# Scaling in Modeling of Core Losses in Soft Magnetic Materials Exposed to
Nonsinusoidal Flux Waveforms and DC Bias Condition

Adam Ruszczyk<EMAIL_ADDRESS>ABB Corporate Research, Starowiślna
13a, 31-038 Kraków, Poland  Krzysztof Sokalski<EMAIL_ADDRESS>Institute of Computer Science, Czȩstochowa University of Technology, Al. Armii
Krajowej 17, 42-200 Czȩstochowa, Poland  Jan Szczygłowski
<EMAIL_ADDRESS>Institute of Power Engineering, Czȩstochowa University
of Technology, Al. Armii Krajowej 17, 42-200 Czȩstochowa, Poland

###### Abstract

Assuming that core loss data of Soft Magnetic Materials obey scaling
relations, models describing the power losses in materials exposed to
nonsinusoidal flux waveforms and DC Bias conditions have been derived. In
order to test these models, the measurement data for two materials have been
collected and the core losses calculated. Agreement between the experimental
data and the model predictions is satisfactory.

###### pacs:

44.05.+e,75.90.+w

## Introduction

The usual waveforms in power electronics are square waves or a superposition
of square waves rather than sine waves. Some magnetic components also operate
under a DC bias, which has a significant influence on the core losses. During
the last two ldecades significant progress in modeling the dependences of core
losses both due to square wave parameters and DC bias has been achieved. The
derived methods for handling the core losses in soft magnetic materials (SMM)
exposed to nonsinusoidal flux waveforms have been based on the Steinmetz
Equation (bib:Steinmetz, ). These methods include the Modified Steinmetz
Equation (MSE) (bib:Albach, ), (bib:Reinert, ), Generalized Steinmetz Equation
(GSE) (bib:Li, ), Improved Generalized Steinmetz Equation (iGSE) (bib:Venka, )
(bib:Boss1, ) and Improved-improved Generalized Steinmetz Equation (i2GSE)
(bib:Ecklebe, ). Methods and models exist, however, based on the assumption
that the shape of the waveform does not matter and as a result only look at
peaks: Scaling of power Losses in SMM (bib:Sokal1, ),(bib:Sokal2,
),(bib:Sokal3, ) and Field-Extrema Hysteresis Model (FHM), (bib:Cale, ). Both
do not capture the effects of the waveform. Is this feature a disadvantage or
an advantage? All the models we are talking about have been derived for
prediction of losses for the designing processes. If one assumes that the
waveform does not vary in such a process then the model is universal and can
be applied to any waveform possessing a constant period. Moreover, if
experimental data obey a scaling law then the data collapse is possible. In
practise the collapse leads to a reduction of the number of independent
variables (bib:Sokal1, ) \- (bib:Sokal3, ) which makes a model or a method
much simpler. For these reasons we will apply the scaling theory in order to
describe the core losses in soft magnetic materials exposed to any periodic
flux waveform. In order to take into account DC Bias we apply the method
derived by Van den Bossche, et al. (bib:Boss1, ), (bib:Boss2, ).
This paper is organized in the following way. In Section I we present the
Scaling approach to modeling of core losses for any periodic flux waveform. In
order to take into account DC bias we combine the scaling with the Van den
Bossche method. Section II is devoted to experiment, measurement data,
estimation of the model parameters and comparison of theoretical model with
the measurement data. Section III contains conclusions.

##  I Approach to scaling of core losses data

Instead of an analysis based on Maxwell’s equations (bib:GB, ),(bib:GB2, ) we
have assumed that the SMM is a complex system in which the function of power
losses obeys the scaling law. This assumption leads to the formula for total
power loss in the form of general homogenous function (bib:Stan1, ),
(bib:Stan2, ):

$\displaystyle\exists\hskip 5.69054pta,b,c\in{\mathbf{R}}:\hskip 2.84526pt$
(1) $\displaystyle\forall\lambda\in{\mathbf{R}}^{+}\hskip
5.69054ptP_{tot}(\lambda^{a}f,\lambda^{b}(\triangle
B))=\lambda^{c}P_{tot}(f,(\triangle B)),$

where $P_{tot}$ \- total loss of power’s density, $f$ \- frequency, $\triangle
B$ \- peak to peak magnetic induction. Substituting $\lambda=(\triangle
B)^{-\frac{1}{b}}$ we have derived the general form of $P_{tot}$:

$P_{tot}(f,(\triangle B))=(\triangle B)^{\beta}F\left(\frac{f}{(\triangle
B)^{\alpha}}\right),$ (2)

where $F(\cdot)$ was an arbitrary function, $\alpha=\frac{a}{b}$, and
$\beta=\frac{c}{b}$. This function depends on the features of the phenomena to
be described. Since our measurement data are unable to consider quasi-static
losses we choose for $F(\cdot)$ the power series as a rough description of the
power losses $P_{tot}$ (bib:Sokal1, )-(bib:Sokal3, ):

$\displaystyle\frac{P_{tot}(f,\triangle B)}{(\triangle
B)^{\beta}}=[\Gamma_{1}\,\frac{f}{(\triangle
B)^{\alpha}}+\Gamma_{2}\,\left(\frac{f}{(\triangle B)^{\alpha}}\right)^{2}+$
(3) $\displaystyle\Gamma_{3}\,\left(\frac{f}{(\triangle
B)^{\alpha}}\right)^{3}+\Gamma_{4}\,\left(\frac{f}{(\triangle
B)^{\alpha}}\right)^{4}+...].$

Note that the left hand side of (3) is a function of the one effective
variable $\frac{f}{(\triangle B)^{\alpha}}$. It is easy to recognize in (3)
the hysteresis losses $P_{h}$ and the eddy current losses $P_{c}$ by the
powers $f$ and $f^{2}$, respectively. All higher terms correspond to excess
losses $P_{ex}$. The formula (3) is the mathematical model, ready to apply.
Values of $\alpha$, $\beta$ and amplitudes $\Gamma_{n}$ have been estimated
for the ten selected soft magnetic materials (bib:Sokal1, )-(bib:Sokal3, ).
Very recently (3) has been applied to Soft Magnetic Composities (bib:Slus, ).
Obtained results confirm the good quality of this model. Unfortunately, the
model fails when a DC Bias is present. In this situation we propose to keep
(3) as a limit for $H_{DC}\rightarrow 0$ and add the second term of a similar
form to (3) with $\Gamma$-s dependent on $H_{DC}$. These we take from Van den
Bossche, et al. (bib:Boss1, ), (bib:Boss2, ). They have proposed mapping
$H_{DC}$ to the corresponding primary magnetization curve which works very
well. In the case of a lack of appropriate experimental data it is possible to
use a mathematical model for $H_{DC}$. For instance:

$H_{DC}\rightarrow tanh(H_{DC}\cdot c),$ (4)

or more generally:

$H_{DC}\rightarrow\\{tanh(H_{DC}\cdot c_{1}),tanh(H_{DC}\cdot
c_{2}),\dots,tanh(H_{DC}\cdot c_{p})\\},$ (5)

where $c_{i}$ are free parameters to be determined (bib:Boss1, ). By this way
we derive the final model’s form:

$\displaystyle\frac{P_{tot}(f,\triangle B,H_{DC})}{(\triangle
B)^{\beta}}=\Sigma_{i=1}^{5}\Gamma_{i}\left(\frac{f}{(\triangle
B)^{\alpha}}\right)^{i\,(1-x)}+$
$\displaystyle\Sigma_{i=1}^{3}\Gamma_{i+5}\left(\frac{f}{(\triangle
B)^{\alpha}}\right)^{(i+4)\,(1-x)}tanh(H_{DC}\cdot c_{i}),$ (6)

where $x$ is a small phenomenological tuning parameter. This parameter is
derived in the following way. (3) arises as the Maclaurin series of (2). This
is why expansion of (3) posseses only integer exponents. However, in nature
this needs not be so. Therefore by the small $x$ we introduce a natural
correction of the series’ exponents.
Equation (6) has been successfully applied to SIFERRIT N87, however playing
with data for VITROPERM 500F 18K and model (6) we have got an expirence which
suggested the following modifications: 1) to split the $x$ parameter into two
independent tuning parameters $x$ and $y$, where $x$ tunes the bias-
independent term, whereas $y$ tunes the part describing the influence of the
bias on the losses, and 2) to distribute functions (5) along the $H_{DC}$
axis. These requirements lead to the following model:

$H_{DC}\rightarrow\\{tanh(H_{DC}\cdot c_{1}-r_{1}),\dots,tanh(H_{DC}\cdot
c_{p}-r_{p})\\},$ (7) $\displaystyle\frac{P_{tot}(f,\triangle
B,H_{DC})}{(\triangle
B)^{\beta}}=\Sigma_{i=1}^{4}\Gamma_{i}\left(\frac{f}{(\triangle
B)^{\alpha}}\right)^{i\,(1-x)}+$
$\displaystyle\Sigma_{i=0}^{2}\Gamma_{i+5}\left(\frac{f}{(\triangle
B)^{\alpha}}\right)^{(i+y)(1-x)}{tanh(H_{DC}\,c_{i+1}-r_{i+1})},$ (8)

where $r_{i}$ describes the position of the $i$-th function (5).

