The current density of an ideal p-n junction under illumination can be described by:
J(V)=Jph−J0(e^qV/kT−1)
where Jph is the photocurrent density, J0 the saturation current density, q the elementary charge, V the voltage, k the Boltzmann's constant, and T the temperature.
A crystalline silicon solar cell generates a photocurrent density of Jph=40mA/cm2 at T=300K. The saturation current density is J0=1.95∗10−10mA/cm2.
Assuming that the solar cell behaves as an ideal p-n junction, calculate the open-circuit voltage Voc (in V).
To calculate the open-circuit voltage (Voc), we need to find the value of V when J(V) = 0. In other words, we're looking for the value of V where the current density in the equation becomes zero.
Using the given equation:
J(V) = Jph - J0(e^(qV/kT) - 1)
We can set J(V) = 0 and solve for V:
0 = Jph - J0(e^(qV/kT) - 1)
Jph = J0(e^(qV/kT) - 1)
Inserting the values given:
40mA/cm^2 = (1.95 * 10^-10)mA/cm^2 * (e^(qV/(k * 300)) - 1)
To simplify the equation, we can divide both sides by J0:
(40mA/cm^2) / (1.95 * 10^-10)mA/cm^2 = e^(qV/(k * 300)) - 1
20.5 * 10^19 = e^(qV/(k * 300)) - 1
Next, isolate the exponential term by adding 1 to both sides:
20.5 * 10^19 + 1 = e^(qV/(k * 300))
Taking the natural logarithm (ln) of both sides:
ln(20.5 * 10^19 + 1) = qV/(k * 300)
Now, we can solve for V by multiplying both sides by (k * 300)/q:
V = (k * 300)/q * ln(20.5 * 10^19 + 1)
Inserting the values for k, T, and q:
V = (1.38 * 10^-23 J/K * 300 K) / (1.60 * 10^-19 C) * ln(20.5 * 10^19 + 1)
Calculating the expression:
V = 0.864 V
Therefore, the open-circuit voltage (Voc) of the ideal p-n junction in the given crystalline silicon solar cell is approximately 0.864 V.