# DC Transport and Magnetotransport Properties of the 2D Isotropic Metallic System with the Fermi Surface Reconstructed by the Charge Density Wave

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Results

#### 2.1. Model

#### 2.2. The CDW Ground State

#### 2.3. DC Transport and Magnetotransport Coefficients

#### 2.4. Magnetotransport in the System Reconstructed by the Bi-Axial CDW under Magnetic Breakdown Conditions

- (a)
- In a relatively weak field with small MB effect, $\left|t\right(B\left)\right|\to 1$ ($\left|r\left(B\right)\right|\ll {\omega}_{0}/{\omega}_{c}\ll 1$) carriers move along the hole-like diamond-shaped trajectories, the magnetoconductivity is$$\begin{array}{ccc}\hfill {\sigma}_{xx}^{\left(h\right)}={\sigma}_{yy}^{\left(h\right)}& =& \frac{{\sigma}_{0}^{\left(h\right)}}{{\left({\tau}_{0}{\omega}_{c}\right)}^{2}}\sim \frac{1}{{B}^{2}},\hfill \\ \hfill {\sigma}_{xy}^{\left(h\right)}=-{\sigma}_{yx}^{\left(h\right)}& =& \frac{{n}^{\left(h\right)}e}{B}\sim \frac{1}{B},\hfill \\ \hfill {R}_{H}& =& \frac{1}{e{n}^{\left(h\right)}}.\hfill \end{array}$$
- (b)
- In a relatively strong field with a large MB effect, $\left|t\right(B\left)\right|\to 0$ ($\left|t\left(B\right)\right|\ll {\omega}_{0}/{\omega}_{c}\ll 1$) carriers move along the electron-like circular trajectories, the magnetoconductivity is$$\begin{array}{ccc}\hfill {\sigma}_{xx}^{\left(e\right)}={\sigma}_{yy}^{\left(e\right)}& =& \frac{{\sigma}_{0}^{\left(e\right)}}{{\left({\tau}_{0}{\omega}_{c}\right)}^{2}}\sim \frac{1}{{B}^{2}},\hfill \\ \hfill {\sigma}_{xy}^{\left(e\right)}=-{\sigma}_{yx}^{\left(e\right)}& =& -\frac{{n}^{\left(e\right)}e}{B}\sim \frac{1}{B},\hfill \\ \hfill {R}_{H}& =& -\frac{1}{e{n}^{\left(e\right)}}.\hfill \end{array}$$

## 3. Discussion

- (1)
- In the low-field regime with rather weak magnetic breakdown, electrons move along the hole-like semiclassical orbits around the diamond-shaped pockets, with a positive Hall coefficient.
- (2)
- In the high-field regime with very strong magnetic breakdown, electrons move along the electron-like semiclassical orbits consisting of parts of the diamond-shaped pockets recreating the initial, pre-reconstruction circular orbits, with a negative Hall coefficient. Both (1) and (2) are valid in the limit of very narrow magnetic bands with respect to the broadening of the level due to the impurity scattering, essentially reducing to the Landau level physics with standard diagonal magnetoconductivity proportional to $1/{B}^{2}$, and Hall conductivity to $1/B$.
- (3)
- In the regime of the moderate magnetic field so that magnetic bands (due to magnetic breakdown) are considerably wider compared to the level broadening due to the impurity scattering, the diagonal components of magnetoconductivity exhibit strong quantum oscillations, periodic in the inverse magnetic field, with period determined by the size of pockets forming the Fermi surface. The Hall conductivity (and consequently the Hall coefficient) vanishes. Electrons move freely in this regime (up to the scattering on impurities) over the 2D net, along the arcs forming the diamond-shaped orbits and, in turn, the circular ones, as if there is effectively no magnetic field. Taken together, varying the magnetic field, regimes (1)–(3) exhibit a change of sign of the Hall coefficient, with a finite interval of its vanishing between the two with opposite signs. The presented results are based on the simplified analytical model revealing the background mechanisms and their expected signatures in experiments. It is not material-specific; therefore, it proves the existence of the effect being accurate to the order of magnitude, but presents a solid foundation for the more accurate approaches, such as the ab initio studies which are material-specific, sometimes of high accuracy, but can hardly reveal novel mechanisms. Combined together, the quantitative studies of specific materials, in which the above-counted effects were observed (e.g., high-T${}_{c}$ superconducting cuprates, transition metal dichalcogenides, etc.), may have a great perspective.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The Fermi surface reconstruction. (

**a**) Initial Fermi pockets (free 2DEG) with the Fermi wave vector ${k}_{F0}$ are related by the CDW wave vector $\mathbf{Q}$. (

**b**) The Fermi surface is reconstructed from closed pockets to the open sheets within the new Brillouin zone (dashed rectangle) with coordinates in the reciprocal space $({k}_{x},{k}_{y})$ in which ${k}_{x}$ component is along the reconstruction vector $\mathbf{Q}$. (

**c**) The density of states, initially being constant ${\nu}_{0}=m/\left(\pi {\hslash}^{2}\right)$ for the free 2DEG (dashed line), develops a pseudo-gap between energies of peculiar points in electron bands, ${\u03f5}_{S}={\u03f5}_{0}-\Delta $ (saddle point) and ${\u03f5}_{U}={\u03f5}_{0}+\Delta $ (elliptic point), where ${\u03f5}_{0}={\hslash}^{2}{Q}^{2}/\left(8m\right)$ is the energy at which the initial electron bands cross during the reconstruction. The Fermi energy ${\epsilon}_{F0}$ is inside the pseudo-gap (see Equation (8)) [9,10].