##  II Experimental data and parameters’ estimations

The B-H Loop measurements have been performed for two materials: SIFERRIT N87
and VITROPERM 500F 18K. The Core Losses per unit volum have been calculated as
the enclosed area of the B-H loop, multiplied by the frequency $f$. The
following factors influence the accuracy of measurements: 1) Phase Shift Error
of Voltage and Current $\leq 4\%$, 2) Equipment Accuracy $\leq 5,6\%$, 3)
Capacitive Couplings $negligible$ (capacitive currents are relatively lower
compared to inductive currents), and 4) Temperature $\leq 4\%$. For details of
the applied measurement method and the errors of factors we refer to
(bib:Ecklebe, ),(bib:Ecklebe1, ). In order to enable readers to compare their
own results with this work we enclose measurement data for SIFERRIT N87 (TABLE
1).
The parameter values of (3) have been estimated for each sample’s measurement
data by minimization of $\chi^{2}$ using the Simplex method of Nelder and Mead
(recipes, ), see TABLE 3 and TABLE 2.

Figure 1:  SIFERRIT N87. Projection of measurement points and scaling theory
points (6) in $((f/(\triangle B)^{\alpha})^{1-x},P_{tot}/(\triangle
B)^{\beta})$ plane. Figure 2: SIFERRIT N87. Projection of measurement points
and scaling theory points (6) in $(tanh(H_{DC}\cdot c_{3}),P_{tot}/(\triangle
B)^{\beta})$ plane.  Figure 3:  VITROPERM500F 18K. Projection of measurement
points and scaling theory points (8) in $((f/(\triangle
B)^{\alpha})^{1-x},P_{tot}/(\triangle B)^{\beta})$ plane. Figure 4:
VITROPERM500F 18K. Projection of measurement points and scaling theory points
(8) in $(tanh(H\,c_{2}-r_{2}),P_{tot}/(\triangle B)^{\beta})$ plane.

Both measurement series consist of 30 points. All measurement points have been
taken in temperature $27^{o}C\pm 0.5^{o}C$. Standard deviations per point are
equal $12[\frac{W}{m^{3}T^{\beta}}]$ and $15[\frac{W}{m^{3}T^{\beta}}]$ for
VITROPERM 500F 18K and SIFERRIT N87, respectively. Applying the formulae (6)
and (8) and the estimated values of parameters we have drawn scatter plots
which compar estimated points with the experimental ones. Each sample is
presented in the two projections. Note that the estimated model parameters are
universal, which means that they work for each $f$ and $\triangle B$.
Attention, in order to prevent generation of large numbers in the estimation
process the unit of frequency was kHz while other magnitutes were expressed in
SI unit system.

Table 1: Measurement data for SIFERRIT N87 $\triangle B[T]$ | $f[kHz]$ | $H_{DC}[\frac{A}{m}]$ | $P_{tot}[\frac{W}{m^{3}}]$
---|---|---|---
0,201 | 50 | 0,070 | 13155,0
0,202 | 50 | 4,699 | 13384,6
0,204 | 50 | 10,70 | 14043,9
0,201 | 50 | 14,75 | 14338,9
0,204 | 50 | 19,83 | 15295,3
0,414 | 50 | 4,920 | 52552,5
0,411 | 50 | 9,96 | 54106,4
0,416 | 50 | 15,11 | 56641,6
0,411 | 50 | 20,18 | 60464,5
0,618 | 50 | 4,662 | 118206
0,600 | 50 | 10 | 121104
0,619 | 50 | 15,03 | 127222
0,613 | 50 | 19,85 | 135366
0,807 | 1 | 4,810 | 582,66
0,814 | 1 | 10,29 | 582,38
0,812 | 1 | 15,29 | 618,87
0,804 | 1 | 19,81 | 682,53
0,810 | 1 | 25,26 | 829,74
0,804 | 2 | 0,067 | 1127,81
0,802 | 2 | 4,780 | 1036,27
0,805 | 2 | 10,04 | 1010,43
0,805 | 2 | 14,78 | 1059,07
0,803 | 2 | 19,81 | 1161,91
0,808 | 5 | 4,818 | 4066,07
0,808 | 5 | 15,27 | 4098,72
0,807 | 5 | 19,59 | 4418,86
0,806 | 10 | 5,077 | 12875,9
0,806 | 10 | 10,16 | 12931,8
0,805 | 10 | 14,78 | 13593,5
0,806 | 10 | 19,70 | 14891,1
Table 2: The set of estimated model’s parameters for VITROPERM 500F 18K, formula (8) $\alpha$ | $\beta$ | $x$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{5}$ | $\Gamma_{6}$
---|---|---|---|---|---|---|---|---
$\Gamma_{7}$ | $y$ | $c_{1}$ | $c_{2}$ | $c_{3}$ | $r_{1}$ | $r_{2}$ | $r_{3}$ | -
7,848601 | 13,40684 | 0,3529973 | -101,7297 | 2390,43 | -334,3942 | 121,0903 | 7,88E+00 | 1,29E+03
-7,66917 | 9,77E-01 | 4,668659 | 9,52E-02 | 43,98191 | 0,2228664 | 3,042717 | -2,016617 | -
Table 3: The set of estimated model’s parameters for SIFERRIT N87, formula (6) $\alpha$ | $\beta$ | $x$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$
---|---|---|---|---|---|---
$\Gamma_{5}$ | $\Gamma_{6}$ | $\Gamma_{7}$ | $\Gamma_{8}$ | $c_{1}$ | $c_{2}$ | $c_{3}$
-7,7413 | -11,375 | -0,1712 | 285,292 | 118,459 | -27,1697 | -2,2167
-5,35E-01 | -3,03E+01 | 5,8086 | -5,750 | -0,05529 | 8,0579 | 0,1102

##  III Conclusions

Agreement between the models of core losses ((6),(8)) and the measurement data
confirms the hypothesis that these data obey scaling relations. Now we can
construct estimators for core loss in soft magnetic materials exposed to
nonsinusoidal periodic flux waveforms and DC Bias condition. Moreover,
thanking to data collapse the space of independent variables is reduced from
the three-dimensional one spaned by $[f,\triangle B,H_{DC}]$ to the two-
dimensional one spaned by $[\frac{f}{(\triangle
B)^{\alpha}}$,$tanh(H_{DC}\cdot c_{i})]$. The consequence which results from
this reduction is very significant: both models (6) and (8) describe the core
loss problem completely (for each $f$, for each $\triangle B$ and for each
$H_{DC}$). The question is then how many model parameters must one introduce
in order to completely describe the core loss in the 3-dimensional space.

