ABSTRACT
The difference in sign between the two diffusion currents arises because, although diffusion takes place down the concentration gradient regardless of charge, the resulting current is in opposite directions for electrons and holes. Hence, the electron and hole current densities in a given direction consist of a drift component caused by an applied field and a diffusion component due to the carrier concentration gradient, that is
e e e dnj e n E D dx
µ⎛= +⎜⎝ ⎠ ⎞⎟ (43.3)
h h h dpj e p E D dx
µ⎛= −⎜⎝ ⎠ ⎞⎟ (43.4)
where the electric field is applied parallel to the x-axis. A relationship between mobility and the diffusion constant for both electrons and holes may be derived from the thermal equilibrium condition, assuming no applied electric field and no current flow. Since
must separately be equal to zero, Eq.(43.3) reduces to he jj and
dx dnDEn ee −=µ (43.5)
This implies that diffusion results in an internal electric field which produces an electrostatic potential , given by Eq.(11.9) as ( )xV ( ) ./ dxxdVE −= We have seen in Chapter 42 that the energy distribution of carriers follows the Boltzmann distribution law (26.25), so that the variation of n with ,x as a result of the motion of electrons in the electrostatic potential, will be given by ( ) TkxeVC BeNn /= (43.6) or
( ) ( ) ( ) dx
xdV Tk
ene dx
xdV Tk
eN dx dn
B == / (43.7) Hence, Eq.(43.5) yields
( ) ( ) dx
xdV Tk
enD dx
xdVn B
ee −=− µ or eBe e Tk
D µ= (43.8) Similarly, from Eq.(43.4) we may obtain
h e TkD µ= (43.9)
Equations (43.8) and (43.9) are known to be the Einstein relations which, on substitution into Eqs.(43.3) and (43.4), lead to
⎟⎠ ⎞⎜⎝
⎛ += dx dn
e Tk
nEej Bee µ (43.10)
⎟⎠ ⎞⎜⎝
⎛ −= dx dp
e Tk
pEej Bhh µ (43.11) In the absence of an external field E there will be no net electron or hole current flow, so that, substituting for n from Eq.(42.46) in Eq.(43.10), we obtain
dx dx ε ε εµ −⎡ ⎤= = =⎣ ⎦ µ (43.12)
and similarly, from Eqs.(42.47) and (43.11), we have
dx
d pj Fhh
εµ== 0 (43.13) In other words, the Fermi energy is constant throughout an inhomogenous semiconductor in thermal equilibrium as
0= dx
d Fε (43.14)
EXAMPLE 43.1. Energy-band diagrams It is often convenient to represent inhomogeneous semiconductors by energy-band diagrams which visualize the electrical behaviour described by the carrier transport equations. Most phenomena can be understood if we use a one-dimensional effective energy-band model (Figure 43.1), where Cε is the lowest energy in the conduction band, and Vε is the highest energy in the valence band for any possible crystallographic direction. In other words
VCg εεε −= is the minimum thermal energy required to excite electrons from the valence band into the conduction band and the x-axis represents an averaged direction through the crystal. This convention is illustrated in Figure 43.1 for homogenous semiconductors.
