ECE 875: Electronic Devices Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University

VM Ayres, ECE875, S14 Hw 04: FRI: Pr Chp. 01 – Chp. 02 Experimental measurements for concentration: Hall effect – Chp. 01: material C-V – Chp. 02: pn junction Lecture 14, 10 Feb 14

VM Ayres, ECE875, S14

I gen = ? OR I rec = ? Which: are you in forward or reverse bias? What happens to the depletion region W D ? VM Ayres, ECE875, S14

= everything that’s left in U In Pr. 2.07:  g is given. VM Ayres, ECE875, S14

Reminder: Lec 13: When trying to turn a pn junction OFF, a substantial generation current makes this difficult Similarly, when trying to turn a pn junction ON, a substantial recombination current makes this difficult VM Ayres, ECE875, S14

Hw 04: FRI: Pr Chp. 01 – Chp. 02 Experimental measurements for concentration: Hall effect – Chp. 01: material C-V – Chp. 02: pn junction Lecture 14, 10 Feb 14

Add a Magnetic field B to a doped semiconductor with a current flowing Operator facing same direction as the historic current F = q (v x B) Consider majority holes: F = +e-(+v x x B z ) F +y = |e- v x B z | on holes to RHS Displaced electrons go to LHS V AB = positive F = q (v x B) Consider majority electrons: F = -e-(-v x x B F +y = |e- v x B z | on electrons to RHS Displaced holes go to LHS V AB = negative Hall Effect: VM Ayres, ECE875, S14

I I Hall Effect: no scattering: p is majority carrier: n is majority carrier: V AB positive negative VM Ayres, ECE875, S14

I I Hall Effect: no scattering: p is majority carrier: n is majority carrier: V AB positive negative VM Ayres, ECE875, S14

I I Hall Effect: ECE 875: with scattering: V AB p is majority carrier: n is majority carrier: positive negative VM Ayres, ECE875, S14

 m : time between scattering events Called the mean free lifetime, also called the momentum relaxation time The mean free lifetime depends on the energy the electron has: Mean free lifetime also depends on the type of scatterer VM Ayres, ECE875, S14

Mean free lifetime can be equivalently described as a mean free length (momentum relaxation length): l also called l m and m in Sze VM Ayres, ECE875, S14

Generally: Low temp T: impurity scattering: N D +, N A - : s = ½ High temp T: phonon scattering: s = 3/2 (further info in Chp. 01 eq’s (49) and (50)) VM Ayres, ECE875, S14

Example: choose the semiconductor with a spherical constant energy surface: Ge, Si, or GaAs

VM Ayres, ECE875, S14 Answer: choose the semiconductor with a spherical constant energy surface: Ge, Si, or GaAs

, depend on the definition of average : Stated without proof: eq’n (72): For a Boltzmann distribution of carriers in a non-degenerate semiconductor: (Note: normalization: Pr. 10: KE) VM Ayres, ECE875, S14

Hall Effect: evaluated for you: where: VM Ayres, ECE875, S14 All you need to know is: s

VM Ayres, ECE875, S14 Hall mobility  H from Hall factor r H : Related to:

 n and  p :

Example: relate eq’n (54) to the result of Pr. 1.10:

Answer:

VM Ayres, ECE875, S14 Example:

VM Ayres, ECE875, S14 Example: Carrier density = ? Mobility = ?

VM Ayres, ECE875, S14 Example: R H not Hall factor r H Carrier density = n OR p = ? Mobility = ?

VM Ayres, ECE875, S14 Example: R H not Hall factor r H Carrier density = n since R H = negative and only one type of carrier is present Mobility = ?

VM Ayres, ECE875, S14 Example: Carrier density = n since R H = negative and only one type of carrier is present Mobility =  n OR  Hall

VM Ayres, ECE875, S14 Example: Carrier density = n since R H = negative and only one type of carrier is present Mobility =  H Note: different s

VM Ayres, ECE875, S14

2

Scattering also depends on the type of scatterer: Low temp T: N D +, N A -: s = ½ High temp T: phonons: s = 3/2 VM Ayres, ECE875, S14

Phonon model: 1D vibrational modes for a linear chain with unequal masses: 1D: m1m1 m2m2 Symmetric and anti-symmetric motion: ± Low frequency (acoustic) and high frequency (optical) solution Equation of motion from F = ma is variation on a harmonic oscillator with multiple solution given by integer k ph = 0, 1, 2, … Frequency ± :

Phonon model: 1D vibrational modes for a linear chain with unequal masses: 1D: m1m1 m2m2 Symmetric and anti-symmetric motion: ± Low frequency (acoustic) and high frequency (optical) solution Equation of motion from F = ma is variation on a harmonic oscillator with multiple solution given by integer k ph = 0, 1, 2, … Frequency ± :

High field effects: Drift + Diffusion Current Densities

High field effects: ECE 474:

Linear slope CurvedZero slope

E to e- then e- to acoustic phonons Interaction with acoustic phonons Interaction with optical phonons

High field effects: E to e- then e- to acoustic phonons Feeding energy to acoustic phonons => more interactions with acoustic phonons Interactions with phonons become significant: when  0 E becomes comparable with speed of sound c s

High field effects: Electron interactions with optical phonons Multiple mechanisms for energy feeding and electron-phonon interactions possible, not simple balance Stated without proof: Empirical relationship for v d for all 3 regimes:

Phonons: Stated without proof: 3D: The total number of acoustic modes = dimension X number of atoms per primitive cell Example: Si: Dimension: 3D Number of atoms per primitive cell: Number of acoustic modes = 6 P. 50: “three acoustic and three optical”: degeneracy

Phonons k = 0 k =  /a Degeneracy in these compact diagrams too: