EE 346: Semiconductor Devices Lecture - 6 02/06/2017 Tewodros A. Zewde 1
DENSTY OF STATES FUNCTON Since current is due to the flow of charge, an important step in the process is to determine the number of electrons and holes in the semiconductor that will be available for conduction. The number of carriers that can contribute to the conduction process is a function of the number of available energy or quantum states since, by the Pauli exclusion principle, only one electron can occupy a given quantum state. Hence, we must determine the density of discrete energy levels, or allowed energy states as a function of energy in order to calculate the electron and hole concentrations. As the energy of this free electron becomes small, the number of available quantum states decreases.
A general expression for the density of allowed electron quantum states can be derived using the model of a free electron with mass m bounded in a three-dimensional infinite potential wall. The same general model can be extended to a semiconductor to determine the density of quantum states in the conduction band and the density of quantum states in the valence band. The density of allowed electronic energy states in the conduction and valance band are given as follow: where m * is effective mass that takes into account the effect of internal forces in the crystal. As the energy of the electron in the conduction band decreases, the number of available quantum states also decreases.
The plot of the density of quantum states as a function of energy. Quantum states do not exist within the forbidden energy band, so g(e)=0 for E v <E <E c. f the electron and hole effective masses were equal, then the functions g c (E) and g v (E) would be symmetrical about the energy midway between E c and E v.
Now that we know the number of available states at each energy, or n dealing with large numbers of particles, we are interested only in the statistical behavior of the group as a whole rather than in the behavior of each individual particle. n a crystal, the electrical characteristics will be determined y the statistical behavior of a large number of electrons. Electrons in a crystal follow the probability function, i.e., one particle is permitted in each quantum state. 1 f (E) = where k Boltzman constant,t Temperature in Kelvin ( E E F ) kt 1+e and E F Fermi energy the energy below which all states are filled with electrons and above which all states are empty at T = 0 K. The function, f(e) is called the Fermi-Dirac distribution or probability function, and it gives the probability that a quantum state at the energy E will be occupied by an electron. f(e) is the probability that a state at energy E is occupied
Hence, the number density N(E) is the number of particles per unit volume per unit energy, i.e., filled quantum states, at any energy E becomes N(E) = g(e) f(e) where the function g(e) is the number of quantum states per unit volume per unit energy, and f(e) is the probability that a state at energy E is occupied. At T=0K, no occupation of states above E F and complete occupation of states below E F At T > 0K, occupation probability is reduced with increasing energy. f(e=e F ) = 1/2 regardless of temperature. At higher temperatures, higher energy states can be occupied, leaving more lower energy states unoccupied (1-f(E)).
For temperatures above absolute zero, there is a nonzero probability that some energy states above E F will be occupied by electrons and some energy states below E F will be empty. This result again means that some electrons have jumped to higher energy levels with increasing thermal energy. Exercise Given that T = 300K, determine the probability that an energy level (a) 1kT above the Fermi energy is occupied by an electron. (b) 2kT above the Fermi energy is occupied by an electron. (c) 3kT above the Fermi energy is occupied by an electron. Note that k is Boltzmann constant, and k=1.381x10-23 J/K.
f(e) 1.20 1.00 0.80 0.60 0.40 0.20 f ( E) ( E EF ) kt 1 e 1 For E < (E f -3kT): f(e) ~ 1-e -(E-Ef)/kT ~1 +/-3 kt 3 kt 3 kt 3 kt 3 kt E f =0.55 ev For E > (E f +3kT): f(e) ~ e -(E-Ef)/kT ~0 T=10 K, kt=0.00086 ev T=300K, kt=0.0259 T=450K, kt=0.039 0.00 0 0.2 0.4 0.6 0.8 1 1.2 E [ev]
The Semiconductor in Equilibrium Equilibrium, or thermal equilibrium, implies that no external forces such as voltages, electric fields. magnetic fields, or temperature gradients are acting on the semiconductor. n this section, we will use the density the density of quantum states in the conduction band and the density of quantum states in the valence band along with the Fermi-dirac probability function to determine the concentration of electrons and holes in the conduction and valence bands, respectively. We will initially consider the properties of an intrinsic semiconductor, that is, a pure crystal with no impurity atoms or defects. We will see that the electrical properties of a semiconductor can be altered in desirable ways by adding controlled amounts of specific impurity atoms. called dopant atoms, to the crystal.
