1. Energy Band Theory C◆ky00 kz001] Ec Ec(km) kx[100 k[111 0 B ky010 ME(k) kz001 Ec(km+dk) Ec(k Figure 1. 19: A: Values of k of equal energy. A: one-dimensional case, B: two-dimensional case; C: three-dimensional case(silicon); D approximation of ellipsoids by spheres(silicon) 1.1.8. Density of states in energy bands The density of permitted states in a three-dimensional crystal is given by (1. 1.33). Its value n(k)=18n3 (11.41) per crystal unit volume. If we define f(k)as the probability that these states are occupied, then the electron density, n, in an energy band En(k can be calculated by integrating the product of the density of states by the occupation probability over the first Brillouin zone 「n(k)k)k (1.1.42) Similarly, the density of holes within an energy band is given by p= Jn(k)[1-f(k)) dk (1.1.43) The function n (k)represents the density of permitted states in an energy band. The function f(k) is a statistical distribution function which is a
1. Energy Band Theory 25 1.1.8. Density of states in energy bands The density of permitted states in a three-dimensional crystal is given by (1.1.33). Its value is: per crystal unit volume. If we define f(k) as the probability that these states are occupied, then the electron density, n, in an energy band can be calculated by integrating the product of the density of states by the occupation probability over the first Brillouin zone: Similarly, the density of holes within an energy band is given by: The function n(k) represents the density of permitted states in an energy band. The function f(k) is a statistical distribution function which is a
function of the energy, En(k). Under thermodynamic equilibrium conditions,fik)is the Fermi-Dirac distribution function defined as: [III Fermi-Dirac Distribution f (k) 7+ expl(En(k -EF)/kT (11.44a) or fe)= 1+ exp/(E-EF)/KT (1.1.44b where EF is an energy value called the "Fermi level", k is the Boltzmann constant,and T is the temperature in Kelvin. The Fermi-Dirac function is plotted in Figure 1.1.20 for T>OK. It is worthwhile noting that f(E)=0.5 if E= EF, regardless of temperature. Therefore, a second definition of the Fermi level is that it is the energy level which has a 50%o probability of being occupied In order to integrate Expressions 1. 1. 42 or 1. 1.43 easily, the depende of n and f on k must be transformed into a dependency on the energy, E. To do this, let us consider a unit cell of the reciprocal crystal lattice where kx, ky and kz are given by Relationship 1. 1. 13 with nx=ny=nz- the volume of this cell is equal to krk,kz=8/L3. If the crystal has unit volume, then L=I and the volume of a unit cell of a unit-volume crystal in k-space is equal to 8Z3. In this crystal the volume of a spherical shell with a thickness dk in k-space is given by(volume of a shell of thickness dk in Figure 1. 19.D): 4π dk)3-k3三4nk2dk (1.1.45) The number of unit cells in that volume is given by the volume of the shell divided by the unit volume of the cell 4r k2 dk k2 (1.1.46) The number of k vectors(and thus the number of energy levels, since there is an energy level for each k vector) is equal to the number of unit cells. Using the Pauli exclusion principle (which states that there can be only 2 electrons for each k vector), the number of electrons is given by 2 (k)dk=- dk (1.1.47) Using the parabolic band approximation, E(k)=hk4/2m* and using a constant effective mass one obtains
26 Chapter 1 function of the energy, Under thermodynamic equilibrium conditions, f(k) is the Fermi-Dirac distribution function defined as:[11] where is an energy value called the "Fermi level", k is the Boltzmann constant, and T is the temperature in Kelvin. The Fermi-Dirac function is plotted in Figure 1.1.20 for T > 0K. It is worthwhile noting that f(E) = 0.5 if regardless of temperature. Therefore, a second definition of the Fermi level is that it is the energy level which has a 50% probability of being occupied. In order to integrate Expressions 1.1.42 or 1.1.43 easily, the dependency of n and f on k must be transformed into a dependency on the energy, E. To do this, let us consider a unit cell of the reciprocal crystal lattice where and are given by Relationship 1.1.13 with the volume of this cell is equal to If the crystal has unit volume, then and the volume of a unit cell of a unit-volume crystal in k-space is equal to In this crystal the volume of a spherical shell with a thickness dk in k-space is given by (volume of a shell of thickness dk in Figure 1.19.D): The number of unit cells in that volume is given by the volume of the shell divided by the unit volume of the cell: The number of k vectors (and thus the number of energy levels, since there is an energy level for each k vector) is equal to the number of unit cells. Using the Pauli exclusion principle (which states that there can be only 2 electrons for each k vector), the number of electrons is given by: Using the parabolic band approximation, and using a constant effective mass, one obtains:
