2.2 Basic Semiconductor Physics

2.2.1 Intrinsic Properties in Equilibrium

Fermi Energy and Carrier Concentration

	In this chapter we deal with basic semiconductor properties and simple devices like pn-junctions on a somewhat simplified, but easy to understand base. The chapter serves to give a good understanding, if not "gut feeling" to what happens in semiconductors, leaving more involved formal theory for later.
		However: Intrinsic semiconductors are theoretical concepts, requiring an absolutely perfect infinite crystal. Finite crystals with some imperfections may have properties that are widely different from their intrinsic properties.
		As a general rule of thumb: If you cannot come up with a material that is at least remotely similar to what it should be in its "intrinsic" state, it is mostly useless because then you cannot manipulate its properties by doping.
		That is the major reason, why we utilize so few semiconductors - essentially Si, GaAs, GaP, InP, GaN, SiC and their relatives - and tend to forget that there is a large number of "intrinsically" semiconducting materials out there. For a short list activate the (German) link.
	Silicon crystals are pretty good and thus are closest to truly intrinsic properties. But even with the best Si, we are not really close to intrinsic properties, see exercise 3.1-1 for that. Nevertheless: This chapter always refers to Silicon, if not otherwise stated!
		A few very basic topics about semiconductors, including some specific expressions and graphical representations will be taken for granted; in case of doubt refer to the link with an alphabetical list of essential topics.
	In this first subchapter we review the properties of intrinsic semiconductors. We make two simplifying assumption at the beginning (explaining later in more detail what they imply):
		The semiconductors is "perfect", i.e. it contains no crystal defects whatsoever.
		The effective density of states in the conduction and valence band, the mass, mobility, lifetime, and so on of electrons and holes are identical.
	All we need to know for a start then is the magnitude of the band gap EG. The Fermi energy then is exactly in the middle of the forbidden band, we can deduce that in two ways:
		Simply by looking at a drawing schematically showing the concentration of electrons in the valence band.
		For ease of drawing, the Fermi distribution is shown with straight lines instead of the actual curved shape.
	The concentration of electrons, ne, in the conduction band is given exactly by
		ne  =  E ' ó õ Ec D(E) · f(E,T) · dE
		With E ' = energy of the upper band edge.
		With the usual approximations:
		Boltzmann distribution instead of Fermi distribution. Substitution of an effective density of states, Neff at the band edge instead of the true, energy-dependent density. Integration from the band edge Ec to infinity we obtain
		ne  =  Neeff   · exp –  EC  –  EF kT
	Neeff  (with the factor two for spin up/spin down included) can be estimated from the free electron gas model in a fair approximation to
		Neeff  =  2 æ ç è 2 pm kT h2 ö ÷ ø 3/2
		How this is done and how some numbers can be generated from this formula (look at the dimensions in the formula above and start wondering) can be found in the link.
		The light blue triangle in the picture symbolizes this concentration!
	All electrons in the conduction band in thermal equilibrium must come from the valence band. The concentration of holes in the valence band, nh thus must be exactly equal to the concentration of electrons in the conduction band, or
		ne  =  nh  =  ni  =  intrinsic concentration
		The dark blue triangle in the picture then symbolizes the hole concentration!
		Given the assumptions made above and the symmetry of the Fermi distribution, the unavoidable conclusion is that the Fermi energy must be exactly in the middle of the band gap.
	The carrier densities are decisive for the conductivity (or resistivity) of the material. If you are not familiar (or forgot) about conductivity, mobility, resistivity, and so on and how they connect to the average properties of an electron gas in thermal equilibrium, go through the basic modules
		Ohms Law and Materials Properties
		Ohms Law and Classical Physics
	We thus have the concentration of mobile carriers in both bands and from that we can calculate the conductivity s via the standard formula
		s  =  e · (µe · ne + µh · nh)
		provided we know the mobilities µ of the electrons and holes, µe and µh, respectively.
	Again, simplifying as much as sensibly possible, with µe = µh = µ we obtain
		s  =  2eµ · Ne eff · exp –  EC  –  EF kT    =   2eµ · Neeff · exp –  Eg 2kT
		because we have EC – EF = Eg/2 for the intrinsic case as discussed so far.
		This gives us already a good idea about the comparable magnitudes and especially the temperature dependences of semiconductors, because the exponential term overrides the pre-exponential factor which, moreover, we may expect not to be too different for perfect intrinsic semiconductors of various kinds.

© H. Föll (Semiconductor - Script)