Solid State Device Fundamentals
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1 Solid State Device Fundamentals ES 345 Lecture Course by Alexander M. Zaitsev Tel: Oice 4101b 1
2 The ree electron model o metals The ree electron model o metals assumes that electrons are ree to move within the metal but are conined to the metal by potential barriers. The minimum energy W needed to extract an electron rom the metal equals ef M, where F M is the work unction. F M may vary rom 2.1 ev (or Cs) to 5.9 ev (Pt). This model ignores the periodic potential due to atoms and as such it does not work well or semiconductors. 5.3 ev 4.1 ev 2
3 Electrons in periodic potential Uniorm potential Periodic potential E E Energy o electron versus wavevector k (linear momentum p=ħk) or ree electron. Energy o electron versus wavevector k in monoatomic crystal lattice o lattice constant a. The energy gap E g is associated with the irst Bragg relection at k=±π/a ( = 2a). 3
4 Description o electrons in crystal A complete description o the electrons in a crystal is based on their wave characteristics, not just the particle characteristics. The electron wave unction is the solution o the three-dimensional Schrödinger wave equation: ħ2 2m 0 2 ψ + V(r)ψ = ψe Solution: = A exp(ik r), where k = 2π/ Acceleration = F = ee d 2 E m ħ 2 dk 2 For each k, there is a corresponding energy E. Eective mass, m = ħ 2 / d2 E dk 2 How much is eective mass o ree electron? 4
5 Eective mass o electron in semiconductors 5
6 Formation o energy bands o electrons in silicon hybridization a Energy levels in silicon as a unction o interatomic distance. The core levels (n = 1, 2) are completely illed with electrons. At the actual interatomic distance a o a crystal with atoms, the 2 electrons in the 3s subshell and the 2 electrons in the 3p subshell hybridize (sp 3 hybridization) producing 4 states o lower energy (valence energy band) and 4 states o higher energy (conduction energy band). 6
7 Electron energy bands in crystals electrons holes Possible energy band diagrams o a crystal: a) a hal illed band, b) two overlapping bands, c) an almost ull band separated by a small bandgap rom an almost empty band and d) a ull band and an empty band separated by a large bandgap. 7
8 Density o states The density o states in a semiconductor equals the density per unit volume and energy o the number o solutions to Schrödinger's equation. Assumptions: - Semiconductor is modeled as an ininite quantum well in which electrons have eective mass, m *, and are ree to move. - The energy in the well is set to zero. - The semiconductor is assumed a cube with side L. This assumption does not aect the result since the density o states per unit volume should not depend on the actual size or shape o the semiconductor. x = A exp(k x x), k x = n/l, n = 1, 2, 3, Calculation o the number o states with wavenumber less than k. g E = 8 2π h 3 m 3/2 E Density o states o electrons per unit volume per unit energy. 8
9 Density o states in semiconductor DE E c E v g c 8 mn 2mn E Ec 8 mp 2mp Ev E ( E) 3 h g v ( E) 3 h E c is the energy o the bottom o the conduction band E v is the energy o the top o the valence band 9
10 Electron energy bands in silicon and SiO 2 E c E g=1.1 ev E c Ev E g= 9 ev Si (Semiconductor) E v SiO 2 (Insulator) Totally illed bands and totally empty bands do not allow current low (just as there is no motion o liquid in a totally illed, or totally empty bottle). Semiconductors have smaller E g than insulators. 10
11 Measuring the Band Gap Energy by Light Absorption ree electron photons photon energy: h v > E g E g E c E v ree hole E g can be determined rom the minimum energy (hn) o photons that are absorbed by the semiconductor. Bandgap energies o selected semiconductors Semi-conductor InSb Ge Si GaAs GaP ZnSe Diamond Eg (ev)
12 Temperature dependence o the energy bandgap The temperature dependence o the energy bandgap, E g, has been experimentally determined yielding the ollowing expression or E g as a unction o the temperature, T: 12
