1. SOLID STATE PHYSICS FOR NANOTECHNOLOGY I EEE5425 Introduction to Nanotechnology1

2. 2 Structural Compositions of Solids Periodic with long range order Short range orderSmall crystalline regions called grains. May occur during thin film growth. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

3. 3 MOSFET on Crystalline Si © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

4. 4 Crystalline Structures Typical feature of crystalline solid is periodicity, which lead to long- range-order Crystal structure= Lattice + basis The whole crystal may be generated by repetition of unit cell Size(s) of unit cell = lattice parameter © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

5. 5 Lattice and Base Lattice [lat-is] : A regular, periodic configuration of points, throughout an area or a space, especially in a crystalline solid. Base: The same group of atoms positioned around each and every lattice point of a crystal. The crystal consists of an atomic basis (or atomic cluster) attached to the lattice points. The “basis” can be a single atom or a group of atoms attached to each lattice point. Each lattice point receives an identical basis (or cluster). The (infinite) crystal consists of the collection of these regularly arranged clusters. Crystal structure= Lattice + basis © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

6. 6 2D Lattice Translation of a unit cell by vector r = 3a + 2b © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

7. 7 Primitive cell vs Unit cell Primitive cell: a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2D, 3D, or other dimensions. A lattice can be characterized by the geometry of its primitive cell. Unit cell: small regions of space that, when duplicated, can be translated to fill the entire volume of the crystal. Although there are an infinite number of ways to specify a unit cell, for each crystal structure there is a conventional unit cell, which is chosen to display the full symmetry of the crystal. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible volume one can construct with the arrangement of atoms in the crystal such that, when stacked, completely fills the space. This primitive unit cell does not always display all the symmetries inherent in the crystal. In a unit cell each atom has an identical environment when stacked in 3 dimensional space. In a primitive cell, each atom may not have the same environment. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

8. 8 Bravais Lattices August Bravais ( 1811 – 1863), a French physicist, was the first to count the categories different types of lattices correctly. The 14 Bravais lattices are arrived at by combining one of the seven crystal systems (or axial systems) with one of the six lattice centerings. From the point of view of symmetry, there are fourteen different kinds of Bravais lattices. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

9. 9 Bravais Lattices The six lattice centerings are: Primitive centering (P): lattice points on the cell corners only Body centered (I): one additional lattice point at the center of the cell Face centered (F): one additional lattice point at center of each of the faces of the cell Centered on a single face (A, B or C centering): one additional lattice point at the center of one of the cell faces. The 14 Bravais lattices are arrived at by combining one of the seven crystal systems (or axial systems) with one of the six lattice centerings. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

10. 10 Bravais Lattices The seven crystal systems are: Cubic Tetragonal Orthorhombic Monoclinic Triclinic Trigonal Hexagonal Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total 7 × 6 = 42 combinations, but it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centered lattices can be described either by a C- or P-centering. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

11. 11 Bravais Lattices © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

12. 12 Lattice Constant Lattice constant –noun Crystallography. a measure of length that defines the size of the unit cell of a crystal lattice. Also called lattice parameter. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

13. 13 Diamond Lattice The diamond lattice is composed of two interpenetrating fcc lattices: one displaced 1/4 of a lattice constant in each direction from the other. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

14. 14 Diamond Lattice of Si In the basic unit of a crystalline silicon solid, a silicon atom shares each of its four valence electrons with each of four neighboring atoms. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

15. 15 Zincblende Lattice The zincblende lattice represents the crystal structure of zincblende (ZnS), gallium arsenide (GaAs), indium phosphide (InP), cubic silicon carbide and cubic gallium nitride. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

16. 16 Miller Indices Crystal planes and directions are important to determine the physical and electrical properties. The notation system is a set of three integers and called Miller indices. Miller Indices (h k l) are used to identify planes of atoms within a crystal structure © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

17. 17 Miller Indices 1.Find the intercepts of the plane with the crystal axes and express these intercepts as integral multiples of the basis vectors: (1,2,3) 2. Invert the three integers found in step 1: (1/1, 1/2, 1/3) 3. Using appropriate multiplier, convert the values found on step 2 to the smallest possible set of whole numbers h, k, and l: with 6 as multiplier we have (6,3,2) 4. Label the plane (hkl): (6 3 2) If an intercept is negative the minus sign placed above Miller index (h k l) If a plane passes through the origin, it can be translated to a parallel position. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

