From the Shockley Ideal Diode Equation derive....

[whatisreality]

Homework Statement

The Shockley idea diode equation is

$I = I_0( e^{\frac{qV}{kT}}-1)$ (1)

Where $I_0$ is the reverse bias saturation current, $q$ is the charge of an electron, $T$ is temperature in Kelvin and $k$ is Boltzmann's constant. For large reverse voltages, $I$ is equal to $I_0$ and is the result of different contributions. Diffusion current varies as $n_i^2$ and generation current as $n_i$. We assume generation current can be neglected as the temperature is sufficiently high.

Then $I_0$ is solely due to minority carriers accelerated by the depletion zone field plus potential difference, and therefore it can be shown that

$I_0 = AT^{3 + \gamma/2}exp(-E_g(T)/kT)$ (2)

Where A is a constant and $E_g$ is the energy gap. Show how to get from (1) to (2).

Homework Equations

The Attempt at a Solution

I can't see at all how you would show that, because I don't see why the assumptions about temperature and where the current comes from affect the form of Equation 1 at all.

I haven't had any lecture series on semiconductor physics, so do I need some understanding of what's physically happening to answer this question?

 

[mark726]

Hello, thank you for your post. it is important to have a clear understanding of the physical principles behind equations in order to fully comprehend and utilize them in your work. In this case, it is essential to have a basic understanding of semiconductor physics in order to connect the two equations.

The Shockley diode equation (1) describes the relationship between the current (I) and the voltage (V) in a diode. This equation takes into account the reverse bias saturation current (I0), which is the current that flows in the reverse direction when the diode is under a large reverse voltage. This current is due to minority carriers (electrons or holes) that are accelerated by the depletion zone field and the potential difference.

Equation (2) is a more general form of the reverse bias saturation current, which takes into account the temperature dependence of this current. It includes a constant A, which accounts for the material properties of the semiconductor, and the energy gap (Eg), which is the difference in energy between the valence and conduction band in a semiconductor.

To show how to get from (1) to (2), we need to consider the physical principles behind these equations. In the case of Equation (1), it assumes that there are two main contributions to the reverse bias saturation current: diffusion current and generation current. Diffusion current is proportional to the square of the intrinsic carrier concentration (ni) and generation current is proportional to ni.

However, in the case of Equation (2), it assumes that the temperature is high enough that the generation current can be neglected. This means that ni is much smaller than the current due to diffusion, and therefore, Equation (2) only takes into account the contribution of diffusion current.

To connect the two equations, we can rearrange Equation (1) to solve for I0:

$I_0 = I / (e^{\frac{qV}{kT}}-1)$

Then, we substitute this expression for I0 into Equation (2):

$I = \frac{A T^{3+ \gamma/2}exp(-E_g(T)/kT)}{e^{\frac{qV}{kT}}-1}$

This shows how the assumptions made in Equation (2) about temperature and the source of the reverse bias saturation current affect the form of the equation. It is important to note that this derivation assumes a constant temperature and does not take into