13 Fermi unction Electrons are Fermions. The Pauli exclusion principle postulates that only one Fermion can occupy a single quantum state. Thereore, as Fermions are added to an energy band, they will ill the available states in an energy band just like water ills a bucket. The states with the lowest energy are illed irst, ollowed by the next higher ones. At absolute zero temperature (T = 0 K), the energy levels are all illed up to a maximum energy, which we call the Fermi level, E F. electrons holes E F Fermi unction provides the probability o occupancy o energy levels by electrons: The Fermi unction at three dierent temperatures 13
14 Fermi unction at high energies Boltzmann approximation: ( E) ( EE )/ kt 1 e 1 E E E E + 3kT + 2kT ( E) e EE kt ( E) e EE kt E E + kt E E kt E kt E E 2kT 3kT ( E) 1 e E E kt ( E) 1 e E E kt (E) E E kt 14
15 Homework Fermi unction 1. Calculate probability or an electron to have an energy o 3kT above the Fermi level. 2. Calculate probability or a hole to have an energy o 3kT below the Fermi level. 3. Compare this data with that calculated using Boltzmann approximation. 15
16 The density o occupied states per unit volume and energy, n(e), ), is simply the product o the density o states in the conduction band, g c (E) and the Fermi unction, (E): Density o electrons and holes Correspondingly or the density or holes, p(e), equals: The total density o charge carriers (concentration) is obtained by integrating the product o the density o states and the probability density unction over all possible states. For electrons in the conduction band the integral is taken rom the bottom o the conduction band, labeled, E c, to the top o the conduction band. n electron concentration, p hole concentration. A common unit is cm
17 Homework Density o states 1. Calculate density o states o ree electron and ree holes o an energy o 1 ev in Si, Ge and GaAs. 2. Calculate/Estimate concentration o electrons and holes in Si, Ge and GaAs at temperatures 100, 300 and 1000K. 17
18 on-degenerate semiconductors on-degenerate semiconductors are deined as semiconductors or which the Fermi energy is at least 3kT away rom either band edge. The approximation o non-degenerate semiconductors allows the Fermi unction to be replaced with a simple exponential unction. In many cases, solid state electronic devices operate on the principles o nondegenerate semiconductors 18
19 Electron and Hole Concentrations n c e ( E E c )/ kt c 2m nkt 2 2 h 3 2 c is the eective density o states o the conduction and. p v e ( E E v )/ kt v 2m pkt 2 2 h 3 2 v is the eective density o states o the valence band. The closer Fermi level moves up to E c, the larger n is. The closer Fermi level moves down to E v, the larger p is. For silicon, c = cm -3 and v = cm
20 Eective density o states in germanium Eective conduction band density o states cm -3 Eective valence band density o states cm -3 20
21 Homework Eective density o states 1. Calculate eective density o states oe electrons and holes in Ge and GaAs. 2. Compare the obtained data with c and v in silicon. 21
22 The np product and the intrinsic carrier concentration n c e ( E E c )/ kt p v e ( E E v )/ kt np c v e ( E E kt E c v )/ cve g / kt np n i 2 n i c v e E g / 2kT In an intrinsic (undoped) semiconductor, n = p = n i n i is the intrinsic carrier concentration n i ~10 10 cm -3 or Si at room temperature 22
23 Intrinsic electron and hole concentration versus temperature n c e ( E E c )/ kt p v e ( E E v )/ kt Intrinsic carrier density versus temperature in: GaAs (red), Silicon (blue), Germanium (black). 23
24 Homework Intrinsic carrier concentration 1. Calculate intrinsic carrier concentration or Si, Ge and GaAs at temperatures 100K and 1000K. 2. Compare the obtained data with c and v shown on previous slide Estimate the temperature at which intrinsic carrier concentration or Si, Ge and GaAs become comparable. 4. At the temperature calculated in 3, can Si, Ge and GaAs be considered as non-degenerate semiconductors? 24
25 Position o Fermi level versus temperature n c e ( E E c )/ kt p v e ( E E )/ kt v E F * E C EV 3 me k ln BT * 2 4 mh m e * > m h * m e * < m h * Red line shows the middle o the bandgap In intrinsic semiconductors, in a broad temperature range, the Fermi level remains close to the middle o the bandgap. 25
26 Homework Position o Fermi level 1. Calculate position o Fermi level in Si, Ge and GaAs at temperatures 100K and 1000K. 2. At what temperature Fermi level crosses E c /E v in these semiconductors? 26
27 27
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