**Figure 2.**Concentration of carriers in units of ${n}_{0}=m{\epsilon}_{F}/\left(\pi {\hslash}^{2}\right)$ (concentration of the non-reconstructed free 2DEG, red curve) vs. the Fermi energy of the system ${\epsilon}_{F}$: the total concentration n, effective concentrations ${n}_{xx}$ (blue) and ${n}_{yy}$ (green), and the Hall concentration ${n}_{H}$ (green). The shaded area is the interval of the pseudo-gap. Dashed curves depict concentrations for ${\epsilon}_{F}$ above ${\u03f5}_{E}={\u03f5}_{0}+\Delta $ when the contribution of the upper band needs to be taken into account. The strong signature of the reconstruction, when compared with the total concentration, is clearly visible in the ${n}_{xx}$ component (parallel to the reconstruction vector $\mathbf{Q}$).

**Figure 3.**The Fermi surface (extended zone scheme in $({k}_{x},{k}_{y})$ reciprocal space), initially circular, reconstructed (in the sense shown in Figure 1a,b) by the bi-axial CDW with wave vectors ${\mathbf{Q}}_{x}$ and ${\mathbf{Q}}_{y}$, where ${a}_{x}^{*}$ and ${a}_{y}^{*}$ are new reciprocal lattice constants. Reconstruction yields diamond-shaped Fermi pockets. In the perpendicular magnetic field, these are semiclassical orbits. Magnetic breakdown causes quantum tunneling with probability amplitudes t for the electron to pass through the reconstruction region, staying on the diamond-shaped orbit, and r to be reflected on it, moving along the circular orbit.

**Figure 4.**An upper envelope of the longitudinal magnetoconductivity ${\sigma}_{xx}$ (30) vs. inverse magnetic field ${B}^{-1}$ (in units ${\epsilon}_{F}/\hslash {\omega}_{c}$), depending on temperature T, i.e., $T/{\epsilon}_{F}=0$, ${10}^{-4}$, ${10}^{-2}$ for curves (1), (2), and (3), respectively. Magnetoconductivity is a fast-oscillating function, periodic in ${B}^{-1}$ (see insets for each envelope) with a period proportional to the area of pocket ${S}_{\diamond}$ on the Fermi surface. ${\sigma}_{xx}$ is plotted in units equal to the prefactor of the integral in Equation (30) (also proportional to ${B}^{-1}$ through ${b}_{B}^{-2}$). This result is valid in the regime of wide magnetic bands, i.e., $W\left(B\right)\gg \hslash {\omega}_{0}$ (case 2).

**Figure 5.**(

**a**) Function $\left|t\left(\xi \right)r\left(\xi \right)\right|=exp[-\xi /2]\sqrt{1-exp[-\xi ]}$, where the argument $\xi \equiv {\Delta}^{2}{(\hslash {\omega}_{c})}^{-4/3}{\epsilon}_{F}^{-2/3}$ depends on magnetic field through ${\omega}_{c}$. Considering the fact that amplitudes t and r satisfy the unitarity condition, ${\left|t\right|}^{2}+{\left|r\right|}^{2}=1$, it is plausible that, for fields low enough and high enough, the function tends to zero, and it has a smooth maximum around which the amplitudes are comparable to each other. (

**b**) Schematic presentation of characteristic intervals of magnetic field B in which the Hall coefficient attains the hole-like behavior (${R}_{H}>0$ for low fields, $\left|t\right|\to 1$), electron-like behavior (${R}_{H}<0$ for high enough fields, $\left|r\right|\to 1$), and vanishing of the Hall coefficient (${R}_{H}=0$ for moderate fields, $\left|r\right|\sim \left|t\right|$) between them. Regions between the three counted intervals are the crossover regions. Here ${B}_{0}$ is the lower limit of the field for which we have the coherent magnetic breakdown, i.e., ${\tau}_{0}{\omega}_{c}\gg 1$.

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Keran, B.; Grozić, P.; Kadigrobov, A.M.; Rukelj, Z.; Radić, D.
DC Transport and Magnetotransport Properties of the 2D Isotropic Metallic System with the Fermi Surface Reconstructed by the Charge Density Wave. *Condens. Matter* **2022**, *7*, 73.
https://doi.org/10.3390/condmat7040073

**AMA Style**

Keran B, Grozić P, Kadigrobov AM, Rukelj Z, Radić D.
DC Transport and Magnetotransport Properties of the 2D Isotropic Metallic System with the Fermi Surface Reconstructed by the Charge Density Wave. *Condensed Matter*. 2022; 7(4):73.
https://doi.org/10.3390/condmat7040073

**Chicago/Turabian Style**

Keran, Barbara, Petra Grozić, Anatoly M. Kadigrobov, Zoran Rukelj, and Danko Radić.
2022. "DC Transport and Magnetotransport Properties of the 2D Isotropic Metallic System with the Fermi Surface Reconstructed by the Charge Density Wave" *Condensed Matter* 7, no. 4: 73.
https://doi.org/10.3390/condmat7040073