Homework 4 - part B

ECE 4339
Han Q. Le (copyrighted) U. of Houston

1. Diffusion differential equation

1.1 Solve the differential equation

by obtaining the most general solution     (((d^2)y)/(d(x^2)))=(y/(L^2)) in the form of Sinh or Cosh

Answer given

Since we know that: ((d^2)/(d(x^2)))Sinh[(x/L)]=(1/(L^2))Sinh[(x/L)]
and                           ((d^2)/(d(x^2)))Cosh[(x/L)]=(1/(L^2))Cosh[(x/L)] 
the most general solution can be written as a linear combination:
                                   y(x)=c1 sinh((x/L))+c2 cosh((x/L))
 where c1 and c2 are constants.

1.2 Dirichlet boundary condition

Let y[a]=A and y[b]=B, obtain 2 equations for unknown coefficients. Then, solve for them and substitute to obtain the complete expression for y[x].

1.3 Simplified expression

Do you think this is an expression the solution?  y[x]=B  ((Sinh[((x-a)/L)])/(Sinh[((b-a)/L)]))-A ((Sinh[((x-b)/L)])/(Sinh[((b-a)/L)])) . Prove it.

1.4 Can you show your result in 1.2 agrees with that in 1.3? (skip if you don’t like algebra)

1.5 Numerical illustration

Let L=1, a=0, b=3., A=(10^4); B=(10^4), plot the solution on linear scale and log scale from a to b.
Then do it again for A=(10^8); B=1.,

App: carrier concentration in base

1.6 Diffusion current

If  y[x]=B  ((Sinh[((x-a)/L)])/(Sinh[((b-a)/L)]))-A ((Sinh[((x-b)/L)])/(Sinh[((b-a)/L)]))   represents carrier density, what is the diffusion current?
(The answer is: (Subscript[J, d])[x]=-q (D/L) (((B cosh(((x-a)/L)))/(Sinh[((b-a)/L)]))-((A cosh(((x-b)/L)))/(Sinh[((b-a)/L)])))    )

1.7 Numerical illustration of diffusion current

Let L=1, a=0, b=3., A=(10^8); B=1., D=1 and charge q=1, plot the diffusion current solution on linear scale and log scale from a to b.

App: Diffusion current in base

2. Optically pumped semiconductor: pumping on one half side.

2.0 Review

See Lecture Chap 4, part 2, section 5.4.1

Consider the semiconductor as infinite. Let the semiconductor be excited for just the right half. Assume that the semiconductor layer is very thin and the carrier generation is uniform in horizontally and vertically. Assume also that the carrier is intrisic and its intrinsic carrier density is very low, and we can neglect it. (In this problem, we are interested only in excess carriers that are generated by external excitation). 
Assume for simplicity that electrons and holes have the same diffusion coefficient and they diffuse together (ambipolar diffusion). What is the carrier distribution and its flux at steady state?

We start by solving the steady state diffusion equation. Let n[x] be the carrier density:
                  ((d^2)/(d(x^2)))n[x]=(1/((Subscript[L, D])^2)) (n[x]-(Subscript[n, P]) )              (2.0.1)
where (Subscript[n, P])=P τ , P is pump rate or the number of carriers generated per second, per unit volune, and τ is the carrier lifetime. Recall the definition of diffusion length above: (Subscript[L, D])=(Sqrt[D τ]) where D is the diffusion coefficient.

2.1 Obtain a most general solution of Eq. (2.1) above

Answer (given, but you are welcome to obtain your own answer)

DSolve[n''[x]== (n[x]-nP)/((Subscript[L, D])^2),n[x],x]

Hence, the most general solution is:
              n(x)=(Subscript[c, 1]) (e^(-(x/(Subscript[L, D]))))+(Subscript[c, 2]) (e^(x/(Subscript[L, D])))+nP                    (2.1.1)
where (Subscript[c, 1]) and (Subscript[c, 2]) are arbitrary coefficients.

2.2 Apply solution of 2.1 just for the dark half (left half)

In the dark half, there is no pumping, which means nP=0, what is the most general solution that does NOT violate any physical limit (it should NOT go to infinite at any point or at infinity). Rename your coefficient(s) with subscript L to indicate that they are only for the left half and apply subscript L to carrier density to indicate that it is the solution for left half.

Answer (given, but you are welcome to obtain your own answer)

Since nP=0, the solution in the dark half from (2.1.1) is:
              (Subscript[n, L])(x)=(Subscript[c, L;1]) (e^(-(x/(Subscript[L, D]))))+(Subscript[c, L;2]) (e^(x/(Subscript[L, D])))                        (2.2.1)

For the left half, term (Subscript[c, L;1]) (e^(-(x/(Subscript[L, D])))) will become infinite at x -> -∞ unless (Subscript[c, L;1])=0. Since this is not physically possible, hence:                    (Subscript[c, L;1])=0. 
Thus, the solution is:
                (Subscript[n, L])(x)=(Subscript[c, L;2]) (e^(x/(Subscript[L, D])))                                       (2.2.2) 
since there is ONLY ONE coefficient, we can drop the subscript 2:
                  (Subscript[n, L])(x)=(Subscript[c, L]) (e^(x/(Subscript[L, D])))                                       (2.2.2a)

2.3 What is the value of carrier density at x= -∞?

2.4 Obtain a solution for carrier density in the left half.

Let the boundary between the dark and light region be x=0. Denote (Subscript[n, L;0]) be the carrier density at x=(SuperMinus[0]) on the dark half, obtain a solution for carrier density.

