Semiconductor Physics and Devices

Transcription

1 EE321 Fall 2015 September 28, 2015 Semiconductor Physics and Devices Weiwen Zou ( 邹卫文 ) Ph.D., Associate Prof. State Key Lab of advanced optical communication systems and networks, Dept. of Electronic Engineering, SEIEE, SJTU wzou@sjtu.edu.cn Office: SEIEE buildings #5-203 ( ) Course site: Laboratory P. 0

2 Semiconductor Physics and Devices Highlights of Previous story Chapter 2 (Part I) Laboratory P. 1

3 Photoelectric effect & Duality principle Light (Photons) photoelectrons Quantum picture (Photoelectric) At a constant incident intensity, the maximum kinetic energy of the photoelectrons varies linearly with frequency. There is A limiting frequency γ 0, below with no photoelectron is produced. Wave-particle duality principle Waves exhibit particle behavior (photoelectric effect); It is expected that Particles have wave-like properties. Energy & Momentum: p = h λ & E = hγ Laboratory P. 2

4 Chapter 2: Quantum Mechanics Basic introduction Duality (wave & particle) The Uncertainty principle Schrodinger s wave equation Various typical functions (for semiconductors) One-electron atom Pauli exclusion principle Laboratory P. 3

5 The Uncertainty principle (1927, Heisenberg) It is stated that we cannot describe the behavior of particles with absolute accuracy. A fundamental relationship between conjugate variables Position vs. Momentum Energy vs. Time Mathematical expression: p x ħ E t ħ Laboratory P. 4

6 Consequence of the uncertainty principle It is not possible to know the exact position of an electron. Probability denotes the percent of finding an electron at a particular position Electron Cloud Laboratory P. 5

7 Probability density function Wave function or wavefunction {Ψ(x, t)} A complex function. Not represent a real physical quantity. It is a mathematical tool that describes the quantum state of a particle (electron) or system of particles (electrons). Probability density function Ψ(x, t) Ψ (x, t) = Ψ(x, t) 2 =Ψ x, t Ψ x, t It means the probability of finding an electron at a particular position and/or with a particular energy. Laboratory P. 6

8 Chapter 2: Quantum Mechanics Basic introduction Duality (wave & particle) The Uncertainty principle Schrodinger s wave equation Various typical functions (for semiconductors) One-electron atom Pauli exclusion principle Laboratory P. 7

9 Time-independent Schrodinger s equation Boundary conditions ( Ψ(x, t) 2 = φ(x) 2 : probability function) Probability over infinite space should be unity: + φ(x) 2 dx = 1 φ(x): finite, single-valued, continuous φ(x) x 2 ( x) 2 m 2 2 ( E V ( x)) ( x) 0 x h Time-related part: ϕ t = exp{ j E ħ t} : finite, single-valued, continuous (if E & V are finite) Laboratory P. 8

10 Chapter 2: Quantum Mechanics Basic introduction Duality (wave & particle) The Uncertainty principle Schrodinger s wave equation Various typical functions (for semiconductors) One-electron atom Pauli exclusion principle Laboratory P. 9

11 Electron in free space 2 ( x) 2 m 2 2 ( E V ( x)) ( x) 0 x h In free space, there is no force acting on the electrons. So, the potential function V(x) is a constant and E>V(x). Assuming V(x)=0, φ x = Aexp jx 2mE ħ 2 + Bexp jx 2mE ħ 2 Use (time-related part): ϕ t = exp{ j E ħ t} Ψ x, t = Aexp j(x 2mE ħ 2 E 2mE t) + Bexp j( x ħ ħ 2 If the travelling wave is in +x direction, B=0 Ψ x, t = Aexp j(kx ωt). E ħ t) Laboratory P. 10

12 Electron in free space (cont.) Ψ x, t = Aexp j(x 2mE ħ 2 E ħ t) Ψ x, t = Aexp j(kx ωt) Since k = 2π λ = 2mE ħ 2, then λ = h 2mE The de Broglie wavelength: λ = h p Finally, p 2 = 2mE, which is the same as the classical theory. Conclusions: 1. A free particle with a well-defined energy will also have a well-defined wavelength and momentum. 2. Probability function of Ψ x, t Ψ x, t =A A is a constant free from position. A free particle with well-defined momentum can be found anywhere with equal probability. (Since p = 0, x = ) Laboratory P. 11

13 Electron in the infinite potential wall - + x<0 V ( x)= 0 0 x + x> a a 2 ( x) 2 m 2 2 ( E V ( x)) ( x) 0 x h Laboratory P. 12

14 Electron in the infinite potential wall (cont.) Energy levels Wave functions Probability functions One energy level One wave/probability function (quantum state) Laboratory P. 13

15 Electron in the potential barrier The wave functions through the potential barrier. Tunneling effect - Laboratory P. 14

16 Chapter 2: Quantum Mechanics Basic introduction Duality (wave & particle) The Uncertainty principle Schrodinger s wave equation Various typical functions (for semiconductors) One-electron atom Pauli exclusion principle Laboratory P. 15

17 Hydrogen, one-electron atom Potential function: 2 2 m ( ( )) E V r x h V r = e2 4πε 0 r Solutions are related to three integers (n, l & m), which corresponds to three dimensions (r, θ, φ), respectively. n, l & m: quantum numbers n = 1,2,3 (Principle quantum number) l = n 1, n 2, n 3 0 m = l, l 1 0 Each group of (n, l & m) denotes a quantum state that the electron may occupy. For hydrogen, (1,0 & 0): n = 1, then l = 0 & m = 0 Laboratory P. 16

