The evolution of electron dispersion in the series of rare-earth tritelluride compounds obtained from their charge-density-wave properties and susceptibility calculations

TL;DR

This study investigates how electron dispersion evolves in rare-earth tritelluride compounds by analyzing their charge-density-wave properties and susceptibility, providing insights into the relationship between atomic number and electronic structure.

Contribution

It introduces a method to predict the evolution of electron-dispersion parameters based on susceptibility calculations and experimental data in RTe₃ compounds.

Findings

Predicted electron-dispersion parameter trends with rare-earth atomic number

Correlated susceptibility calculations with experimental transition temperatures

Identified evolution patterns of charge-density-wave wave vectors

Abstract

We calculated electron susceptibility of rare-earth tritelluride compounds RTe$_3$ as a function of temperature, wave vector and electron-dispersion parameters. Comparison of results obtained with the available experimental data on the transition temperature and on the wave vector of a charge-density wave in these compounds allowed us to predict values and the evolution of electron-dispersion parameters with the variation of atomic number of rare-earth element R.

Tables

Table 1. Table 1: List of parameters describing the dispersion and the CDW transition temperatures T C ​ D ​ W ​ 2 subscript 𝑇 𝐶 𝐷 𝑊 2 T_{CDW2} and T C ​ D ​ W ​ 2 subscript 𝑇 𝐶 𝐷 𝑊 2 T_{CDW2} for R = Gd, Dy, Ho, Er, Tm.

Compound	$T_{CDW1}$NRu2008 , K	$T_{CDW2}$NRu2008 , K	Lattice parameter NRu2008 , $\AA$	$t_{\parallel }$, eV	$t_{⟂}$, eV	$E_F$, eV
GdTe3	377	-	4.320	$\approx$1.37	¿ 0.35	0.95
DyTe3	306	49	4.302	1.85Brouet2008	0.35Brouet2008	1.28
HoTe3	284	126	4.290	1.96	¡ 0.35	1.35
ErTe3	267	159	4.285	2.06	¡ 0.35	1.42
TmTe3	244	186	4.275	2.20	¡ 0.35	1.52

Equations

$$\begin{array}{rl}{\epsilon }_1\left(k_x,k_z\right)=& -2t_{\parallel }\cos \left[\left(k_x+k_z\right)a/2\right]-2t_{\perp }\cos \left[\left(k_x-k_z\right)a/2\right]-E_F,\\ {\epsilon }_2\left(k_x,k_z\right)=& -2t_{\parallel }\cos \left[\left(k_x-k_z\right)a/2\right]-2t_{\perp }\cos \left[\left(k_x+k_z\right)a/2\right]-E_F,\end{array}$$

$$\chi \left(\omega \right)=\frac{i}{\hslash }{\int }_0^{\infty }⟨\left[\hat{A}\left(t\right),\hat{B}\left(0\right)\right]⟩e^{i\omega t}dt.$$

$$\chi \left(\omega \right)=\sum _{ml}{A}_{ml}{B}_{lm}\frac{n_F\left(E_m\right)-n_F\left(E_l\right)}{E_l-E_m-\omega -i\delta },$$

$$\chi \left(\mathbf{Q}\right)=\sum _{\alpha ,{\alpha }^{\prime }}\int \frac{4d^d\mathbf{k}}{{\left(2\pi \right)}^d}\frac{n_F\left(E_{\mathbf{k},\alpha }\right)-n_F\left(E_{\mathbf{k}+\mathbf{Q},{\alpha }^{\prime }}\right)}{E_{\mathbf{k}+\mathbf{Q},{\alpha }^{\prime }}-E_{\mathbf{k},\alpha }},$$

