Photonics Essentials: From Diffusion Current to Equation 3.5

Source

Thomas P. Pearsall, Photonics Essentials: An Introduction with Experiments (McGraw-Hill, 2003), Chapter 3, Photodiodes, printed pages 38–40.

Question

The chapter first gives a diffusion-current equation containing one spatial derivative,

\[J(x)=-qD_e\frac{\partial n}{\partial x}.\]

Why does the carrier equation later contain the second derivative \(\partial^2 n/\partial x^2\)?

Important

Current \(J\) does not turn directly into concentration. Equation 3.5 asks whether more carriers enter a small region than leave it. That requires the spatial derivative of \(J\), introducing one more derivative.

The small-slab picture

Carrier accumulation equals inflow minus outflow, plus optical generation, minus recombination.

The symbols used below are:

Symbol	Meaning	Units
\(\Delta n_p(x,t)\)	Excess minority-electron concentration in p-type material	\(\mathrm{cm^{-3}}\)
\(J(x)\)	Diffusion current density, using the book’s sign convention	\(\mathrm{A\,cm^{-2}}\)
\(D_e\)	Electron diffusion coefficient	\(\mathrm{cm^2\,s^{-1}}\)
\(\tau_e\)	Minority-electron recombination lifetime	\(\mathrm s\)
\(G_L\)	Optical generation rate per unit volume	\(\mathrm{cm^{-3}\,s^{-1}}\)

Step 1: A concentration gradient produces diffusion

Carriers diffuse from high concentration toward low concentration. The book writes the resulting current density as

\[J(x)=-qD_e\frac{\partial\Delta n_p}{\partial x}.\]

The chapter initially writes \(n(x)\) instead of \(\Delta n_p(x)\). In the neutral p-region,

\[n_p(x)=n_{p0}+\Delta n_p(x).\]

The equilibrium concentration \(n_{p0}\) is spatially constant, so

\[\frac{\partial n_p}{\partial x} =\frac{\partial\Delta n_p}{\partial x}.\]

The two notations therefore give the same diffusion term.

Step 2: Apply carrier conservation to a thin slab

Consider a slab extending from \(x\) to \(x+dx\).

• Current density entering the slab is \(J(x)\).

• Current density leaving it is \(J(x+dx)\).

Expand the outgoing current to first order:

\[J(x+dx) \approx J(x)+\frac{\partial J}{\partial x}dx.\]

Yes: this is a Taylor series

The general Taylor expansion of a smooth function about \(x\) is

\[J(x+dx) = J(x) +\frac{\partial J}{\partial x}dx +\frac{1}{2}\frac{\partial^2J}{\partial x^2}(dx)^2 +\cdots.\]

The book keeps terms only through first order in the very small slab width \(dx\):

\[J(x+dx) = J(x) +\frac{\partial J}{\partial x}dx +O\!\left((dx)^2\right).\]

After subtracting the entering and outgoing currents and dividing by \(dx\), the omitted part is \(O(dx)\). It approaches zero as the slab is made infinitesimally thin. This is why the first-order expansion is sufficient for the local conservation equation.

The net carrier inflow per unit volume and time is consequently

\[\begin{split}\begin{aligned} \text{net inflow} &=\frac{J(x)-J(x+dx)}{q\,dx}\\ &=-\frac{1}{q}\frac{\partial J}{\partial x}. \end{aligned}\end{split}\]

This is the crucial extra derivative.

Step 3: Add generation and recombination

Three processes change the excess-carrier concentration:

\[\text{rate of change} = \text{net inflow} -\text{recombination} +\text{optical generation}.\]

In symbols,

\[\frac{\partial\Delta n_p}{\partial t} = -\frac{1}{q}\frac{\partial J}{\partial x} -\frac{\Delta n_p}{\tau_e} +G_L.\]

Step 4: Substitute the diffusion current

Insert \(J=-qD_e\,\partial\Delta n_p/\partial x\):

\[\begin{split}\begin{aligned} -\frac{1}{q}\frac{\partial J}{\partial x} &=-\frac{1}{q}\frac{\partial}{\partial x} \left( -qD_e\frac{\partial\Delta n_p}{\partial x} \right)\\ &=D_e\frac{\partial^2\Delta n_p}{\partial x^2}, \end{aligned}\end{split}\]

where \(D_e\) is assumed constant. The complete result is

\[\frac{\partial\Delta n_p}{\partial t} = D_e\frac{\partial^2\Delta n_p}{\partial x^2} -\frac{\Delta n_p}{\tau_e} +G_L. \tag{3.5}\]

Why it is second order

The derivative chain is

\[\boxed{ \Delta n_p \xrightarrow{\ \partial/\partial x\ } J \xrightarrow{\ -\partial/\partial x\ } \text{carrier accumulation} }.\]

Therefore:

• Equation 3.5 is second order in position \(x\).

• It is only first order in time \(t\).

• The second spatial derivative measures the curvature of the concentration profile, not simply its slope.

A useful physical reading is:

• constant concentration gives no diffusion current;

• a straight concentration profile gives constant current but no local accumulation;

• a curved concentration profile makes current vary with position, so some regions gain or lose carriers.

Unit check

Every term in Equation 3.5 must have units of concentration change per time:

\[\left[ D_e\frac{\partial^2\Delta n_p}{\partial x^2} \right] = \frac{\mathrm{cm^2}}{\mathrm s} \frac{\mathrm{cm^{-3}}}{\mathrm{cm^2}} = \mathrm{cm^{-3}\,s^{-1}}.\]

The recombination and generation terms have the same units:

\[\left[\frac{\Delta n_p}{\tau_e}\right] = [G_L] = \mathrm{cm^{-3}\,s^{-1}}.\]

Steady-state Equation 3.6

At steady state the concentration profile no longer changes with time:

\[\frac{\partial\Delta n_p}{\partial t}=0.\]

Equation 3.5 then becomes

\[\frac{d^2\Delta n_p}{dx^2} = \frac{\Delta n_p}{D_e\tau_e} -\frac{G_L}{D_e}. \tag{3.6}\]

In the dark, \(G_L=0\). Define the electron diffusion length

\[L_e=\sqrt{D_e\tau_e}.\]

The equation reduces to

\[\frac{d^2\Delta n_p}{dx^2} -\frac{\Delta n_p}{L_e^2}=0,\]

whose decaying solution in a long neutral region is

\[\Delta n_p(x)=\Delta n_p(0)e^{-x/L_e}.\]

Because the spatial equation is second order, two boundary conditions are needed. Typically the junction supplies \(\Delta n_p(0)\) and the excess concentration approaches zero far from the junction.

Source correction

Warning

The paragraph below Equation 3.5 says that Equation 3.4 is a second-order differential equation. Equation 3.4 is algebraic. The sentence should refer to Equation 3.5.

Continue interactively

Use the Photonics Essentials: Chapter 3 Interactive Physics Lab to change the diffusion coefficient, carrier lifetime, generation rate, and the other Chapter 3 variables while the corresponding curves update.