## References

  * (1) C.P. Steinmetz, On the law of hysteresis, Trans. Amer. Inst. Elect. Eng., 9 , 3-64 (1892).
  * (2) M. Albach, T. Durbaum, and A. Brockmeyer, Calculating core losses in transformers for arbitrary magnetizing currents a comparison of different approaches. , IEEE Power Electronics Specialists Conference, 1996, pp. 1463 8.
  * (3) J. Reinert, A. Brockmeyer J. Reinert, A. Brockmeyer, and R.W. De Doncker, Calculation of losses in ferro- and ferrimagnetic materials based on the modified Steinmetz equation , Annual Meeting of the IEEE Industry Applications Society, 1999.
  * (4) Jieli Li, T. Abdallah, and C. R. Sullivan, Improved calculation of core loss with nonsinusoidal waveforms , in Annual Meeting of the IEEE Industry Applications Society, 2001, pp. 2203 2210.
  * (5) K. Venkatachalam, C. R. Sullivan, T. Abdallah, and H. Tacca, Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters IEEE Workshop on Computers in Power Electronics (COMPEL), 2002.
  * (6) Alex Van den Bossche, Vencislav Valchev, Georgi Georgiev, Measurement and loss model of ferrites in non-sinusoidal waves , IEEE Power Electronics Specialists Conference, 2004.
  * (7) Jonas Mühlethaler, Jürgen Biela, Johann Walter Kolar and Andreas Ecklebe, Improved Core-Loss Calculation for Magnetic Components Employed in Power Electronic Systems, IEEE TRANSACTIONS ON POWER ELECTRONICS, 27, pp.964-973 (2012).
  * (8) J. Mühlethaler, J. Biela, J.W. Kolar, A. Ecklebe, ”Core Losses Under the DC Bias Condition Based on Steinmetz Parameters,” IEEE Transactions on Power Electronics, 27, pp.953-963, (2012).
  * (9) H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena , Oxford: Clarendon Press 1971.
  * (10) H.E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Rev. Mod. Phys.,71, S358-S366 (1999).
  * (11) K. Sokalski, J. Szczygłowski, M. Najgebauer and W. Wilczyński, Thermodynamical Scaling of Eddy Current Losses in Magnetic Materials, Proc. 12th IGTE Symposium, ( 2006) pp. 83-86.
  * (12) K. Sokalski, J. Szczygłowski, M. Najgebauer and W. Wilczyński, Losses scaling in soft magnetic materials, COMPEL: Int. J. Comput. Math. Electr. Electron. Eng.,26, 640-649 ( 2007).
  * (13) K. Sokalski, J. Szczygłowski, and W. Wilczyński, Scaling conception of power loss’ separation in soft magnetic materials, arXiv:1111.0939v1 [cond-mat.mtrl-sci] 3 Nov 2011.
  * (14) J. Cale, S.D. Sudhoff, S. D. and R.R. Chan, A Field-Extrema Hysteresis Loss Model for High-Frequency Ferrimagnetic Materials, IEEE Transactions on Magnetics,44, issue 7, pp. 1728-1736 (2008).
  * (15) A. Van den Bossche, et al., Ferrite Losses of Cores with Square Wave Voltage and DC bias, J. Appl. Phys., 99, p. 08M908 (2006).
  * (16) G. Bertotti, General Properties of Power Losses in Soft Ferromagnetic Materials, IEEE Trans. Magn. 24, p.621 (1988).
  * (17) G. Bertotti, F. Fiorillo and G. P. Soardo, THE PREDICTION OF POWER LOSSES IN SOFT MAGNETIC MATERIALS, JOURNAL DE PHYSIQUE, Colloque C8, Supplement au no 12, 49 (1988).
  * (18) G. Bertotti, A general statistical approach to the problem of eddy current losses, J. Mag. Mag. Mater. 41, p.253 (1984).
  * (19) B. Ślusarek, B. Jankowski, K. Sokalski, J. Szczyg owski, Characteristics of Power Loss in SMC a Key for Designing the Best Values of Technological Parameters, Journal of Alloys and Compounds, 581, pp. 699-704 (2013), http://dx.doi.org/10.1016/j.jallcom2013.07.084.
  * (20) W.H.Press, B.P.Flannery, S.A.Teukolsky, and W.T.Vetterling, Numerical Recipes. The Art of Scientific Computing, Cambridge Univ. Press 1987, pp. 289-293.


# A non-uniformly inner amenable group

Isaac Goldbring  Department of Mathematics
University of California, Irvine, 340 Rowland Hall (Bldg.# 400), Irvine, CA
92697-3875<EMAIL_ADDRESS>http://www.math.uci.edu/ isaac

###### Abstract.

We provide an example of two elementarily equivalent countable ICC groups $G$
and $H$ such that $G$ is amenable and $H$ is not inner amenable. As a result,
we provide the first example of elementarily equivalent groups whose group von
Neumann algebras are not elementarily equivalent, answering a question asked
by many researchers.

The author was partially supported by NSF grant DMS-2054477.

##  1\. Introduction

If $G$ is a group and $X$ is a set, an action $G\curvearrowright X$ of $G$ on
$X$ is called amenable if there is a finitely additive $G$-invariant
probability measure on $X$. $G$ is called amenable if the action of $G$ on
itself by left multiplication is amenable while $G$ is called inner amenable
if the action of $G$ on $G\setminus\\{e\\}$ by conjugation is amenable.
Throughout this note, we assume basic facts about amenable groups; we
recommend [9] as a good reference. We note that amenable groups are inner
amenable; see, for example, [9, Theorem 2.20].

A group $G$ is called ICC if all nontrivial conjugacy classes are infinite.

Two groups $G$ and $H$ are called elementarily equivalent if they satisfy the
same first-order sentences in the language of groups. Equivalently, by the
Keisler-Shelah theorem, two groups are elementarily equivalent if and only if
they have isomorphic ultrapowers.

The main result of this note is the following:

######  Theorem 1.1.

There are elementarily equivalent countable ICC groups $G$ and $H$ such that
$G$ is amenable and $H$ is not inner amenable.

In Section 3, we show how our main result settles a question in the model
theory of operator algebras while in Section 4 we explain how to interpret the
main result as the existence of an inner amenable but not uniformly inner
amenable group.

We thank David Jekel, Yash Lodha, and Jennifer Pi for many useful discussions
around this project.

##  2\. Proof of the main result

Throughout this note, $K:=\mathbb{F}_{2}^{alg}$ denotes the algebraic closure
of the field of two elements while $L$ denotes any algebraically closed field
of characteristic $2$ and positive transcendence degree (for example,
$L=\mathbb{F}_{2}(t)^{alg}$, where $t$ is an indeterminate).

Recall that for any field $F$, $\operatorname{SL}_{2}(F)$ denotes the group of
$2\times 2$ matrices over $F$ of determinant $1$.

$\operatorname{SL}_{2}(K)$, being a locally finite group, is amenable. In
fact, for any field $F$, $\operatorname{SL}_{2}(F)$ is amenable if and only if
$F$ is algebraic over a finite field (see, for example, [1, Proposition 11]).
Consequently, we have that $\operatorname{SL}_{2}(L)$ is _not_ amenable.

We actually claim that $\operatorname{SL}_{2}(L)$ is not even _inner_
amenable. To see this, we will need the following fact (for a proof, see, for
example, [9, Theorem 2.21]):

######  Fact 2.1.

Suppose that $G\curvearrowright X$ is an amenable action and that the
stabilizer subgroup $G_{x}$ is amenable for each $x\in X$. Then $G$ is
amenable.

By considering the conjugation action of $G$ on $G\setminus\\{e\\}$, one has,
as a corollary, the following:

######  Fact 2.2.

If $G$ is inner amenable and the centralizer $C(g)$ of each $g\in
G\setminus\\{e\\}$ is amenable, then $G$ is amenable.

Recall that a group $G$ is called commutative transitive (CT) if $C(g)$ is
abelian for all $g\in G\setminus\\{e\\}$. (The nomenclature is due to the fact
that CT groups are those groups for which, on nontrivial elements, commutation
is a transitive relation.) Since abelian groups are amenable, the previous
fact implies:

######  Corollary 2.3.

An inner amenable CT group is amenable.

Thus, to prove that $\operatorname{SL}_{2}(L)$ is not inner amenable, it
suffices to quote the following:

######  Fact 2.4.

For any field $F$ of characteristic $2$, $\operatorname{SL}_{2}(F)$ is a CT
group.

Fact 2.4 has a slippery history. In [15], it was mentioned that
$\operatorname{PSL}_{2}(\mathbb{F}_{2^{n}})$ is a CT group for any $n\geq 1$.
(Recall that for any field $F$,
$\operatorname{PSL}_{2}(F):=\operatorname{SL}_{2}(F)/\\{\pm 1\\}$; when $F$
has characteristic $2$, $\operatorname{PSL}_{2}(F)$ coincides with
$\operatorname{SL}_{2}(F)$.) Strangely enough, no proof of this fact is given,
but the author claims “That this group possesses the property in question
follows from its analysis.” Then a reference is given to [3, pages 262-265].
From this fact and some basic model theory, one can prove Fact 2.4; see [7,
Theorem 3.7(1)]). Given that this “proof” of Fact 2.4 is somewhat incomplete,
we prefer to give a detailed proof here; we thank David Jekel for providing us
with this proof and for his permission to include it here. (Not that we will
need to know this but, building on work of Suzuki [14] and Wu [16], it follows
that $\operatorname{PSL}_{2}(F)$ is _not_ a CT group when $F$ has positive
characteristic different from $2$; see [7, Theorem 3.7(2)].)