Full text

I Introduction

In the last two decades the rare-earth tritelluride compounds RTe3 (R=rare-earth elements) were actively studied, both theoreticallyYao06 ; Hong2006 and experimentally by various techniquesDiMasi95 ; Brouet2008 ; Moore2010 ; Schmitt2011 ; NRu2008 ; Ru2008 ; Zocco2015 ; Lavagnini10R ; Fang07 ; BB ; Hu2014 ; Kogar2019 ; Sinchenko2014 ; SinchPRB2012 ; SinchPRB2017 ; SinchJETPL2018 . A very rich electronic phase diagram and the interplay between different types of electron ordering,NRu2008 ; Ru2008 ; Zocco2015 as well as amazing physical effects in electron transport even at room temperatureSinchenko2014 ; SinchPRB2012 ; SinchPRB2017 stimulate this interest. These compounds undergo a transition to a unidirectional charge-density wave (CDW) state with wave vector $\bm{Q}_{CDW1}\approx(0,0,2/7c^{\ast})$. The corresponding transition temperature $T_{CDW1}$ decreases with the atomic number of rare-earth element R:NRu2008 $T_{CDW1}$ drops from over $600$K in LaTe3 Kogar2019 to $T_{CDW1}=244$K in TmTe3. However, the CDW energy gap does not completely cover the Fermi surface (FS), as can be seen from the ARPES measurements Brouet2008 ; Moore2010 ; Schmitt2011 , and the electronic properties below $T_{CDW1}$ remain metallic with a reduced density of electron states at the Fermi level. In RTe3 compounds with heaviest rare-earth elements the second CDW emergesNRu2008 with the wave vector $\bm{Q}_{CDW2}\approx(2/7a^{\ast},0,0)$ and the transition temperature $T_{CDW2}$ increasing with the atomic number of the rare-earth element R NRu2008 from $T_{CDW2}=52$K in DyTe3 to $T_{CDW2}=180$K in TmTe3. After the second CDW the RTe3 compounds remain metallic, similarly to NbSe3. A third CDW has been proposedHu2014 from the optical conductivity measurements, but not yet confirmed by the X-ray studies. At lower temperature the RTe3 compounds become magnetically ordered.Ru2008 In addition to all this, at high pressure the RTe3 compounds become superconducting.Zocco2015

To understand the richness of this phase diagram and the physical properties in each phase it is very helpful to have information about the evolution of electronic structure of RTe3 compounds with the change of the atomic number of rare-earth element R. Unfortunately, the ARPES data are available only for very few compounds of this family and, in spite of a notable progress in instrumentation, still have a large errorbar. The electron transport measurements are much more sensitive but they only give indirect information about the electronic structure, because of a large number of electron scattering mechanisms.Sinchenko2014 ; SinchPRB2012 ; SinchPRB2017 In this paper we use the extensive experimental data on the evolution of the CDW1 wave vector $Q_{CDW1}$ and transition temperature $T_{c}$ to study the evolution of the electronic structure of $RTe_{3}$ compounds. We calculate the electron susceptibility, responsible for CDW1 instability, as a function of the wave vector and temperature at various parameters, which determine the electron dispersion. The comparison of the results obtained with available experimental data allows us making predictions about the evolution of these electron-structure parameters with the atomic number of rare-earth element R.

II Calculation

At temperatures $T>T_{CDW1}$ the in-plane electron dispersion in RTe3 is described by a 2D tight binding model of the Te plane as developed in Brouet2008 , where the square net of Te atoms in each conducting layer forms two orthogonal chains created by the in-plane $p_{x}$ and $p_{z}$ orbitals. Correspondingly, $x$ and $z$ are the in-plane directions. In this model $t_{\parallel}$ and $t_{\perp}$ are the hopping amplitudes (transfer integrals) parallel and perpendicular to the direction of the considered $p$ orbital. The resulting in-plane electron dispersion can be written down as

[TABLE]

where the calculated parameters for DyTe3 are $t_{\parallel}=1.85$ eV, $t_{\perp}=0.35$ eV Brouet2008 and the in-plane lattice constant $a\approx 4.305$\mathrm{\SIUnitSymbolAngstrom}$$ Ru2008 . The Fermi energy $E_{F}$ is determined from the electron density, namely, from the condition of 1.25 electrons for each $p_{x}$ and $p_{z}$ orbitals.Brouet2008 This condition gives us $E_{F}=-2t_{\parallel}\cos(\pi(1-\sqrt{3/8}))$. It is slightly (by 10%) less than the originally used Fermi-energy value $E_{F}=-2t_{\parallel}\sin(\pi/8)$, inaccurately determined Brouet2008 from the same condition (see Appendix). The resulting expression shows the relation between these two parameters $t_{\parallel}$ and $E_{F}$, which is important because they both affect the electron susceptibility.

For calculation we use the Kubo formula for the susceptibility of quantity $A$ with respect to quantity $B$ (see §126 of Landau ):

[TABLE]

For the free electron gas in the terms of matrix elements it becomes

[TABLE]

where $m$ and $l$ denote the quantum numbers $\left\{\bm{k},s,\alpha\right\}$, which are the electron momentum $\bm{k}$, spin $s$, and the electron band index $\alpha$. In the CDW response function the quantities $A$ and $B$ are the electron density, so that Eq. (2) is a density-density correlator. To study the CDW onset one needs the static susceptibility at $\omega=0$ but at a finite wave vector $\bm{Q}$. Electron spin only leads to a factor $4$ in susceptibility, but the summation over band index $\alpha$ must be retained if there are more than one band crossing the Fermi level. As a result we have for the real part of electron susceptibility

[TABLE]

where $n_{F}(\varepsilon)=1/\left(1+\exp\left[(\varepsilon-E_{F})/T\right]\right)$ is the Fermi-Dirac distribution function, d is the dimension of space. Since the dispersion in the interlayer $y$-direction is very weak in RTe3 compounds, we can take $d=2$. And each of the band indices $\alpha$ and $\alpha^{\prime}$ may take any of two values $1,2$, because in RTe3 two electron bands cross the Fermi level. Here we assume that the matrix elements $A_{ml}$ and $B_{lm}$ do not depend on the band index. This means that due to the e-e interaction the electrons may scatter to any of the two bands with equal amplitudes. This assumption has virtually no effect on both temperature and $\bm{Q}$-vector dependence of the electron susceptibility, because the latter is determined mainly by the diagonal (in the band index) terms, which are enhanced in RTe3 by a good nesting.