2.5 What is the carrier flux (Subscript[F, L])[x] in the dark half (left half) from the result in 2.3?

Hint

Take derivative of (2.4.1)     (Subscript[F, L])[x]=-D  (d/(dx))(Subscript[n, L])[x]

2.6 Apply solution of 2.1 for the optically pumped half (right half)

Here, nP ≠ 0. Determine a solution that does NOT violate any physical limit (it should NOT go to infinite at any point or at infinity). Then, use subscript R (right) for all symbols to distinguish from those of left half.

Hint

For the right half, the general solution from Eq. (2.1.1) is:
                                 (Subscript[n, R])(x)=(Subscript[c, R;1]) (e^(-(x/(Subscript[L, D]))))+(Subscript[c, R;2]) (e^(x/(Subscript[L, D])))+nP        (2.6.1)
However, one must drop one of the 2: either (Subscript[c, R;1]) (e^(-(x/(Subscript[L, D])))) or (Subscript[c, R;2]) (e^(x/(Subscript[L, D]))) (which one and why?)

2.7 What is the value of carrier density at x= +∞?

2.8 Let the carrier density at x=(SuperPlus[0]) be (Subscript[n, R;0]), obtain the solution of carrier density in the right half.

Hint

The correct answer is
                                (Subscript[n, R])(x)=((Subscript[n, R;0])-nP) (e^(-(x/(Subscript[L, D]))))+nP                (2.8.1)

2.9 Calculate the carrier flux for the left half based on the solution above.

2.10 Graphical solution

Based on the results obtained from 2.4, 2.8, we can plot the carrier density solution for each half if given (Subscript[n, L;0]), (Subscript[n, R;0]) and nP. We can also plot the fluxes. The APP below lets us choose ratios ((Subscript[n, L;0])/nP) and  ((Subscript[n, R;0])/nP) to see the carrier density on each half. However, can we arbitrarily choose any values for ((Subscript[n, L;0])/nP) and  ((Subscript[n, R;0])/nP)?

In other words, does nature pick specific solution with unique values of  ((Subscript[n, L;0])/nP) and  ((Subscript[n, R;0])/nP) or does nature leave it to us to pick  ((Subscript[n, L;0])/nP) and  ((Subscript[n, R;0])/nP) to our heart delight?

Run the APP, select values of ((Subscript[n, L;0])/nP) and  ((Subscript[n, R;0])/nP) that you think give correct solution, copy and paste in and explains why.

APP: Carrier distribution in optically pumped semiconductor

2.11 Obtain 2 graphical results for the lowest and highest mobility. Compare and discuss about density and flux

3. Analysis of paper on p-n diode electroluminescence

Read the paper: “Time-resolved imaging of radiative recombination in 4H-SiC p-i-n diode” by A. Galeckas et al., Appl. Phys. Lett. Vol. 74, pp. 3398-3400, 1999. It is available on the last section of lecture Chapter 4, part 2.

To answer the questions below, you need to read the paper. Looking at a figure alone will NOT be enough. Read those paragraphs pertaining to each figure to answer them. You do NOT need to understand 100% of what in the paper to answer all these questions below.

3.1

Label on Fig. 1 (copy and use some software such as ppt drawing to show) where the substrate is, the epilayers and their thickness, the mesa and its dimension.

3.2

What does figure 2 describe? What color do you think it looks like?

3.3

Look at Fig. 3. Look at the black-white inset photo. What is the bright streak at the top? Why does it get darker as you go to the bottom?

3.4

Look at Fig. 3 again. What does it plot? Is the vertical axis on log scale or linear scale? If we assume that the emitting light intensity is proportional to the electron-hole (EH) density, how do you express the spatial variation of the EH density along these layers? Look at the curve for 100 A/cm2, if you approximate the carrier density profile as (e^(-x/(Subscript[L, D]))) where x is the distance and (Subscript[L, D]) is the diffusion length, what are the diffusion length values for n- and n+ region?

Help to answer this question:

(see HW 4A,  APP  How to find an exponential coefficient  in problem 3)

APP: How to find an exponential coefficient - applied to diffusion length

3.5

Recalling that diffusion length (Subscript[L, D])=(Sqrt[D τ]) where D is the diffusivity and τ is the effective carrier lifetime, let’s assume that the D is the diffusivity is approximately the same for both n- and n+ region, which region, n- or n+, has a longer carrier lifetime? Can you speculate why so?

3.6

Recalling that the diffusivity D is also proportional to the mobility. Which region, n- or n+ do you think the mobility is higher? Do you think that both mobility and carrier lifetime contribute to the difference in the diffusion length between the two regions?

3.7

Look at Fig. 4. What does the top photo (Fig. 4(a)) show? Copy and paste the figure and draw a arrow to show where you think there is an excess electron-hole population. Do you think that this is a spatially uniform diode along the horizontal direction? In Fig. 4(a), which part, left or right, is more efficient as an LED? What does Fig. 4(b) show? Is the vertical scale logarithmic or linear?

3.8

Look at Fig. 5. What do the researchers claim the effective carrier lifetime they measured for each region, n- and n+? How did they measure these values?

3.9

Go back to questions (3.4) and (3.5), use the lifetime given in the paper you that you find in question (3.8) above, use the value of diffusion length you find in (3.4), infer what you think the diffusivity for each region are, using appropriate relations.

3.10

Comment on what you learn from this paper