###### Proof of Fact 2.4.

Without loss of generality, we may assume that $F$ is algebraically closed.
Fix $g\in\operatorname{SL}_{2}(F)\setminus\\{1\\}$. Since $F$ is algebraically
closed, $g$ is similar to a matrix $g^{\prime}$ in $\operatorname{SL}_{2}(F)$
that is in Jordan canonical form (see [12, Chapter XIV, Corollary 2.5]). Since
$g$ and $g^{\prime}$ have isomorphic centralizers in
$\operatorname{SL}_{2}(F)$, we may assume that $g$ itself is in Jordan
canonical form.

First suppose that $g$ is diagonal. Then $g=\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}$ for some $a\in F$. Fix $h=\begin{pmatrix}c&d\\\
e&f\end{pmatrix}\in C(g)$. Since $gh=\begin{pmatrix}ac&ad\\\
a^{-1}e&a^{-1}f\end{pmatrix}$ while $hg=\begin{pmatrix}ac&a^{-1}d\\\
ae&a^{-1}f\end{pmatrix}$, we see that $(a-a^{-1})d=(a-a^{-1})e=0$; since
$a\not=1$ (else $g=1$), we have that $a-a^{-1}\not=0$ (recall that $F$ has
characteristic $2$), whence $d=e=0$. It follows that $C(g)$ is the set of
diagonal matrices in $\operatorname{SL}_{2}(F)$, which is certainly abelian.

Now suppose that $g$ is not diagonal. In this case, we have that
$g=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ for some $a\in F\setminus\\{0\\}$.
Once again, fix $h=\begin{pmatrix}c&d\\\ e&f\end{pmatrix}\in C(g)$. This time
we have $gh=\begin{pmatrix}c+e&d+f\\\ e&f\end{pmatrix}$ while
$hg=\begin{pmatrix}c&c+d\\\ e&e+f\end{pmatrix}$. We thus see that $e=0$ and
$c=f$. It follows that $C(g)$ is the set of upper triangular matrices in
$\operatorname{SL}_{2}(F)$ whose diagonal elements are the same; this group is
easily checked to be abelian. ∎

Recalling that $\operatorname{SL}_{2}(L)$ is not amenable, Corollary 2.3 and
Fact 2.4 imply that:

######  Corollary 2.5.

$\operatorname{SL}_{2}(L)$ is not inner amenable.

Finally, we record:

######  Lemma 2.6.

$\operatorname{SL}_{2}(K)$ and $\operatorname{SL}_{2}(L)$ are elementarily
equivalent.

###### Proof.

Since $K$ and $L$ are algebraically closed fields of the same characteristic,
we have that $K$ and $L$ are elementarily equivalent (see [13, Proposition
2.2.5]). It remains to quote a standard fact: if two fields $F_{1}$ and
$F_{2}$ are elementarily equivalent, then so are the groups
$\operatorname{SL}_{2}(F_{1})$ and $\operatorname{SL}_{2}(F_{2})$. (There are
many ways to see this, but perhaps the quickest is to use the Keisler-Shelah
theorem to conclude that $K$ and $L$ have isomorphic ultrapowers and then use
the fact that taking $\operatorname{SL}_{2}$ of a field commutes with
ultrapowers.) ∎

We note the following well-known fact.

######  Proposition 2.7.

For any algebraically closed field $F$ of characteristic $2$,
$\operatorname{SL}_{2}(F)$ is an ICC group.

###### Proof.

Fix $g\in\operatorname{SL}_{2}(F)\setminus\\{e\\}$; as before, we may suppose
that $g$ is in Jordan canonical form. If $g=\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}$, then it remains to note that

$\begin{pmatrix}b&1\\\ 0&b^{-1}\end{pmatrix}^{-1}\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}\begin{pmatrix}b&1\\\
0&b^{-1}\end{pmatrix}=\begin{pmatrix}a&(a+a^{-1})b\\\ 0&a^{-1}\end{pmatrix}.$

Since $a+a^{-1}\not=0$ (lest $g=e$, recalling that the characteristic of $F$
is $2$), we see that by letting $b$ vary over $F\setminus\\{0\\}$, we find
infinitely many conjugates of $g$.

The other case is that $g=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ with $a\in
F\setminus\\{0\\}$. In this case, we note that, for $b,c\in F$ with $bc=1$, we
have

$\begin{pmatrix}b&0\\\ 0&c\end{pmatrix}^{-1}\begin{pmatrix}1&1\\\
0&1\end{pmatrix}\begin{pmatrix}b&0\\\
0&c\end{pmatrix}=\begin{pmatrix}1&c^{2}\\\ 0&1\end{pmatrix}.$

Once again, letting $c$ range over $F\setminus\\{0\\}$, we find infinitely
many conjugates of $g$. ∎

The previous proposition holds in greater generality:
$\operatorname{PSL}_{2}(F)$ is ICC for any algebraically closed field of
positive characteristic; we assumed that the field had characteristic $2$ just
to give a simpler proof and since this is the only case that we need.

##  3\. An application to operator algebras

Theorem 1.1 above allows us to provide a negative solution to a question asked
by many researchers in the model theory of operator algebras, namely, does
elementary equivalence of groups imply the elementary equivalence of their
group von Neumann algebras?

######  Proposition 3.1.

There are elementarily equivalent countable ICC groups $G$ and $H$ such that
their group von Neumann algebras $L(G)$ and $L(H)$ are not elementarily
equivalent.

###### Proof.

The groups $G:=\operatorname{SL}_{2}(K)$ and $H:=\operatorname{SL}_{2}(L)$
yield the desired example. Indeed, since $G$ is a countable, amenable, ICC
group, by Connes’ landmark result in [2], $L(G)$ is the hyperfinite II1 factor
$\mathcal{R}$. However, by a result of Effros [5], since $H$ is not inner
amenable, $L(H)$ does not have property Gamma. Since $\mathcal{R}$ has
property Gamma and property Gamma is an axiomatizable property of tracial von
Neumann algebras (see [6, 3.2.2]), we have that $L(G)$ and $L(H)$ are not
elementarily equivalent. ∎

##  4\. Uniformly inner-amenable groups

If $G$ is a group, $S\subseteq G$ is finite, and $\epsilon>0$, then a
$(S,\epsilon)$-Følner set in $G$ is a finite set $T\subseteq G$ such that
$|gT\triangle T|<\epsilon|T|$ for all $g\in S$. A group $G$ is amenable if and
only if $(S,\epsilon)$-Følner sets in $G$ exist for all finite $S\subseteq G$
and all $\epsilon>0$. $G$ is said to be uniformly amenable if there is a
function $f:\mathbb{N}\to\mathbb{N}$ such that, for all finite $S\subseteq G$,
if $|S|\leq n$, then there is a $(S,\frac{1}{n})$-Følner set $T\subseteq G$
with $|T|\leq f(n)$. In [10], Keller studied uniformly amenable groups,
showing that $G$ is uniformly amenable if and only if some (equivalently any)
nonprincipal ultrapower of $G$ is amenable. (Keller used the language of
nonstandard extensions rather than ultrapowers, but the arguments are
identical in either case.) It then follows that $G$ is uniformly amenable if
and only if $H$ is amenable whenever $H$ is elementarily equivalent to $G$.
Indeed, one direction follows from the fact that an ultrapower of $G$ is
elementarily equivalent to $G$, while the other direction follows from the
fact that any group elementarily equivalent to $G$ embeds (elementarily) into
some ultrapower of $G$ and the fact that subgroups of amenable groups are
amenable.

There are plenty of uniformly amenable groups; for example, any solvable group
is uniformly amenable. An example of an amenable group that is not uniformly
amenable is the locally finite group $S_{\infty}:=\bigcup_{n\geq 1}S_{n}$ as
it is fairly straightforward to find a copy of a nonabelian free group inside
of an ultrapower of $S_{\infty}$.

In [11], an appropriate notion of Følner set for inner amenable groups was
discussed (but not named):

######  Definition 4.1.

For nonempty finite subsets $S,T\subseteq G$ and $\epsilon>0$, we say that $T$
is a $(S,\epsilon)$-c-Følner set if $|gTg^{-1}\triangle T|<\epsilon|T|$ for
all $g\in S$.

The following fact is mentioned in [11, Section 6]:

######  Fact 4.2.

A group $G$ is inner amenable if and only if: for all finite $S\subseteq G$
and all $\epsilon>0$, there are arbitrarily large $(S,\epsilon)$-c-Følner sets
in $G$.