18 The radial probability density function Lowest energy state (n=1) Next-higher energy state (n=2) Laboratory P. 17

19 Chapter 2: Quantum Mechanics Basic introduction Duality (wave & particle) The Uncertainty principle Schrodinger s wave equation Various typical functions (for semiconductors) One-electron atom Pauli exclusion principle Laboratory P. 18

20 Pauli exclusion principle In any given system (an atom, molecule, or crystal), no two electrons may occupy the same quantum state. In an atom, no two electrons may have the same set of quantum numbers For instant, in Hydrogen (one electron atom), the electron have two possible states (electron spin, s = 1 2 or 1 2 ) Quantum numbers (n, l, m & s) Quantum states (ns 1~2 np 1~6 nd 1~10 ) s state (n 1): l = 0, m = 0, s = 1 or 1 (2 states) 2 2 p state(n 2): l = 1, m = 0, 1 or 1, s = 1 or 1 (6 states) 2 2 d state (n 3): (10 states) For example: Carbon (6 protons): 1s 2 2s 2 2p 2 Laboratory P. 19 Silicon (14 protons): 1s 2 2s 2 2p 6 3s 2 3p 2

21 Homework (till Oct. 12) 2.10, 2.20, 2.26 Laboratory P. 20

22 Semiconductor Physics and Devices Chapter 3: Introduction of the Quantum Theory of Solids Overview: This chapter generalizes previous theory to the electron in a crystal lattice. Two major points: 1. Determine the properties of carriers in a crystal lattice. 2. Determine the statistical characteristics of the very large number of electrons in a crystal. Laboratory P. 21

23 Object of Chapter 3 The final target of this course s study is aimed to determine the current-voltage characteristics of semiconductor devices E Note: It is necessary to master the significant concepts and conclusions although some complicated mathematical derivations are not required. Laboratory P. 22

24 Outline Formation of energy bands The k-space diagram Difference among Metals, semiconductors, vs. insulators Laboratory P. 23

25 Splitting of discrete quantized level In a single atom (e.g., one electron, hydrogen), the energy of a bound electron is quantized and only discrete values of electron energy are allowed. As atoms are moved closer together, the wavefunctions of the electrons start to overlap. Laboratory P. 24

26 Splitting of discrete quantized level (cont.) This electron interaction results in the discrete quantized energy level splitting into multiple (for hydrogen, 1 -> 2) but closely separated levels. Basis: Pauli exclusion theory The splitting of n=1 state. Laboratory P. 25

27 Energy bands As a large number of atoms with their electrons get close together, the allowed energies form a band of energies. Laboratory P. 26

28 Band gap At certain interatomic distances, these energy bands may be separated by regions where electron energies are not allowed. These are the forbidden energies called the band gap (E g ) Recall (chapter 2): Silicon (14 protons): 1s 2 2s 2 2p 6 3s 2 3p 2 The splitting of the 3s and 3p states of silicon into the allowed (valencelower band, conduction-higher band) and forbidden energies. Laboratory P. 27

29 Example 1 Question: The forbidden energy band of GaAs is 1.42 ev. Determine the minimum frequency of an incident photon that can interact with a valence electron and elevate the electron to the conduction band. Laboratory P. 28

30 Example 1 (cont.) What is the corresponding wavelength? Laboratory P. 29

31 Outline Formation of energy bands The k-space diagram Difference among Metals, semiconductors, vs. insulators Laboratory P. 30

32 The k-space diagram Recall (chapter 2): Wave-particle Duality principle De Broglie (1924) Waves exhibit particle behavior (photoelectric effect); It is expected that Particles have wave-like properties. De Broglie wavelength: Wave-particle duality principle Momentum of a photon: p = h λ = ħ k k = 2π e n m er λ ħ= h/ 2π: a modified Planck s constant E = p2 2m = k2 ħ2 2m The following figure schematically shows this parabolic relationship between the energy E and the momentum p (or k) for a particle in free space. De Broglie wavelength: λ = h p Energy & Momentum: p = h λ & E = hγ OFS 2 : Optical Fiber Switching & Sensing System Laboratory P. 9 Laboratory P. 31

33 k vs. E diagram (in the crystal) The energy of the particle in the crystal lattice as a function of the wave number k shows discontinuities. Bound particles (not free space ) These discontinuities are the forbidden energies for the particles in the crystal. The principle results: - Electrons in the crystal occupy certain allowed energy bands - Electrons are excluded from the forbidden energy bands. Laboratory P. 32

34 A reduced k-space diagram E The reduced-zone representation E Allowed Forbidden Allowed Forbidden Allowed 3 a 2 a a a a a a 0 a Quantum states k Laboratory P. 33

35 Example (silicon): 2D model (covalent bonding) Silicon lattice at ideal T=0 K: Each atom is surrounded by eight valence electrons All electrons are in their lowest-energy states In a word, the valence band is fully filled with electrons, but the conduction band is empty without any electron occupied. Conduction band (empty) E g Valence band (full) Laboratory P. 34

36 As the temperature increases above 0 K Thermal energy is gained by valence electrons Break the covalent bonding and jump into the conduction band Laboratory P. 35

37 From the viewpoint of the E-k diagram T=0 K T>0k Laboratory P. 36

38 Semiconductor Physics and Devices End of Day #3 Weiwen Zou ( 邹卫文 ) Ph.D., Associate Prof. State Key Lab of advanced optical communication systems and networks, Dept. of Electronic Engineering, SEIEE, SJTU wzou@sjtu.edu.cn Office: SEIEE buildings #5-203 ( ) Course site: Laboratory P. 37