EE 529 Semiconductor Optoelectronics – Semiconductor Basics EE529 Semiconductor Optoelectronics Semiconductor Basics 1. Semiconductor materials 2. Electron and hole distribution 3. Electron-hole generation and recombination 4. p-n junction Reading: Liu, Chapter 12, Sec. 13.1, 13.5 Reference: Bhattacharya, Sec. 2.1-2.2, 2.5-2.6, 4.2 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Energy Bands in Semiconductors Origin: Periodic lattice structure in the crystal. E-k diagram details the band structure. k : Electron wave vector k = 2 π / λ Indirect bandgap Si Near the band edge: Direct bandgap + 2 2 k = E E c * 2 m e − 2 2 k = E E v * 2 m h * m , : e h Effective masses of electrons/holes 2 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Semiconductor Materials Elementary: e.g., Si and Ge Binary Ternary Quaternary Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Semiconductor Basics III-V Compound Semiconductors Solid curve: Direct bandgap Dashed curve: Indirect bandgap Al x Ga 1-x As closely lattice matched to GaAs In 1-x Ga x As y P 1-y lattice matched to InP for 0 ≤ y ≤ 1 and x = 0.47y Matching lattice constant is important when depositing one semiconductor on another. 4 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Semiconductor Basics Electron-Hole Generation How do electrons get to the conduction band (and leave holes in the valence band)? Free electrons and holes can be generated by: • Thermal excitation (thermal equilibrium) • Optical excitation (quasi-equilibrium) • Current injection (quasi-equilibrium) It’s important to understand the carrier (electrons and holes) distribution as a function of energy. 5 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Semiconductor Basics Carrier Distribution in the Energy Band Concentration of electrons (holes) versus energy in the conduction (valence) band = Density of states (density of allowed energy levels) Analogy: Density of available office spaces X Probability of occupancy (probability that each of these levels is occupied) Analogy: People’s desire of occupying these spaces A skyscraper without elevators 6 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Semiconductor Basics Probability of Occupancy 1 Fermi-Dirac function = f ( E ) − E E + f exp 1 k T B f ( E ) = probability of occupancy by an electron 1 - f ( E ) = probability of occupancy by a hole (in valence band) (Electrons like to sink to the bottom, holes like to float to the top.) 7 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Density of States ― Bulk Material 2 k ρ = π ( ) k 2 * 3/2 (2 m ) ρ = − 1/2 e ( ) E ( E E ) π C C 2 3 2 * 3/2 (2 m ) ρ = − 1/2 h ( ) E ( E E ) π V V 2 3 2 8 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Density of States ― Quantum Wells Green laser diode Nature Photonics 3 , 432 - 434 (2009) Discrete energy levels ( ) π 2 2 / q d = = , 1 , 2, ... E q q 2 m Density of states * m > + π e , E E E ρ = C q 1 2 d ( ) E 1 C < + 0, E E E C q 1 9 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Quantum Wires and Quantum Dots Quantum wires π * (2 / d d )( m / 2 ) , > + + 1 2 e E E E E ρ = − − − C q 1 q 2 ( ) E 1/2 ( E E E E ) C C q 1 q 2 0, otherwise (Gudiksen et al., Nature, 2002) Quantum dots 1 ρ = δ − − − − ( E ) 2 ( E E E E E ) c c q 1 q 2 q 3 d d d 1 2 3 10 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Semiconductor Basics Thermal-Equilibrium Carrier Concentration Bulk Semiconductor − ∞ E E = ρ = ∫ n ( ) ( ) E f E dE N ( ) T F F C (a) (b) (c) (d) 0 C C 1/2 k T E c B g ( E ) ∝ ( E – E c ) 1/2 − E E E E E E v = ρ − = ∫ V F p ( )[1 E f E dE ( )] N ( ) T F E c + χ [1– f ( E )] 0 V V 1/2 k T −∞ B CB Area = ∫ = For