October 25th, 2010
Class 14
Phonon Heat Capacity
Where U is the energy. The different contributions to the heat capacity can be obtained by taking the derivative with respect to temperature of the corresponding energy. Particularly, the contribution to the heat capacity of the phonon modes, is known as the lattice heat capacity Clat.
Density of States in One Dimension
We found that,
Thus,
Density of States in Three Dimensions
By applying periodic boundary conditions in the same way that we did for 1-D, the number of states inside a sphere of radius K is:
where L3=V
Thus the density of states is then given by:
The problem is now reduced to finding the group velocity (or the dispersion relation) for a particular crystal and to apply the equations above.
Debye Model for the Density of States
In this model, valid for small values of T, the velocity of sound is taken as a constant for each polarization type, thus
and
where
What leads to
This is known as the Debye T3 approximation. The agreement with this law worsens as T approaches the Debye temperature, the law is most approximately met for T within 1-2% of the Debye temperature. Thus the Debye approximation is good for very low temperature (compare to the Debye temperature).
Einstein Model of the Density of States
This model considers N oscillators all vibrating at the same frequency o and thus
D()=Nδ(-o)
For large T, CV~3NkB, the Dulong and Petit Value.
In summary, the Einstein model accounts for the high temperature limit while the Debye model accounts for the low temperature regime.
Anharmonic Crystal Interactions
The harmonic approximation is just that, an approximation and many of the disagreement between the theoretical predictions (within the harmonic approximation) and the experiment can be tracked down to the limitations of this approach. For instance, the thermal expansion predicted by the harmonic model is zero. To see this, the average displacement is
Where the Boltzmann distribution was used to weight the displacements according to their thermodynamical probability. However, the integral on the top is the integral of an odd functions (a function for which f(x)=-f(-x)) integrated is a symmetrical interval, and thus it is zero. The denominator is the intergral of a non-zero, positive defined function what warranties that the denominator does not go to zero. So the average displacement is zero at ALL temperatures and the material is NOT predicted to expand with temperature.
If anharmonic terms are added to the energy (terms of power larger than 2 of the displacement), this prediction changes. For instance
then
If we assume the cubic and quartic terms are very small compared to kBT, we can expand in series to get
and
Leading to
Thermal Conductivity
Without anharmonicity there is no interaction between phonons, without this interaction thermal conductivity would be infinite as shown below.
The steady state flow of heat is given by
This diffusion process requires a tortuous transport as opposed to ballistic, otherwise the flux would only depend on the difference in temperature between the ends of the sample and not on the gradient, (like electrons accelerated by an electric field, their kinetic energy change only depends on the voltage difference and no on how far they move). So this behavior requires “heat carriers” to often collide and scatter.
Using kinetic theory we can find a model for K.
Consider a gas of particles (like phonons) with n particles per unit of volume moving with average velocity in the x direction of , in equilibrium, half will move in the positive direction and half in the negative direction. Thus, the flux of particle is where there is a flux of equal magnitude on each direction.
If c is the heat capacity per particle, each particle will lose an amount of energy of cΔT when it moves from a place at temperature T+ΔT to a place at temperature T. If lx is the distance between collisions, then
where τ is the average time between collision
So, the flux of energy jU
Where nc=C (heat capacity per unit volume), and vτ=l
So
Thus, if there is no interaction with the lattice, l is infinite and the thermal conductivity is infinite.
The interaction occurs because of the third order anharmonic term presence of one phonon causes the modulation in space and time of the lattice elastic constant, a second phonon sees that strain and it is scattered producing then a third phonon in a three-phonon process.
K1+K2=K3
Picture from wikipedia
This process, although it explains phonon collision and finite thermal conductivity, it is still not enough to account for phonon local thermal equilibrium as the total momentum of the phonon gas remains unchanged (thus one end of a metal cannot get in equilibrium with a temperature source) any momentum lost by one phonon goes to another. An interaction with the lattice as a whole is necessary, thus the process must be of the form
K1+K2=K3+G
These processes are known as Umklapp processes. However this is not a forced phenomena, remember that all physically meaningful values for the phonon wave number Kmust be in the first brillouin zone, it is possible that the K1+K2 leads to a phonon with wave number K3 outside the zone, but that is not a physically meaningful state, thus a G vector must be used to bring the phonon back into the first Brillouin zone. Looking at this issues from a different point of view, when two colliding phonons that have a wave number so large that the sum will lead to a momentum outside the first Brillouin zone, the process is not possible as a phonon with such a momentum cannot exist and the lattice as a whole must absorb some momentum, thus the total momentum of the phonon gas changes and the phonons can thermalize. At high T, the population of high momentum states is very large and Umklapp processes dominate. In this regime the number of phonon is proportional to T (See Einstein model) and the number of collision is proportional to the population, the mean free path is proportional to 1/T, thus thermal conductivity is proportional to 1/T. At low T, most phonons have low enough momentum that it is very unlikely that in a collision a phonon is created with momentum outside the first Brillouin zone and thus heat conduction is dominated by normal processes.
