Lecture_3

# Lecture_3 - electrons in thermal equilibrium have...

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

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”

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

View Full Document
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
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 ( )

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## Lecture_3 - electrons in thermal equilibrium have...

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

View Full Document
Ask a homework question - tutors are online