Area = n ( ) E dE n n E (E)dE = n E electrons n E ( E ) E c E c E F E F E v p E ( E ) E v Area = p For holes VB We can simplify this more … 0 fE ) g ( E ) n E ( E ) or p E ( E ) (a) Energy band diagram. (b) Density of states (number of states per unit energy per unit volume). (c) Fermi-Dirac probability function (probability of occupancy of a state). (d) The product of g ( E ) and f ( E ) is the energy density of electrons in the CB (number of electrons per unit energy per unit volume). The area under n E ( E ) vs. E is the electron concentration. 11 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Carrier Concentration and Mass Action Law − ≥ − ≥ For non-degenerate semiconductors, ( E E ) / k T 3.6 ( E E ) / k T 3.6 F V B C F B − ( E E ) = − C F n N exp 0 C k T B = π * 2 3/2 N 2(2 m k T h / ) : Effective density of states at the conduction band edge C e B − ( E E ) = − F V exp p N 0 V k T B = π * 2 3/2 N 2(2 m k T h / ) : Effective density of states at the valence band edge V h B The location of the Fermi level energy E F is the key. π 3 E 2 k T ( ) = 3/2 − = * * g 2 Mass action law: n p 4 B m m exp n 0 0 e h i 2 h k T B → Knowing one carrier concentration, you can determine the other (no matter intrinsic or extrinsic) 12 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Intrinsic and Extrinsic Semiconductor + E E 1 N = − C V C E k T ln Intrinsic: Fi B 2 2 N V n-type semiconductor p-type semiconductor Electron energy B atom sites every 10 6 Si atoms Electron Energy E c x Distance into crystal CB As + h + E c e– ~0.05 eV B– E d B – B – B – As + B – As + As + As + E a ~ 0.05 eV h + E v VB E v x 6 i i 13 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Exercise: Fermi Levels in Semiconductors a) Where is the Fermi level of intrinsic bulk Si at room temperature? b) What kind of dopants can make it n-type? c) If the donor concentration N d is 10 16 cm -3 , where will the Fermi level be? d) If the wafer is compensation-doped with boron (N a = 2 x 10 17 cm -3 ), where will the Fermi level be? 14 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Semiconductor Basics Quasi Equilibrium ― What happens to the Fermi level s during photon absorption 1 = f ( ) E − C Probability of occupancy for electrons: E E + Fc exp 1 k T B 1 = f ( ) E Probability of occupancy for holes: − V E E + exp Fv 1 k T B How to calculate the quasi Fermi levels? 15 Lih Y. Lin
EE 529 Semiconductor Optoelectronics EE 529 Semiconductor Optoelectronics – Semiconductor Basics – Semiconductor Basics Quasi-Fermi Levels The figure below shows positions of quasi-Fermi levels as a function of photo- generated electron-hole pair density. The semiconductor is n-type GaAs with N D = 10 15 cm -3 . Q: Why does ε fp decrease gradually with increasing density while ε fn shows a sudden increase? 16 Lih Y. Lin
EE 529 Semiconductor Optoelectronics Exercise: Carrier Lifetime and Internal – Semiconductor Basics Quantum Efficiency in GaAs = × − 18 3 A n-type ( n 5 10 m ) GaAs is under optical excitation generating excess carrier 0 = − = − concentration . It has the followsing recombination coefficients: N n n p p 0 0 = × = × = + = × − − − − − 42 6 1 5 1 17 3 1 5 10 s , 8 10 m s , and C C C 5 10 m s . Assume that A B e h = = C C C / 2 . (1) Find the range of N where each of the three different recombination e h processes (Shockley-Read, bimolecular, Auger) dominates. (2) Plot the spontaneous τ as a function of N for ≤ ≤ − 18 26 3 carrier lifetime 10 N 10 m . (3) Assume only the s bimolecular recombination process is radiative, plot the internal quantum efficiency vs N . 17 Lih Y. Lin
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