Using Eq. (4) we calculate the electron susceptibility $\chi$ as a function of CDW wave vector $\bm{Q}$ and temperature for various parameters $t_{\parallel}$ and $t_{\perp}$ of the bare electron dispersion (1). The CDW phase transition happens when $\chi\left(\bm{Q},T\right)U=1$, where the interaction constant $U$ only weakly depends on the rare-earth atom R in RTe3 family. The position of susceptibility maximum $\chi\left(\bm{Q}\right)$ gives the wave vector $\bm{Q}_{CDW1}$ of CDW instability as a function of the band-structure parameters $t_{\parallel}$ and $t_{\perp}$. The value of susceptibility in its maximum as a function of temperature $\chi_{max}\left(T\right)$ gives the evolution of CDW transition temperature $T_{CDW1}$ as a function of $t_{\parallel}$ and $t_{\perp}$.

III Results and Discussion

First we analyze the evolution of CDW1 wave vector. The experimentally observed dependence of $Q_{CDW1}$ on the atomic number of rare-earth atom R can be taken, e.g., from Ref. Kogar2019 : $Q_{CDW1}$ monotonically increases by $\approx 10$% with the increase of R-atom number from $Q_{CDW1}\approx 0.275$ r.l.u. in LaTe3 to $Q_{CDW1}\approx 0.303$ r.l.u. in TmTe3. The dependence of CDW wave vector $c$-component, $\bm{Q}_{CDW1}=(0,0,Q_{CDW1})$, on the perpendicular hopping term $t_{\perp}$, calculated using Eq. (4), is shown in Fig.1. As we can see from this graph, $Q_{CDW1}(t_{\perp})$ demonstrates approximately linear dependence. The value $t_{\perp}$=0.35 eV, proposed in Ref. Brouet2008 from the band structure calculations, is located in the middle of this plot. The obtained $Q_{CDW1}(t_{\perp})$ dependence is rather weak: while $t_{\perp}$ increase dramatically, from $0.2$ to $0.5$eV, $Q_{CDW1}$ changes by only $\sim 8\%$ in \mathrm{\SIUnitSymbolAngstrom}$^{-1}$. In the reciprocal lattice units (r.l.u.) this variation is slightly stronger, as the lattice constant $c$ decreases with the atomic number from $c\approx 4.407$\mathrm{\SIUnitSymbolAngstrom} in LaTe3 to $c\approx 4.28$\mathrm{\SIUnitSymbolAngstrom}$in ErTe3 and TmTe3, and the r.l.u. correspondingly increases in$\mathrm{\SIUnitSymbolAngstrom}$^{-1}$. However, just the $Q_{CDW1}(t_{\perp})$ dependence cannot explain the observed evolution of the CDW1 wave vector with the R-atom number, because it is too weak.

The dependence $\chi$($t_{\perp}$) is shown in Fig.2. The electron susceptibility varies within one percent of its’ maximum value and thus remains almost constant. The $\chi_{CDW1}$ values are calculated on the wave vectors $Q_{CDW1}$, obtained for each value of $t_{\perp}$ as a position of the susceptibility maximum. From this plot we conclude that the parameter $t_{\perp}$ has almost no effect on the CDW1 transition temperature. Hence, to interpret the evolution of CDW1 transition temperature $T_{CDW1}$ and of its wave vector $Q_{CDW1}$ with the rare-earth atomic number, one needs to consider their $t_{\parallel}$-dependence.

The dependence $Q_{CDW1}(t_{\parallel})$ is shown in Fig. 3. The interval of this plot comprises the values $t_{\parallel}=1.7eV$ and $t_{\parallel}=1.9$ eV, obtained in Ref. Brouet2008 from the band structure calculations for the lightest and heaviest rare-earth elements. $Q_{CDW1}$($t_{\parallel}$) demonstrates sublinear monotonic dependence, but $Q_{CDW1}$ there increases with the increasing of parameter $t_{\parallel}$. It is opposite to the dependence $Q_{CDW1}(t_{\perp})$. Comparing Fig. 3 with the experimental data on $Q_{CDW1}$, summarized in Ref. Kogar2019 , we may conclude that the parameter $t_{\parallel}$ increases with the atomic number of the rare-earth element. According to the band structure calculations in Ref. Brouet2008 this transfer integral indeed increases from $t_{\parallel}=1.7eV$ in LaTe3 to $t_{\parallel}=1.9$ eV in LuTe3. Thus, our conclusion qualitatively agrees with the band-structure calculations in Ref. Brouet2008 . However, according to our calculation the variation of $t_{\parallel}$ with the atomic number of the rare-earth element must be stronger in order to account for the observed $Q_{CDW1}$ dependence.

