We must now pray to the neutron gods.
Camille Bernal and Claire Saunders
Phonon Anharmonicity in Germanium 3
from 296 to 1203 K
on volume. For any given temperature, a material will expand if its free energy is lower at a larger volume. Thermal expansion creates a force against the hypothetical springs keeping atoms together, increasing the E term in the free energy. This expansion will force against the springs, increasing the E term in the free energy. However, there may be a gain in the temperature multiplied by entropy (TS), which allows for the free energy to be reduced. To find the optimal volume we minimize∂V∂F.
We can further write,
∂F
∂V =−P= ∂E
∂V −T∂S
∂V. (3.1)
Therefore, ifP=0,
T∂S
∂V =T∂E
∂V. (3.2)
The left hand side is approximately the elastic pressure and the right hand side is often called a thermal pressure. To findV(T)we end up having to balance the elastic pressure with the entropy change withV(andT).
We are therefore very interested in understanding how the entropy of a material varies with VandT. We had previously noted in Chapter 2 that most of the entropy stems from the vibrational entropy of phonons. We had also noted that the phonon entropy contribution, and the phonon contribution to other relevant thermodynamic quantities, can be calculated from the density of states, which is essentially a spectrum of phonon frequencies. These phonon frequencies shift in both the quasiharmonic and anharmonic model, as described in the previous chapter.
Understanding how and why this spectrum of phonon frequencies varies as a function of volume and temperature not only allows us to calculateSvib(V,T), but also gives us a deeper
understanding of the origins of thermal expansion. Let us examine the phonon frequencies more closely.
We can write the frequency,ω, as a function of volume and temperature. Volume tells us generally how far apart the atoms are, which of course, affects the potential in the hypothet- ical springs between atoms. Temperature tells us how far the atoms move about their initial equilibrium positions. Expandingω=ω(V,T)to first order yields
dω = ∂ω
∂V
T
dV+ ∂ω
∂T
V
dT (3.3)
The first term in the expansion is related to the Gruneisen parameter, which we discussed before in Chapter 2. We rewrite the Gruneisen parameter,γ, here as,
γ≡ − V
ω
∂ω
∂V
T
(3.4) The next term is related to the anharmonicity parameter, which is essentially the fractional change in frequency as a function of temperature. We define the anharmonicity parameter as,
A≡ − 1
ω
∂ω
∂T
V
(3.5) These two definitions allow us to separately think about the two theories we discussed in the previous chapter. In a purely quasiharmonic theory, the phonon modes are harmonic, but their frequencies must be shifted with volume. The only role of temperature is to expand the volume. The frequencies would therefore shift the same way if the volume change was induced by varying the ambient pressure instead of the temperature.
A purely anharmonic theory, on the other hand, has frequencies that are only depen- dent on the temperature. Even if the volume remains fixed, the frequencies shift due to the increased excursion distance of the atoms about their equilibrium positions, leading to phonon-phonon interactions.
Usually, however, we have both quasiharmonic and anharmonic contributions to the shift- ing frequencies, so we writeω(V(T),T). The big question is then, for any material, what are the relative effects to the phonon frequencies of quasiharmonicity and anharmonicity? Can we characterize a material as more quasiharmonic-like or more anharmonic-like?
In an ideal world, we would like to be able to say that materials with certain properties exhibit high levels of anharmonicity, while others with other properties exhibit high levels of quasiharmonicity instead. This is not the case in the real world. It is often hard to get a sense of how a material will behave before doing computational or experimental studies.
In this thesis we examine the deviations from harmonicity of Germanium, an element with atomic number 32. Ge is a hard-brittle, lustrous, grayish-white metalloid. It is in group 14 and period 4 on the periodic table. It is chemically similar to its two neighbors, tin and silicon. The thermal properties of Ge are important for uses in fiber-optic systems, solar cell applications, light emitting diodes, and infrared optics.
Both the quasiharmonic and anharmonic approximation predict shifts in phonon fre- quencies. We would like to understand the role of both quasiharmonic and anharmonic effects in generating these frequency shifts.
Before proceeding, it is important to note a common misconception about phonon the- ory. It is frequently stated that we can sufficiently use a harmonic model at low tempera- tures, a quasiharmonic model at moderate temperatures and an anharmonic model at high
temperatures. The issue here is, usually, the frequency shifts from quasiharmonicity, due to expansion of the volume, grow linearly with temperature. In anharmonic theory the pre- dicted frequency shifts are also linear with temperature. This means that if anharmonicity dominates at high temperature, then it will dominate at all temperatures (fig 3.1). It is there- fore not so simple as using a specific theory at a specific temperature. The real, fundamental question is which one dominates at all temperatures.
Figure 3.1: Illustration idea from Prof. Brent Fultz. Note that if anharmonicity dominates the contribution to the frequency shifts at high temperatures, it does so at all temperatures as well.
There is one redeeming quality, however, about the prior misconception. It is that anhar-
monic effects, if they exist, will definitely be more noticeable at higher temperatures. In fig 3.1 it is clear that the difference in delta omega between the two theories is greater at higher temperatures. High temperature studies of phonon behavior are therefore critical to our un- derstanding of the role of anharmonicity in materials.
High temperature studies are also important because most of the entropy of materials stems from phonons. We can recover the phonon contribution to thermodynamic quanti- ties from the density of states. For example, in eq. 2.29 we used it to calculate the vibrational entropy. We would like to determine how this phonon contribution varies as a function of temperature. Recall that eq. 2.29 was actually derived using the harmonic approximation.
It assumed, however, that we would have accurate densities of states (spectrum of phonon frequencies), even at higher temperatures.
A naive method of calculatingSvibat higher temperatures would therefore be to keep the density states constant as temperature increases, and rely solely on the Plank distribution to generate the changes inSvib. A better method would be to actually experimentally determine the phonon DOS at higher temperatures, and substitute that in for g(e) in equation 2.29.
This would allow us to determine the accurate contribution ofSvibto the total entropy. This has indeed been done for several materials at lower temperatures, including germanium. It is therefore well known, at least at low to moderate temperatures [7], that most of the entropy of materials stems fromSvib. This was discovered partly through previous experimental work that employed INS to study the low to moderate temperature lattice dynamics, but to our knowledge, there has not been an INS study of phonons in Ge up and through the melting point.
Our understanding of the phonon spectra at high temperatures is still emerging. In this
section of the thesis we present an analysis of the Ge phonon spectra at temperatures up to 1203 K. To reiterate, the merits of this high temperature analysis are two-fold. It will 1: allow us to better differentiate and elucidate the quasiharmonic and anharmonic contributions to the softening of phonon frequencies (because these differences are more apparent at high temperature) and 2: allow us to test whether or not the statement that most of the total entropy in Ge can be accounted for by harmonic-theory-derived vibrational entropy remains valid at higher temperatures.