ECE440 Nanoelectronics. Lecture 08 Review of Solid State Physics

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1 ECE440 Nanoelectronics Lecture 08 Review of Solid State Physics

2 A Brief review of Solid State Physics Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal bonds Basic concepts of semiconductors

3 Contents Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal bonds Basic concepts of semiconductors

4 Crystalline, polycrystalline, and amorphous A crystal is a solid, where the constituent atoms are arranged in a certain periodic fashion. A solid without any periodicity is called amorphous solid. A solid where only small regions are of a single crystal material is called polycrystalline solid. (a) Crystalline (b) Amorphous (c) Polycrystalline Three types of solids: (a) ordered crystalline and (b) amorphous materials are illustrated by microscopic views of the atoms, whereas (c) polycrystalline structure is illustrated by a more macroscopic view of adjacent single-crystalline regions, each of which has a crystalline structure as in (a).

5 Crystal Structure 7 Crystal Systems 14 Bravais Lattice Triclinic 1 Monoclinic Orthorhomic 4 Tetragonal Cubic 3 (sc, fcc, bcc) Trigonal 1 Hexagonal 1 (3 point groups) (30 space groups) (i.e., 30 crystal structures) Symmetry = rotation only rotation + translation

6 SC, BCC, FCC Lattices (a) (b) (c) Three types of cubic lattices: (a) simple, (b) body centered, and (c) face centered.

7 Lattice and reciprocal lattice (,, ) (,, ) Lattice Reciprocal lattice ᴨ Ω, ᴨ Ω, ᴨ Ω with Ω.. δ.ᴨ

8 SC Lattice Basis Vectors a 3 a a 1

9 Reciprocal Lattice of Simple Cubic (S.C.),,, Ω... ᴨ Ω ᴨ ᴨ ᴨ Ω ᴨ ᴨ ᴨ Ω ᴨ ᴨ The reciprocal lattice of SC is still SC!

10 BCC Lattice Basis Vectors z a a 1 x a 3 y

11 Reciprocal Lattice of Body-Centered Cubic (B.C.C.) Ω

12 ᴨ Ω ᴨ ᴨ Ω ᴨ ᴨ ᴨ The reciprocal lattice of BCC is FCC! ᴨ Ω ᴨ ᴨ

13 FCC Lattice Basis Vectors y a 1 a 3 a x z

14 Reciprocal Lattice of Face-Centered Cubic (F.C.C.) Ω

15 ᴨ Ω ᴨ 4 4 ᴨ Ω ᴨ 4 ᴨ Ω ᴨ ᴨ ᴨ ᴨ The reciprocal lattice of FCC is BCC!

16 Review of Semiconductor Physics Lattice S.C.,, ; B.C.C. F.C.C. ᴨ, Reciprocal Lattice ᴨ ᴨ ᴨ ᴨ ᴨ ᴨ S.C. ᴨ, F.C.C. B.C.C. ᴨ

17 Diamond Structure The basic lattice structure for diamond, silicon, and germanium is the diamond structure. The diamond structure is a face centered cubic structure with an extra atom placed a distance a 1 /4 + a /4 + a 3 /4 from each of the original face centered atoms, as shown below. Thus, a diamond lattice contains twice as many atoms per unit volume as a facecentered cubic lattice. Four nearest neighbor atoms to each atom are shown by complimentary color for easier visualization. The lattice of volume V = a 3 consists of 8 atoms. Diamond structure is of facecentered type of cubic lattices. The tetrahedral bonding arrangement of neighboring atoms is clear. a

18 Besides the translations, the crystal symmetry contains other symmetry elements, for example, specific rotations around high symmetry axes. In cubic crystals, axes directed along the basis vectors are equivalent and they are the symmetry axes. Rotation symmetry in cubic crystals. Rotation Symmetry C 3 C 3 z C 4 C 3 C 3 C 4 y C 4 x

19 Translational Symmetry Translational symmetry is the property of the crystal to be carried" into itself under parallel translation in certain directions and certain distances. For any three dimensional lattice it is possible to define three fundamental noncomplanar primitive translation vectors (basis vectors) a 1, a, and a 3, such that the position of any lattice site can be defined by the vector R = n 1 a 1 + n a + n 3 a 3,wheren 1, n, and n 3 are arbitrary integers. If we construct the parallelepiped using the basis vectors, a 1, a,anda 3, we obtain just the primitive cell. Two-dimensional lattice. Three unit cells are illustrated by A, B, and C. Two basis vectors are illustrated by a 1 and a. B a a C a 1 a a 1 A a 1