We say that $G$ is uniformly inner amenable if there is a function
$f:\mathbb{N}\to\mathbb{N}$ such that, for all finite subsets $S\subseteq G$
with $|F|\leq n$, there is a finite subset $T\subseteq G$ with $n\leq|T|\leq
f(n)$ that is a $(S,\frac{1}{n})$-c-Følner set for $G$. One can prove an
analogue of Keller’s result for amenable groups as follows:

######  Proposition 4.3.

For a group $G$, the following are equivalent:

  1. (1)

$G$ is uniformly inner amenable.

  2. (2)

If $H$ is elementarily equivalent to $G$, then $H$ is inner amenable.

  3. (3)

Every ultrapower of $G$ is inner amenable.

Before beginning the proof of Proposition 4.3, we record one fact. Recall that
if $H$ is a subgroup of $G$, then $H$ is existentially closed in $G$ if: for
every existential sentence $\varphi$ with parameters in $H$, if $\varphi$ is
true in $G$, then $\varphi$ is true in $H$. Note that elementary subgroups are
in particular existential subgroups. Recall also that, unlike the case of
amenable groups, subgroups of inner amenable groups need not be inner
amenable. For example, $G\times\mathbb{Z}$ is inner amenable for _any_ group
$G$.

######  Lemma 4.4.

If $G$ is an inner amenable group and $H$ is a subgroup of $G$ that is
existentially closed in $G$, then $H$ is inner amenable.

###### Proof.

Fix a finite set $S\subseteq H$, $\epsilon>0$, and $n\geq 1$. Since $G$ is
inner amenable, there is a finite set $T\subseteq G$ that is a
$(S,\epsilon)$-c-Følner set with $|T|\geq n$. The existence of this
$(S,\epsilon)$-c-Følner set fact can be expressed by an existential sentence
with parameters from $H$ and is thus true in $H$ since $H$ is existentially
closed in $G$. ∎

###### Proof of Proposition 4.3.

(1) $\Rightarrow(2)$: Suppose that $G$ is uniformly inner amenable as
witnessed by a function $f$. Then for any given $n\geq 1$, the fact that any
finite subset of $G$ of size at most $n$ has a c-Følner set with error
$\frac{1}{n}$ of size in between $n$ and $f(n)$ can be expressed by a first-
order sentence true of $G$ and thus of any group elementarily equivalent to
$G$.

(2) $\Rightarrow(1)$: Suppose that $G$ is not uniformly inner amenable.
Without loss of generality, we may suppose that $G$ is inner amenable. Then
there is some $n\geq 1$ such that, for all $m\geq 1$, there exist finite
subsets $S_{m}\subseteq G$ with $|S_{m}|\leq n$ such that all
$(S_{m},\frac{1}{n})$-c-Følner sets in $G$ of size at least $n$ have
cardinality at least $m$. By the compactness theorem, there is an elementary
extension $H$ of $G$ which contains a subset $S\subseteq H$ with $|S|\leq n$
for which there is no $(S_{m},\frac{1}{n})$-c-Følner set in $H$ of cardinality
at least $n$. Then $H$ is not inner amenable.

(2) $\Rightarrow$ (3) follows from the fact that ultrapowers of $G$ are
elementarily equivalent to $G$.

(3) implies (2): Suppose that $H$ is elementarily equivalent to $G$ and embed
$H$ elementarily in an ultrapower $G^{\mathcal{U}}$ of $G$. By (3),
$G^{\mathcal{U}}$ is inner amenable whence so is $H$ by Lemma 4.4. ∎

Theorem 1.1 can thus be stated:

######  Corollary 4.5.

$\operatorname{SL}_{2}(K)$ is an amenable group that is not uniformly inner
amenable.

In an earlier version of this paper, we asked if there is an inner amenable,
_nonamenable_ group that is not uniformly inner amenable. In response to this
question, Jesse Peterson informed us of the following fact:

######  Fact 4.6.

If $G$ and $H$ are groups, then $G\times H$ is inner amenable if and only if
at least one of $G$ or $H$ is inner amenable.

The previous fact appears to be folklore but we were unable to find it
explicitly stated in the literature. One direction is easy: if $G$ is inner
amenable and $\mu$ is a finitely additive measure on $G\setminus\\{e_{G}\\}$
witnessing that $G$ is inner amenable, then $\mu\times\delta_{e_{H}}$ is a
measure on $(G\times H)\setminus\\{(e_{G},e_{H})\\}$ that witnesses that
$G\times H$ is inner amenable. For a proof of the other direction, see [4,
Proposition 2.4].

As a result of this fact, we see that, for any non-inner amenable group $G$,
we have that $G\times\operatorname{SL}_{2}(K)$ is inner amenable, nonamenable,
and not uniformly inner amenable. To see that
$G\times\operatorname{SL}_{2}(K)$ is not uniformly inner amenable, note that
$(G\times\operatorname{SL}_{2}(K))^{\mathcal{U}}\cong
G^{\mathcal{U}}\times\operatorname{SL}_{2}(K)^{\mathcal{U}}$, which is not
inner amenable as neither $G^{\mathcal{U}}$ nor
$\operatorname{SL}_{2}(K)^{\mathcal{U}}$ are inner amenable ($G^{\mathcal{U}}$
is not inner amenable by Lemma 4.4).

## References

  * [1] B. Bekka, _Infinite characters on $\operatorname{GL}_{n}(\mathbb{Q})$, on $\operatorname{SL}_{n}(\mathbb{Z})$, and on groups acting on trees_, Journal of Functional Analysis 227 (2019), 2160-2178.
  * [2] A. Connes, _Classification of injective factors. Cases II 1, II∞, IIIλ, $\lambda\not=1$_, Ann. of Math. 104 (1976), 73–115.
  * [3] L. Dickson, _Linear groups, with an exposition of the Galois field theory_ , Leipzig B.G. Tuebner (1901).
  * [4] B. Duchesne, R. Tucker-Drob, and P. Wesolek, _CAT(0) cube complexes and inner amenability_ , to appear, Groups, Geometry, and Dynamics. arXiv 1903.01596.
  * [5] E. Effros, _Property Gamma and Inner Amenability_ , Proceedings of the American Mathematical Society 47 (1975), 483-486.
  * [6] I. Farah, B. Hart, and D. Sherman. Model theory of operator algebras III: Elementary equivalence and II1 factors. Bulletin of the London Mathematical Society, vol. 46 (2014), 609-628.
  * [7] B. Fine, A. Gaglione, G. Rosenberger, and D. Spellman, _On CT and CSA groups and related ideas_ , Journal of Group Theory 19 (2016), 923-940.
  * [8] I. Goldbring, S. Kunnawalkam Elayavalli, and Yash Lodha, _Generic algebraic properties in spaces of enumerated groups_ , to appear in Transactions of the American Mathematical Society.
  * [9] K. Juschenko, _Amenability of discrete groups by example_ , Mathematical Surveys and Monographs 266 (2022), American Mathematical Society.
  * [10] G. Keller, _Amenable groups and varieties of groups_ , Illinois J. Math. 16 (1972), 257-268.
  * [11] R. Tucker-Drob and D. Kerr, _Dynamical alternating groups, stability, property Gamma, and inner amenability_ , to appear, Annales Scientifiques de l’Ecole Normale Superieure arXiv
  * [12] S. Lang, _Algebra_ , Graduate Texts in Mathematics 211 (2002), Springer-Verlag.
  * [13] D. Marker, _Model theory: an introduction_ , Graduate Texts in Mathematics 217 (2002), Springer-Verlag.
  * [14] M. Suzuki, _The nonexistence of a certain type of simple groups of odd order_ , Proc. Amer. Math. Soc. 8 (1957), 686–695.
  * [15] L. Weisner, _Groups in which the normalizer of every element but the identity is abelian_ , Bull. Amer. Math. Soc. 31 (1925), 413-416.
  * [16] Y. F. Wu, _Groups in which commutativity is a transitive relation_ , Journal of Algebra 207 (1998), 165-181.


