Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 1 Chap 4. Semiconductor in Equilibrium  Carriers in Semiconductors  Dopant Atoms and Energy Levels  Extrinsic Semiconductor  Statistics of Donors and Acceptors  Charge Neutrality  Position of Fermi Energy

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 2 Equilibrium Distribution of Electrons and Holes  The distribution of electrons in the conduction band is given by the density of allowed quantum states times the probability that a state will be occupied. The thermal equilibrium conc. of electrons n o is given by  Similarly, the distribution of holes in the valence band is given by the density of allowed quantum states times the probability that a state will not be occupied by an electron.  And the thermal equilibrium conc. Of holes p o is given by

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 3 Equilibrium Distribution of Electrons and Holes

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 4 The n o and p o eqs.  Recall the thermal equilibrium conc. of electrons  Assume that the Fermi energy is within the bandgap. For electrons in the conduction band, if E c -E F >>kT, then E-E F >>kT, so the Fermi probability function reduces to the Boltzmann approximation,  Then  We may define, (at T =300K, N c ~10 19 cm -3 ), which is called the effective density of states function in the conduction band

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 5 The n o and p o eqs.  The thermal equilibrium conc. of holes in the valence band is given by  For energy states in the valence band, E >kT,  Then,  We may define, (at T =300K, N v ~10 19 cm -3 ), which is called the effective density of states function in the valence band

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 6 n o p o product  The product of the general expressions for n o and p o are given by  for a semiconductor in thermal equilibrium, the product of n o and p o is always a constant for a given material and at a given temp.  Effective Density of States Function

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 7 Intrinsic Carrier Concentration  For an intrinsic semiconductor, the conc. of electrons in the conduction band, n i, is equal to the conc. of holes in the valence band, p i.  The Fermi energy level for the intrinsic semiconductor is called the intrinsic Fermi energy, E Fi.  For an intrinsic semiconductor,  For an given semiconductor at a constant temperature, the value of n i is constant, and independent of the Fermi energy.

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 8 Intrinsic Carrier Conc.  Commonly accepted values of n i at T = 300 K Siliconn i = 1.5x10 10 cm -3 GaAsn i = 1.8x10 6 cm -3 Germaniumn i = 1.4x10 13 cm -3

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 9 Intrinsic Fermi-Level Position  For an intrinsic semiconductor, n i = p i,  E midgap =(E c +E v )/2: is called the midgap energy.  If m p * = m n *, then E Fi = E midgap (exactly in the center of the bandgap)  If m p * > m n *, then E Fi > E midgap (above the center of the bandgap)  If m p * < m n *, then E Fi < E midgap (below the center of the bandgap)

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 10 Dopant and Energy Levels

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 11 Acceptors and Energy Levels

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 12 Ionization Energy  Ionization energy is the energy required to elevate the donor electron into the conduction band.

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 13 Extrinsic Semiconductor  Adding donor or acceptor impurity atoms to a semiconductor will change the distribution of electrons and holes in the material, and therefore, the Fermi energy position will change correspondingly.  Recall

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 14 Extrinsic Semiconductor  When the donor impurity atoms are added, the density of electrons is greater than the density of holes, (n o > p o )  n-type; E F > E Fi  When the acceptor impurity atoms are added, the density of electrons is less than the density of holes, (n o < p o )  p-type; E F < E Fi

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 15 Degenerate and Nondegenerate  If the conc. of dopant atoms added is small compared to the density of the host atoms, then the impurity are far apart so that there is no interaction between donor electrons, for example, in an n-material.  nondegenerate semiconductor  If the conc. of dopant atoms added increases such that the distance between the impurity atoms decreases and the donor electrons begin to interact with each other, then the single discrete donor energy will split into a band of energies.  E F move toward E c  The widen of the band of donor states may overlap the bottom of the conduction band. This occurs when the donor conc. becomes comparable with the effective density of states, EF  E c  degenerate semiconductor

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 16 Degenerate and Nondegenerate

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 17 Statistics of Donors and Acceptors  The probability of electrons occupying the donor energy state was given by where N d is the conc. of donor atoms, n d is the density of electrons occupying the donor level and E d is the energy of the donor level. g =2 since each donor level has two spin orientation, thus each donor level has two quantum states.  Therefore the conc. of ionized donors N d + = N d –n d  Similarly, the conc. of ionized acceptors N a - = N a –p a, where

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 18 Complete Ionization  If we assume E d -E F >> kT or E F -E a >> kT (e.g. T= 300 K), then that is, the donor/acceptor states are almost completely ionized and all the donor/acceptor impurity atoms have donated an electron/hole to the conduction/valence band.

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 19 Freeze-out  At T = 0K, no electrons from the donor state are thermally elevated into the conduction band; this effect is called freeze-out.  At T = 0K, all electrons are in their lowest possible energy state; that is for an n-type semiconductor, each donor state must contain an electron, therefore, n d = N d or N d + = 0, which means that the Fermi level must be above the donor level.

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 20 Charge Neutrality  In thermal equilibrium, the semiconductor is electrically neutral. The electrons distributing among the various energy states creating negative and positive charges, but the net charge density is zero.  Compensated Semiconductors: is one that contains both donor and acceptor impurity atoms in the same region. A n-type compensated semiconductor occurs when N d > N a and a p-type semiconductor occurs when N a > N d.  The charge neutrality condition is expressed by where no and po are the thermal equilibrium conc. of e - and h + in the conduction band and valence band, respectively. N d + is the conc. Of positively charged donor states and N a - is the conc. of negatively charged acceptor states.

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 21 Compensated Semiconductor

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 22 Compensated Semiconductor  If we assume complete ionization, N d + = N d and N a - = N a, then  If N a = N d = 0, (for the intrinsic case),  n o = p o  If N d >> N a,  n o = N d  If N a > N d, is used to calculate the conc. of holes in valence band

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 23 Compensated Semiconductor

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 24 Position of Fermi Level  The position of Fermi level is a function of the doping concentration and a function of temperature, E F (n, p, T).  Assume Boltzmann approximation is valid, we have

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 25 E F (n, p, T)

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 26 E F (n, p, T)

Solid-State Electronics Chap. 4 Instructor: Pei-Wen Li Dept. of E. E. NCU 27 Homework  4.18  4.20  4.24