EEE209/ECE230 Semiconductor Devices and Materials Lecture 2: Carrier Concentration and the Fermi-Dirac Distribution Dr. Mosaddequr Rahman Room No: UB50513 Email: mosaddeq@bracu.ac.bd
Carrier Concentration How to Find the Carrier Concentration? Current is the rate at which charge flows. The charge carriers in a semiconductor are electrons and holes. To determine the electrical properties and analyze device behavior, we often need to know the carrier concentration i.e., number of electrons and holes per cm3 of the material. For heavily doped extrinsic semiconductor, (for n-type) and (for p-type) However, we don’t know anything about the minority carrier concentration. Nor do we know about the temperature dependence of the carriers. In this section, we wish to determine the concentration of electrons and holes in the conduction and valence bands. Electrons and holes are distributed over the empty conduction band and valence band states, respectively, as function of energy. The distribution depends upon the temperature. So it is important to find the electron and hole distribution over the required energy range, in order to determine the carrier concentration.
The Carrier Concentration Equilibrium Electron and Hole Concentrations To calculate the number of electrons in the CB and holes in the VB we must know 1) How many states are available 2) What is the probability of occupancy We thus introduce g(E)dE, DENSITY OF STATES (cm-3), available for occupancy in the energy range dE.
Carrier Concentration Electron Distribution The distribution (with respect to energy) of electrons in the conduction band is given by the density of allowed quantum states, gc(E), times the probability, f(E), that an electron occupies a state. where f(E) is the Fermi-Dirac probability function and gc(E) is the density of quantum states in the conduction band. Hole Distribution The distribution (with respect to energy) of holes in the band is given by the density of allowed quantum states, gv(E), times the probability, 1-f(E), that a state is Not occupied by an electron. Both g(E) and f(E) are in terms of per unit volume per unit energy.
Density of States Density of States h is the Planck’s constant Exercise: Determine the total number of energy states in Si between Ec and Ec + kT at T=300 K. [ ].
The Carrier Density Distribution The Carrier Density Integral
The Fermi-Dirac Distribution The Fermi-Dirac Distribution Function The Fermi-Dirac distribution function, which provides the probability of occupancy of energy levels by electrons, is based on following premises: - Particles (electrons) are indistinguishable, and only one particle is allowed in each quantum state. - At absolute zero temperature (T = 0 K), the energy levels are all filled up to a maximum level, called the Fermi level. Beyond the Fermi level, all states are empty. - At higher temperature, there is a gradual transition between the completely filled states and the completely empty states. Mathematically, Fermi-Dirac distribution function is given by, and gives the probability that an available state at energy E will be occupied by an electron at absolute temperature T. The quantity EF is called the Fermi level, and k is Boltzmann’s constant (k = 1.38x10-23 J/K = 8.62x10-5 eV/K )
Fermi Probability function for T = 0 K. The Fermi-Dirac Distribution The Fermi-Dirac Distribution Function Consider the situation where T = 0 K. For E < EF, For E > EF, Thus for T = 0 K, For energies E < EF, all states are filled. For energies E > EF, all states are empty. Fermi Probability function for T = 0 K.
The Fermi-Dirac Distribution The Fermi-Dirac Distribution Function Consider the situation where T > 0 K. For E = EF, thus an energy state at the Fermi level has a probability of ½ of being occupied by an electron. In general, for temperatures T > 0 K, there is a nonzero probability that some energy states above EF will be occupied by electrons and some energy states below EF will be empty.
Fermi Probability function vs. Energy for T > 0 K. The Fermi-Dirac Distribution The Fermi-Dirac Distribution Function As the temperature climbs, we notice that more states below EF are empty and more states above EF are filled. Fermi Probability function vs. Energy for T > 0 K. For E - EF >> kT, Fermi-Dirac distribution function can be approximated to,
The Fermi Level The Fermi Level – Intrinsic Semiconductor For intrinsic semiconductor, since electrons and holes are generated in pairs, n0 = p0 for all temperature. This leads to, i stands for intrinsic, and EFi is the Fermi level for intrinsic semiconductor. This shows that for intrinsic semiconductor, Fermi level lies exactly halfway between the CB and the VB.
The Fermi Level The Fermi Level – Intrinsic Semiconductor EC EF = EFi Energy band diagram of an intrinsic semiconductor. Fermi level lies exactly halfway between the CB and the VB. 12
The Density of States and the Carrier Concentration Carrier Concentration – Intrinsic Semiconductor Ec Ev The figure shows a possible distribution of density of states and electron and hole concentrations for an EF near the mid gap energy (Fermi probability function superimposed): The shaded areas under the curves in the figure is then the total density of electrons in the conduction band (CB) and the total density of holes in the valence band (VB).