18. 18 Miller Indices Step1: ( 2, 4, 1) Step2: (1/2, 1/4, 1/1) Step3: 4 x (1/2, 1/4, 1/1) Step4: (2 1 4) © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

19. 19 Miller Indices (001) (111)(221) ? ? ? © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

20. 20 Miller Indices © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

21. 21 Miller Indices Equivalence of the cube faces ( {100} planes ) by rotation of the unit cell within the cubic lattice. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

22. 22 Miller Indices Crystal directions in the cubic lattice. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

23. 23 Miller Indices Visualization and Miller indices of common crystalline directions © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

24. 24 Miller Indices ConventionInterpretation ( hkl )Crystal plane { hkl }Set of equivalent planes [ hkl ]Crystal direction Set of equivalent directions © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

25. 25 Comments on Crystal Planes mechanical, electrical, chemical and optical properties depend on crystal orientation. crystals can be cleaved along atomic planes  very flat, smooth surfaces mirror facets for semiconductor lasers wet chemical etchants etch along crystal planes © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

26. 26 Orientation-dependent Etching KOH Si +OH - + 2H 2 O → SiO 2 (OH) 2 2- + 2H 2 (g) Etch rate : {110} > {100} >> {111} Etch rate of Si in KOH depending on crystallographic plane © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

27. 27 Orientation-Dependent Mobility Hole and electron mobility in CMOS devices fabricated on (100)-, (111)-, and (110)- orientated silicon substrates for HfO (lines) and oxynitride (dots) gate dielectrics. Nominal current flow on (100), (111), and (110) surfaces are in [110], [112] and [110] directions, respectively (solid lines and open dots). Yang et al. IEEE Elect Dev Lett. 24 331 (2003) © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

28. Electrons in Periodic Potential © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology28 When we wan to examine the properties of an electron in a periodic lattice, we need to consider Schrödinger's; equation such that the potential energy term V(r) reflects the fact that the electron sees a periodic potential. In a one dimensional lattice, ionized atoms form a periodic potential distribution for electrons due to Coulomb potential. x U(x) Schrodinger’s equation can be written as where the potential energy term is periodic V(r)=V(r+T) and where T is the crystal translation vector.

29. Bloch’s Theorem © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology29 Bloch’s theorem applies to waves in periodic structures in general. In a periodic potential distribution, wave function solutions of Schrodinger’s equation can be written as the product of a plane wave and a periodic function. where k is the wavevector to be determined (called the Bloch wavevector) and where u is periodic Thus It is important to note that the Bloch theorem shows that the electronic can propagate through a perfect periodic medium without scattering(i.e. without hitting the atoms).

30. Kronig – Penny Model of Band Structure -1 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology30 As an approximate model for one dimensional crystal can be given as where a=a1+a2 is the period of the lattice. This is known as Kronig-Penny model. V=V 0 V=0 -a 2 a1a1 0 a In the region -a 2 ≤ x ≤ 0, the potential V=V 0, and the solution of Schrödinger's equation is where In the region 0 ≤ x ≤ a 1, the potential V=0, and the solution of Schrödinger's equation is where

31. Kronig – Penny Model of Band Structure -2 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology31 Using Bloch theoremand therefore Using these relationships the wavefunction in the period a 1 ≤ x ≤ a 1 +a can be written as Enforcing the continuity of Ψ and Ψ’ at x=0 and x=a1 leads to the eignevalue equation, if 0 < E < V 0 and if E > V 0

32. Kronig – Penny Model of Band Structure -3 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology32 In the preceding equations, the energy E is the only unknown parameter. For a solution to exist, we must have Therefore the right hand side of the last two equations denoted as r(E) must obey this condition. A typical plot of r(E) vs E is as follows: The figure makes it that there are certain allowed values of energy, called allowed energy bands and certain unallowed values of energy, called band gaps. That is if E is in an allowed energy band, Schrodinger’s equation has solution, and if E is in an allowed energy band there is no solution. Within an allowed band, energy can take any value (i.e. it is not discretized). Note that as energy increases, the allowed energy bands increase in width, and so the forbidden bands decrease in width.

