge recombination and generation rates (RG) Let’s put it all together !!! So…

34 ECE 663 Relation between current and charge

35 ECE 663 Continuity Equation

36 ECE 663 The equations At steady state with no RG .J = q .(nv) = 0

37 Let’s put all the maths together… Thinkgeek.com

38 ECE 663 All the equations at one place (n, p) E J ∫ 

39 Simplifications 1-D, RG with low-level injection r N =  p/  p, r P =  n/  n Ignore fields E ≈ 0 in diffusion region J N = qD N dn/dx, J P = -qD P dp/dx

40 ECE 663 Minority Carrier Diffusion Equations ∂np∂np ∂2np∂2np ∂t ∂x 2 npnp nn = D N -+ G N ∂pn∂pn ∂2pn∂2pn ∂t ∂x 2 pnpn pp = D P -+ G P

41 ECE 663 Example 1: Uniform Illumination ∂np∂np ∂2np∂2np ∂t ∂x 2 npnp nn = D N -+ G N Why?  n(x,0) = 0  n(x,∞) = G N  n  n(x,t) = G N  n (1-e -t/  n )

42 ECE 663 Example 2: 1-sided diffusion, no traps ∂np∂np ∂2np∂2np ∂t ∂x 2 npnp nn = D N -+ G N  n(x,b) = 0  n(x) =  n(0)(b-x)/b

43 ECE 663 Example 3: 1-sided diffusion with traps ∂np∂np ∂2np∂2np ∂t ∂x 2 npnp nn = D N -+ G N  n(x,b) = 0  n(x,t) =  n(0)sinh[(b-x)/L n ]/sinh(b/L n ) L n =  D n  n

44 Numerical techniques 2

45

46 ECE 663 At the ends…

47 ECE 663 Overall Structure

48 ECE 663 In summary While RG gives us the restoring forces in a semiconductor, DD gives us the perturbing forces. They constitute the approximate transport eqns (and will need to be modified in 687) The charges in turn give us the fields through Poisson’s equations, which are correct (unless we include many-body effects) For most practical devices we will deal with MCDE


Download ppt "ECE 663 The first few chapters showed us how to calculate the equilibrium distribution of charges in a semiconductor np = n i 2, n ~ N D for n-type The."

Similar presentations


Ads by Google