Physics and Engineering of Radiation Detection

5.1. Semiconductor Detectors 251

5.1.B ChargeCarriersDistribution

The free charges in the bulk of a semiconductor crystal can occupy different en-
ergy levels with an occupancy that can be described by the so calledBoltzmann
distribution

f(E)=

1

1+e(E−EF)/kBT

, (5.1.1)

whereEis the energy of the electron,kBis the familiar Boltzmann constant,Tis
the absolute temperature, andEFis the Fermi level.
The Fermi functionf(E) actually gives the probability at which an available
energy state E can be occupied by an electron. For intrinsic semiconductors, which
have equal number of positive and negative charge carriers, the Fermi level lies
exactly in the middle of the band gap. This is the level at which the probability of
electron occupancy is exactly 1/2, or in other words, half of the states are filled by
electrons (see example below).
As can be inferred from the relation 5.1.1, the occupancy of charge carriers is a
function of the absolute temperature. Of course the reason for this can be traced
back to the few electron volt wide band gaps of semiconductors that are comparable
to the energy of thermal agitations even at room temperatures. The temperature
dependence is so strong that even small fluctuations in temperature can produce
significant changes in the number of free charge carriers. We will see later that this
effect is a serious problem in semiconductor detectors since it may cause nonlinear
changes in the response of the detector.

Example:

Compute the probability for an electron to occupy the Fermi level in an

intrinsic semiconductor.

Solution:

The required probability can be computed from the Fermi function 5.1.1

f(E)=

[

1+e(E−EF)/kBT

]− 1

Since we have to find the probability at the Fermi level, therefore we substitute

E=EFin the above equation to get

f(E)=

[

1+e(EF−EF)/kBT

]− 1

=

1

2

5.1.C Intrinsic, Compensated, and Extrinsic Semiconductors

The energy band structure shown in Fig.5.1.1 represents an ideal semiconductor.
A crystal in which the impurities are either non-existent or they do not affect its
conduction properties significantly is said to beidealorintrinsic. The electrons and
holes in such a material are in equilibrium with each other. This state of equilibrium
is actually a consequence of similar temperature dependences of the density of states