# Enhanced Superconductivity and Electron Correlations in Intercalated ZrTe3

Yu Liu,1,§,¶ Xiao Tong,2 V. N. Ivanovski,3 Zhixiang Hu,1,4 Denis Leshchev,5
Xiangde Zhu,1,∗ Hechang Lei,1,† Eli Stavitski,5 Klaus Attenkofer,5,‡ V.
Koteski,3 and C. Petrovic1,4,∥ 1Condensed Matter Physics and Materials Science
Department, Brookhaven National Laboratory, Upton, New York 11973, USA
2Center of Functional Nanomaterials, Brookhaven National Laboratory, Upton,
New York 11973, USA
3Department of Nuclear and Plasma Physics, Vinca Institute of Nuclear Sciences
- National Institute of the Republic of Serbia, University of Belgrade,
Belgrade 11001, Serbia
4Materials Science and Chemical Engineering Department, Stony Brook
University, Stony Brook, New York 11790, USA
5National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA

###### Abstract

Charge density waves (CDWs) with superconductivity, competing Fermi surface
instabilities and collective orders, have captured much interest in two-
dimensional van der Waals (vdW) materials. Understanding of CDW suppression
mechanism, its connection to emerging superconducting state and electronic
correlations provides opportunities for engineering the electronic properties
of vdW heterostructures and thin film devices. Using combination of the
thermal transport, X-ray photoemission spectroscopy, Raman measurements, and
first-principle calculations, we observe an increase in electronic
correlations of the conducting states as CDW is suppressed in ZrTe3 with 5% Cu
and Ni intercalation in the vdW gap. As superconductivity emerges,
intercalation brings decoupling of quasi-one-dimensional conduction electrons
with phonons as a consequence of intercalation-induced lattice expansion but
also a drastic increase in Zr2+ at the expense of Zr4+ metal atoms. These
observation demonstrate the potential of atomic intercalates in vdW gap for
ground state tuning but also illustrate the crucial role of Zr metal valence
in formation of collective electronic orders.

##  I INTRODUCTION

Electronic correlations associated with charge density wave (CDW) and
superconductivity (SC), both the Fermi surface (FS) instabilities and low-
temperature collective orders in solids, attract considerable attention Gruner
; CastroNeto ; Schmitt ; Kivelson . In particular, layered van der Waals (vdW)
transition metal chalcogenides raised interest due to possible quantum
critical phenomena BarathH ; JoeY ; yet due to relatively easy exfoliation,
heteroepitaxial growth, and device fabrication of vdW crystals they can be
used to investigate confinement of correlated electronic wave functions
involved in CDW and SC and also to engineer new quantum states GeremewA ;
ZhangZ ; ConstantCB ; ZhouX ; HanGH .

ZrTe3 provides a good platform due to low-dimensional crystal structure with
$P2_{1}/m$ unit cell symmetry [Fig. 1(a)]: quasi-two-dimensional (2D) ZrTe3
prisms aligned along the $c$-axis are separated by the vdW gap with quasi-one-
dimensional (1D) ZrTe6 chains running along the $b$-axis; in addition there
are Te2-Te3 chains along the $a$-axis Bra ; Fur ; Wieting . ZrTe3 shows not
only CDW ($T_{\textrm{CDW}}$ $\sim$ 63 K) with Peierls modulation
$\emph{{q}}\approx(\frac{1}{14},0,\frac{1}{3})$ but also a filamentary SC
($T_{\textrm{c}}\sim$ 2 K) that arises on cooling from a quasi-2D electronic
conduction (resistivity $\rho_{a}\approx\rho_{b}\sim\frac{\rho_{c}}{10}$)
Eaglesham ; Takahash ; Nakajima ; Yamaya . A local-pairs-induced SC mechanism
was proposed Tsu . Band structure calculation and angular resolved
photoemission (ARPES) measurements revealed that the FS consists mainly of a
three-dimensional (3D) FS sheet centered at the Brillouin zone (BZ) center and
the quasi-1D FS sheets parallel to the inclination of the BZ boundary Stowe ;
Fesler ; Star ; Hoesch ; Lyu . A pseudogap feature and CDW fluctuations
persist far above $T_{\textrm{CDW}}$ Per ; Yokoya ; Chinotti . Kohn anomaly
associated with a soft phonon mode was identified Hoesch1 , while the Raman
experiment also emphasized the important role of electron-phonon (e-p)
coupling at ambient or high pressure Hu ; Gleason . Additionally, mixed
valence of Zr in ZrTe3 nanoribbons was reported Yu . Valence segregation with
SC in transition metal oxides with different electron correlation strength can
be connected to SC mechanism and the e-p coupling GoodenoughJB ; VarmaCM ;
RiceTM ; KimM .

Bulk SC can be induced in ZrTe3 by applying physical pressure, intercalation,
doping, and disorder Yomo ; Hoesch2 ; Gu ; HC ; XD1 ; Yadav ; Zhu ; XD2 ; Cui
; Ganose ; Yan ; Yue . A pressure-induced re-entrant SC in ZrTe3 implies a
possible unconventional SC mechanism Yomo , while the ultra-low-temperature
thermal conductivity indicates multiple nodeless gaps in ZrTe3-xSex Cui .
Light intercalation of $\sim$ 5% Cu and Ni can induce bulk SC with
$T_{\textrm{c}}\sim$ 3.8 and 3.1 K in Cu0.05ZrTe3 and Ni0.05ZrTe3,
respectively, that coexists with the CDW order HC ; XD1 . The CDW-related
resistivity anomaly for the current flow along the $a$-axis shifts to lower
temperatures with reduced amplitudes, however, the optical absorption spectra
show an increase of the CDW gap size in (Cu,Ni)0.05ZrTe3 Mirri .

Figure 1: (Color online) (a) Crystal structure of ZrTe3 with quasi-1D ZrTe6
chains along the $b$-axis and the Te2/Te3 rectangular network layer viewed
from the $c$-axis. (b) Normalized Zr (left), Cu and Ni (right) $K$-edge XANES
spectra at room temperature. (c) Temperature dependence of the normalized
resistivity $\rho$(T)/$\rho$(300 K) for ZrTe3 and (Cu,Ni)0.05ZrTe3 with the
current flow along the $a$-axis (left) and the $b$-axis (right). Insets show
the low-temperature superconducting transitions and current flow direction in
crystals.

Thermopower measurement is suitable to characterize the nature and sign of
carries as well as the correlation strength in superconductors whereas X-ray
photoemission spectroscopy (XPS) and Raman measurements are good probes of the
valence state and phonon vibrations in transition metal compounds Collignon ;
Kefeng ; Behnia ; Pourret ; Demeter . Herein we examine the electron
correlation strength and the Zr valence in superconducting (Cu,Ni)0.05ZrTe3
single crystals. Thermal transport and Raman measurements show a suppressed
CDW in (Cu,Ni)0.05ZrTe3 when compared to ZrTe3. An obvious anomaly around
$T_{\textrm{CDW}}$ in thermal conductivity $\kappa$(T) was observed in ZrTe3,
indicating strong e-p coupling. The anomaly is suppressed with weaker
amplitudes in (Cu,Ni)0.05ZrTe3. A clear slope change in thermopower $S$(T)
corresponds to a FS reconstruction across the CDW transition. Both the ratio
of $T_{\textrm{c}}$ to the Fermi temperature $T_{\textrm{F}}$ and the
effective mass $m^{*}$ rise in (Cu,Ni)0.05ZrTe3 when compared to
ZrTe2.96Se0.04 imply increase in electronic correlations of the bulk
conducting states due to contribution of Cu and Ni, in agreement with the
first-principle calculations. Whereas intercalation in the vdW gap stabilizes
bulk SC at the expense of CDW order, it coincides with drastic reduction of Zr
mixed valence and the expansion of unit cell along the $a$\- and
$c$-directions as well as a rapid increase in atomic concentration ratio of
Zr2+ to Zr4+ induced by Cu,Ni intercalation. This affects the Zr 4$d$-derived
electronic states near the $\Gamma$ point in the Brillouen zone below the
Fermi level.

##  II EXPERIMENTAL DETAILS

High quality single crystals of ZrTe3 with $\sim$ 5% Cu,Ni intercalation were
fabricated by chemical vapor transport method HC ; XD1 . X-ray absorption near
edge spectroscopy measurement was performed at 8-ID beamline of the National
Synchrotron Light Source II at Brookhaven National Laboratory in the
fluorescence mode. Thermal transport was measured in quantum design PPMS-9
with standard four-probe technique. Thermopower was measured by using one-
heater-two-thermometer setup with hooked copper leads using silver paint
contact directly on crystals. Continuous measuring mode was adopted for
thermopower measurement with the maximum heater power and period set as 50 mW
and 1430 s along with the maximum temperature rise of 3$\%$. The relative
error in our measurement for thermopower was below 5% based on Ni standard
measured under identical conditions. Sample dimensions were measured by an
optical microscope Nikon SMZ-800 with 10 $\mu$m resolution. Specific heat was
measured on warming procedure by the heat pulse relaxation method in PPMS-9.

