1. Lecture 2 OUTLINE Semiconductor Fundamentals (cont’d) – Energy band model – Band gap energy – Density of states – Doping Reading: Pierret 2.2-2.3, 3.1.5; Hu 1.3-1.4,1.6, 2.4

2. Potential Energy Profiles Discrete allowed energy levels When two atoms are in close proximity, the upper energy levels are shifted to bonding and anti-bonding levels. Lecture 2, Slide 2EE130/230A Fall 2013 V(r)  1/r is mostly a coulombic potential btwn the positive nucleus & negative electrons. N atoms 1 atom 2 atoms many bonding/anti-bonding levels

3. Energy states in Si atom  energy bands in Si crystal Si: From Atom to Crystal The highest nearly-filled band is the valence band The lowest nearly-empty band is the conduction band Lecture 2, Slide 3EE130/230A Fall 2013 R.F. Pierret, Semiconductor Fundamentals, Figure 2.5

4. Energy Band Diagram Simplified version of energy band model, showing only the bottom edge of the conduction band (E c ) and the top edge of the valence band (E v ) E c and E v are separated by the band gap energy E G EcEc EvEv electron energy distance Lecture 2, Slide 4EE130/230A Fall 2013

5. Electrons and Holes (Band Model) Conduction electron = occupied state in the conduction band Hole = empty state in the valence band Electrons & holes tend to seek lowest-energy positions  Electrons tend to fall and holes tend to float up (like bubbles in water) Lecture 2, Slide 5 Increasing hole energy Increasing electron energy EcEc EvEv electron kinetic energy hole kinetic energy E c represents the electron potential energy. EE130/230A Fall 2013

6. Electrostatic Potential, V and Electric Field, E The potential energy of a particle with charge -q is related to the electrostatic potential V(x): Lecture 2, Slide 6 Variation of E c with position is called “band bending.” 0.7 eV EE130/230A Fall 2013

7. E G can be determined from the minimum energy of photons that are absorbed by the semiconductor Measuring the Band Gap Energy Band gap energies of selected semiconductors Lecture 2, Slide 7 SemiconductorGeSiGaAs Band gap energy (eV)0.671.121.42 EcEc EvEv photon h > E G EE130/230A Fall 2013

8. g(E)dE = number of states per cm 3 in the energy range between E and E+dE Near the band edges: Density of States for E  E c Lecture 2, Slide 8 EcEc EvEv dE E density of states, g(E) EcEc EvEv for E  E v SiGeGaAs m n,DOS */m o 1.080.560.067 m p,DOS */m o 0.810.290.47 Electron and hole density-of-states effective masses EE130/230A Fall 2013

9. When an electron is moving inside a solid material, the potential field will affect its movement. For low kinetic energy where p is the crystal momentum i.e. a conduction electron behaves as a particle but with an effective mass m* Schrödinger equation: E : total energy  : wave function ħ : reduced Planck constant Effective Mass, m* Lecture 2, Slide 9EE130/230A Fall 2013

10. E G and Material Classification Neither filled bands nor empty bands allow current flow Insulators have large E G Semiconductors have small E G Metals have no band gap (conduction band is partially filled) silicon Lecture 2, Slide 10 E G = 1.12 eV E G = ~ 9 eV silicon dioxide EcEc EvEv EcEc EvEv metal EvEv EcEc EE130/230A Fall 2013

11. By substituting a Si atom with a special impurity atom (Column V or Column III element), a conduction electron or hole is created. Doping Donors: P, As, Sb N D ≡ ionized donor concentration (cm -3 ) Lecture 2, Slide 11 Acceptors: B, Al, Ga, In N A ≡ ionized acceptor concentration (cm -3 ) EE130/230A Fall 2013 http://inventors.about.com/library/inventors/blsolar5.htm

12. Doping Silicon with a Donor Example: Add arsenic (As) atom to the Si crystal The loosely bound 5th valence electron of the As atom “breaks free” and becomes a mobile electron for current conduction. Lecture 2, Slide 12EE130/230A Fall 2013 As Si

13. Doping Silicon with an Acceptor The B atom accepts an electron from a neighboring Si atom, resulting in a missing bonding electron, or “hole”. The hole is free to roam around the Si lattice, carrying current as a positive charge. Lecture 2, Slide 13 Example: Add boron (B) atom to the Si crystal EE130/230A Fall 2013 B Si

14. Solid Solubility of Dopants in Si ATOMS PER CUBIC CENTIMETER F. A. Trumbore, Bell Systems Technical Journal, 1960 Lecture 2, Slide 14EE130/230A Fall 2013

15. Doping (Band Model) Ionization energy of selected donors and acceptors in silicon Lecture 2, Slide 15 DonorsAcceptors DopantSbPAsBAlIn Ionization energy (meV) E c -E D or E A -E v 3945544567160 EcEc EvEv Donor ionization energy EDED EAEA Acceptor ionization energy EE130/230A Fall 2013

16. Dopant Ionization Lecture 2, Slide 16EE130/230A Fall 2013 R.F. Pierret, Semiconductor Fundamentals, Figure 2.13

17. Charge-Carrier Concentrations Charge neutrality condition:N D + p = N A + n At thermal equilibrium, np = n i 2 (“Law of Mass Action”) Note: Carrier concentrations depend on net dopant concentration! Lecture 2, Slide 17EE130/230A Fall 2013

18. n-type Material (n > p) N D > N A (more specifically, N D – N A >> n i ): Lecture 2, Slide 18EE130/230A Fall 2013

19. p-type Material (p > n) Lecture 2, Slide 19 N A > N D (more specifically, N A – N D >> n i ): EE130/230A Fall 2013

20. Carrier Concentration vs. Temperature Lecture 2, Slide 20EE130/230A Fall 2013 R.F. Pierret, Semiconductor Fundamentals, Figure 2.22

21. Terminology donor: impurity atom that increases n acceptor: impurity atom that increases p n-type material: contains more electrons than holes p-type material: contains more holes than electrons majority carrier: the most abundant carrier minority carrier: the least abundant carrier intrinsic semiconductor: n = p = n i extrinsic semiconductor: doped semiconductor such that majority carrier concentration = net dopant concentration Lecture 2, Slide 21EE130/230A Fall 2013

22. Summary Allowed electron energy levels in an atom give rise to bands of allowed electron energy levels in a crystal. – The valence band is the highest nearly-filled band. – The conduction band is the lowest nearly-empty band. The band gap energy is the energy required to free an electron from a covalent bond. – E G for Si at 300 K = 1.12 eV – Insulators have large E G ; semiconductors have small E G Lecture 2, Slide 22EE130/230A Fall 2013

23. Summary (cont’d) E c represents the electron potential energy Variation in E c ( x )  variation in electric potential V Electric field E - E c represents the electron kinetic energy Lecture 2, Slide 23EE130/230A Fall 2013

24. Summary (cont’d) Dopants in silicon: – Reside on lattice sites (substituting for Si) – Have relatively low ionization energies (<50 meV)  ionized at room temperature – Group-V elements contribute conduction electrons, and are called donors – Group-III elements contribute holes, and are called acceptors Dopant concentrations typically range from 10 15 cm -3 to 10 20 cm -3 Lecture 2, Slide 24EE130/230A Fall 2013