20 Crystal Structure 7 Crystal Systems 14 Bravais Lattice Triclinic 1 Monoclinic Orthorhomic 4 Tetragonal Cubic 3 (sc, fcc, bcc) Trigonal 1 Hexagonal 1 (3 point groups) (30 space groups) (i.e., 30 crystal structures) Symmetry = rotation only rotation + translation Si, Ge = Diamond GaAs = Zinc blende structures GaN = Wurtzite

21 Crystal structure = Bravais lattice + Basis = + Crystal Lattice Basis

22 Contents Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal bonds Basic concepts of semiconductors

23 Crystal directions and planes z [001] [010] 0 [010] y x [001] Crystal directions in cubic crystals.

24 Miller Index Planes: A plane, All equivalent planes Direction: A direction, All equivalent directions Brillouin Zone Wigner Seitz Cell Constructed by drawing perpendicular bisector planes in the reciprocal lattice from the chosen center to the nearest equivalent reciprocal lattice sites.

25 Contents Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal bonds Basic concepts of semiconductors

26 E 1 E 1 E 1 E 1 Large number of atoms E 1 Energy band Formation of energy band. Ground energy level E 1 of a single atom evolves into energy band when large number of single atoms are interacting with each other.

27 Energy levels Electrons can only have discrete values of energy Electrons in the outermost shell, called Valence Electrons. With large number of atoms in solid, energy levels form bands Important bands are the valence band, conduction band and energy gap

28 Energy level E 3 3 Energy level E Energy level E 1 1 Overlap Energy bands E 1, E, and E 3 Schematics of energy bands 1,, and 3 that correspond to single atom energy levels E 1, E, and E 3, respectively.

29 Case of dielectric: filled valence band and empty conduction band; E g > 5 ev. Case of a semiconductor: filled valence band and empty conduction band (at low temperature); E g <5eV.

30 Bandgap of Insulator, Semiconductor, and Conductor

31 Contents Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal bonds Basic concepts of semiconductors

32 Direct bandgap and indirect bandgap semiconductors Light wavelengths λ are much greater than electron de Broglie wavelengths and the photon wavevectors q =π / λ, in turn, are much smaller than the electron wavevectors ( k 1, k >> q ). Under light absorption and emission the electrons transferred between the valence and conduction bands practically preserve their wavevectors. k1 k k and Ec( k) Ev( k) 3 Si 3 GaAs E (ev) 1 E g E (ev) 1 E g -1-1 L [111] Γ [100] X L [111] Γ [100] X

33 Direct vs. Indirect Bandgap Semiconductors Figure 1 Energy band diagrams and major carrier transition processes in InP and silicon crystals. In a direct band structure (such as InP, left), electron-hole recombination almost always results in photon emission, whereas in an indirect band structure (such as Si, right), free-carrier absorption, Auger recombination and indirect recombination exist simultaneously, resulting in little photon emission. D. Liang and J. E. Bowers, Nature Photonics 4, 511 (010) Internal quantum efficiency η i (defined as the ratio of the probability that an excited e-h pair recombine radiatively and the probability of e-h pair combination.) i ( rad ( rad ) 1 ) ( 1 nonrad ) 1 rad nonrad nonrad Figure Simplified energy diagram of Si. The most relevant transitions are shown. Black: absorption of a photon via a phonon-assisted indirect transition; red: emission of a photon via a phonon-assisted indirect transition; orange: free carrier absorption; green: Auger recombination process. L. Pavesi, Routes toward silicon-based lasers. Materials Today 8, 18 (005)

34 Contents Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal bonds Basic concepts of semiconductors

35 Ionic crystals Ionic crystals are made up of positive and negative ions. The ionic bond results primarily from attractive electrostatic interaction of neighboring ions with opposite charges. There is also a repulsive interaction with other neighbors of the same charge. Attraction and repulsion together result in a balancing of forces that leads to the atoms being in stable equilibrium positions in such an ionic crystal. As for the electronic configuration in a crystal, it corresponds to a closed (completely filled) outer electronic shell. A good example of an ionic crystal is NaCl (salt). Neutral sodium, Na, and chlorine, Cl, atoms have the configurations Na 11 (1s s p 6 3s 1 ) and Cl 17 (1s s p 6 3s 3p 5 ), respectively. That is, the Na atom has only one valence electron, while one electron is necessary to complete the shell in the Cl atom. It turns out that the stable electronic configuration develops when the Na atom gives one valence electron to the Cl atom. Both of them become ions with opposite charges and the pair has the closed outer shell configuration (like inert gases such as helium, He, andneon,ne). The inner shells are, of course, completely filled both before and after binding of the two atoms. In general, for all elements with almost closed shells, there is a tendency to form ionic bonds and ionic crystals. These crystals are usually dielectrics (insulators).

