4.1 Charge Carriers in Semiconductor

The distribution（with respect to energy）of electrons in the conduction band is given by the density of allowed quantum states times the probability that a state is occupied by an electron. Similarly，the distribution（with respect to energy）of holes in the valence band is the density of allowed quantum states in the valence band multiplied by the probability that a state is not occupied by an electron. These statements are written in equation forms as

n（E）=gc（E）fF（E）） （4.1）

p（E）=gv（E）［1－fF（E）］ （4.2）

where fF（E）is the Fermi-Dirac probability function and gc（E）is the density of quantum states in the conduction band. The total electron concentration per unit volume in the conduction band is then found by integrating Equation（4.1）over the entire conduction-band energy. The total hole concentration per unit volume is found by integrating Equation（4.2）over the entire valence-band energy.

To find the thermal-equilibrium electron and hole concentrations，we need to determine the position of the Fermi energy EF with respect to the bottom of the conduction-band energy Ec and the top of the valence-band energy Ev. To address this question，we will initially consider an intrinsic semiconductor. An ideal intrinsic semiconductor is a pure semiconductor with no impurity atoms and no lattice defects in the crystal（e. g.，pure silicon）. We have argued that for an intrinsic semiconductor at T=0K，all energy states in the valence band are filled with electrons and all energy states in the conduction band are empty of electrons. The Fermi energy must，therefore，be somewhere between Ec and Ev.（The Fermi energy does not need to correspond to an allowed energy.）

As the temperature begins to increase above 0K，the valence electrons will gain thermal energy. A few electrons in the valence band may gain sufficient energy to jump to the conduction band. As an electron jumps from the valence band to the conduction band，an empty state，or hole，is created in the valence band. In an intrinsic semiconductor，then，electrons and holes are created in pairs by the thermal energy so that the number of electrons in the conduction band is equal to the number of holes in the valence band.

Fig.4.1（a）shows a plot of the density of states function in the conduction band gc（E），the density of states function in the valence band gv（E），and the Fermi-Dirac probability function for T＞0K when EF is approximately half way between Ec and Ev. If we assume，for the moment，that the electron and hole effective masses are equal，then gc（E）and gv（E）are symmetrical functions about the midgap energy（the energy midway between Ec and Ev）. We noted previously that the function fF（E）for E > EF is symmetrical to the function 1－fF（E）for E＜EF about the energy E=EF. This also means that the function fF（E）for E=EF+dE is equal to the function 1－fF（E）for E=EF－dE.

Fig.4.1（a）Density of states functions，Fermi-Dirac probability function，and areas representing electron and hole concentrations for the case when EF is near the midgap energy；（b）expanded view near the conduction band energy；and（c）expanded view near the valence band energy.

Fig.4.1（b）is an expanded view of the plot in Fig.4.1（a）showing fF（E）and gc（E）above the conduction band energy Ec. The product of gc（E）and fF（E）is the distribution of electrons n（E）in the conduction band given by Equation（4.1）. This product is plotted in Fig.4.1（a）. Fig.4.1（c）is an expanded view of the plot in Fig.4.1（a）showing |1－fF（E）| and gv（E）below the valence band energy Ev. The product of gv（E）and |1－fF（E）|is the distribution of holes p（E）in the valence band given by Equation（4.2）. This product is also plotted in Fig.4.1（a）. The area under these curves is then the total density of electrons in the conduction band and the total density of holes in the valence band. From this we see that if gc（E）and gv（E）are symmetrical，the Femi energy must be at the midgap energy in order to obtain equal electron and hole concentrations. If the effective masses of the electron and hole are not exactly equal，then the effective density of states functions gc（E）and gv（E）will not be exactly symmetrical about the midgap energy. The Fermi level for the intrinsic semiconductor will then shift slightly from the midgap energy in order to obtain equal electron and hole concentrations.

We have argued that the Fermi energy for an intrinsic semiconductor is near midgap. In deriving the equations for the thermal-equilibrium concentration of electrons n0 and the thermal-equilibrium concentration of holes p0，we will not be quite so restrictive. We will see later that，in particular situations，the Fermi energy can deviate from this midgap euergy. We will assume initially，however，that the Fermi level remains within the bandgap energy.

The equation for the thermal-equilibrium concentration of electrons may be found by integrating Equation（4.1）over the conduction band energy，or

n0=∫gc（E）f fF（E））dE （4.3）

The thermal-equilibrium electron concentration in the conduction band can be written as

The parameter Nc is called the effective density of states function in the conduction band . If we were to assume that m*n=m0，then the value of the effective density of states function at T=300K is Nc=2.5×1019cm-3，which is the order of magnitude of Nc for most semiconductors. If the effective mass of the electron is larger or smaller than m0，then the value of the effective density of states function changes accordingly，but is still of the same order of magnitude.

The thermal-equilibrium concentration of holes in the valence band may be written as

Nv is called the effective density of states function in the valence band. The magnitude of Nv is also on the order of 1019cm-3 at T=300K for most semiconductors.

The effective density of states functions，Nc and Nv，are constant for a given semiconductor material at a fixed temperature. Tab.4.1 gives the values of the density of states function and of the effective masses for silicon，gallium arsenide，and germanium. Note that the value of Nc for gallium arsenide is smaller than the typical 1019cm－3 value. This difference is due to the small electron effective mass in gallium arsenide.

Tab.4.1 Effective density of states function and effective mass values