Free Electron Theory
Many of the properties of metals can be understood if we consider the electrons in the metal as particles that are free to move in a box. This model it is clearly an approximation since it assumes two completely wrong hypothesis, 1) electrons do not interact with each other, 2) electron do not interact with the lattice. Both are totally wrong even in classical terms since electrons and the lattice are charged, electrons are negative and the lattice bears a net positive charge. Surprisingly, this theory explain acceptably well some of the properties of metals as we will see below.
Consider alkali metals, Li, Na, K, Rb, Cs. All of them have the structure of an inert gas plus 1 electron on a s level. For these metals it is not hard to share that electron and in treating these system theoretically, it is assumed each atom contribute with 1 electron to the electron gas. All those electrons now navigate freely around the entire metal.
Classical theory of metals, produced long before quantum mechanics (Drude’s model) surprisingly predict fairly well conduction properties, ohms law, thermal conduction and the relationship between them (although the proportionality constant was a little bit off compared to the experiment). However it overestimates by about 2 orders of magnitudes the electronic contribution of the heat capacity. How can this be possible? The answer is “cancellation of errors”, the heat capacity is proportional to the number of electrons that are able to absorb thermal energy, while conduction properties is proportional to that and to the average kinetic energy of those electrons. The classical theory (as we will see later) overestimate the number of electrons that can actually absorb thermal energy in about 2 orders of magnitude (thus the heat capacity is overestimated) but it underestimates, by about the same factor, the energy of those electrons and thus, by coincidence, conductive properties are very well predicted.
Electrons in a one-dimensional infinite box
The closest problem to the one in hand is that of electrons in a box, a typical problem in quantum mechanics.
Any particle is characterized by a kinetic energy and a potential energy. Classically that means that the total energy of the system is given by:
Where H is known as “the Hamiltonian” or energy operator, p is the particle’s momentum, m is its mass and V is the potential acting on the particle.
In quantum mechanics, p can be represented by the operator p=–iħ d/dx where the action of this “operator” p is to take the derivative of the function and then multiply that by the constant–iħ. p2 means applying this operator twice, thus
The above is known as the Schrödinger equation and all that there is to know about the electron can be obtained from the equation above (except that this equation does not have an exact solution for most potential functions)
For the electron in a box case, the potential energy is defined as V=0 inside, V=infinite outside
What lead to the equation:
With boundary conditions that force the wave function to be zero at z=0 and z=L (where L is the box width)
The condition for that the wave function needs to be zero for z=0 is satisfied proposing a solution of the form:
The condition that the wave function must be zero at L is satisfied if k=nπ/L or,
=> k=2/n
Thus the solutions are quantized and the allowed energy values are given by
Normalization of the wave function
Energy difference between two consecutive levels increases as n increases
Transition between levels leads to absorption or emission of a photon with energy equal to the difference between levels.
The above solutions correspond to a single electron in an infinite well. If we consider now that the metal have N electrons (assuming 1 per atom), and invoke the Pauli exclusion principle, then we can only accommodate 2 electrons per energy level and thus only N/2 levels will be filled with electrons. The energy level with the maximum energy that holds electrons is known as the Fermi energy and it is given by
Class activity 5.
Effect of Temperature
The picture described above, where energy levels are being filled one by one (with two electrons each) until we run out of electrons, is only valid at zero Kelvin. As the temperature increases, some electrons with energies of the order of kBT around the Fermi energy can be promoted to higher energy levels and thus some levels above the Fermi level are occupied while some levels below the Fermi level are empty. The Fermi-Dirac distribution accounts for that effect:
where is the energy and is the chemical potential (we will define it in a little bit). Notice that as T tends to zero, the equation above tends to two different values, 0 or 1, depending on the relative values of and, for the exponential goes to zero while for the exponential goes to infinity. As the temperature increase, this function decreases smoothly as increases.
The Fermi-Dirac distribution is a probability distribution; it indicates the probability of a given energy level to be occupied. At 0K, the probability of occupation of energy levels below the chemical potential is 1 (all are occupied) while the probability of occupation of energy levels above the chemical potential is zero (all empty). The chemical potential is an empirical parameter defined as the energy where the probability of occupation is exactly ½. For very low temperature kBT<, f
We can already see here the reasons for the discrepancy between the experiment and the Drudes model. Due to the Pauli Exclusion Principle, not all the electrons can increase their energy due to temperature, only those near the Fermi level, this account for the 2 orders of magnitude in the heat capacity. By virtue of the Pauli Exclusion Principle also, those electrons that can absorb energy have a much higher energy than expected by the classical theory since they are forced to be in higher energy levels, both effects, small number of electrons with higher energy approximately cancel each other given conduction properties in reasonable agreement with the experiment
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