The Carrier Density and the Effecttive Density of States Equilibrium Electron and Hole Concentrations The electron concentration can be calculated by integrating the carrier density distribution from EC is called the effective density of states in the CB. Where, And f (EC) is the occupation propability of available states at the conduction band edge EC by an electron. Likewise, the equilibrium hole concentration in the VB is given by, is called the effective density of states in the VB. where
The Carrier Concentration Revisited Equilibrium Electron and Hole Concentrations From slide 6, the equilibrium electron concentration in CB is For EC – EF >> KT, the electron concentration n0 can be written as, Likewise, the equilibrium hole concentration in the VB is given by,
The Fermi Level: Extrinsic Semiconductor Fermi Level for Extrinsic Semiconductor It can be concluded that for n-type material EF is closer to EC and as (EC-EF) n p Similarly for p-type material, EF is closer to EV and as (EF-EV) p n EF moves toward EC for n-type EF moves toward EV for p-type EC EF = EFi, for intrinsic semiconductor EV 16
The Effecttive Density of States Effective Density of States
The Probability Calculation Exercise: Assume the Fermi energy level is 0.30 eV below the conduction band energy and T = 300 K. Determine the probability of a state being occupied by an electron at Ec. Exercise: Assume the Fermi energy level is exactly in the center of the bandgap energy of a semiconductor at T = 300K.. Calculate the probability that an energy state in the bottom of the conduction band (at Ec) is occupied by an electron for Si.. For Si, Eg = 1.12 eV. Since, E - EF = EC - EF = 0.56 >> kT, we can use the Boltzmann approximation: Hence,
The Fermi Level: Extrinsic Semiconductor Exercise: Calculate the thermal equilibrium electron and hole concentration in silicon at T = 300K, where the Fermi energy level is 0.22 eV below the conduction band energy Ec. (EF = Ec – 0.22 eV). Assume Nc=2.8x1019 cm-3 and Nv=1.04x1019 cm-3 For p-type material, EC EF EV (EC-EF) (EF-EV) (EC-EV) Equilibrium hole concentration
The Fermi Level: Extrinsic Semiconductor Fermi Level Position: Extrinsic Semiconductor We can now determine the position of the Fermi energy level as a function of the doping concentrations. For n-type semiconductor n-type semiconductor p-type semiconductor For p-type semiconductor
The Fermi Level: Extrinsic Semiconductor Carrier Concentration and the Fermi Level Position EC For n-type EV For p-type 21
Mass Action Law Mass Action Law The product of equilibrium electron and hole concentrations can be written as: For Intrinsic semiconductor: Which is same for the extrinsic semiconductor. So for any semiconductor, at thermal equilibrium, This is known as the mass action law.
Mass Action Law Equilibrium Carrier Concentration Revisited The mass action law is a powerful relation, which enables to find quickly the hole density if the electron density is known or vice versa. Using the mass action law, we can write: Further, for an extrinsic semiconductor, electron concentration can be written as: Similarly, it can be shown that,
Mass Action Law
Temperature Dependence of Carrier Concentration Complete Ionization At room temperature, the donor states are essentially completely ionized which says that almost all donor impurity atoms have donated an electron to the conduction band. n ≈ Nd For the valence band, at room temperature, acceptors atoms are also essentially completely ionized. This means that each acceptor atom has accepted an electron from the valence band. Energy band diagrams at room temperature for (a) n-type and (b) p-type semiconductors.
Temperature Dependence of Carrier Concentration Freeze-Out The opposite of complete ionization occurs at T = 0 K. At absolute zero degrees, all electrons are in their lowest possible energy state. For n-type, each donor state must contain an electron. nd = Nd No electrons from the donor state are thermally elevated to the conduction band; this effect is called freeze-out. For p-type, each acceptor site will be empty. No electrons from the valence band are thermally elevated to the acceptor states; this effect is also called freeze-out. Energy band diagrams at T = 0 K for (a) n-type and (b) p-type semiconductors.
Temperature Dependence of Carrier Concentration Low Temp. High Temp. Freeze out At high enough temperatures, the material returns to its intrinsic properties. B=5.4x1031 for Si Electron concentration vs. Temperature for an n-type material
Temperature Dependence of Carrier Concentration Exercise: Find the intrinsic carrier concentration in silicon at T = 200 K. Assume Nc = 2.8×1019 cm-3, Nv = 1.04×1019 cm-3, and Eg = 1.12 eV. Ans. ni = 7.68×104 cm-3
Space Charge Neutrality In thermal equilibrium, the semiconductor is electrically neutral. The electrons are distributed among the various energy states, creating negative and positive charges, but the net charge density is zero. A semiconductor can be doped with both donors (that donate electrons) and acceptors (that accept electrons). In a bulk semiconductor, there must be charge neutrality everywhere because any excess charge will immediately set up an electric field. The electrons will flow towards the positive charge until the value Q is reduced to 0. Then E→0. Then the attractive force, F→0.
Compensated Semiconductor Let a semiconductor be doped with Nd no. of donors and Na no. of acceptor atoms. (Such materials are known as compensated semiconductors) When completely ionized, the no. of positive and negative charges due to ionized donor and acceptor atoms will be and , respectively. If n0 is the total electron concentration and p0 is the total hole concentration, then the concentrations of negative and positive charges due to electrons and holes will be n0 and p0, respectively. The condition of space charge neutrality requires that the total no. of positive charges is equal to the total no. of negative charges at any time. Thus we can write, ++++++++ neglecting p0, as for n-type n0 >> p0.
Compensated Semiconductor Exercise: Determine the position of the Fermi level with respect to the valence band energy in p type GaAs at T = 300 K. Assume Nd= 4 x 1015 cm-3 and Na = 5 x 1016 cm-3 and Nv = 7 x 1018 cm-3.
Compensated Semiconductor Exercise: Consider a compensated GaAs semiconductor at T = 300 K. Nd= 5 x 1015 cm-3 and Na = 5 x 1016 cm-3. Calculate the thermal equilibrium electron and hole concentrations.