Fermi surface nesting and the origin of charge density waves in metals

The concept of a charge density wave (CDW), which is induced by Fermi-surface nesting, originated from the Peierls idea of electronic instabilities in purely one-dimensional metals and is now often applied to charge ordering in real low-dimensional materials. The idea is that if Fermi surface contours coincide when shifted along the observed CDW wave vector, then the CDW is considered to be nesting derived. We show that, in most cases, this procedure has no predictive power, since Fermi surfaces either do not nest at the right wave vector or nest more strongly at the wrong vector. We argue that only a tiny fraction, if any, of the observed charge ordering phase transitions are true analogs of the Peierls instability because electronic instabilities are easily destroyed by even small deviations from perfect nesting conditions. By using prototypical CDW materials ${\text{NbSe}}_2$, ${\text{TaSe}}_2$, and ${\text{CeTe}}_3$, we show that such conditions are hardly ever fulfilled and that the CDW phases are actually structural phase transitions driven by the concerted action of electronic and ionic subsystems, i.e., a $\mathbf{q}$-dependent electron-phonon coupling plays an indispensable part. We also mathematically show that the original Peierls construction is so fragile that it is unlikely to apply to real materials. We argue that no meaningful distinction between a CDW and an incommensurate lattice transition exists.

Figure 1

(Color online) A comparison of ${\chi }^{\prime }\left(q\right)$ under ideal 1D conditions with perfect nesting at $T=0$ to ${\chi }^{\prime }\left(q\right)$ under various nonideal conditions. Even moderate deviations from the ideal (such as those found in real materials) rapidly reduce the divergence to a relatively weak enhancement.

Figure 2

(Color online) A schematic showing the original energy bands of our model system in comparison to the bands after a transition has caused a gap at the Fermi energy. The energy associated with a gap formation is shaded (red) and continues down to the bottom of the band. Representative energy states connected by the nesting vector are shown $\left(\Delta E_k\right)$ along with infinitesimal energy gain $\left(\delta E_k\right)$ associated with the gap opening at each $\mathbf{k}$.

Figure 3

(Color online) The Fermi surface of ${\text{TaSe}}_2$ shown with (a) no shift of the Fermi level and (b) a downward shift of the Fermi level $\left(-40\text{meV}\right)$. The topology of the Fermi surface, especially the light (yellow) sheets changes appreciably with a small change in the Fermi energy, and brings the surface into good agreement with experiment.

Figure 4

(Color online) The imaginary (left) and real (right) parts of the susceptibility for ${\text{TaSe}}_2$. The nesting peaks (imaginary part) do not correspond to the observed CDW wave vector, while the very weak peaks in the real part do.

Figure 5

(Color online) (a) The structure of ${\text{CeTe}}_3$ showing the Te (large spheres) planes interposed with Ce (small spheres) and Te staggered units. One of the two nonequivalent Te ions in the Te-only planes is shaded lighter to distinguish it. (b) A schematic of the quasi-1D tight-binding model employed to illustrate the origin of the Fermi surface. (c) The quasi-1D Fermi sheets resulting from the nearest-neighbor tight binding model. The new BZ corresponding to the larger unit cell in which the two different Te ions are distinguished is shown as a darker (blue) diamond.

Figure 6

(Color online) A diagram showing the imaginary (left) and real (right) parts of the susceptibility as a function of $q_x,q_y$, with $q_z=0$. The arrow connects the strongest peak in the imaginary part, located at ${\mathbf{q}}_{\text{nest}}$, to its corresponding position in the real part. In the real part, the nesting peak is absent and ${\mathbf{q}}_{\text{CDW}}$ dominates the spectrum, indicating the importance of states away from $E_F$.

Figure 7

(Color online) (a) The full, calculated Fermi surface of ${\text{CeTe}}_3$ (thin solid lines, green and brown), with the one-dimensional bands obtained from a nearest neighbor tight-binding model overlayed on top (thick dashed crimson lines). The Fermi surface in essence can still be thought of as intersecting 1D Fermi sheets, even after the bands have been folded down into the lower symmetry cell (see text). (b) A shift of one of the quasi-1D Fermi sheets (dashed lines represent the shifted FS) along the (1,1,0) direction produces nearly perfect nesting. (c) A shift along the observed CDW wave vector direction (1,0,0) produces imperfect nesting.

Figure 8

(Color online) The relaxed configuration of an initially one-dimensional chain of Na ions. If the dimensionality of the chain is restricted, no distortion at all occurs.