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Next: Junctions, depletion regions, band Up: From Semi-conductivity to Micro-electronics Previous: Fermi statistics, charge carrier

Diffusion and drift of charge carriers

The next step in understanding the operation of semiconductors is the concept of diffusion and drift of the charge carriers. Note that the nature, amount and regional location of these charge carriers can be be manipulated according to the intentions of the chip designers.

Diffusion
Whenever there is a concentration gradient, material will move in the direction of the highest rate of decrease of the gradient. (As if to remove the concentration difference - see figure 26.)

Figure 26: Movement of material down a concentration gradient.
\includegraphics[width=0.65\textwidth]{conc-grad.eps}

Fick's Law

The flux of particles is proportional to the gradient of particles.
(Flux is the number of particles /second /area.)

In the case of charge carriers, the flux is the current density with the appropriate sign. For example, in the case of electrons as charge carriers :

\begin{displaymath}
J_f = -(-e)D\frac{dn_v}{dx}
\end{displaymath} (106)

where $n_v$ is the number of charge carriers (now per unit volume) and $D$ is the diffusion coefficient with units :
\begin{displaymath}[D]= L^2T^{-1}.
\end{displaymath} (107)

Drift
When an electric field is applied to a semiconductor, the carriers will move at a velocity that is proportional to the magnitude of the field. This velocity is called the drift velocity $v_d$.

\begin{displaymath}
v_d = \mu \xi
\end{displaymath} (108)

where $\mu$ is called the mobility, with units
\begin{displaymath}[\mu]= L^2V^{-1}T^{-1}.
\end{displaymath} (109)

and $\xi$ is the electric field. This constitutes a drift current
\begin{displaymath}
J_d = v_d n_v e = \mu n_ve\xi
\end{displaymath} (110)

At equilibrium, the drift and diffusion currents are equal.

\begin{displaymath}
J_d = J_f
\end{displaymath} (111)

so
\begin{displaymath}
\mu n_ve\xi = eD\frac{dn_v}{dx}
\end{displaymath} (112)

Figure 27: Equilibrium condition for drift and diffusion currents.
\includegraphics[width=0.65\textwidth]{drift.eps}

We may write the electric field as the gradient of the potential

\begin{displaymath}
\xi = -\frac{d\phi}{dx}.
\end{displaymath} (113)

Therefore
\begin{displaymath}
-\mu n_ve\frac{d\phi}{dx} = eD\frac{dn_v}{dx}
\end{displaymath} (114)

so
\begin{displaymath}
\frac{dn_v}{d\phi} = - \frac{\mu n_v}{D}.
\end{displaymath} (115)

This differential equation has the solution
\begin{displaymath}
n_v = n_{v0} e^{-\frac{\mu}{D}\phi}
\end{displaymath} (116)

for the electron distribution per unit volume. But because this non-uniform electron distribution has to be maintained against the potential $\phi$ using thermal energy, we must also have
\begin{displaymath}
n_v = n_{v0} e^{-\frac{e\phi}{kT}}.
\end{displaymath} (117)

We therefore find the Einstein relations

$\displaystyle \frac{\mu_n}{D_n}$ $\textstyle =$ $\displaystyle \frac{e}{kT} \qquad \mbox{for electrons}$ (118)
$\displaystyle \mbox{and similarly}$      
$\displaystyle \frac{\mu_p}{D_p}$ $\textstyle =$ $\displaystyle \frac{e}{kT} \qquad \mbox{for holes}$  

That is, we can express the diffusion coefficient as
\begin{displaymath}
D = \frac{\mu kT}{e},
\end{displaymath} (119)

a result which we will soon use.

Note
Carrier mobility is very important. It is affected by temperature, doping concentration and the magnitude of the applied field. It also depends on the effective mass. Carriers with small effective masses have large mobilities. As a result, holes are significantly less mobile than electrons.


next up previous
Next: Junctions, depletion regions, band Up: From Semi-conductivity to Micro-electronics Previous: Fermi statistics, charge carrier
Simon Connell 2004-10-04