36 Covalent crystals Covalent bonding is typical for atoms with a low level of outer shell filling up. An excellent example is provided by a Si crystal. As we discussed in the previous Chapter, the electron configuration of Si can be presented as: core + 3s 3p.Tocompletetheouter3s 3p shell, a silicon atom in crystal forms four bonds with four other neighboring silicon atoms. The central Si atom and each of its nearest neighbor Si atoms share two electrons. This provides socalled covalent (chemical) bonds (symmetric combinations of the sp 3 orbitals) in the Si crystal. The four bonding orbitals form an energy band completely filled by the valence electrons. This band is called the valence band. Si Si Si Si Si Fig Four sp 3 -hybrid bonding orbitals in Si crystal.

37 Binding energies for different types of crystals. Type of crystal coupling Crystal Energy per atom (ev) Ionic NaCl LiF Covalent Diamond, C Si Ge Metallic Molecular and Inert gas crystals Na Fe Al CH 4 Ar The ionic and covalent crystals typically have binding energies in the range of 1 10 ev, while molecular and inert gas crystals are weakly coupled systems. Metals have intermediate coupling. The binding energy of a crystal is an important parameter, since it determines the stability of the crystal, its aging time, applicability of different treatment processes, etc

38 Contents Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal bonds Basic concepts of semiconductors

39 Semiconductor materials

40 Covalence Bonds Atoms of solid materials form crystals, which are 3D structures held together by strong bonds Between atoms covalence Bonds like Silicon. Silicon forms a covalent crystal. The shared electrons are not mobile. Therefore there is a energy gap between the valence band and the conduction band.

41 Holes An electron hole is the conceptual and mathematical opposite of an electron Holes do not travel like electrons Only electrons can move from atom to atom

42 Doping in Silicon Pure silicon wafers are called intrinsic silicon Intrinsic silicon doesn t have enough free electrons for current conduction. So impurities are introduced to increase conductivity The introduction of these impurities is called doping

43 Donors and Acceptors Donors are dopant atoms that added to a semiconductor to provide electrons Acceptors are dopant atoms that provide holes

44 Type of doping N (negative) type: doped with donors and has extra free electrons. P (positive) type: doped with acceptors and has extra holes.

45 Electronic properties of doped silicon qualitative picture.

46 Methods for doping Diffusion impurities move by a difference in concentration gradients Conducted at very high temperatures Gas sources are the most common but liquids and solids are also used The sources react with silicon to form dopant oxide which then diffuses into the rest of the substrate by the increase of temperature

47 Methods for doping Ion implantation: Alternative to high temperature diffusion A beam of highly energetic dopant ions is aimed at the semiconductor surface Collision with ion distorts the crystal structure Annealing has to be performed to correct the damage

48 Electronic properties: Silicon in general. E F E C E V E G = 1.1 ev Boltzman constant: k = ev/k Fundamental materials property: n N c e ( E E c F )/ kt Where n = concentration of negative (electron) carriers (typically in cm -3 ) E c is the energy level of the conduction band E F is the Fermi level. N c is the intrinsic density of states in the conduction band (cm -3 ). Similarly, p N V e ( E E F V )/ kt Where p = concentration of positive (hole) carriers (typically in cm -3 ) E V is the energy level of the valence band N V is the intrinsic density of states in the valence band (cm -3 ).

49 Electronic properties: intrinsic (undoped) silicon. Density of states in conduction band, N C (cm -3 ) 3.E+19 Density of states in valence band, N V (cm -3 ) 1.83E19 Note: without doping, n = p n i where n i is the intrinsic carrier concentration. For pure silicon, then n i N c N V exp( E Thus n i = 9.6 x 10 9 cm -3 G / kt) Similarly the Fermi level for the intrinsic silicon is, E i E V ( EC EV ) / (1/ ) kt ln( NV / NC ) Where we have used E i to indicate intrinsic Fermi level for Si.

50 Energy Bands Bloch function ɸ ħ, Valance band, Conduction band, Bandgap Insulator, Semiconductor, Conductor Direct bandgap vs. Indirect bandgap 1.4 ~ Effective mass, 1 1 ħ

51 Carrier Concentration Donor Acceptor n-type p-type Fermi level exp exp when, ᴨ ħ ᴨ ħ ln exp exp