In Fig. 4 we plot the calculated $\chi(t_{\parallel})$ dependence, which is approximately linear. Similarly to our calculations of $\chi(t_{\perp})$, the susceptibility value is taken in its maximum as a function of the wave vector $Q_{CDW1}$. $\chi$ changes significantly – about 35% of its maximum value in the full range of parameter $t_{\parallel}$ change. The $CDW_{1}$ transition temperature $T_{c}$ is given by the equation Gruner2000 $|U\chi(Q_{CDW1},T_{c})|=1$. Since the susceptibility increases with the decrease of temperature, the largest value of $\chi$ corresponds to the highest value of CDW transtition temperature. We assume that the electron-electron interaction constant $U$ remains almost the same for considered series of RTe3 compounds, because they have very close electronic structure. The result obtained (see Fig. 4) is comparable to the change of transition temperature $T_{CDW1}$ observed in the RTe3 seriesRu2008 . The value $t_{\parallel}=1.85eV$ in $DyTe_{3}$ is the reference point. The experimentally observed transition temperature to $CDW_{1}$ state in TmTe3 is $T_{CDW1}=245K$, while for GdTe3 it is $T_{CDW1}=380K$ and for DyTe3 it is $T_{CDW1}=302K$.Ru2008 This transition temperature is reduced by 35% of its maximum value from GdTe3 to TmTe3. Thus, we may assume that this range of $t_{\parallel}$ describes the whole series of compounds from TmTe3 to GdTe3. Moreover, basing on our calculations, we predict the values $t_{\parallel}\approx 1.37eV$ in GdTe3, $t_{\parallel}\approx 1.96eV$ in HoTe3, $t_{\parallel}\approx 2.06eV$ in ErTe3, and $t_{\parallel}\approx 2.20eV$ in TmTe3.

The dependence $\chi(t_{\perp})$ calculated at temperature above the transition is shown in Fig.5. It is important to note there that with the decrease of temperature the wave vector does not shift and thus does not change its value, as it shown in Fig.6: the position of maximum of susceptibility is almost the same for two different temperatures. Thus, the electronic susceptibility in Fig.5 is calculated on the same $Q_{max}$ wave vectors in Fig.1, but has lower value with the increase of temperature from 240 K to 400 K.

The transition temperatures and conducting band parameters for varies RTe3 compounds are summarized in Table.1. The $t_{\parallel}$ increases with the increase of the atomic number of rare-earth element R; the $t_{\perp}$ decreases with the increase of the atomic number of rare-earth element R according to $Q_{max}$($t_{\perp}$), but since the electronic susceptibility does not depend on $t_{\perp}$ we cannot calculate the values.

Our suggested values of the transfer integrals $t_{||}$ and $t_{\perp}$ assume that the effective electron-electron interaction at CDW wave vector does not depend considerably on the rare-earth element R.

IV Conclusions

To summarize, we have calculated the electron susceptibility on CDW1 wave vector in the rare-earth tritelluride compounds as a function of temperature, wave vector, and two tight-binding parameters ($t_{\|}$ and $t_{\perp}$) of the electron dispersion. From these calculations we have shown that the parameter $t_{\perp}$ has almost no effect on the CDW1 transition temperature $T_{CDW1}$ and weakly affects the CDW1 wave vector $Q_{CDW1}$. On contrary, the variation of parameter $t_{\|}$ with the atomic number $n$ of rare-earth element drives the variation of both $T_{CDW1}$ and $Q_{CDW1}$. Note that the increase of $t_{\parallel}$ and of $t_{\perp}$ has opposite effects on $Q_{CDW1}$. Using the experimentally measured transition temperatures $T_{CDW1}$ , we estimated the values of $t_{\parallel}$ from our calculations for the whole series of RTe3 compounds from TmTe3 to GdTe3.

Acknowledgements.

The work was partially supported by the RFBR grants No. 19-02-01000, 17-52-150007 and 18-02-00280, by the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST ”MISiS”, and by the Foundation for Advancement of Theoretical Physics and Mathematics ”BASIS”. P.D.G. thanks the State assignment 0033-2019-0001 ”The development of condensed-matter theory”. V.V.K. acknowledges Act 211 Government of the Russian Federation, contract No. 02.A03.21.0011.