Chapter 1 Review of Basic Semiconductor Physics

1 Semiconductor Optoelectronics (Farhan Rana, Cornell University) Chapter 1 Review of Basic Semiconductor Physics semiconductors This Review is not meant to teach you Semiconductor Physics only to refresh your memory. Most semiconductors are formed from elements from groups II, III, VI, V, VI of the periodic table. The most commonly used Semiconductor is silicon or Si. In a Si crystal each Si atom forms a covalent bond with 4 other Si atoms. Si has 4 electrons in its valence (or outer most shell) and therefore it can bond with 4 other Si atoms. A cartoon depiction of Si crystal is then as shown in the Figure. A silicon lattice (diamond lattice) Semiconductor Optoelectronics (Farhan Rana, Cornell University) In a Si crystal each Si atom bonds with 4 other Si atoms in a tetrahedral geometry, as shown.

2 This structure is called a diamond Lattice (since diamond crystals consisting of C atoms also have the same structure). The diamond lattice is essentially an FCC lattice (face centered cubic) with a single-atom basis. The lattice constant a is also shown in the figure. Note that a is not the actual distance between the nearest Si atoms. a is the length of one side of the diamond unit cell (not the wigner-seitz cell) that has the cubic symmetry. semiconductors are also formed by combining elements from group III and group V of the periodic table. This is possible since group III elements have 3 electrons in their outer most shell and group V elements have 5 electrons in their outermost shell.

3 So a III-V covalent bond is possible. Most common III-V semiconductors are GaAs and PIn. Each Ga atom is surrounded by 4 Asatoms and each As atom is surrounded by 4 Ga atoms in a tetrahederal geometry. GaAslattice is an example of zinc blende lattice . The difference between zinc blende and diamond lattices is that in diamond lattice all atoms are the same. InP also has a zinc blende lattice. GaAs and InP are examples of compound semiconductors . Si, C, and Ge are examples of elemental semiconductors . Not all compound semiconductors have the zinc blende lattice. For example, III-Nitrides ( GaN, AIN,InN) can also have the wurtzite lattice structure show below. A GaAs lattice (zincblende lattice)A GaN lattice(wurtzite lattice) Semiconductor Optoelectronics (Farhan Rana, Cornell University) Just like the zinc blende lattice is a FCClattice with a single-atom basis, the wurtzite lattice is a HCPlattice (hexagonal close packed) with a single-atom basis.

4 For ideal HCPlattice 38 ac. There is one thing common in zinc blende and wurtzite lattices; both have tetrahederal coordination. Group II elements and group VI elements also combine to give II-VI compound Semiconductor like ZnSe, CdTe, CdSe,ZnO etc. Most of these have zine blende or wurtzite lattices (but some do have rock salt lattic structures). Most of the IV VI semiconductors ( PbS, PbSe,PbTe) called lead salts have the rock salt structure (similar to a NaCl crystal). Semiconductor Bandstructure In a solid the electronic energy levels are obtained by solving the Schrodinger Equation rErrVm 222 (1) where rV is the periodic potential from the atoms sitting on the lattice sites.

5 The solutions of (1) can be written as, ruerrknrkikn,, The eigenfunctions rkn , are called Bloch functions and satisfy, rkErrVmknnkn,,22 2 The vector k can take values belonging to the first Brillouin zone (FBZ) and ''n takes integral values. Therefore, all the possible energy levels of the solid can be labeled by the set of values kn,. If one plots kEn as a function of k for different integral values of n one obtains the bandstructure of the solid. The first Brillouin zone (FBZ) corresponding to a FCC lattice (or a diamond or a zinc blende lattice) is shown below. FBZ of an FCC lattice Semiconductor Optoelectronics (Farhan Rana, Cornell University) The bandstructures of Si, Ge, and GaAs are shown below.

6 Only few chosen bands are shown. A particular feature of all Semiconductor is that electrons in semiconductors fill all the low lying energy bands (called the valence bands). There are four valence bands, but only the highest three are shown in the figure. The highest energy in the valence bands is denoted by vE. In pure semiconductors the conduction bands are all empty on electrons. The lowest energy in the conduction bands is denoted by cE. There are also four conduction bands and all four are shown in the figure. The difference gvcEEE is called the band gap of the Semiconductor . Near the bottom of the lowest conduction band and the top of the highest valence band one may Taylor expand the energy kEn.

7 Assuming isotropic parabolic bands, conduction band dispersion near the band bottom can be written as, cceccKkKkmEkE 22 And for the valence band one can write, vvhvvKkKkmEkE 22 where em and hm are electron and hole effective masses and the vectors cK and vK are the locations in k-space of conduction band minimum and valence band maximum. 0 vK for all semiconductors that we will consider. 0 cK for most III-V and II-VI semiconductors . semiconductors for which vcKK are called direct gap or just direct ( GaAs, InP, GaN, ZnSe, CdSe, ZnO). semiconductors for which vcKK are called indirect gap or just indirect ( Si, Ge, C, SiC, GaP, AlAs). As we will see later in the course, all optically active semiconductors are direct gap.

8 When 0 vcKK, and assuming isotropic parabolic bands, Conduction band: eccmkEkE222 Silicon bandstructureGermanium bandstructure GaAs bandstructureSemiconductor Optoelectronics (Farhan Rana, Cornell University) Valence band: hvvmkEkE222 More generally, one can write for the conductions band with minimum at cK (assuming parabolic bands), cecccKkMKkEkE 122 where eM is the effective mass matrix, zzzyzxyzyyyxxzxyxxemmmmmmmmmM Physical considerations demand that eM be symmetric. Similarly, for the valence band one get, vhvvvKkMKkEkE 122 counting Electronic States in semiconductors In a solid of volume V, the number of energy levels is one band in volume kd3 of the FBZ is 3322 kdV (The multiplies 2 accounts for the two spin states).

9 So all summations of the from FBZk where values of k are restricted to the FBZ can be replaced by the integral, FBZ332 kdVk The number of energy levels per band in a crystal of volume Vis given by, FBZ of volume 22222333 FBZ VkdVFBZk But, cell unit primitive the of volume2 FBZ of volume3 The number of energy levels per band in a crystal of volume Vis then given by, crystal the in cells primitive of number 2cell primitive the of volume2 V Therefore each primitive cell contributes two states or energy levels to each band. Linear Combination of Atomic Orbitals Approach to Energy Bands Semiconductor Optoelectronics (Farhan Rana, Cornell University) Linear combination of atomic orbitals is another way to understand energy band formation in semiconductors .

10 In semiconductors , the atomic states of the outermost shell ( the single 3s and the three 3p in a Si atom, and the single 4s and the three 4p in a Ga atom and the same in a Asatom) combine or hybridize with the states of the neighboring atoms to result in the four valence bands and the four conditions bands, as shown in the figure below. In this hybridization process the total number of energy levels of all the atoms is conserved. Suppose N Si atoms from a crystal then the total number of energy levels before hybridization is N42 . Now lets find the total number of energy levels in the resulting crystal. As found earlier, the number of energy levels per band in a crystal of volume Vis then given by, crystal the in cells primitive of number 2cell primitive the of volume2 V So the total number of energy levels in eight bands (four conduction bands and four valence bands) is, bandsof#crystal the incells primitive of #2 Since each Si primitive cell has two Si atoms (diamond lattice is an FCC lattice with a two-atom basis) we get, NN4822bandsof#crystal the incells primitive of #2 And we get the same answer as before.