33. Kronig – Penny Model of Band Structure -4 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology33 Since we can generate an important figure called dispersion diagram, which is a diagram of energy versus wavenumber (E vs. k). To generate the dispersion diagram, start at E=0 and compute r(E). If |r(E)|=|cos ka|>1, we are at a forbidden energy (i.e. in a bandgap) and we need to increase E a bit and try again. If |r(E)|≤ 1, we are at an allowed energy (i.e. in an energy band) and in this case the corresponding wavenumber is Since cosine is an even function –k will also be a solution. By increasing E by a small amount and checking the value of r(E), we can generate the plot of allowed and unallowed energy bands. One form of the result will look like as follows. This depiction is known as the extended zone scheme.

34. Kronig – Penny Model of Band Structure -5 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology34 The various sections of wavenumber space are divided into what are called Brillouin zones with the range Denoting the important first Brillouin zone. The second Brillouin zone is the range and so on for higher zones.

35. Kronig – Penny Model of Band Structure -6 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology35 Another depiction arises from noting that the energy bands in the higher Brillouin zones can be all translated to the first Brillouin zone by shifts of n2π/a. This results in what is called reduced zone scheme as shown below. This is the most common format for describing the band structure.

36. Effective Mass -1 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology36 An electron in empty space has a well defined constant mass. However, sometimes, it is useful to view mass merely a proportionality constant between force and acceleration, remembering the Newton’s second law F=ma. This is particularly appropriate when studying electrons in crystals where they appear to ac as if their mass is different from the free space value. Quantum mechanically an electron can be represented by e wave packet with the electron velocity being its group velocity: The influence of electron’s environment is contained in the energy relation E(k). For an electron in free space: However determining E(k) can not be easy for realistic crystals.

37. Effective Mass -2 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology37 On the other hand, complete band structure of material does not need to be known, since only electrons in certain regions of band are of interest. We typically are more interested in the behavior of electrons near the band edge (conduction band or valence band) since they contribute to the conduction most. In this case only the local behavior of E-k curve is important.

38. Effective Mass -3 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology38 From Newton’s law f motion And from the group velocity definition Hence In a real three dimensional crystal, effective mass is

39. Effective Mass -4 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology39 Mathematically, the effective mass is inversely proportional to the curvature of an E vs k plot. m* is positive near the bottoms of all bands. m* is negative near the tops of all bands. A negative effective mass simply means that, in response to an applied force, the electron will accelerate in a direction opposite to that expected from purely classical considerations. The negative effective mass helps to describe, conceptually and mathematically, charge transfer (and therefore conduction) in a partially filled band. We define an empty state left behind an electron as a positive charge and a positive effective mass and call it hole.

40. 40 Effective Mass –5 In any calculation involving the mass of the charge carriers, we must use effective mass values for the particular material involved. m n *–the electron effective mass m p *–the hole effective mass The n subscript indicates the electron as a negative charge carrier, and the p subscript indicates the hole as a positive charge carrier. Materialm n */m 0 m p */m 0 Si1.180.81 Ge0.550.36 GaAs0.0660.52 m 0 –mass of electron in rest m 0 = 9.10938188×10 -31 kg © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

41. Bloch Oscillations -1 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology41 Remembering the group velocity we can calculate and plot E vs k and E vs v g When a DC bias is applied, electron will accelerate and its k will increase towards the value k=π/2a. When a DC bias is applied, the electron will accelerate and its k will increase towards the value k=π/2a. At k=π/2a there is an inflection point at E-k curve and electron velocity reaches to maximum. With further increase of k, electron decelerates and finally as k reaches the Brillouin zone boundary at π/a. Electron velocity goes to 0 indicating that the electron wavefunction is a standing wave rather than travelling wave.

42. Bloch Oscillations -2 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology42 As a result, when we apply DC field the electron, due to the band structure, oscillates back and forth. These are called Bloch Oscillations. In practice this behavior is not seen. Electrons are scattered before they reach to the amplitude of the Bloch oscillations. Bloch oscillations, however, can be observed in superlattices where band structure can be precisely controlled.

43. Band Theory of Solids -1 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology43 We concluded that, due to the periodic potential associated with the crystalline lattice, there are allowed and disallowed energy bands. Let us look at how carrier transport is affected if a band is filled with electrons or not. 1.If an allowed band is completely empty of electrons, obviously there are no electrons in the band to participate in electrical conduction. This can happen, for example, in a high-energy band where the energies of the band are above the energies of the systems electrons. 2. Similarly, and surprisingly, if an allowed band is completely filled with electrons, those electrons can not contributive to electric conduction either. 3.Only electrons in a partially filled energy band can contribute to conduction.

