ECE 4339 L. Trombetta ECE 4339: Physical Principles of Solid State Devices Len Trombetta Summer 2007 Chapter 2: Carrier Modeling Goal: To understand what this figure means.

ECE 4339 L. Trombetta Electrons bound to a Si atom can orbit the nucleus at certain radii, and with certain (quantized) energies. In Si, there are electrons in the first three “shells”. Every atom has an infinite number of energy levels into which an electron could go. The “ground state” of the atom has the electrons in the lowest levels, as shown here for Si. Note there are six allowed levels in the p-subshell, but only two are occupied by electrons. The s-subshell, which has only two allowed levels, is full.

ECE 4339 L. Trombetta A set of quantum numbers n, m, l, s for an electron indicate everything we need to know about the electron, including energy and wave function. The quantum numbers “index” the allowed solutions to the Schrödinger equation for the H atom. That is, the solutions are functions that depend on these values of n, l, m, s.

ECE 4339 L. Trombetta Preliminaries: Bonding Types Ionic Bonding: electrons leave one atom for another in the molecule. The resulting Coulomb attraction keeps the molecule together. Example: NaCl [1] [1] Let’s look at these elements…

ECE 4339 L. Trombetta Covalent Bonding: electrons spend some time near the other atom; kind of a time-sharing. Example: Cl 2 [2] [2]

ECE 4339 L. Trombetta Metallic Bonding: Outer electrons “delocalize” from the atom cores; they are “everywhere”. The electron and nuclear charges hold each other together. Example: Na [3] [3]

ECE 4339 L. Trombetta Another important questions we can ask about the electrons in any molecule or solid is this: What energy levels are available to them? The same question can be asked about individual atoms: what energy levels are available? The answer is that there are discrete energy levels corresponding to the various orbits, as we have seen.

ECE 4339 L. Trombetta The curves represent the potential experienced by an electron attached to a positive atom core: V(r) ~ 1/r. The lines represent possible electron energy levels in the atom. To answer that question about solids, we take another view of electrons: What happens when we make a solid out of many of these? energy

ECE 4339 L. Trombetta The outer-most electron levels are now associated with the entire crystal rather than with a single atom. These levels have broadened into energy bands. The inner electrons are still bound to the individual atoms, although they will also spread a little. Here is a solid…

ECE 4339 L. Trombetta The quantized atomic energy levels “broaden” into energy bands when the atoms combine to form a solid. Pauli Exclusion Principle: only 1 electron with a given set of quantum numbers can be in a “state” (2 with opposite spin). N = number of atoms in the solid Let’s look more closely at this…

ECE 4339 L. Trombetta Recall the isolated Si atom…

ECE 4339 L. Trombetta conduction band valence band conduction band edge valence band edge For our purposes we can simplify this picture: ECEC EVEV EGEG Many closely spaced levels

ECE 4339 L. Trombetta

Donors and Acceptors Bonding model Energy Band model

ECE 4339 L. Trombetta Summary We can describe the energy levels available to electrons in solids in terms of “energy bands”. Energy bands are populated by electrons; the absence of an electron in a mostly filled band is best described in terms of “holes”. Some electrons and holes exist in “intrinsic” semiconductors, but we can dramatically increase their numbers by “doping” with donors and acceptors. Flow of electrons and/or holes is a current. Some numbers Intrinsic carrier density: n i = p i ~ 1 x cm -3 (Si, E g = 1.12 eV) ~ 2 x 10 6 cm -3 (GaAs, E g = 1.42 eV) Dopant densities in Si: < N D, N A < cm -3 N D = #/cm 3 of donors N A = #/cm 3 of acceptors

ECE 4339 L. Trombetta Density of States: The Concept Professor’s Offices/Labs basement 1 st floor 2 nd floor 3 rd floor 4 th floor The chart represents the number of offices and labs for professors in ECE on each floor. It is a “density of states” for offices and labs. Note that any given office/lab may or may not actually contain a professor.

ECE 4339 L. Trombetta The Density of States g(E) dE answers the question “How many electron energy levels are there per unit volume (cm 3 ) in the energy range E to E + dE?” This is what the density of states looks like for semiconductors. Here we are plotting energy vs. number of states per cm 2 per eV. A simple “derivation” of the density of states from QM is presented in my notes, for 1-D and for 3-D. You will not be responsible for knowing this.

ECE 4339 L. Trombetta Probability of Occupancy: The Concept Professor’s Offices/Labs basement 1 st floor 2 nd floor 3 rd floor 4 th floor The chart shows the number of offices and labs that actually have professors in them. Note that the vending machines are on the first floor.

ECE 4339 L. Trombetta The Fermi-Dirac Distribution Function f(E) answers the question “What is the probability of finding an electron at an energy level E?” Most of the change takes place over a range of +/- 3 kT about E F. A simple “derivation” of the Fermi function from statistical mechanics is presented in my notes. You will not be responsible for knowing this.

ECE 4339 L. Trombetta

Q: What exactly is the “Fermi Level”? A: (hmmm….) An Italian surveying instrument. A: (ahhhh…) A parameter of the Fermi distribution function: At E = E F, f(E) = ½ At absolute zero T, all the electrons are at or below E F. At higher T, electrons just below E F are moved to just above E F.

ECE 4339 L. Trombetta Putting it all together… Energy band diagram showing Fermi Level Density of states: how many levels as a function of E? Fermi function f(E): probability of finding an electron The product g(E)*f(E) Area under the curve is the number of electrons f(E) 1 - f(E)

ECE 4339 L. Trombetta Here we have multiplied the Fermi function by the density of states and integrated over energy. We have superimposed the result on the our simplified energy band diagram. For the valence band we have to multiply (1 – f) by the density of states because we are dealing with holes.

ECE 4339 L. Trombetta Here we have multiplied the Fermi function by the density of states and integrated over energy. We have superimposed the result on the our simplified energy band diagram. For the valence band we have to multiply (1 – f) by the density of states because we are dealing with holes.

ECE 4339 L. Trombetta Figure 2.20 Intrinsic carrier concentration n Ge, Si, and GaAs as a function of temperature. Values of ni for Si and GaAs from the graph at a few convenient temperatures are also indicated.

ECE 4339 L. Trombetta