Short Base Pn Junction Diode Total Current Density Equation in steady state
Now consider when voltage \( V_D \) is applied to the diode. Consider the forward biased case and observe the following figure
Recall from previously due to minority carrier injection into the QNR(outside the depletion region)
- \( n_p(-x_p)\gg n_{po}(-x_{po})=n_{po} \)
- \( p_n(x_n)\gg p_{no}(x_{no})=p_{no} \)
This is essentially the concentration difference that drives the net diffusion into the QNR. See below for more
Note that at the ohmic contacts in this figure we ensure that
- \( p_n(W_n)=p_{no} \)
- \( n_p(-W_p)=n_{po} \)
that is the minority carrier concentrations are at their thermal equilibrium concentrations. Essentially we are saying that all excess carries injected starting from the edge of SCR must all recombine at the ohmic contact after diffusing across the negative concentration gradient
assuming the balance between the very large diffusion and drift current densities in the depletion region in thermal equilibrium is not perturbed significantly by the forward basis(i.e the carriers are still in near equilibrium with each other across the junction), we may then apply the following relation derived previously for thermal equlibrium:
$$p_{no} = p_{po} e^{-\phi_B / V_{th}}, \quad n_{po} = n_{no} e^{-\phi_B / V_{th}}$$
noting that \( n_p(-x_p) \) is the minority electron concentration on the p side edge of depletion region while \( p_n(x_n) \) is the minority hole concentration on the n side edge of depletion region
\[ p_n (x_n) = p_p (-x_p) e^{-\phi_j / V_{th}} = p_p (-x_p) e^{-(\phi_B - V_D) / V_{th}} \] and \[ n_p (-x_p) = n_n (x_n) e^{-\phi_j / V_{th}} = n_n (x_n) e^{-(\phi_B - V_D) / V_{th}} \] where \( \phi_j=\phi_B-V_D \)Under conditions of low level injection which requires
- \( n_p (-x_p) \ll p_{po} \approx N_a, \quad p_n (x_n) \ll n_{no} \approx N_d \)
- \( p_p (-x_p) \approx p_{po} = N_a, \quad n_n (x_n) \approx n_{no} = N_d \)
Essentially we can neglect the slight increase in majority carrier concentration due to minority carrier injection
Our 2 equations relating the minority and majority carriers under bias the edges of the depletion region can now be rewritten as
\[ p_n (x_n) = N_d e^{-\phi_B / V_{th}} e^{V_D / V_{th}} \] and \[ n_p (-x_p) = N_a e^{-\phi_B / V_{th}} e^{V_D / V_{th}} \]therefore we have just derived what is known as
We have
\[ p_n (x_n) = p_{no} e^{V_D / V_{th}}, \quad n_p (-x_p) = n_{po} e^{V_D / V_{th}} \]pn Junction Currents under forward bias
During steady state diffusion where forward biased junction injects minority carriers continuously which diffuse across the bulk region to the ohmic contacts where they continuously recombine at the same rate.
The bulk regions are short enough that we can neglect the loss of minority carriers due to recombination outside the depletion region(this situation is known as the short-base diode case)
Then concentration of minority charge carriers linearly decreases from the edge of the space charge reion to the ohmic contacts on both the n and p sides for forward-biased short-base diode
Proof
By Fick's first law of diffusion, the diffusive flux \( J \) is given by:
\[ J = -D \frac{dC}{dx}, \]where \( D \) is the diffusion coefficient (assumed constant), and \( C(x) \) is the concentration as a function of spatial coordinate \( x \).
Under steady-state conditions, there is no accumulation or depletion of mass at any point in the medium, implying that the concentration does not change with time:
\[ \frac{\partial C}{\partial t} = 0. \]Using Fick’s second law,
\[ \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2} = 0. \]Since \( D \) is constant and nonzero, this simplifies to the ordinary differential equation:
\[ \frac{d^2 C}{dx^2} = 0. \]Integrating once,
\[ \frac{dC}{dx} = A, \]where \( A \) is an integration constant. Integrating again,
\[ C(x) = A x + B, \]where \( B \) is another integration constant.
The constants \( A \) and \( B \) are determined by boundary conditions. Since this equation is linear in \( x \), the concentration profile is spatially linear under steady-state diffusion.
Now with this understanding of why we have a linear profile, we attempt to find an equation given the figure above which we get using basic knowledge of linear graphs to be
\[ p_n (x) = p_n (x_n) - \left( \frac{p_n (x_n) - p_{no}}{W_n - x_n} \right) (x - x_n) \quad \text{for} \quad x_n \leq x \leq W_n \] and \[ n_p (x) = n_p (-x_p) + \left( \frac{n_p (-x_p) - n_{po}}{W_p - x_p} \right) (x + x_p) \quad \text{for} \quad -W_p \leq x \leq -x_p \] and \[ p_p (x) = N_a + n_p (x) \quad \text{for} \quad -W_p \leq x \leq -x_p \] and \[ n_n (x) = N_d + p_n (x) \quad \text{for} \quad x_n \leq x \leq W_n \]Next we express for the total current density
\[ J = J_p^{diff} (x_n) + J_n^{diff} (-x_p) = -q D_p \frac{dp_n}{dx} \Bigg|_{x = x_n} + q D_n \frac{dn_p}{dx} \Bigg|_{x = -x_p} \]Note that we seemed to have ignored the contributions by drift currents or majority carriers. This is because in forward bias we recall there is a net diffusion current which is due to the injection of minority charge carriers. Therefore, we only need to consider the diffusive current due to these minority charge carriers inserted which can be found by subtracting the edge minority carrier concentration with that of the thermal equilibrium as below. Finally, because we know the relation of the diffusion profile will be linear, we can express the gradients in our expression for \( J \) like so
\[ J = -q D_p \left[ \frac{p_n (W_n) - p_n (x_n)}{W_n - x_n} \right] + q D_n \left[ \frac{n_p (-x_p) - n_p (-W_p)}{W_p - x_p} \right] \]Now using the fact that \( p_n(W_n)=p_{no} \) and \( n_p(-W_p)=n_{po} \) (recall the above boundary conditions) as well as our expressions for \( p_n(x_n) \) and \( n_p(-x_p) \) above, we substitute them in to get
\[ J = \left( \frac{-q D_p}{W_n - x_n} \right) (p_{no} - p_n e^{V_D / V_{th}}) + \left( \frac{q D_n}{W_p - x_p} \right) (n_{po} e^{V_D / V_{th}} - n_{po}) \]
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