880:174 Physics of Modern Materials Chapter 6: Semiconductors 1. Intrinsic Semiconductors As we saw in the previous chapter whether a material is categorized as an insulator or semiconductor depends on the band gap. For small band gaps, the material will be an insulator at T = 0 K but a fair conductor at room temperature. Such a material is a semiconductor. An intrinsic semiconductor is one without any impurities. Thus, its behavior is determined by intrinsic properties (e.g., lattice constant, band gap, etc.) alone. Silicon and germanium are the closest approximations to intrinsic semiconductors because of decades of improvement in processing technology. (GaAs is also produced with high purity for the IT industry.) However, even the purest Si and Ge have impurity concentrations of approximately 1 in 1010. Thus, an intrinsic semiconductor is an approximation, albeit a very good one. We will assume (because of ease of calculation) that the carriers of current are electrons near the bottom of the conduction band (CB) and holes near the top of the valence band (VB). Under these conditions, the bands are nearly parabolic so that the electrons and holes act to good approximation like free particles with their masses taken to be the effective masses. (See Solymar & Walsh text for justification.) Thus, we can use the density of states expression previously derived. Let the top of the VB be at E= 0. Thus, the bottom of the CB will be at E = Eg. To find the number of electrons in the CB, we need to integrate Z(E)F(E)dE over all the states in the CB. However, we wish to assume that only electrons close to the bottom of the band are relevant, which would be in violation of the extended range of integration. We are rescued by the fact that the Fermi energy is usually near the midpoint of the energy gap at temperatures of interest, so that relevant energies in the conduction and valence bands are far from EF. Thus, at room temperature and below, the condition E ! EF >> kBT holds for electrons in the CB and EF ! E >> kBT holds for holes in the VB. Under these conditions, the semiconductor is said to be non-degenerate. Under these conditions, the product Z(E)F(E) is only significant in the region near the band extrema because F(E) falls off exponentially with energy. The upper limit on the integral can be extended to infinity because at higher energies, F(E) ! 0. With our chosen origin for the energy, for the CB, we have 4!(2m" )3/2 Z(E) = e (E # E )1/2 . (6.1) h3 g For E ! EF >> kBT , the FD function approximates to 1 F(E) = e!(E ! EF )/kBT . (6.2) With all our approximations, we can now calculate the number of electrons in the CB: ! 4#(2m$ )3/2 ! N = Z(E)(F(E)dE = e (E % E )1/2 e%(E % EF )/kBT dE. (6.3) e " h3 " g Eg Eg Using the substitutionu = (E ! Eg ) / kBT , one find finds that the integral is converted to the " form # x1/2e! xdx, which evaluates to ! / 2. Thus, we obtain 0 !(Eg ! EF )/kBT Ne = Nce , (6.4) where 3/2 # 2!m"k T & N = 2 e B . (6.5) c $% h2 '( One can employ similar tactics to calculate the number of holes in the VB. In this case, the number of holes in the VB equals the number of electrons removed from the VB. Thus, we need the probability that an electron state is not occupied, i.e., 1 – F(E). Now, 1 e(E ! EF )/kBT 1 1! F(E) = 1! = = " e(E ! EF )/kBT . (6.6) 1+ e(E ! EF )/kBT 1+ e(E ! EF )/kBT 1+ e(EF ! E)/kBT The approximation at the end of Eq. (6.6) applies for EF ! E >> kBT. The density of states for the VB is given by 4!(2m" )3/2 Z(E) = h (#E)1/2 . (6.7) h3 Thus, the number of holes in the VB is given by 0 4$(2m% )3/2 0 N = Z(E)[1! F(E)]dE = e (!E)1/2 e(E ! EF )/kBT dE, (6.8) h # 3 # !" h !" which yields !EF /kBT Nh = Nve , (6.9) where 3/2 # 2!m"k T & N = 2 h B . (6.10) v $% h2 '( For an intrinsic semiconductor, all electrons in the CB are excited there from the VB, i.e., Ne = Nh . (intrinsic semiconductor) (6.11) Equating Eqs. (6.4) and (6.9) yields !(Eg ! EF )/kBT !EF /kBT Nce = Nve , (6.12) which can be easily solved for EF to give E 3 " m! % E = g + k T ln h . (6.13) F B $ ! ' 2 4 # me & Thus, at T = 0 K, the Fermi level is exactly in the middle of the band gap. As the temperature increases, EF may increase or decrease depending on the ratio of the effective masses. For example for Si, EF is slightly (<1% of Eg) below mid-gap at 300 K and for GaAs, it is slightly above mid-gap. 2 We should note that if we substitute for EF in the equations for Ne and Nh, we find that both quantities are proportional toe!Eg /2kBT . This is more easily seen if we calculate the product of the two quantities: !