Charge Density Wave

Introduction

Charge Density Wave (CDW) is a spatial modulation of the conduction electron density in a metal and an associated modulation of the lattice atom positions. In the relation to the symmetry breaking, CDW is a broken symmetry state of metals brought by the electron-phonon interactions whose ground states are the coherent superposition of electron-hole pairs, and it displays spatial variation.

CDW states was first discussed by Frohlich in 1954 and by Peierls in 1955. The latter pointed out that a one-dimensional metal coupled to the underlying lattice is not stable at low temperatures. The ground state of the coupled electron-phonon system is characterized by a gap in the single-particle excitation spectrum and by a collective mode formed by electron-hole pairs involving a wave vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = 2k_F \!} with the charge density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(\vec{r})=\rho_0 + \rho_1 cos(2\vec{k_F} \cdot \vec{r}+\phi)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_0 \!} is the unperturbed electron density of the metal and the condensate is called the charge density wave. As with the superconductivity, CDW has a complex order parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta e^{i\phi} \!} with the magnitude of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \!} determines the size of the electronic gap and the amplitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta u \!} of the atomic displacements. The phase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \!} determines the position of the CDW relative to the underlying lattice.

Materials

CDW primarily exists in the low-dimensional metals, that is metals whose Fermi surfaces exhibit high anisotropy (metal is defined as an object in which the Fermi surface exists). This anisotropy is the answer to the question why the conductivity along one axis is higher than it is in any other axis, hence although the material itself appears to be 3 dimensional, the conductivity exist only on a certain plane or a certain axis. The materials are typically conducting polymers, or also known as synthetic metals.

As a simple picture of how the CDW comes into the system, let's consider a thought experiment about making a polyacetylene out of polyethylene (although not so realistic) as shown in figure 1.

Figure 1. Dehydrogenating a polyethylene

This configuration with many free electrons is unstable because, of course, energetically unfavoured. For now, let's assume it is possible to make such a reaction and hold it "stable" for a while for observation. What we have after dehydrogenating the polyethylene is basically a metal (just looking at the free electrons). That being a metal, as we shall see later exhibits the so called nesting of the Fermi surface which gives rise to many interesting phenomena in the low dimensional metals, for example Kohn anomaly, Peierls transition, and density waves. Peierls transition deforms the lattice and an alternation of the double and single bonds will occur, producing polyacethylene.

(add polyacethylene picture here)

The double bonds is, naturally, stronger than the single bonds which results in the double bonds being shorter than the single bonds. The difference in the bond length implies that the charge density changes when one is moving along the axis. The alternation of the bonds gives the spatial variation.

Therefore, with this rather oversimplified picture, we see how the CDW develops and how some other phenomenon play their part.

Density Of States (DOS) of a 1-D Electron Gas (1DEG)

Fermi surface of a 1DEG consists of 2 sheets with a distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2k_F \!} apart, located at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm k_F} . The k-space DOS for 1DEG is just Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L/2\pi \!} and the number of states is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L/2\pi \times 2k_F = L k_F / \pi \!} . The 2 orientations of the spin, when taken into account, gives a factor of 2, so the total number of states (or the number of electrons) is : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \frac{2 L k_F}{L}}

And the Fermi wave vector is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F = \frac{N \pi}{2L}}

The Fermi energy is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}_F = \frac{\hbar^2}{2m} k_F^2 = \frac{\hbar^2}{2m} \left(\frac{N \pi}{2L}\right)^2\!}

The density of states of one spin is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} n(\mathcal E) &= \frac{L}{2 \pi} \int dk \delta \left(\mathcal{E} - \frac{\hbar^2 k^2}{2m}\right) \\ &= \frac{L}{2\pi} \sum_{i=1}^2 \int dk \frac{\delta \left(k - \sqrt{\frac{2m \mathcal{E}}{\hbar^2}} \right)}{\left|\frac{\hbar^2 k_i}{m}\right|}\\ &= \frac{L}{2\pi}\left|\frac{m}{\hbar^2 k}\right|\\ &= \frac{L}{\pi \hbar}\sqrt{\frac{m}{2 \mathcal{E}}} \end{align} \!}

The result of our calculation shows that

1. We have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F = \frac{N \pi}{2L} \!} if N = 2 we will have a full band. Therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F = \pi / L \!} gives an insulator (or intrinsic semiconductor). At this point, we shall consider Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n < 2/L} so the system is metallic. Periodic potential contribution will be considered later.

2. The DOS is proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E^{-1/2} \!} , while it is constant and proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E^{1/2} \!} in the 2D and 3D system, respectively. Therefore, some quantities having a factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d n(\mathcal{E})/d\mathcal{E} \!} changes sign going from 3D to 1D.

3. 1DEG Fermi wave vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F ~ n \!} , while in 2D and 3D it is proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{1/2} \!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{1/3} \!} , respectively.

4. When the electron changes state by interacting with phonons, the only possible transition is from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F \!} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -k_F \!} and vice versa by the conservation of energy and momentum.

