The first law is common knowledge to most people and the classical form of the second law is clearly consistent with everyday observation. The third law can be further explained (but not proven) based on the definition provided above. As the temperature approaches zero Kelvin, the thermal energy approaches zero as well. As particles have less thermal energy, they will be more likely to occupy the lowest possible energy states of a given system. This reduces the number of possible configurations since fewer and fewer states can be occupied. As the temperature approaches zero Kelvin, the number of configurations and hence the entropy becomes constant.
1.4.3. The thermodynamic identity
The thermodynamic identity states that a change in energy can be caused by adding heat, work or particles. Mathematically this is expressed by:
where U is the total energy, Q is the heat and W is the work. mis the energy added to a system when adding one particle without adding either heat or work. This energy is also called the electro-chemical potential. N is the number of particles.
The Fermi energy, EF, is the energy associated with a particle, which is in thermal equilibrium with the system of interest. The energy is strictly associated with the particle and does not consist even in part of heat or work. This same quantity is called the electro-chemical potential, m, in most thermodynamics texts.
1.4.5. Some useful thermodynamics results
Listed below are two results, which will be used while analyzing semiconductor devices. The actual derivation is beyond the scope of this text.
- The thermal energy of a particle, whose energy depends quadratically on its velocity, equals kT/2 per degree of freedom, where k is Boltzmann's constant. This thermal energy is a kinetic energy, which must be added to the potential energy of the particle, and any other kinetic energy. The thermal energy of a non-relativistic electron, which is allowed to move in three dimensions, equals 3/2 kT.
- Consider an electron occupying an energy level at energy, E,which is in thermal equilibrium with a large system characterized by a temperature T and Fermi energy EF. The probability that this electron occupies such energy level is given by:
The function f(E) is called the Fermi function and applies to all particles with half-integer spin. These particles, also called Fermions, obey the Pauli exclusion principle, which states that no two Fermions in a given system can have the exact same set of quantum numbers. Since electrons are Fermions, their probability distribution also equals the Fermi function.
|Calculate the energy relative to the Fermi energy for which the Fermi function equals 5%. Write the answer in units of kT. |
The problems states that:
which can be solved yielding:
Finally, we need to introduce the thermal voltage, Vt, the potential an electron needs to traverse to gain an energy equal to the thermal energy kT. This voltage equals the thermal energy divided by the electronic charge, q, of the electron:
The numeric value of thermal voltage in Volt also equals the thermal energy in units of electron-Volt. At 300K (27oC) Vt equals 25.86 mV.
Boulder, August 2007