44. Band Theory of Solids -2 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology44 There is another way to view band structures that is particularly helpful in understanding how two systems interact when brought together. It turns out that if a quantum system has energy levels E 1, E 2, E 3, then if two such identical systems (e.g. two atoms) are brought together, it can be shown that each energy level ill split into two levels. Where E n ± is an energy level slightly above or below the energy value E n of the isolated system. E2E2 E1E1 System 1 E ~ E1E1 System 2 E2E2 x E2+E2+ E2-E2- System 1 E ~ E1E1 System 2 E2E2 x E1+E1+ E1-E1-

45. Band Theory of Solids -3 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology45 The splitting is due to the overlap of each system's wavefunctions (orbitals). For example, in the case of two atoms that can come together to form a molecule, the atomic orbitals associated with each atom begin to overlap as atoms are brought together. This can be seen by considering a simplified liner model of forming a lithium (Li) molecule. Lithium has the electronic configuration 1s 2 2s 1 and in forming the molecule Li 2 the s shell atomic orbitals form antibonding and bonding molecular orbitals as depicted. In the ground configuration the bonding molecular state is filled with two 2s 1 electrons (one from each atom) and the antibonding state is empty. If N identical atoms are brought together, each energy level of an isolated atom, E 1, E 2, E 3 will split into N levels forming quasi-continous bands.

46. 46 Band Theory of Solids -4 When isolated atoms are brought together to form a solid, various interactions occur between neighboring atoms. The forces of attraction and repulsion between atoms will find a balance at the proper interatomic spacing for the crystal. In this process, important changes occur in the electron energy level configurations, and these changes result in the varied electrical properties of solids. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

47. 47 Band Theory of Solids -5 when two atoms are completely isolated from each other, they can have identical electronic structures as the spacing between the two atoms becomes smaller, electron wave functions begin to overlap. The Pauli exclusion principle dictates that no two electrons in a given interacting system may have the same quantum state; thus there must be a splitting of the discrete energy levels of the isolated atoms into new levels belonging to the pair rather than to individual atoms. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

48. 48 Band Theory of Solids -6 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

49. 49 Band Theory of Solids -7 Linear combinations of atomic orbitals (LCAO): The LCO when 2 atoms are brought together leads to 2 distinct “normal modes – a higher energy antibonding orbital and a lower energy bonding orbital. Note that the electron probability density is high in the region between the ion cores (covalent “bond”), leading to lowering of the bonding energy level and the cohesion of the crystal. If instead of 2 atoms, one brings together N atoms, there will be N distinct LCAO, and N closely spaced energy levels in a band. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

50. 50 Band Theory of Solids -8 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

51. 51 Band Theory of Solids -9 Energy levels in Si as a function of atomic spacing. The core levels (n=1,2) in Si are completely filled with electrons. At the actual atomic spacing of the crystal, the 2N electrons in the 3s subshell and the 2N electrons in the 3p subshell undergo sp3 hybridization, and all end up in the lower 4N states (valence band) while the higher lying 4N states (conduction band) are empty separated by a band gap. 1s 2 ⇨ 2N states2p 6 ⇨ 6N states 3p 6 ⇨ 6N states 2s 2 ⇨ 2N states3s 2 ⇨ 2N states © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

52. 52 Band Theory of Solids -10 Electrons must occupy different energies due to Pauli Exclusion principle. Two atoms Six atoms Solid of N atoms © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

53. 53 Intrinsic Materials –1 A perfect semiconductor crystal with no impurities or lattice defects is called an intrinsic semiconductor. In such material there are no free charge carriers at T= 0 K, since the valence band is filled with electrons and the conduction band is empty. At higher temperatures electron-hole pairs are generated as valence band electrons are excited thermally across the band gap to the conduction band. These EHPs are the only charge carriers in intrinsic material. If one of the Si valence electrons is broken away from its position in the bonding structure such that it becomes free to move about in the lattice, a conduction electron is created and a broken bond (hole) is left behind. The energy required to break the bond is the band gap energy E g. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