1. Energy Band Theor n(E)dE 2m*)312E1/2dE (1.1.48) This equation yields the density of states for a particle of mass m* having an energy ranging between E and E+dE. In the case of electrons with a mass me located near the bottom of the conduction band, the energy is referenced to the minimum of the conduction band (Ec), which yields m(E)dE=2-2A(2me 3/2(E-Ed/dE (11.49) In the case of holes with a mass mh located near the top of the valence band, the energy is referenced to the maximum of the valence band(Ev) and one n(E)dE=22A (2mn/(EyE)dE (1.1.50) Integration of Equations 1. 1.42 and 1. 1.43 can now be performed. The integration can be further simplified by approximating the Fermi-Dirac (FD)distribution by the Maxwell-Boltzmann(MB)distribution. Both distributions are almost identical provided that E-EF is large enough, which is the case in typical semiconductors (i.e 1+ exp(u) when u >> 1(see Problem 1.10): Maxwell-Boltzmann Distribution f(E/+exp((E-EF)kT] exp/- Er (11.51) Fermi-Dirac Maxwell-boltzmann To calculate the electron density, n, in the conduction band( CB)we replace the integral over k-values in Relationship 1. 1. 42 by an integral over energy. n=「n(k(klk=Jn(E∥EdE(cm-3) E-EF 2x20(2m32|EE)1叫67」dE (11.52) In a typical semiconductor the vast majority of the electrons in the conduction band have an energy close to Ec. Therefore, the lower and pper bound of the integral can thus be replaced by Ec and infinity
1. Energy Band Theory 27 This equation yields the density of states for a particle of mass m * having an energy ranging between E and E+dE. In the case of electrons with a mass located near the bottom of the conduction band, the energy is referenced to the minimum of the conduction band which yields: In the case of holes with a mass located near the top of the valence band, the energy is referenced to the maximum of the valence band and one obtains: Integration of Equations 1.1.42 and 1.1.43 can now be performed. The integration can be further simplified by approximating the Fermi-Dirac (FD) distribution by the Maxwell-Boltzmann (MB) distribution. Both distributions are almost identical provided that is large enough, which is the case in typical semiconductors when u >> 1 (see Problem 1.10): To calculate the electron density, n, in the conduction band (CB) we replace the integral over k-values in Relationship 1.1.42 by an integral over energy: In a typical semiconductor the vast majority of the electrons in the conduction band have an energy close to Therefore, the lower and upper bound of the integral can thus be replaced by and infinity
Chapter l respectively. To integrate, a change of variables can be used wherey =(E- Ec/kT, which yields: (2m32|E-E E-E 22小3 kt dE Ec-E 2n2h3(2m kT)3/2 exP:kT 2 exp(y)dy 2r2A ( kT)3/exp . Ec-EE kT」2 (11.53) 27cm kT n=N kT th Nc (cm-3)(1.1.54) B n(E)f(E) citrons) E -+- n(E)(1-f(E) holes) Density of states, n(E)0 f(E) I Occupied levels Figure 1.20: Density of states near the bottom of the conduction band and the top of the valence band(A), Fermi-Dirac function(B), and density of holes and electrons in the conduction and valence bands(C) for T*0K. Note that at T=OK, f( e)=l for E<EF and f(e)=0 for E>EF At T=OK the valence band is completely filled with electrons(empty of holes)and there are no electrons in the conduction band.[2] Nc is called the "effective density of states in the conduction band".It represents the number of states having an energy equal to Ec which, when multiplied by the occupation probability at Ec, yields the number of electrons in the conduction band. Likewise the total number of holes in
28 Chapter 1 respectively. To integrate, a change of variables can be used where which yields: is called the "effective density of states in the conduction band". It represents the number of states having an energy equal to which, when multiplied by the occupation probability at yields the number of electrons in the conduction band. Likewise the total number of holes in
1. Energy Band Theory the valence band can be calculated using this technique, based on Equation (1. 1.43). The effective density of states for holes in the valence band is: 3/2 EF-E 2h P k7」wihN h (cm-3)(1.1.55) The density of holes and electrons in the conduction and valence bands shown in Figure 1. 20. C for a Fermi level EF at midpoint of Ec and ey 1.2. Intrinsic semiconductor By virtue of Expressions 1. 1.54 and 1. 1.55 the product of the electron concentration and hole concentration in a semiconductor under thermodynamic equilibrium conditions is given by Ec-EF pn=Nc exp -kT Ny EF-Ev -NcNy expf-Eg/kt) 兀2k2n emh T exp(-Eg/kr) (1.2.la) where ni is called the intrinsic carrier concentration pn Product under Thermodynamic Equilibrium (1.21b) A semiconductor is said to be intrinsic" if the vast majority of its free carriers(electrons and holes) originate from the semiconductor atoms themselves. In that case if an electron receives enough thermal energy to jump"from the valence band to the conduction band, it leaves a hole behind in the valence band. Thus, every hole in the valence band conduction electrons is exactly equal to the number of valence holes. r of corresponds to an electron in the conduction band, and the numbe ni (12.2) Ec+Eν3 2+kT In ≡E (12.3) or, if me-h(simplifying approximation):where 4=+E (1.2.4)
1. Energy Band Theory 29 the valence band can be calculated using this technique, based on Equation (1.1.43). The effective density of states for holes in the valence band is: The density of holes and electrons in the conduction and valence bands is shown in Figure 1.20.C for a Fermi level at midpoint of and 1.2. Intrinsic semiconductor By virtue of Expressions 1.1.54 and 1.1.55 the product of the electron concentration and hole concentration in a semiconductor under thermodynamic equilibrium conditions is given by: where is called the intrinsic carrier concentration. and or, if (simplifying approximation): where A semiconductor is said to be "intrinsic" if the vast majority of its free carriers (electrons and holes) originate from the semiconductor atoms themselves. In that case if an electron receives enough thermal energy to "jump" from the valence band to the conduction band, it leaves a hole behind in the valence band. Thus, every hole in the valence band corresponds to an electron in the conduction band, and the number of conduction electrons is exactly equal to the number of valence holes: