# Solution to Exercise 8.1-6

# Constructing Quantitative Logarithmic IV Characteristics

| First we get a few important relations and numbers.
|  | If 1/kT = 40 eV–1 at 300 K, we have 1/kT = 40 · 300/400 = 30 eV–1 at 400 K
|  | The current densities j1 and j2 can always be written as
j1 | = | c1 · ni 2 | = | j1' · exp–(Eg/kT)
--- | --- | --- | --- | ---
j2 | = | c2 · ni | = | j2' · exp–(Eg/2kT)
j1' | = | j1 · exp(Eg/kT)
--- | --- | ---
j2' | = | j2 · exp(Eg/2kT)
| This gives us the following numbers:
| Theory | Measured
--- | --- | ---
| j1 | j2 | j1 | j2
Calculated ji' | 2.06 · 105 A/cm2 | 5.74 · 10–1 A/cm2 | 1.29 · 1010 A/cm2 | 3.58 · 102 A/cm2
| T = 300 K
Starting values ji U = 0 V | 1.6 · 10–14 A/cm2 | 1.6 · 10–10 A/cm2 | 10–9 A/cm2 | 10–7 A/cm2
Calculated ji U = 0.5 | 7.76 · 10–6 A/cm2 | 3.52 · 10–7 A/cm2 | 0.46 A/cm2 | 2.2 · 10–3 A/cm2
| T = 400 K
Starting values ji U = 0 V | 9.60 · 10–10 A/cm2 | 3.92 · 10–8 A/cm2 | 6.01 · 10–5 A/cm2 | 2,44 · 10–5 A/cm2
Calculated ji U = 0,5 V | 9.67 · 10–3 A/cm2 | 1,5 · 10–4 A/cm2 |  |
| Now to the questions: Question 1. Construct rather quantitatively the logarithmic IV characteristics (= log j - eU plot) of two solar cells with the j2 and j2 values as given in the table. Question 2: Determine the open circuit voltage UOC for room temperature and for 400 K and discuss your finding.
| Constructing the graph is easy now; here is the result:
|  | We note that the "-1" term can be neglected as soon as we have current density values about 10 times larger then the starting values, i.e. below U » 0.1 V. At lower values this term dominates the characteristics by forcing the currents to zero, i.e. to –¥ in a log plot, but that is of no interest here.
|  | The addition of both curves only introduces a slight "rounding" at the intersection point.
|  | The open circuit voltage follows from the intersection of the j(U) curves with a straight line at j = – jPh. It is immediately clear that only the j1 part is of interest here.
| The effect of temperature is shown in a separate graph and only for the "theoretical" set of the jPh:
|  | While the decreasing slope of the curves would increase UOC, the large increase in the starting value of j1 has a much stronger effect and causes a substantial decrease of UOC with temperature.

With frame

Basic IU-Characteristics of Solar Cells

Exercise 8.3-1

Exercise Basic 8.1-1

Exercise 8.1-6 Constructing Quantitative Logarithmic IV Characteristics

© H. Föll (Semiconductor Technology - Script)