54. 54 Intrinsic Materials –2 Since the electrons and holes are created in pairs, the conduction band electron concentration n (electrons per cm 3 ) is equal to the concentration of holes in the valence band p (holes per cm 3 ). Thus for intrinsic material n= p= n i where n i is concentration of EHPs in intrinsic material or intrinsic concentration. n i depends on temperature (!) Obviously, if a steady state carrier concentration is maintained, there must be recombination of EHPs at the same rate at which they are generated. Recombination occurs when an electron in the conduction band makes a transition (direct or indirect) to an empty state (hole) in the valence band, thus annihilating the pair. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

55. 55 Intrinsic Materials –3 Band diagram for intrinsic semiconductor © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

56. 56 Extrinsic Materials –1 In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductors by purposely (controllably) introducing impurities into the crystal. This process is called doping. By doping, a crystal can be altered so that it has a predominance of either electrons or holes. There are two types of doped semiconductors: p-type (mostly holes) n-type (mostly electrons) © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

57. 57 Extrinsic Materials –2 When a crystal is doped such that the equilibrium carrier concentrations n 0 and p 0 are different from the intrinsic carrier concentration n i, the material is said to be extrinsic. Donors Dopants increasing electron concentration Acceptors Dopants increasing electron concentration P (phosphorus)B (Boron) As (Arsenic)Ga (Gallium) Sb (Antimony)In (Indium) Al (Aluminum) © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

58. 58 Extrinsic Materials –3 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

59. 59 Extrinsic Materials –4 When impurities or lattice defects are introduced into an otherwise perfect crystal, additional levels are created in the energy band structure, usually within the band gap. n-type semiconductors An impurity from V-column of the periodic table (P, As, and Sb) introduces an energy level very near the conduction band in Si or Ge. This level is filled with electrons at T= 0 K, and very little thermal energy is required to excite these electrons to the conduction band. At T about 50–100 K virtually all of the electrons in the impurity level are "donated" to the conduction band. Such an impurity level is called a donor level. Thus semiconductors doped with a significant number of donor atoms will have n 0 >> n i or p 0 at room temperature. This is n-type material. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

60. 60 Extrinsic Materials –5 Semiconductor Si (Z= 14): 1s 2 2s 2 2p 6 3s 2 3p 2 Dopant (donor) As (Z= 33): 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 3 n-type © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

61. 61 Extrinsic Materials –6 Band diagram for n-type semiconductor © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

62. 62 Extrinsic Materials –7 Atoms from III-column (B, Al, Ga, and In) introduce impurity levels in Ge or Si near the valence band. These levels are empty of electrons at 0 K. At low temperatures, enough thermal energy is available to excite electrons from the valence band into the impurity level, leaving behind holes in the valence band. Since this type of impurity level "accepts" electrons from the valence band, it is called an acceptor level, and the column III impurities are acceptor impurities in Ge and Si. Doping with acceptor impurities can create a semiconductor with a hole concentration p 0 much greater than the conduction band electron concentration n 0 or n i ( this type is p- type material) © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

63. 63 Extrinsic Materials –8 Semiconductor Si (Z= 14): 1s 2 2s 2 2p 6 3s 2 3p 2 Dopant (acceptor) B (Z= 5): 1s 2 2s 2 2p 1 p-type © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

64. 64 Extrinsic Materials –9 Band diagram for p-type semiconductor © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

65. 65 Extrinsic Materials –10 Example: Calculate the approximate energy required to excite the extra electron of As donor atom into the conduction band of Si (the donor binding energy). Solution: Let’s assume for rough calculations that the As (1s 2 2s 2 2p 2 3s 2 3p 6 3d 10 4s 2 4p 3 ) atom has its four covalent bonding electrons rather tightly bound and the fifth “extra” electron loosely bound to the atom. We can approximate this situation by using the Bohr model results, considering the loosely bound electron as ranging about the tightly bound "core" electrons in a hydrogen-like orbit. We have to find energy necessary to remove that “extra” electron from As atom. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

66. 66 Extrinsic Materials –11 We can approximate As dopant atom in Si lattice by using the Bohr model: the loosely bound “fifth” electron is ranging about the tightly bound "core" electrons in a hydrogen-like orbit. The magnitude of the ground-state energy (n= 1) of such an electron in the Bohr model is Constant K in this case is K=4πε r ε 0 where ε r is relative dielectric constant of Si Approximation of As dopant atom in Si lattice. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

