Charge Density Waves

So far the description and the discussion of electronic properties of Ni-Mn-Ga focused on the austenitic state of Ni2MnGa and the importance of the electronic contribution as the driving force for the (pre)martensitic transition. Since the structural phase transition of Ni-Mn-Ga is often accompanied by a periodic modulation of the lattice, it should be taken into account that a periodic mod-ulation of the lattice’s ions is always accompanied by the rearrangement of the electron distribution as well [102]. As the new arrangement of the lattice has to represent an energetic minimum of the system, both the ionic and the electronic systems are contributing to the new equilibrium configuration. This scenario leads to a strong electron-phonon coupling in the system.

In order to demonstrate the consequences of the described scenario an ex-ample of a linear ion chain and a 1D electron gas performing a transition to a modulated state shall serve as a simplified example. Figure 2.16(a) schemat-ically depicts the band structure of an ion chain with the lattice parametera for the case of a half-filled conduction band. Let us assume that the lattice

per-EF

Figure 2.16.: (a)Lattice structure of a linear ion chain, the corresponding charge density and the band structure of a half-filled conduction band.(b)Periodic modulation of the lattice results in a spatially modulated charge densityρ(x). The band structure changes in a way that at new Brillouin zone boundaries a gap of width∆opens. This lowers the energy of the energy states in the vicinity ofEF.

forms a transition to a periodically modulated configuration (see Fig. 2.16(b)).

Consequently the unit cell increases and the Brillouin zone reduces. If the wave vector of the lattice modulation coincides with the Fermi wave vectork_F, then the new Brillouin zone boundary will be shifted tok_F and the energy of the oc-cupied electronic states in the vicinity of the Fermi level will be lowered. This happens due to the opening of an energy gap∆at the new BZ boundaries.

This is due to the fact that the free electrons are transformed into Bloch states in an underlying modified lattice potential. For the presented case of a half filled conduction band withk_F =^2π/athe periodicity of the ion chain will be doubled to2a, but depending on the position of the Fermi level other modu-lation periodicities will occur. This transition is known as thePeierls transition and leads to an instability of the electron-phonon system.

The consequence that arises for the lattice is the renormalization of the

2.3. Electronic Properties and instability of Ni-Mn-Ga

Figure 2.17.: Kohn anomaly of a phonon mode at a wave vectorq= 2k_F accord-ing to the mean field (MF) theory.(a)Acoustic phonon dispersion relation of a one dimensional metal at various temperatures above the transition tempera-ture.(b)The image illustrates the strong dependency on the dimensionality of the systemT =T0. (Image from [89]).

phonon frequency, which is dependent on the susceptibilityχ(q, T)(see [89]

for a detailed description). The Fermi surface topology has a leading influence on χ(q, T). The susceptibility has its maximum for the nesting wave vector, e.g. the wave vector which connects two parts of the FS, and is dependent on the strength of the nesting behavior. The temperature dependance ofχ(q, T) in turn defines the transition temperatureT₀. The resulting phenomenon of phonon softening for temperaturesT > T₀is known as Kohn anomaly⁸and is shown for different temperatures and dimensionalities in Fig. 2.17.

After the transition belowT0the ions and the electronic system form a new equilibrium state, where both the position of ions and the electronic charge densityρ(x) are periodically modulated (see Fig. 2.16(b)). In this new state, which is referred to as the charge density wave (CDW) state, the lattice and the electronic subsystems are strongly coupled and stabilize the new equilib-rium [103]. If the system is disturbed in this CDW state, the strong correlation

8 The reader should note that this phenomenon was already mentioned in connection with the discussion of the FS of Ni2MnGa in the previous section.

leads to both single particle excitations at the electronic gap∆and a collective excitation of the two subsystems. These excitations are described by a phason mode and an amplitudon mode, respectively.

CDW in the context of Ni-Mn-Ga

As it was already pointed out in section 2.1, numerous experiments revealed a significant softening of the [ξξ0] TA2 phonon branch at a wave vector of

2π

a[0.33,0.33,0]for the austenitic phase when approaching the the premarten-sitic transformation temperatureT_PM [16, 77]. The origin of this anomaly has been intensively studied and its importance for the martensitic phase transition and the appearance of the lattice modulation along[ξξ0]was the subject of de-bate over the past years. In their theoretical work, Velikokhatnyi and Naumov were the first to maintain that the martensitic transformation is of the Peierls type [23]. Other groups identified the instability through examination of the FS nesting and electron-phonon coupling as a Kohn anomaly [24, 85]. Further experiments connected the anomalous thermal properties of Ni-Mn-Ga alloys nearT_M to the nesting of the Fermi level DOS, which is appropriate for the Peierls transition [104].

In more recent experiments the observation of phasons for the three-layered martensitic phase of Ni2MnGa by means of inelastic neutron scattering was reported [25]. Using time-resolved optical reflectivity Mariageret al.measured coherent optical phonons in the pre-martenstic phase and characterized them as an amplitudon [105]. UPS measurements have demonstrated the presence of a pseudogap below the Fermi energy, which appears at the onset of the premartensitic phase and survives in the martensitic phase [26, 27]. These ob-servations were interpreted in terms of a CDW state formation which persists also in the martensitic state. These findings indicate the Ni2MnGa alloy as a 3-D metallic CDW compound and a system where a CDW and ferromagnetic order coexist [27].

2.3. Electronic Properties and instability of Ni-Mn-Ga