Response Function of One-Dimensional Electron Gas (1DEG)

Electron in a material is attracted to the ions and feels repulsive force from any other electrons. As the result, in many-electron system there will be a spatial variation of the charge or known also as screening. One common example is electron will attract positive charged particles and creates a surplus of positive charge surrounding it and screens its field. Therefore we write the total charge density as a sum of the external (initial) and the induced charge density.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(\vec{r}) = \rho_{ext}(\vec{r}) + \rho_{ind}(\vec{r}) \!}

From the theory of the dielectric media, the external potential is assumed to be linearly related to the total physical potential

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi^{ext}(\vec{r}) = \int dr' \epsilon(\vec{r} - \vec{r'}) \phi(\vec{r'}) \!}

By using the Fourier transformation we can write this in the k-space as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi^{ext}(\vec{q}) = \epsilon(\vec{q}) \phi(\vec{q}) \!}

And the induced charge density satisfies the relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{ind}(\vec{q}) = \chi(\vec{q}) \phi(\vec{q}) \!}

By using the perturbation theory and assuming linear response only, we get

INSERT THE STEPS FROM A&M HERE

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(\vec{q}) = -e^2 \int \frac{d^D k}{(2\pi)^D} \frac{f_{\vec{k}} - f_{\vec{k+q}}}{\epsilon_{\vec{k}}-\epsilon_{\vec{k}-\vec{q}}} \!}

Now consider T = 0 (Lindhard response function), we see that each term does no vanish only when either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{\vec{k}} < \epsilon_F \!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{\vec k + \vec q} > \epsilon_F \!} (or to say that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\vec{k}} = 1 \!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\vec k + \vec q} = 0} ) or the opposite. What it means is the nonvanishing value comes from the electron-hole pairs (occupied-unoccupied states).

Now by evaluating the integral (for 1D case)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(q) = -\frac{e^2}{2\pi} \int_{k_F}^{k_F} dk \frac{1}{\frac{-\hbar^2}{2m} (2kq +q^2)} \!}

The resulting prefactor is not that important, we are interested in the q dependence of the result, which is.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(q) \propto ln\left|\frac{q+2k_F}{q-2k_F}\right| \!}

INSERT PICTURE OF \CHI vs Q HERE

So, we see that the response function for 1DEG diverges at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = 2k_F \!} , what it means is that an external perturbation creates a divergent charge redistribution. By self consistency, at T = 0 the 1DEG is unstable with respect to the formation of a periodically varying electron charge or spin density. The period is related to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_F \!} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_0 = \pi / k_F \!} .

Nesting of the Fermi Surface

The divergence arises only in low-dimensional cases, this is due to a particular topology of the Fermi surface. In 2D case, the dispersion relation can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon(\vec{k}) = \epsilon_0 + 2t_a cos(k_x a) + 2t_b cos(k_y b) \!}

Where a and b are the lattice constants in the x and y direction and t's are the transfer integrals along those directions. For the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_a >> t_b \!} (quasi 1D case), expanding the dispersion relation close to the Fermi surface yields

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon(\vec{k}) = \epsilon_0 + v_F (k_x - k_F) + 2t_b cos(k_y b) \! }

With this, the Fermi surface being the surface of a constant energy satisfies the condition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_x = k_F - \frac{2t_b cos(k_y b)}{v_F} \!}

which leads to the sinusoidal Fermi surface in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_x - k_y \!} plane

INSERT PICTURE OF THE FERMI SURFACE HERE

and we get a large number of electron-like pairs with similar energies, the condition for this particular case is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{Q} = (2k_F, \pi / b) \!}

Looking at the picture above, we see that each part of the Fermi surface can be translated perfectly into another using the reciprocal lattice vector, the same way as translating a unit cell of a crystal. The result is there is a large number of the overlapping of the Fermi surface. Fermi surface which has this kind of properties is said to exhibit a nesting. In 1D case, the Fermi surface is perfectly flat, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = 2k_F \!} gives the nesting condition and the number of the pair of states which satisfies this condition blows up.

Transition Temperature to Instabilities in 1DEG

Looking back at the equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(\vec{q}) \!} and evaluating near Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = 2k_F \!} at a finite temperature gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(q=2k_F, T) = -e^2 n(\epsilon_F) ln \left(\frac{1.14 \epsilon_0}{k_B T}\right) \!}

With this, consider an external potential which induces a charge density which in turn creates an induced potential. Assuming linear relation between the induced density and the potential

$\phi ^{ind}({\vec {q}})=-g\rho ^{ind}({\vec {q}})\!$,

where g is a coupling constant, then using the linear relation between the total potential and the induced charge

$\rho ^{ind}({\vec {q}})=\chi ({\vec {q}},T)\phi ({\vec {q}})\!$,

we can write the relation between the induced charge and the external potential only.

${\begin{aligned}\rho ^{ind}&=\chi ({\vec {q}},T)(\phi ^{ext}({\vec {q}})+\phi ^{ind}({\vec {q}})\\&=\chi ({\vec {q}},T)(\phi ^{ext}({\vec {q}})-g\rho ^{ind}({\vec {q}}))\\\end{aligned}}$

Grouping the induced charge density together, we obtain the relation

$\rho ^{ind}({\vec {q}})={\frac {\chi ({\vec {q}},T)\phi ^{ext}({\vec {q}})}{1+g\chi ({\vec {q}},T)}}\!$,

whose divergence happens when the denominator vanishes. This condition and substituting the susceptibility at a finite temperature gives

$1+gn(\epsilon _{F})ln{\frac {1.14\epsilon _{0}}{k_{B}T_{MF}}}=0\!$ with all factor absorbed into g. subscript MF is to indicate that the transition temperature is given by the mean field approach.

$T_{MF}={\frac {1.14\epsilon _{0}}{k_{B}}}exp\left({\frac {-1}{gn(\epsilon _{F})}}\right)\!$

Transition to the CDW

So far we considered only the electron, neglecting any interaction with the ions. However it appears that the lattice contributions cannot be neglected, not only that the electron interacts with the ions, but in 1D case the lattice is easily distorted. In this part, we shall consider a coupled electron-phonon system in 1D, the electron being 1DEG while the ions form a linear chain (remember the dehydrogenated polyethylene).