67. 67 Extrinsic Materials –12 Besides relative dielectric constant of Si, we have to use conductivity effective mass of electron m n * in Si in the formula for energy E: © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

68. 68 Extrinsic Materials –13 Generally, the column-V donor levels lie approximately 0.01 eV below the conduction band in Ge, and the column-III acceptor levels lie about 0.01 eV above the valence band. In Si the usual donor and acceptor levels lie about 0.03-0.06 eV from a band edge. When a semiconductor is doped n-type or p-type, one type of carrier dominates. For example, when we introduce donors, the number of electrons in conduction band is much higher than number of the holes in the valence band. In n-type material: holes –minority carriers electrons –majority carriers In p-type material: holes –majority carriers electrons –minority carriers © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

69. Excitons -1 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology69 We have considered band transitions in semiconductors and implicitly assumed that process of absorption (of a photon, thermal energy etc.) created a free electron and a free hole, each of which can contribute conduction. There is another effect that is worth mentioning, primarily because its importance in quantum-confined structures such as carbon nanotubes and quantum dots. The basic idea is that after an electron transition, it is possible for the electron and the created hole to be bound together by their mutual Coulomb attraction, forming a quasi particle called an exciton.

70. Excitons -2 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology70 The two-particle electron-hole exciton can be modeled like the two-particle hydrogen atom. However unlike the hydrogen atom, which consists of one proton and one electron (having different masses) in empty space, here the bound electron-hole pair moves through a material characterized by relative permittivity. Considering the formulas for energy and Bohr radius of the Hydrogen atom and using the reduced mass Then the binding energy and radius of the ground sate exciton are given by Where R Y is the Rydeberg energy and a 0 is the Bohr radius.

71. Excitons -3 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology71 As an example, for GaAs(ε r =13.3) using an average of the heavy and light hole masses m r =0.0502me, we find that E=-3.86 meV r ex =265a 0 =14nm The binding energy of the pair, E, can be easily overcome by thermal effects (e.g. kT=25meV at room temp.), thus breaking the exciton into free electrons and holes. Therefore in bulk materials exciton effects are usually only observed at very low temperatures and for relatively pure samples since impurities such as dopants tend to screen the Coulomb interaction much as occurs in conductors.

72. Excitons -4 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology72 To demonstrate the concept of optical absorption by bandgap transition and excitons the absorption coefficient versus incident photon energy for GaAs is shown n below for various temperatures. At room temperature one can observe the lack of absorption below the bandgap (Eg=1.43 eV). At low temperatures, the peaks at the onset of absorption are due to the creation of excitons.

73. Charge Carrier Statistics -1 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology73 In previous quantum well problems the allowed energy states were obtained, but here was no way to say which states would actually be “filled” by electrons. Here we will examine how many allowed states are near an energy of interest, and the probability that those states will actually be filled with electrons. Density of states and particle statistics concepts are indispensible in study of bulk materials as well as small material systems. The density of states is required as the first step in determining the carrier concentrations and energy distributions of carriers within a semiconductor. Integrating the density of states function g(E) between two energies E 1 and E 2 tells us the number of allowed states available to electrons in the cited energy range per unit volume of the crystal. In principle, the density of states could be determined from band theory calculations for a given material. Such calculations however, would be rather involved and impractical. Fortunately, an excellent approximation for the density of states near the band edges can be obtained a simple and familiar approach of “particle in a box” problem.

74. Charge Carrier Statistics -2 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology74 Remember the solution of Schrodinger’s equation for particle in a 3D box where Each solution can be uniquely associated with a k-space vector k = (n x π/L x )a x +(n y π/L y ) a y +(n z π/z) a z where a x,a y,a z are unit vectors directed along k-space coordinate axes. In the figure, each point represents one solution of Schrodinger’s equation.

75. Charge Carrier Statistics -3 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology75 Taking note of lattice arrangements of the solution dots, we can deduce that a k-space “unit cell” of volume (π/L x )(π/L y ) (π/L z ) contain one allowed solution. And therefore: Considering that there is no physical difference between wavefunction solutions which differ only in sign, the total number should be divided to 8. On the other hand, for electrons, two allowed spin states (spin up and spin down ) must be associated with each independent solution. We therefore obtain

76. Charge Carrier Statistics -4 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology76 The next step is to determine the number of states with a k-value between arbitrarily chosen k and k+dk. This is equivalent to adding up the states lying between the two k-space spheres shown in the figure. Considering that the large dimensions of the system and close-packed density of k-space state s, the desired result is simply obtained by multiplying the k- space volume between the two spheres, 4πk 2 dk, times the last equation for the allowed states per unit k-space volume.

