3DDensityofStatesThedensityofstatesVIP专享

3D Density of States The density of states refers to the number of quantum states per unit energy. In other words, the density of states, denoted by ( )g E , indicates how densely packed quantum states in a particular system. So, what is the importance of the density of states? Consider the expression ( )g E dE . Integrating the density of the quantum states over a range of energy will produce a number of states. ( )( )EEN Eg E dE  Thus ( )g E dE represents the number of states between and EdE . The number of quantum states is important in the determination of optical properties of a material such as a semiconductor (i.e. carbon nanotubes as well as quantum dots). From the Schrodinger equation, we know that the energy of a particle is quantized and is given by 222kEm The variable k is related to the physical quantity of momentum. A particle’s energy is 22221222m vpEmvmm Relating the previous two equations yields 22222kppEkmm The momentum is a vector which has components in the x, y, and z directions. Therefore, k must also have direction components, , and xyzkkk . Since energy is not a vector, the more accurate expression for energy is 222kEm In a 3D system, then, the total energy is given by 22222xyzEkkkm Recall the result of the analysis of the 1D potential well. An electron can only exist in the well, and the wave function is given by ( )cos()sin()where:xAkxBkxnka where a is the width of the barrier. For the cosine term in the wave function, n must be an odd integer, and for the sine term, n must be an even integer. Therefore, the wave function is only valid for all integers greater than zero. In three dimensions,...