Solar Cells 2
shoomoosh
class06SolarCells2020-08.Densityofstatesandchargecarrierenergydistributions.pptxDensity of states and
charge carrier energy distributions
Prof. Richard R. King
Solar Cells
EEE 565
Arizona State University
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Thank you !
Many thanks to
Prof. Steve Goodnick
for his slides on solar cells which appear throughout this class
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For a general semiconductor crystal with dimensions x = L, y = L, z = L,
eventually the crystal size imposes a finite number of values on the possible k-values an electron may assume, where k is the electron wavevector.
We are interested in:
If we consider that the system is bounded within a volume, V, and apply periodic boundary conditions
Thus, k takes on discrete values, with a volume per cell in k-space of
Density of states
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Each state can have either spin up or spin down so an allowed state in k-space can hold 2 electrons. The volume in k-space per state is:
Volume of k-space between k and k + dk is given by:
Density of states
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Energy vs. wavevector (E vs. k) diagrams
Direct gap
Indirect gap
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For parabolic bands
Therefore, the density of states between E and E + dE is found by
Or finally:
Conduction band
Valence band
Density of states
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The probability that a state is occupied by an electron is given by the 1 particle distribution function f(k)
In equilibrium, f(k) is only depends on energy, and for electrons (i.e. particles obeying the Pauli exclusion principle, or Fermions), is given by the Fermi-Dirac distribution
where Ef is the Fermi energy, kB is Boltzmann’s constant, and T is the absolute temperature
Semiconductor statistics
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To find the electron density in the conduction band (CB), we multiply the density of states by the probability of the state being occupied, and integrate over all states in the CB
Semiconductor statistics
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Density of States and Electron Energy Distribution
Density of states g(E)
Fermi-Dirac distribution f(E)
Electron energy distribution n(E)
T = 0
Electron energy distribution n(E)
T > 0
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For holes in the Valence Band (VB), the probability of an unoccupied state is (or hole) is
Semiconductor statistics
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If the Fermi energy is more than several kT from the band edge, the Fermi-Dirac distribution is approximated by a Maxwell-Boltzmann distribution:
The expressions for n and p become
is the effective density of states
Semiconductor statistics
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For an undoped semiconductor in equilibrium:
The intrinsic concentration then becomes
The intrinsic Fermi level, Ei is found to be basically at mid-gap, and is used as a reference level
small
Semiconductor statistics
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• The bottom of the conduction band and the top of the valence band represent the lowest energy states for electrons and holes respectively. In real space, we often draw band diagram showing Ec and Ev as a function of position.
VL
• The vacuum level, VL serves as a reference
• is the electron affinity, and is characteristic of each semiconductor
Energy bands in position space
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• Impurity atoms in the crystal lattice introduce localized states in the middle of the gap
• Impurities with a valence higher than the net valence of crystal behave as donors, giving up a free electron to the CB
• Impurities with a valence lower than the net valence of crystal behave as acceptors, taking an electron from the VB, leaving a free hole
donor
acceptor
Donor band (high density)
Impurity levels: Donors and Acceptors
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For shallow donors, complete ionization occurs at room temperature, and the electron concentration in the conduction band is determined by the external doping
Likewise, for shallow acceptors, they are easily ionized by accepting and electron from the valence band, leaving behind free holes
In the general case, in a bulk material, charge neutrality must hold
Doped semiconductors: n- and p-type
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Amorphous materials have substantially different electronic properties than the crystalline form: different band gap, mobility, etc.
Crystal structure of a-Si has no long range order, changing its material properties
Eg = 1.7 eV, mobilities are lower than crystalline silicon
Bandgap in an amorphous material is given by mobility gap
Absorption coefficient is closer to direct band gap than indirect
Amorphous materials
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16
Core shell nanocrystal
A. P. Alivisatos, J. Phys. Chem. 100, 13226 (1996)
Semiconductor Nanocrystals
Quantization becomes important when the nanocrystal radius is less than the effective Bohr radius
Simple model: Hard spherical shell, V = 0 inside, infinite outside, hydrogen like
CdSe Nanoparticle
Nanoparticles and nanocrystals
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A. P. Alivisatos, J. Phys. Chem. 100, 13226 (1996)
Nanoparticles and nanocrystals
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Classified as either small molecule or polymeric
organic semiconductors
States are described in terms of molecular orbitals
Essentially amorphous materials, no long range order
Photoexcited charge carriers are excitons, i.e., bound electron-hole ‘molecules’
Organic materials
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