Topic 11-1: Heat and Light for Intrinsic Semiconductors Summary: In this video we aim to discover how intrinsic semiconductors respond to heat and light. We first look at the response of semiconductors as the temperature is increased, in terms of the density of states and the Fermi Dirac distribution. We then look at the different types of band gaps and conservation laws related to each type of transition, direct and indirect. From this we discover that while a direct transition can occur without aid an indirect transition requires a third particle, namely a phonon. We end this video by looking at absorption coefficients for each type of transition. Things to remember o Started with a single electron in a flat bottomed box o This gave a decent approximation for properties like electrical conductivity which was found to be! =!!!!!! o Discovered nuclei create potential wells and resolved Schrodinger s equation o Found!!! =!!!!!!! Got a new dispersion that is no longer parabolic From here we define metals, semimetals, semiconductors and insulators
We will now focus on intrinsic semiconductors Intrinsic semiconductors: intrinsic materials have no significant doping, they are pure, this means their properties are intrinsic, that is to say their properties don t arise from impurities Goal: describe how intrinsic semiconductors respond to heat and light We already know how the density of states of a semiconductor is filled at 0 kelvin with the Fermi level in the middle of the gap As we raise the temperature the Fermi Dirac function will smear meaning the electrons will be thermally excited into the next available state The bubble below the gap is a collection of empty states as the electrons are thermally excited Now we need to introduce some vocabulary to make more sense of our band diagrams Both the valence and conduction band contribute to conduction
Band gap is the lowest energy transition o Example shown in the diagram of silicon below Band gap for silicon is indirect because the maxima and minima don t line up on the same k value Direct band gaps occur when the maxima and minima line up on the same k value, as in the GaAs band structure diagram below Now look at optical transitions For a direct band gap o Send in a photon of energy E γ o Can only excite an electron from the valence band to the conduction band if E γ is greater than or equal to the energy of the band gap Indirect case is a little more complicated Electron must change both energy and crystal momentum
To understand this lets go back to energy and momentum conservation laws For the direct gap transition o Energy conservation:!!! =!!"#$%&$ +!! o Momentum conservation:!!! =!!"#$%&$ +!! Example: Green light o!! =!!!""!" Where!!! is the final state and!!"#$%&$ is the initial state o!!" =!! for a 5 angstrom cell!.!!" o!!!!" ~!!""" o Photons provide little momentum to the electron even when they have significant energy For the indirect case we need a third particle o Use a phonon as the third particle o Can create or destroy them at will Go back to silicon example Phonon provides change in crystal momentum o If we use a phonon going in the direction of the induced momentum we destroy a phonon o Could create a phonon by going in the other direction
o Both satisfy the momentum equation Conservation laws for indirect transition Phonon destruction o!! +! +!!!!"#$%!& o Electron eats up phonon as well as photon energy Phonon creation o!! +!!!!"#$%!& +! o Photon creates phonon while exciting electron In either case we have a three particle process with a low probability of occurring To determine a materials band gap we simply shin light on it and see what it absorbs Use Beer s law to go from absorbance to an absorption coefficient as a function of the energy of incoming light o!!,! =!!!!!!! o z= depth, λ=wavelength, I o = initial intensity, α= absorption coefficient
Absorption coefficient for direct and indirect gaps (see figure below) o Direct transition has a sharp edge where the bad gap begins because any photon with energy below this won t be able to excite electrons o Absorption coefficient increases with energy because as energy increases more above states are available for the electron to jump into o For indirect band gaps at sufficiently high energies we can excite electrons across the gap and we see direct behavior o Before this we are relying on the low probability three particle process and therefore have a low absorption coefficient o Direct! h!!! o Indirect! (h!!! )! ignoring phonon energy Example: Solar cell materials o Silicon Low absorption coefficient To capture sunlight need a thick film (~100 micrometers) o GaAs Strong absorption coefficient Only ~ 5 micrometer film thickness required to absorb sunlight
Questions to Ponder 1. If we have intrinsic silicon and hit it with intense light what happens to the effective optical gap as you increase the intensity of the light? 2. What happens to electrons excited into the conduction band when you apply an electric field? 3. Consider a semiconductor with a band gap of 1 ev. Plot the thermal smearing of electrons for three different temperatures. How deep into the conduction band are electrons excited?