X-ray photoemission spectroscopy (XPS) measurement was carried out in an
ultrahigh-vacuum (UHV) system with $3\times 10^{-10}$ Torr base pressure,
equipped with a SPECS Phoibos 100 spectrometer and a non-monochromatized Al-Kα
X-ray source ($h\nu$ = 1486.6 eV). The XPS peak positions were calibrated
using adventitious C $1s$ at 284.8 eV. Single selected point unpolarized Raman
spectrum experiment was performed using WITec confocal Raman microscope alpha
300 equipped with an red laser ($\lambda$ = 633 nm), an electron multiplying
CCD detector and an 100$\times$/0.9NA objective lens. The Raman scattered
light was focused onto a multi-mode fiber and monochromator with a 1800
line/mm grating. Samples for the room temperature XPS were exfoliated in air
then be sputtered in UHV by 2$\times$10-5 Torr of Ar+ ions with kinetic energy
of 2500 eV for 60 min in order to remove surface oxygen contamination. Samples
for Raman experiments were freshly exfoliated.

First-principle calculations were performed using the Vienna ab initio
simulation package (VASP) by relaxing the atomic positions simultaneously with
changing of the unit cell volume and shape KresseG . The energy convergence
criterion was 10-6 eV, while the interatomic forces were less than 0.02 eV/Å.
Energy cutoff for plane waves was 480 eV, the $k$-point network was
6$\times$9$\times$4 with the PBEsol as an exchange-correlation functional
PerdewJ . After lattice relaxation, the spin-orbit coupling (SOC) was included
for both the density of states (DOS) and the band structure calculations. The
band unfolding and the DOS figures were obtained by the Vaspkit program
VASPKIT . The high-symmetry points of the Brillouin zone were chosen on the
same manner as in Ref. Ganose

##  III RESULTS AND DISCUSSIONS

The crystal structure of ZrTe3 [Fig. 1(a)] consists of ZrTe3 trigonal prisms
building blocks stacked along the $b$-axis and forming an infinite quasi-1D
chain Bra ; Fur . The monoclinic unit cell contains two neighboring chains
bound via the nearest inter-chain Zr-Te1 to form layers in the $ab$-plane. On
the other hand, Te2-Te3 atoms form a rectangular network with alternate
distances of 2.79 and 3.10 Å along the $a$-axis, and single distance of 3.93 Å
along the $b$-axis. The 5% Cu,Ni intercalations induce expansion of the $a$\-
and $c$-axis lattice parameters, $\sim$ 0.2-0.3% for Cu0.05ZrTe3 and 0.6-0.7%
for Ni0.05ZrTe3, respectively HC ; XD1 . However the $b$-axis lattice constant
is nearly unchanged. Figure 1(b) shows the normalized Zr, Cu and Ni $K$-edge
X-ray absorption near edge spectroscopy (XANES) spectra measured at room
temperature (RT). The Zr $K$-edge absorption energy at $\sim$ 18.008 keV for
ZrTe3 indicates a dominant Zr4+ state Zahir ; it shifts to lower energies with
Cu,Ni intercalation towards the Zr2+ state. A weak pre-edge feature was
observed in the Cu,Ni spectra, which is absent in that of Zr, along with the
similar shape of Cu,Ni XANES, confirms that Cu,Ni atoms are well hybridized
and intercalated in the vdW gap rather than substituted at Zr sites Westre .

A clear resistivity upturn was observed at $T_{\textrm{CDW}}$ = 63 K for
ZrTe3, defined by the minima point in $d\rho/dT$, with the current parallel to
the $a$-axis [Fig. 1(c)]. $T_{\textrm{CDW}}$ is suppressed to 58 and 41 K for
Cu0.05ZrTe3 and Ni0.05ZrTe3, respectively, with a reduced amplitude. The
residual resistivity ratio (RRR) of $\sim$ 12.7 for ZrTe3 also decreases to
5.3 and 2.4 for Cu0.05ZrTe3 and Ni0.05ZrTe3, respectively, caused by increased
disorder scattering. An abrupt resistivity drop is clearly seen, signaling the
onset of SC, with zero resistivity at $T_{\textrm{c}}$ $\thickapprox$ 3.6 and
3.1 K for Cu0.05ZrTe3 and Ni0.05ZrTe3, respectively. For the current parallel
to the $b$-axis, no CDW anomaly was observed but SC remains for
(Cu,Ni)0.05ZrTe3; consistent with refs.HC ; XD1 .

Figure 2(a) shows the temperature-dependent thermal conductivity $\kappa$(T)
with the heat pulse applied in the $ab$-plane. The $\kappa$(300 K) shows a
relatively low value of 1.63 W/K$\cdot$m for ZrTe3, caused by the combination
of low crystal symmetry and chemical composition with heavy elements. A clear
kink in $\kappa$(T) is observed around $T_{\textrm{CDW}}$ for ZrTe3,
indicating strong e-p coupling. The value of $\kappa$(300 K) increases to
$\sim$ 1.87 and 1.77 W/K$\cdot$m for Cu0.05ZrTe3 and Ni0.05ZrTe3,
respectively. In addition, $\kappa$(T) is weakly temperature-dependent at high
temperatures; the absence of a commonly observed maximum in $\kappa$(T)
indicates a significant acoustic phonon scattering Bar .

Figure 2: (Color online) Temperature-dependent (a) thermal conductivity $\kappa$(T) and (b) thermopower $S$(T) for indicated samples. Inset in (b) shows linear fit of $S$(T) from 5 to 15 K. Table 1:  A set of parameters derived from the thermopower $S$(T) and specific heat $C_{p}$(T) for indicated single crystals. Samples |  $S/T$($\mu$V/K2) |  $\gamma$(mJ/mol$\cdot$K2) | $q$ |  $T_{\textrm{c}}$(K) |  $T_{\textrm{F}}$($10^{3}$K) |  $T_{\textrm{c}}$/$T_{\textrm{F}}$($\times 10^{-3}$) | $m^{\ast}/m_{e}$ |  $\Theta_{\textrm{D}}$(K) |  $T_{\textrm{peak}}$(K) |  $T_{\textrm{CDW}}$(K)
---|---|---|---|---|---|---|---|---|---|---
ZrTe3 | 0.114(3) |  |  | 2 | 3.7(1) | 0.54(1) |  |  | 19(3) | 60(2)
Cu0.05ZrTe3 | 0.09(1) | 2.6(1) | 3.3(2) | 3.6 | 4.7(5) | 0.77(8) | 2.2(1) | 186(1) | 30(5) | 55(2)
Ni0.05ZrTe3 | 0.188(4) | 2.7(1) | 6.7(1) | 3.1 | 2.3(1) | 1.35(6) | 2.8(1) | 192.4(1) | 26(3) | 58(2)

Figure 2(b) presents the temperature-dependent thermopower $S$(T) for ZrTe3
and (Cu,Ni)0.05ZrTe3 crystals. Above 100 K, there are weak changes in $S$(T).
With decreasing temperature, the $S$(T) of ZrTe3 changes its slope at
$T_{\textrm{CDW}}$, reflecting a FS reconstruction across the CDW transition.
Then $S(T)$ changes its sign from negative to positive inside the CDW state
featuring a peak value at $\sim$ 19(3) K, indicating a crossover behavior from
dominant electron-to-hole-like conduction. Similar behavior is observed in
(Cu,Ni)0.05ZrTe3 but with reduced absolute values of $S$ at high temperatures,
showing an increase of the hole band contribution in (Cu,Ni)0.05ZrTe3. It can
be seen that $T_{\textrm{CDW}}$ is suppressed to lower temperatures for
(Cu,Ni)0.05ZrTe3 when compared to ZrTe3, in line with the resistivity data
[Fig. 1(c)]. It is plausible to consider both the electronic diffusion
$S_{\textrm{diff}}$ and phonon drag $S_{\textrm{drag}}$ contribution to
thermopower in ZrTe3. The $S_{\textrm{drag}}$ term usually gives $\propto
T^{3}$ for $T\ll\Theta_{\textrm{D}}$, $\propto 1/T$ for
$T\gg\Theta_{\textrm{D}}$, and a peak feature at
$(\frac{1}{4}\sim\frac{1}{5})\Theta_{\textrm{D}}$. The peak feature in ZrTe3
can not be simply attributed to only the phonon-drag effect since the peak
temperature is lower than $\Theta_{\textrm{D}}/5\approx$ 36.6(1) K; this is
also applicable for (Cu,Ni)0.05ZrTe3 [see Table I]. However, the phonon drag
should diminish by $1/T$ at high temperatures which is not seen here, pointing
to the presence of diffusion contribution as well. At low temperatures, the
diffusive Seebeck response of Fermi liquid dominates and is expected to be
linear in $T$. In a single-band system, $S$(T) is given by Barnard ; Miyake ,