77. Charge Carrier Statistics -5 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology77 Therefore Then by definition where V is the volume of the crystal and N(E) is the density of states. Thus, finally

78. Charge Carrier Statistics -6 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology78 To obtain the conduction and valence band densities of states near the band edges in real materials, the mass m of the particle in the forgoing derivation is replaced by the appropriate carrier effective mass. Also if E C is taken to be the minimum electron energy in the conduction band and E V the maximum hole energy in the valence band the E in the last equation must be replaced by E-E C in treating conduction band states and by E V -E in treating valence band states. Introducing the subscripts c and v to identify the conduction and valance band densities of states, respectively, we can then write in general

79. Charge Carrier Statistics -6 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology79 To gain appreciation of the last equations assuming m*=m we have For example if E=0.1 eV and V 0 =0 then N(E)=2.15x10 21 1/eVcm 3 Thinking physically, if there are enough electrons to fill the various states, then the density of states N(E) is the density of electrons having energy E. However, this is not case in reality and therefore we should find the probability for electrons to have the energy E.

80. 80 Statistical Distributions –1 Given: System of N particles (air molecules, for example) in thermal equilibrium at temperature T. Question: How is the total energy E distributed over the particles? or: How many particles have the energy E 1, E 2, etc.? How can we answer these questions? © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

81. 81 Fermi-Dirac statistics identical particles odd half-integral spin (fermions) close together (overlapping ψ) Statistical Distributions – 2 Maxwell-Boltzmann statistics identical particles “far” apart (no overlap of ψ) Bose-Einstein statistics identical particles integral spin (bosons) close together (overlapping ψ)) Distinguishable particles (e.g. molecules in a gas) Indistinguishable particles - Bosons (e.g. photons) Indistinguishable particles - Fermions (e.g. electrons) © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

82. 82 Statistical Distributions – 3 Maxwell-Boltzmann statistics Bose-Einstein statistics Fermi-Dirac statistics © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

83. 83 The Fermi Level –2 The function f(E),-the Fermi-Dirac distribution function,- reflects probability that an available energy state at energy level E will be occupied by an electron at temperature T. The quantity E F is called the Fermi level Important property of the Fermi level: for an energy E equal to the Fermi level energy E F,the occupation probability is Fermi level is an energy state which has always a probability of ½ of being occupied by an electron. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

84. 84 For any energy value E> E F : The Fermi Level –3 Let’s take a look on what happens with Fermi-Dirac distribution function when temperature changes. At T= 0 K, every energy state up to the Fermi level E F is filled with electrons and all states above E F are empty. 1. T = 0 For any energy value E< E F : At T = 0 K the Fermi-Dirac distribution function f(E) takes the simple rectangular form what means: © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

85. 85 The Fermi Level –4 2. T > 0 There is some probability for states above the Fermi level to be filled. At T >0,for E >E F probability that energy states above E F are filled f (E) ≠0 (> 0). At T >0,there is a corresponding probability [1-f (E)]≠0 that states below E F (E > E F ) are empty. Fermi function f (E) is symmetrical about the Fermi level E F for all temperatures: the probability f (E F +ΔE) that a state ΔE above E F is filled is the same as the probability [1-f(EF+ΔE)] that a state ΔE below E F is empty. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

86. 86 The Fermi Level –5 The symmetry of the distribution of empty and filled states about E F makes the Fermi level a natural reference point in calculations of electron and hole concentrations in semiconductors. The Fermi-Dirac distribution function © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

87. 87 The Fermi Level –6 The symmetry of the distribution of empty and filled states about E F makes the Fermi level a natural reference point in calculations of electron and hole concentrations in semiconductors. For intrinsic material we know: the concentration of holes in the valence band is equal to the concentration of electrons in the conduction band. ⇓ the Fermi level E F must lie at the middle of the band gap in intrinsic material. Since f (E) is symmetrical about E F, the electron probability "tail" of f (E) extending into the conduction band is symmetrical with the hole probability tail [1 –f (E)] in the valence band. The Fermi level in intrinsic material is located at energy E i Fermi level in intrinsic material © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

