Lecture_3

Lecture_3 - electrons in thermal equilibrium have...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Prof. J. S. Harris 1 EE243. Semiconductor Optoelectronic Devices (Winter 2010) electrons in thermal equilibrium have Fermi-Dirac distribution, probability f e ( E,T ) of electron in state of energy at temperature is (1.19) μ is the chemical potential (or, the Fermi energy E F ), k B is Boltzmann ʼ s constant Note that k B T ~ 25 meV at room temperature chemical potential (Fermi energy) corresponds to energy for which f e = 1/2 At zero temperature Fermi- Dirac distribution is step function, all states up to E F totally full f e E , T ( ) = 1 1 + exp E μ k B T Semiconductor statistical mechanics - electron distribution Prof. J. S. Harris 2 EE243. Semiconductor Optoelectronic Devices (Winter 2010) Holes correspond to the absence of an electron Hence the probability of ±nding a hole in a given state is the probability that there is NOT an electron in that state Hence f h = 1 - f e and so (1.20) (1.21) f h E , T ( ) = 1 1 1 + exp E μ k B T = exp E μ k B T 1 + exp E μ k B T f h E , T ( ) = 1 1 + exp E μ ( ) k B T Semiconductor statistical mechanics - hole distribution Hence the holes also have a Fermi-Dirac distribution difference - look at hole energies “upside down”
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 Prof. J. S. Harris 3 EE243. Semiconductor Optoelectronic Devices (Winter 2010) For energies far above the chemical potential, Fermi-Dirac distribution behaves like the classical Maxwell-Boltzmann distribution (very low occupancy), f M-B i.e., for electrons, (1.22) Where This is to be expected, since at these high energies, the occupation probabilities are small hence virtually no in±uence of the Pauli exclusion restriction that no more than one electron may occupy a state. f M B E , T ( ) = A exp E k B T A = exp μ k B T Semiconductor statistical mechanics - classical limit Prof. J. S. Harris 4 EE243. Semiconductor Optoelectronic Devices (Winter 2010) Position of Fermi level or chemical potential depends on number N of electrons in system and the energy distribution of available states. In general, the number of electrons, n(E) , per unit energy interval is (1.23) i.e., n ( E ) is the product of the number of electron states per unit energy, g ( E ), and the probability of the states being ²lled, f(E,T) hence, the total number of electrons in the system, N , is (1.24) The chemical potential can be viewed as a normalization parameter chosen so that the integral in Eq. (1.24) gives the correct N n E ( ) = f E , T ( ) g E ( ) N = f E , T ( ) g E ( ) dE Semiconductor statistical mechanics - setting value of μ
Background image of page 2
3 Prof. J. S. Harris 5 EE243. Semiconductor Optoelectronic Devices (Winter 2010) The single most important form of density of states is the parabolic band approximation, (Eq.(1.18)) Fermi energy In general at zero temperature, f(E,0) = 1 from E = 0 up to E = E F , and is zero for all energies E > E F hence, for parabolic bands (integrating Eq. (1.24) gives (1.25) Hence, the Fermi energy, E F , becomes (1.26) N = 1 2 π 2 0 E F 2 m * 2 E dE = 1 3 2 2 m * 2 E F E F = 2 2 m * 3 2 N ( )
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 12

Lecture_3 - electrons in thermal equilibrium have...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online