(Eg ! EF )/kBT !EF /kBT !Eg /kBT NeNh = (Nce )(Nve ) = Nc Nve . (6.14) Since Ne = Nh for intrinsic semiconductors, defining each as Ni, the intrinsic carrier density, we have !Eg /2kBT Ni = Ne = Nh = Nc Nv e . (6.15) 2. Extrinsic Semiconductors An extrinsic semiconductor is one that has impurities in it, usually deliberately introduced to give the semiconductor specific properties. Let us consider silicon as our prototypical intrinsic semiconductor. Si has 4 valence electrons (due to sp3 hybridization). In a Si crystal, each Si atom is covalently bonded to 4 other Si atoms. The bonding requirements are satisfied for all Si atoms, with each essentially having an inert-gas configuration of 8 electrons (except at the surfaces of the crystal, which we do not consider). Suppose that we introduce a pentavalent impurity atom such as phosphorus (P). [Arsenic (As) and antimony (Sb) are others.] The P impurity will give up 4 valence electrons to bonding with 4 Si atoms. This leaves a lone electron without a partner to bond with. This extra electron is rather loosely bound to the parent P atom because of the screening of the nucleus caused by the 4 additional electrons from the Si atoms that form bonds with the P atom. Hence, this electron can be excited into the conduction band with much less energy than is necessary to excite an electron from the valence band across the band gap. Pentavalent impurities are called donors because they "donate" an electron to the CB. A donor atom that has donated an electron is ionized and positively charged. One can use a hydrogen-atom model to estimate the binding energy of an electron bound to the donor atom. The screening by the core electrons and the bonding electrons means that the effective nuclear charge seen by the extra electron is +e. We use the formula for the binding energy of the H atom, with electron mass replaced by the effective mass and the permittivity of free space !0 replaced by the permittivity of the semiconducting material. (The permittivity is equal to the dielectric constant K times !0 .) So, " 4 mee Ebinding ! 2 2 . (6.16) 2(4#$) ! Using parameters for Si, Eq. (6.16) yields 0.05 eV, which is in reasonable agreement with the experimentally measured value of 0.045 eV. The energy of the bound donor electron is usually given the symbol ED. Thus, the energy necessary to ionize a donor atom and promote the electron to the CB is Eg ! ED and so 3 Eg ! ED = Ebinding . (6.17) Since Ebinding is small in comparison with the typical gap size (at least for Si, Ge, and GaAs), the donor levels are close to and just below the bottom of the CB. If the impurity atom is trivalent, such as boron (B), aluminum (Al), or indium (In), then there is one fewer electron than is required to satisfy all the bonding requirements. Hence, one can consider this to be a hole in one of the covalent bonds. An electron near the top of the valence band can easily hop into this hole if it is supplied with a small amount of energy. Thus, the hole has hopped into the top of the valence band, where it can contribute to electric current. Trivalent impurities are called acceptors because they accept electrons from the valence band. (Alternatively, one can say they donate holes to the valence band.) An acceptor atom that has accepted an electron is ionized and negatively charged. The acceptor levels are close to the top of the VB and just above it. The acceptor levels are higher than the VB states because they represent the absence of a bound electron. Real materials usually have both acceptors and donors. However, for electronics applications, extrinsic semiconductors are usually engineered with one type of impurity dominant, whether donor or acceptor. For example, to increase the number of electrons available for conduction at room temperature, Si is doped with donors. The calculation the number of electrons and the number of holes that we performed for intrinsic semiconductors remains valid for extrinsic semiconductors. To calculate the Fermi level for extrinsic semiconductors, we must take into account all sources of electrons and holes.