Energy Bands and Charge Carriers in Semiconductors
Electron and Hole Concentrations at Equilibrium
  • The F-D distribution function can be used to calculate the electron and hole concentrations in semiconductors, if the densities of available states in the conduction and valence bands are known.
  • In equilibrium, the concentration of electrons in the conduction band can be given by

    (2.8)

    where N(E)dE is the density of available states/cm3 in the energy range dE.
  • Note: the upper limit of is theoretically not proper, since the conduction band does not extend to infinite energies; however, since f(E) decreases rapidly with increasing E, the contribution to this integral for higher energies is negligible.
  • Using the solution of 's wave equation under periodic boundary conditions, it can be shown that

    (2.9)
  • Thus, N(E) increases with E, however, f(E) decreases rapidly with E, thus, the product f(E)N(E) decreases rapidly with E, and very few electrons occupy states far above the conduction band edge, i.e., most electrons occupy a narrow energy band near the conduction band edge.
  • Similarly, the probability of finding an empty state in the valence band [1 - f(E)] decreases rapidly below , and most holes occupy states near the top of the valence band.
  • Thus, a mathematical simplification can be made assuming that all available states in the conduction band can be represented by an effective density of states NC located at the conduction band edge and using Boltzmann approximation.

    Thus,(2.10)

    where .
  • Note: as ( - ) decreases, i.e., the Fermi level moves closer to the conduction band, the electron concentration increases.
  • By similar arguments,

    (2.11)

    where is the effective density of states located at the valence band edge .
  • Note: the only terms separating the expressions for and are the effective masses of electrons ( ) and holes ( ) respectively, and since , hence, .
  • Thus, as ( - ) decreases, i.e., the Fermi level moves closer to the valence band edge, and the hole concentration increases.
  • These equations for and are valid in equilibrium, irrespective of the material being intrinsic or doped.
  • For intrinsic material lies at an intrinsic level (very near the middle of the band gap), and the intrinsic electron and hole concentrations are given by
    and (2.12)
  • Note: At equilibrium, the product is a constant for a particular material and temperature, even though the doping is varied,

    i.e.,(2.13)

  • This equation gives an expression for the intrinsic carrier concentration ni as a function of , , and temperature:

    (2.14)

  • These relations are extremely important, and are frequently used for calculations.
  • Note: if were to be equal to , then would have been exactly at mid gap (i.e., - = - =/2).
  • However, since , is displaced slightly from mid gap (more for GaAs than that for Si).
  • Alternate expressions for and :
    and (2.15)

  • Note: the electron concentration is equal to ni when is at , and n0 increases exponentially as moves away from towards the conduction band.
  • Similarly, the hole concentration varies from to larger values as moves from towards the valence band.

EXAMPLE 2.3: A Si sample is doped with B . What is the equilibrium electron concentration n0 at 300 K? Where is relative to ? Assume for Si at 300 K = 1.5 x

SOLUTION: Since B (trivalent) is a p-type dopant in Si, hence, the material will be predominantly p-type, and since >> , therefore, will be approximately equal to, and = . Also, . The resulting band diagram is:



Temperature Dependence of Carrier Concentrations
  • The intrinsic carrier concentration has a strong temperature dependence, given by
    (2.16)
  • Thus, explicitly, ni is proportional to T3/2 and to e 1/T, however, Eg also has a temperature dependence (decreasing with increasing temperature, since the interatomic spacing changes with temperature).

    Fig.2.5 The intrinsic carrier concentration as a function of inverse temperature for Si, Ge, and GaAs.
  • As changes with temperature, so do and .
  • With and T given, the unknowns are the carrier concentrations and the Fermi level position with respect to one of these quantities must be given in order to calculate the other.
  • Example: Si doped with donors ().
  • At very low temperature, negligible intrinsic EHPs exist, and all the donor electrons are bound to the donor atoms.
  • As temperature is raised, these electrons are gradually donated to the conduction band, and at about 100 K (1000/T = 10), almost all these electrons are donated this temperature range is called the ionization region.
Once all the donor atoms are ionized, the electron concentration , since for each donor atom, one electron is obtained.


Fig.2.6 Variation of carrier concentration with inverse temperature clearly showing the three regions: ionization, extrinsic, and intrinsic.
  • Thus, remains virtually constant with temperature for a wide range of temperature (called the extrinsic region), until the intrinsic carrier concentration ni starts to become comparable to .
  • For high temperatures, >> , and the material loses its extrinsic property (called the intrinsic region).
  • Note: in the intrinsic region, the device loses its usefulness => determines the maximum operable temperature range.
Compensation and Space Charge Neutrality
  • Semiconductors can be doped with both donors () and acceptors () simultaneously.
  • Assume a material doped with > predominantly n-type lies above acceptor level Ea completely full, however, with above , the hole concentration cannot be equal to .
  • Mechanism:
    • Electrons are donated to the conduction band from the donor level
    • An acceptor state gets filled by a valence band electron, thus creating a hole in the valence band.
    • An electron from the conduction band recombines with this hole.
    • Extending this logic, it is expected that the resultant concentration of electrons in the conduction band would be instead of .
    • This process is called compensation.
  • By compensation, an n-type material can be made intrinsic (by making = ) or even p-type (for >).

    Note: a semiconductor is neutral to start with, and, even after doping, it remains neutral (since for all donated electrons, there are positively charged ions (); and for all accepted electrons (or holes in the valence band), there are negatively charged ions ().
  • Therefore, the sum of positive charges must equal the sum of negative charges, and this governing relation,
    given by (2.17) is referred to as the equation for space charge neutrality.
  • This equation, solved simultaneously with the law of mass action (given by ) gives the information about the carrier concentrations.
    Note: for , .