Chapter IV: Magnetic quasi-harmonic model for Fe and Fe 3 C
4.2 Standard phonon quasiharmonic approximation
Figure 4.2:Phonon energies of orthorhombic Fe3C (cementite). (a)Partial phonon densi- ties of states of57Fe in57Fe3C obtained through NRIXS at 14 K and 463 K.(b)Mean phonon energies vibrational modes of Fe3C. The dashed curve shows the QHA, and the vertical dot- ted line the Curie transition temperature.
I fitted the NFS spectra using the CONUSS software package [7, 8]. The distribution of the hyperfine magnetic field (HMF) around the57Fe atoms was approximated by two Gaussian distributions, corresponding to the two distinct Fe sites in the Fe3C crystal. Fits are shown as solid curves in Fig.4.3a. The mean of these fitted HMF distributions,Bhf, are plotted in Fig.4.3b below the Curie transition, compared to data from the literature. The HMF, Bhf(T), is proportional to the magnetization, M(T), of the Fe atoms in the material [9].
At room temperature we find a mean HMF ofBhf= 20.8 T, in good agreement with [10] and [11]. A mean-field model was used to extrapolate the decreasing magnetic field and determine a Curie transition temperature of fTC = 460K.
It agrees with [12] and is about 20 K lower than [10].
105
Figure 4.3: Temperature dependence of magnetism in Fe3C (cementite). (a)Nuclear for- ward scattering (NFS) spectra of Fe3C at several temperatures. The spectra were fitted using CONUSS, displayed on a log scale, and are offset for clarity. (b)Mean hyperfine magnetic fieldsBhf, from fitted NFS curves of panel a compared to measurements by Le Caër, et al.
[10], and Xiao, et al. [11]. Solid orange curve is a power law fit, and dashed orange curve is the extrapolation to low temperatures using [10].
harmonic oscillator of energy εi =ℏωi can be written as [13]
Zi =
∞
X
n=0
e−(n+1/2)kεiBT, (4.1) where n is the occupancy of the oscillator. Expanding the sum as a geometric series gives
Zi = e−εi/2kBT
1−e−εi/kBT. (4.2)
The assumption that phonons oscillate at fix energies would result in an un- changing phonon DOS. A quick look at Fig.4.1a is enough to realize that the phonon modes of iron do shift considerably with temperature to lower energies. The quasiharmonic approximation (QHA) relaxes this assumption, allowing the phonon energies to shift as function of the unit cell volume. The temperature dependence is included only implicitly through the volume, i.e.,
ω = ω(V(T)), but this is often enough to capture important thermophysical properties of materials, such as thermal expansion.
In the QHA, a Grüneisen parameter, γi, is typically used for describing the fractional change in frequency of mode i per fractional change in volume V
γi =−V0 ωi
∂ωi
∂V , (4.3)
where V0 is an initial volume. If we simplify the problem by using an average γ for all modes,
ω(V) =ω(V0+ ∆V) = ω0(1−γ∆V /V0) . (4.4) A linear dependence between the phonon energy and volume is typically as- sumed in the QHA, giving a constant Grüneisen parameter.
The harmonic phonon free energy1, F, can be obtained from the partition function of Eq.4.2 asF =−kBT lnZ [13]. For a 3D material with N atoms it becomes
F(T) = Z ∞
0
ℏω
2 3N g(ω) dω+kBT Z ∞
0
ln
1−e−kℏωBT
3N g(ω) dω , (4.5) where the phonon DOS, g(ω), is normalized to 1. In the high T limit, this is simplified to
F0(T) = 3N kBT Z ∞
0
ln ℏω
kBT
g(ω) dω . (4.6) For the QHA we useω(V)from Eq.4.4, and then expand the logarithm to its leading term, giving
F(T) = 3N kBT Z ∞
0
ln
ℏω0(1−γ∆V /V0) kBT
g(ω) dω , (4.7) F(T) = F0(T)−3N kBT
Z ∞
0
γ∆V /V0g(ω) dω , (4.8) F(T) = F0(T)−3N γkBT ∆V /V0 , (4.9) where F0(T) is given by Eq.4.6 and the last line used the normalization of g(ω) and an average γ for all modes. Since we are relying on the formalism of the harmonic model (Eq.4.2), phonons in the QHA are still non-interacting
1We refer to the free energy with anF here to be consistent with our publication [14].
It is, however, equivalent to the Gibbs free energyGused elsewhere in this thesis.
107 and there are no 3-phonon processes that give rise to finite phonon life-times and broadening of phonon modes. There are also no anharmonic effects that arise from a direct temperature dependence, and no coupling between different excitations.
Thermal expansion is efficiently predicted with methods of density functional theory that implement the QHA [15]. But the reliability of the QHA is uncer- tain because with true anharmonicity, phonon frequencies have an explicit de- pendence onT andV asω(V, T)[4,5,16], which differs from theω(V(T))used in the QHA [17, 18]. With a volume coefficient of thermal expansion β and a change of temperature∆T, the crystal expands by the amount∆V =β∆T V0. For computing thermal expansion, the total free energy will include an addi- tional term for the elastic energy as
F(T + ∆T) =F0(T)−3N kBT γ β∆T + 1
2B(β∆T V0)2 , (4.10) where B is the bulk modulus. In a typical case with γ ≃ +2, F is reduced by expanding the crystal as the decreasing phonon free energy (2nd term) competes with the increasing elastic energy (3rd term). Minimizing Eq.4.10 gives an equilibrium value of thermal expansion β = γ3N kB/B. With cV = 3N kB/atom in the classical limit, we obtain the widely-stated result
β = γcV
B . (4.11)
Equation 4.11 has been previously used to define a “magnetic Grüneisen pa- rameter” when β andcV are attributed to magnetism (early examples are [19, 20]).
As shown in Fig.4.1b, the QHA describes well the shift of phonon modes be- low 500 K for iron. As we approach the Curie transition at 1044 K, however, the actual shift of the phonon energies is much larger than predicted by the QHA, suggesting an additional contribution to the thermal shifts. Temper- ature causes more interactions between phonons (anhamonicity), as well as between spins. Additionally, atomic displacements due to phonons can affect the exchange interactions between spins, and vice-versa. Such a spin-lattice coupling could be the cause of the deviation from the QHA as the spins dis- order as the Curie transition is approached.
0.75 0.50
0.25
0.00 M/M
00.20 0.15 0.10 0.05 0.00
/
0(a)
m
= 0 .22
m
= 0.07
m
= 0.07
low transv.
high transv.
longitudinal
0.0 -0.2 -0.4 -0.6
M/M
00.000 0.005 0.010 0.015 0.020
/
0(b)
m
= -0.0 29
Figure 4.4:Fractional deviation of the phonon energy from the QHA (∆ω/ω0) vs. fractional change of the magnetization (∆M/M0) for(a)Fe and(b)Fe3C. Magnetization data for Fe are from [21]. Dashed lines and labeled values are results of linear fits to the data, corre- sponding to magnetic Grüneisen parametersγmof Eq.4.12.