Equilibrium Distribution of Electrons and Holes Since the current in a semiconductor is determined largely by the number of electrons in the conduction band and the number of holes in the valence hand, an important characteristic of the semiconductor is the density of these charge carriers. The distribution (with respect to energy) of electrons in the conduction band is given by n(e) = g c (E) f(e) where g c (E) is the density of quantum states in the conduction band and f(e) if the Fermi-Dirac probability function. Similarly, the distribution (with respect to energy) of holes in the valence band is given by p(e) = g v (E) [1- f(e)]
Energy band Density Occupancy diagram of states factors E e - EF E v LL_ gc( E) 8 v(e) E t 1 - f(e) f(e) E e E V Carrier distributions gc(e)f(e) : 8v(E)[ 1 - /( )] (a) EF above midgap E e 8c(E) - E F gv(e) E v E t Electrons E _.{.-:.. e, E v r' k Holes (b) E p near midgap Ee -EF E v LL- E 8v( E) K 8e(E) t E e (c) E p below midgap
\1-fF(t)J Figure 4.1 (a)density of states fimctions, Fenni-Dirac probability fonction,and areas representing electron and hole concen1ratioa5 for the case when,- is near the midgap energy; (b) expanded vie w near the conduction band energy; and (c) e-xpanded view near the valence band energy.
The thermal-equilibrium concentration of electron per unit volume is found by integrating the density functions over the entire conduction-band energy, or For electrons in conduction band, E>E c, and if E c -E F >>kt where the exponential term in the denominator of f (E) 1 (E E F ) kt 1+e is much greater than unity, so the Femi-Dirac distribution function reduces to the Maxwell-Boltzmann approximation (Boltzmann approximation), which is f (E ) E E F ) kt ē ( N c == is effective density of state in conduction band.
The thermal-equilibrium concentration of holes in the valence band is found by Following similar steps, we have where N v = valence band. is effective density of states function in the For an intrinsic semiconductor, the concentration of electrons in the conduction band is equal to the concentration of holes in the valence band. Let E F =E Fi for the Fermi energy level for the intrinsic semiconductor, and then
which leads to where E g is the bandgap energy. For a given semiconductor material at a constant temperature, the value of n i is a constant, and independent of the Fermi energy. The intrinsic carrier concentration is a very strong function of temperature. The figure shows the intrinsic carrier concentration of Ge, Si, and GaAs as a function of temperature.
The intrinsic Fermi-level position can be determined using the fact that the electron and hole concentrations are equal, i.e., which leads to where f m * p = m* n, then the intrinsic Fermi level is exactly in the center of the bandgap. f m* p > m* n, the intrinsic Fermi level is slightly above the center, and f m* p < m* n, it is slightly below the center of the bandgap. The intrinsic Fermi level must shift away from the band with the larger density of states in order to maintain equal numbers of electrons and holes.
DOPANT ATOMS AND ENERGY LEVELS Without help the total number of carriers (electrons and holes) is limited to 2ni. For most materials, this is not that much, and leads to very high resistance and few useful applications. The intrinsic semiconductor may be an interesting material, but the real power of semiconductors is realized by adding small, controlled amounts of specific dopant, or impurity, atoms. This process is known as doping the crystal. Adding controlled amounts of dopant atoms, either donors or acceptors, creates a material called an extrinsic semiconductor. An extrinsic semiconductor will have either excess electrons (n-type) or excess holes (ptype).
Donor impurity atom: Consider adding a group V element as a substitutional impurity to silicon. Example: P, As, Sb in Si The fifth valence electron is denoted as a donor electron, and a donor impurity atom donates an electron to the conduction band.
The electron in the conduction band can now move through the crystal generating a current, while the positively charged ion is fixed in the crystal.
Concept of a Donor adding extra electrons: Band diagram equivalent view The energy level, E D, is the energy state of the donor electron. The donor impurity atoms add electrons to the conduction band without creating holes in the valence band. The resulting material is referred to as an n-type semiconductor (n for the negatively charged electron).
Acceptor impurity atom: consider adding a group element as a substitutional impurity to silicon. One covalent bonding position appears to be empty. Example: B, Al, n in Si One less bond means the acceptor is electrically satisfied One less bond means the neighboring silicon is left with an empty state.
The "empty" position associated with the boron atom becomes occupied, and other valence electron positions become vacated. These other vacated electron positions can be thought of as holes in the semiconductor material. +
The hole can move through the crystal generating a current, while the negatively charged boron atom is fixed in the crystal. +
The hole can move through the crystal generating a current, while the negatively charged boron atom is fixed in the crystal. +
Concept of an Acceptor adding extra hole : Band diagram equivalent view The electron occupying the "empty" position does not have sufficient energy to be in the conduction band, so its energy is far smaller than the conductionband energy. The energy level, E A, is the energy state of the acceptor electron. The group atom accepts an electron from the valence band and so is referred to as an acceptor impurity atom. This type of material is referred to as a p-type semiconductor (p for the positively charged hole).