Energy density functional description

22 Energy density functional description the aforementioned avoided crossing. This corresponds to the frequency region in which ₁ is negative, and the real part n of the refractive index is zero. Even if it is called Ω_LO, the upper limit of the reststrahlen band does not correspond to a longitudinal mode to which the light is coupling to. As a matter of fact light, propagating in air as a transverse wave, cannot couple to longitudinal modes in a bulk crystal (note that in Fig.

1.7 no line is displayed at Ω_LO). An important feature of the phonon-polariton dispersion relations is the fact that at frequencies close to the reststrahlen band, where the behavior is phonon-like, the group velocity vg =∂ω/∂k is very small. This implies that at those frequencies polaritons can propagate in the material, but very slowly. These aspect has to be taken into account when the optical properties are measured in the time domain, like in the experiments described in this thesis. The electric field reflected from the material at these frequencies it is going to be detected at later times compared to that reflected in reststrahlen band.

Energy density functional description 23

L= 1

2Ω²_{T O}Q²− 1

2₀(ε∞−1)E²−Ω_{T O}p

₀(ε₀−ε∞)QE. (1.36) In this equation, the first term is the energy of the mechanical oscillator with eigenfrequency Ω_{T O} and normalized coordinate Q. The second term accounts for the energy of the electric field in the solid at frequencies far away from the oscillator eigenfrequency. This can be easily shown recalling that from eq. 1.23 follows ε_∞−1 = χ, which is the susceptibility leading to the background polarization introduced in section 1.2. Finally, the third term in eq. 1.36 is taking into account the linear interaction between the oscillator Q and the electric field E.

From this energy density functional it is possible to retrieve the force acting on the mechanical oscillator:

F_Q =−∂L

∂Q =−Ω²_{T O}Q+ Ω_{T O}p

₀(ε₀−ε∞)E. (1.37) Combining the definitions ofε₀ andε∞ from eq. 1.23 it is possible to write:

Ω²_{T O0}(ε₀−ε∞) = N Z²

µ (1.38)

and substituting into 1.37 leads to:

FQ=−Ω²_{T O}Q+p

N/µZE, (1.39)

which is equivalent to Eq. 1.13 derived in section 1.2 for the case of an undamped oscillator.

The polarization P can also be derived from the energy density func-tional via the relation:

P =−∂L

∂E =₀(ε∞−1)E+ Ω_{T O}p

₀(ε₀−ε∞)Q. (1.40)

24 Energy density functional description The first term in this equations gives a polarization equivalent toP =₀χE, which in section 1.2 was referred to as P_∞. The second term in eq. 1.40 leads, substituting eq. 1.38, to:

P =p

N/µZQ=p

N/µp

N µ W Z =N ZW, (1.41) which is equivalent to P_phonon introduced in section 1.2.

Chapter 2

Phonon nonlinearities and amplification

The focus of this thesis is the investigation of the phonon response to very intense laser pulses, capable of displacing the ions by up to a few percent of the equilibrium interatomic distances. Under these circumstances, the lattice response is expected to exhibit a nonlinear behavior that the Lorentz model fails to describe. Therefore, an expansion of such model capable of describing the nonlinear optical properties of an infrared active mode is presented.

First principle DFT calculations are used to explore the nonlinear re-sponse of the lattice to very large applied static electric fields and conse-quently large ionic displacements.

These results are then incorporated in a model describing the dynamic response of the phonon to electric fields oscillating at frequencies close to its resonance and showing that a parametric amplification of the phonon coordinate oscillation is expected.

25

26 Nonlinear expansion of the polarization

2.1 Nonlinear expansion of the polarization

The polarization induced in a material by an applied electric field entails two contributions, namely the resonant contributionP_phonon due to the dis-placement of the charged ions and the background contributionP∞ascribed to the screening of other phonons and electrons. When the electric fields interacting with the solid are very intense and ions displacement Q be-comes large, the polarization bebe-comes non-linear, and both these polariza-tion terms have to be expanded.

Figure 2.1: (a) Resonant phonon contribution to the polarization, P_phonon. For small phonon coordinate displacements, the polarization is linear in Q, and the Born effective charge Z^∗ =∂P/∂Q (b) is constant.

When the displacement increases, the polarization deviates from linear, with a consequent increase ofZ^∗. It is important to note that the Born effective charge increase is quadratic in Q, i.e. it is independent on the direction of the phonon coordinate displacement.

The first contribution to the nonlinear polarization,P_phonon, is proportional to the mode coordinate Q through the effective dipolar charge Z^∗ that embodies the oscillator strength of the phonon. Such Born effective charge is defined as Z^∗ = ∂P_phonon/∂Q and is a constant for small values of Q,

Nonlinear expansion of the polarization 27 where the polarizationP_phonon is linear in the phonon coordinate, as shown in Fig. 2.1. In this case:

P_phonon =Z^∗Q, (2.1)

which is equivalent to eq. 1.41, where the chargeZof the Lorentz model has been re-normalized to account for the mode effective mass µand the num-ber of oscillators per unit volume N, leading to the Born-effective-charge Z^∗ =p

(N/µ)Z. For very high driving electric fields,P_phonon depends non-linearly on the lattice displacement Q, as calculated from first principles DFT calculations and shown in Fig. 2.1a. Correspondingly, the Born ef-fective charge becomes a function of Q as shown in Fig. 2.1b. For SiC Z^∗ depends quadratically on the lattice coordinate Q as

Z^∗ = ∂P

∂Q =Z₀^∗ +αQ². (2.2)

Importantly, the sign of the change in Z^∗ does not depend on the direction of the phonon displacementQ. Hence, an oscillating lattice mode will result in a net average change of the Born effective charge.

The second contribution to the polarization, P_∞, is in linear response de-scribed by:

P∞ =₀χE =₀(ε∞−1)E, (2.3) as detailed in chapter 1. Similarly to the Born effective charge Z^∗, the permittivity ε∞ is constant only for small lattice displacements Q, while it increases for large amplitudes. This effect is captured by Fig. 2.2a, in which the slopeχ=ε∞−1 ofP∞, calculated from DFT, increases with the lattice displacement Q. Just like the Born effective charge, ∞ scales quadratically with Q:

28 Nonlinear expansion of the polarization

Figure 2.2: (a) Non-resonant contribution P_∞ =0χE to the polariza-tion for different lattice displacements. The susceptibilityχ increases as the displacementQ becomes larger. (b) Dependence of the non-resonant permittivityε_∞on the phonon coordinateQ.

ε∞ = 1 +χ= 1 +∂P∞

∂E =ε∞,0+ 2βQ², (2.4) where the pre-factor 2 descends from a formal derivation of the polarization from the energy density functional that will be described in section 2.3. The correction to the permittivity for large values of the phonon coordinate is sketched in Fig. 2.2b, hence oscillations ofQresult in a net average increase of ε∞.

Summarizing, the nonlinear polarization of a strongly driven normal mode includes two corrections quadratic in Q, one to the phonon effec-tive dipolar charge Z^∗ and one to the dielectric constant ε∞. The overall nonlinear polarization can be written as:

P =P_phonon+P∞ = Z₀^∗+αQ²

Q+₀ ε∞,0+ 2βQ²−1

E. (2.5) These two terms are typically neglected in the linear lattice response but must be included when studying the dynamics of an infrared active mode subject to a strong resonant optical field E₀sin (ωt).

Nonlinear equation of motion 29