88. 88 The Fermi Level –7 For intrinsic material at T > 0 K © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

89. 89 The Fermi Level –8 In n-type material there is a high concentration of electrons in the conduction band compared with the hole concentration in the valence band. ⇓ The distribution function f (E) lies above its intrinsic position on the energy scale. n-type material has larger concentration of electrons at E c and correspondingly smaller hole concentration at E v, than intrinsic material. Energy difference (E c -E F ) gives a measure of concentration of electrons in the conduction band. Fermi level in n-type material © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

90. 90 The Fermi Level –9 For n-type material at T > 0 K © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

91. 91 The Fermi Level –10 In n-type material there is a high concentration of holes in the valence band as compared to the electron concentration in the conduction band. ⇓ The distribution function f (E) lies below its intrinsic position on the energy scale. The [1 –f (E)] tail below E v is larger than the f (E) tail above E c in p-type material. The value of (E F - E v ) indicates how strongly p-type the material is. Fermi level in p-type material © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

92. 92 The Fermi Level –11 For p-type material at T > 0 K © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

93. Temperature Dependence of Carrier Concentrations © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology93 Typical temperature dependence of the majority carrier concentration in n- type semiconductor. (Phosphorus doped Si, N D = 10 15 1/cm 2 )

94. 94 Charge Carrier Concentrations at Equilibrium Our goal is to find carrier concentration n 0 and p 0 in semiconductor. We can find concentration of carriers if we know: 1.the distribution function f (E) (probability of carriers to occupy energy state) 2.the densities of states N (E) in the valence and conduction bands. Then concentration of electrons in the conduction band at equilibrium: where f (E) is Fermi distribution function; N (E) is the density of states; N(E)dE is the density of states (cm -3 ) in the energy range dE. The subscript “0”used with the electron and hole concentration symbols (n 0, p 0 ) will indicate equilibrium conditions. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

95. 95 Concentrations at Equilibrium –2 Using quantum mechanics approach, it can be shown that density of states N(E) in the conduction band is proportional to √ E : Then concentration of carriers in the conduction band at equilibrium can be calculated as: (we calculated this before!) © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

96. 96 Concentrations at Equilibrium –3 At room temperature kT is 0.026 eV → (E c -E F ) >>kT → Fermi distribution function f (E) can be simplified: © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology The last integral may be solved more easily by making a change of variable. If we let

97. Concentrations at Equilibrium –3 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology97 Then equation becomes The integral is the gamma function with a value of Then the equation for n 0 becomes which is the concentration of carriers in the conduction band at equilibrium

98. 98 Concentrations at Equilibrium –4 -concentration of carriers in the conduction band at equilibrium. or where N C is the effective density of states located at the bottom of the conduction band E c We have found concentration of electrons in the conduction band at equilibrium. © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

99. 99 Concentrations at Equilibrium –5 The concentration of holes in the valence band at equilibrium is where constant N V is the effective density of states in the valence band: Thus, the concentration of holes in the valence band is © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

100. 100 Concentrations at Equilibrium –6 © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology

101. Position of Fermi Level © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology101 For intrinsic semiconductors n 0 =p 0 =ni which leads to which leads to Therefore the intrinsic Fermi level is near the middle of the bandgap. If the electron effective mass and hole effective mass are equal the intrinsic Fermi level lies precisely in the middle of the bandgap.

102. Position of Fermi Level © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology102 For extrinsic n-type semiconductor, if we assume that the dopant concentration N D is much higher than the intrinsic carrier concentration n i and all the dopant atoms are ionized (the usual situation ) then, Substituting this intowe find In a similar manner, for a p-type semiconductor with an acceptor doping density N A >>pi

103. Exercise -1 The probability that a state is filled at the conduction band edge (E C ) is precisely equal to the probability that a state is empty at the valence band edge (E V ). Show mathematically where the Fermi level is located? © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology103

104. Exercise -2 The probability of a state being filled at Ei+kT is equal to the probability a state is empty at Ei+kT. Where is the Fermi level located? What type of semiconductor is it in terms of carrier type? © Nezih Pala npala@fiu.edu EEE5425 Introduction to Nanotechnology104