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ENE 311. Lecture 5. The band theory of solids. Band theories help explain the properties of materials. There are three popular models for band theory: - Kronig-Penney model - Ziman model - Feynman model. Kronig-Penney Model.
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ENE 311 Lecture 5
The band theory of solids • Band theories help explain the properties of materials. • There are three popular models for band theory: - Kronig-Penney model - Ziman model - Feynman model
Kronig-Penney Model • Band theory uses V 0. The potential is periodic in space due to the presence of immobile lattice ions. Kronig-Penney model Ideal w
Kronig-Penney Model • Ions are located at x = 0, a, 2a, and so on. The potential wells are separated from each other by barriers of height U0 and width w. • From time-independent Schrödinger equation in 1-dimension (x-only), we have (1)
Kronig-Penney Model • For this equation to have solution, the following must be satisfied (2) (3) (4)
Kronig-Penney Model Forbidden band Allowed band
Kronig-Penney Model • We plot the right-hand side of (2) as a function of a and since the left-hand side of the same equation is always between -1 and +1, a solution exists only for the shaded region and no solution outside the shaded region. • These regions are called “allowed and forbidden bands of energy” due to the relation between and E.
Kronig-Penney Model From equation (2), we have • If P increases, allowed bands get narrower and the forbidden bands get wider. • If P decreases, allowed bands get wider and forbidden bands get narrower. • If P = 0, then cos(a) = cos(ka)
Kronig-Penney Model • If P , then sin(a) = 0 • At the boundary of an allowed band cos(ka) = 1, this implies k = n/a for n = 1, 2, 3, …
Kronig-Penney Model How to plot E-k diagram • Choose values between -1 to +1, then find argument of right-hand side (a) which satisfies chosen values. • Likewise to left-hand side (ka). • a = (any number in radian)
Kronig-Penney Model • ka = (any number in radian) • Plot E-k diagram
Brillouin Zone Reduced Brillouin Zone
Number of electrons per unit volume • The total number of electrons per unit volume in the range dE (between E and E + dE) is given by where N(E) = density of states (number of energy levels per energy range per unit volume) F(E) = a distribution function that specifies expectancy of occupation of state or called “probability of occupation”.
Number of electrons per unit volume • The density of states per unit volume in three dimensions can be expressed as
Number of electrons per unit volume • The probability of occupancy is given by the Fermi-Dirac- distribution as where EF = Fermi energy level (the energy at F(E) = 0.5) k = Boltzmann’s constant T = absolute temperature (K)
Number of electrons per unit volume • For T = 0 K: If E > EF, F(E) = 0 F(E) = 1/(e +1) = 0 If E < EF, F(E) = 1 F(E) = 1/( e- + 1) = 1 • For T > 0, F(EF) = 0.5
Number of electrons per unit volume • From equation (5),
Number of electrons per unit volume • For T > 0
Number of electrons per unit volume Characteristics of F(E) • F(E), at E = EF, equals to 0.5. • For (E – EF) >3kT This is called “Maxwell – Boltzmann distribution”.
Number of electrons per unit volume Characteristics of F(E) 3. For (E – EF) < 3kT 4. F(E) may be distinguished into 3 regions for T > 0 as • E = 0 to (E = EF – 2.2kT): F(E) is close to unity. • (E = EF – 2.2kT) to (E = EF + 2.2kT): F(E) changes from nearly 1 to nearly 0. • (E = EF + 2.2kT) to E = : F(E) is close to zero.
Intrinsic carrier concentration • Free charge carrier density or the number of electrons per unit volume • For electrons: E1/2= (E - EC)1/2 and • For holes: E1/2 = (EV - E)1/2 and
Intrinsic carrier concentration • At room temperature, kT = 0.0259 eV and (E – EF) >> kT, so Fermi function can be reduced to Maxwell-Boltzmann distribution.
Intrinsic carrier concentration • Therefore, the electron density in the conduction band at room temperature can be expressed by (6) = effective density of states in the conduction band.
Intrinsic carrier concentration • Similarly, we can obtain the hole density p in the valence band as (7) = effective density of states in the valence band
Intrinsic carrier concentration • Schematic band diagram. • Density of states. • (c)Fermi distribution function. • (d) Carrier concentration
Intrinsic carrier concentration • For intrinsic semiconductors, the number of electrons per unit volume in the conduction band equals to the number of holes per unite volume in the valence band. (8) where ni = intrinsic carrier density
Intrinsic carrier concentration • From (8);
Intrinsic carrier concentration • The Fermi level of an intrinsic semiconductor can be found by equating (6) = (7) as
Intrinsic carrier concentration Ex. Calculate effective density of states NC and NV for GaAs at room temperature if GaAs has and .
Intrinsic carrier concentration Soln We clearly see that the only difference between NC and NV is the values of effective electron and hole mass.
Intrinsic carrier concentration Ex. From previous example, calculate intrinsic carrier density ni for GaAs at room temperature where energy gap of GaAs is 1.4 eV.
Intrinsic carrier concentration We may have a conclusion that • As EF EC, then n increases. • As EF EV, then p increases. • As T = 0 K, then EF is at Eg/2 • If EF > EC or EF < EV, then that semiconductor is said to be “degenerate”.
Donors and Acceptors • When a semiconductor is doped with some impurities, it becomes an extrinsic semiconductor. • Also, its energy levels are changed.
Donors and Acceptors The figure shows schematic bond pictures for n-type and p-type.
Donors and Acceptors • For n-type, atoms from group V impurity release electron for conduction as free charge carrier. • An electron belonging to the impurity atom clearly needs far less energy to become available for conduction (or to be ionized). • The impurity atom is called “a donor”. • The donor ionization energy is EC – ED where ED is donor level energy.
Donors and Acceptors • For p-type, atoms from group III capture electron from semiconductor valence band and produce hole as free charge carrier. • EA is called “acceptor level” and EA – EV is called “acceptor ionization level energy”. • This acceptor ionization level energy is small since an acceptor impurity can readily accept an electron.
Donors and Acceptors • The ionization energy or binding energy, producing a free charge carrier in semiconductor, can be approximately expressed by
Donors and Acceptors Ex. Calculate approximate binding energy for donors in Ge, given that r = 16 and = 0.12m0.
Donors and Acceptors Soln
Donors and Acceptors (a) donor ions and (b) acceptor ions.
Donors and Acceptors • Consider an n-type semiconductor, if ND is the number of donor electrons at the energy level ED, then we define to be the number of free electron carrier (number of ND that have gone for conduction). or ionized donor atom density can be written as
Donors and Acceptors • For a p-type, the argument is similar. Therefore, NA- or free-hole density or ionized acceptor atom density is written as
Donors and Acceptors We can obtain the Fermi level dependence on temperature for three cases: • very low temperature • intermediate temperature • very high temperature.