$\frac{S}{T}=\pm\frac{\pi^{2}}{2}\frac{k_{\textrm{B}}}{e}\frac{1}{T_{\textrm{F}}}=\pm\frac{\pi^{2}}{3}\frac{k_{\textrm{B}}^{2}}{e}\frac{N(\varepsilon_{\textrm{F}})}{n},$
(1)

where $k_{\textrm{B}}$ is the Boltzmann constant, $T_{\textrm{F}}$ is the
Fermi temperature related to the Fermi energy $\varepsilon_{\textrm{F}}$ and
the density of states $N(\varepsilon_{\textrm{F}})$ as
$N(\varepsilon_{\textrm{F}})$ = $3n/2\varepsilon_{\textrm{F}}$ =
$3n/k_{\textrm{B}}T_{\textrm{F}}$, and $n$ is the carrier concentration. In a
multiband system it gives the upper limit of $T_{\textrm{F}}$ of the dominant
band. The derived $S/T$ from 5 to 15 K [inset in Fig. 2(c)] is $\sim$ 0.114(3)
$\mu$V/K2 for ZrTe3; $\sim$ 0.09(1) and 0.188(4) $\mu$V/K2 for Cu0.05ZrTe3 and
Ni0.05ZrTe3, respectively. Then we calculate $T_{\textrm{F}}$ $\approx$
$3.7(1)\times 10^{3}$ K for ZrTe3; $4.7(5)\times 10^{3}$ and $2.3(1)\times
10^{3}$ K for Cu0.05ZrTe3 and Ni0.05ZrTe3, respectively. The ratio of
$T_{\textrm{c}}/T_{\textrm{F}}$ characterizes the correlation strength in
superconductors; $T_{\textrm{c}}/T_{\textrm{F}}$ is $\sim$ 0.1 in
Fe1+yTe1-xSex, pointing to the importance of electron correlation; while
$\sim$ 0.02 in a BCS superconductor LuNi2B2C Pourret . Here the
$T_{\textrm{c}}/T_{\textrm{F}}$ is $\sim$ 0.54(1)$\times 10^{-3}$ for ZrTe3;
$\sim$ 0.77(8)$\times 10^{-3}$ and 1.35(6)$\times 10^{-3}$ for Cu0.05ZrTe3 and
Ni0.05ZrTe3, respectively, indicating weak electron correlation strength but
also a substantial enhancement in (Cu,Ni)0.05ZrTe3 when compared to ZrTe3
[Table I].

The electronic specific heat can also be expressed as:

$\gamma=\frac{\pi^{2}}{2}k_{\textrm{B}}\frac{n}{T_{\textrm{F}}}=\frac{\pi^{2}}{3}k_{\textrm{B}}^{2}N(\varepsilon_{\textrm{F}}).$
(2)

Combining equations (1) and (2) yields: $S/T=\pm\gamma/ne$; the units are V/K
for $S$, J/K2$\cdot$m3 for $\gamma$, and m-3 for $n$, respectively Behnia ;
PtSbSn . In order to compare different materials, it is common to express
$\gamma$ in J/mol$\cdot$K2. Then we can define a dimensionless quantity
$q=\frac{S}{T}\frac{N_{\textrm{A}}e}{\gamma}$, where $N_{\textrm{A}}$ is the
Avogadro number; the calculated $q$ are 3.3(2) and 6.7(1) for Cu0.05ZrTe3 and
Ni0.05ZrTe3, respectively. Given the volume of unit cell $\sim$ 0.23 nm3, we
obtain the carrier density per volume $n\approx 1.3(1)\times 10^{21}$ cm-3 and
$k_{\textrm{F}}\approx 3.4(1)$ nm-1 for Cu0.05ZrTe3; $n\approx 6.5(1)\times
10^{20}$ cm-3 and $k_{\textrm{F}}\approx 2.68(1)$ nm-1 for Ni0.05ZrTe3. The
effective mass $m^{*}$, derived from
$k_{\textrm{B}}T_{\textrm{F}}=\hbar^{2}k_{\textrm{F}}^{2}/2m^{*}$, are 2.2(1)
and 2.8(1) $m_{e}$ for Cu0.05ZrTe3 and Ni0.05ZrTe3, respectively. Both the
ratio of $T_{\textrm{c}}$ to the Fermi temperature $T_{\textrm{F}}$ and the
effective mass $m^{*}$ rise in (Cu,Ni)0.05ZrTe3 when compared to
ZrTe2.96Se0.04 [$T_{\textrm{c}}$/$T_{\textrm{F}}$ = 0.0016(1) and $m^{\ast}$ =
1.53(1) $m_{e}$] APL , indicating the increase of correlation strength with
Cu,Ni-intercalation when compared to Se-substitution.

Figure 3: (Color online) (a) Raman peaks and (b) Zr $3d$ XPS spectra measured
at room temperature for indicated samples.

Figure 3(a) shows the Raman peaks for ZrTe3 ($\sim$ 215 and 155 cm-1) Yang ;
APL , corresponding to $A_{g}$ modes which involve atomic movements in the
$ac$-plane Hu . The mode at 155 cm-1 hosts momenta that have a larger
component along the $a$-axis of Te2-Te3 chains. The conductive electrons
belong to the quasi-1D FS are mainly contributed by the Te 5$p_{x}$ orbitals
of Te2-Te3 chains, which can be only scattered forward or backward along
Te2-Te3 chains. Consequently, phonons on this mode will experience stronger
scattering with conduction electrons of quasi-1D FS, causing the wide peak as
observed due to strong e-p coupling by satisfying
$\emph{{q}}\cdot\emph{{v}}_{\textrm{F}}\approx\omega_{q\nu}$, where
$\emph{{v}}_{\textrm{F}}$ is Fermi velocity and $\omega_{q\nu}$ is the phonon
energy Hu . The mode at 215 cm-1 also involves the Te2-Te3 chains, but does
not exhibit a large linewidth as that at 155 cm-1. This can be explained by
the fact that this mode is not coupled with the quasi-1D conduction electrons
and can only be affected by non-conductive 3D-like electrons Hu . The
sharpness of this peak suggest the good crystallinity of ZrTe3.

For (Cu,Ni)0.05ZrTe3, the mode at 215 cm-1 is almost unaffected; in contrast,
the mode at 155 cm-1 are significantly modulated. The position of highest peak
red shift $\sim$ 12 cm-1 from 155 to 143 cm-1, suggesting tensile lattice
deformation of Te2-Te3 chains along the $a$\- or $c$-axis. This is
quantitatively consistent with the structure analysis that Cu,Ni
intercalations induce expansion of the $a$\- and $c$-axis lattice parameters.
In addition, broad peaks at $\sim$ 125 and 170 cm-1 are resolved for
(Cu,Ni)0.05ZrTe3. Weak Raman peaks at $\sim$ 122 and 177 cm-1 that also belong
to the eight $A_{g}$ modes in the $ac$-plane were observed for pure ZrTe3 at
70 K with a $cc$ polarized laser Hu . Therefore, the lattice modulation caused
by Cu,Ni intercalations enabled these vibration modes become observable at RT
here with an unpolarized laser. More importantly, the major peak at 143 cm-1
with narrow linewidth is much sharper comparing to the corresponding peak for
ZrTe3 at 155 cm-1, which suggests the decoupling of quasi-1D conduction
electrons with phonons at RT. This probably due to the e-p coupling condition
of $\emph{{q}}\cdot\emph{{v}}_{\textrm{F}}\approx\omega_{q\nu}$ being less
satisfied since the phonon energy is reduced as consequence of intercalation-
induced lattice expansion. Thus, the Raman results proved that the suppressed
CDW is structural distortion-driven suppression of the quasi-1D conductive e-p
coupling. Structurally driven CDW suppression due to hydrostatic pressure or
Se doping-induced long range disorder of lattices in the $c$\- or $a$-axis,
respectively, were also reported Gleason ; APL . In these two cases, the
suppression of e-p coupling were evidenced by the reduced Raman intensity
ratio of the broad peak at 155 cm-1 to the peak at 215 cm-1; structurally, the
lattice parameters in the $c$-axis or the size of crystal unit cell were
compressed. While in (Cu,Ni)0.05ZrTe3, in contrast, the suppression of e-p
coupling is evidenced by the reduced-linewidth of the major peak at 155 cm-1;
structurally, the lattice parameters in the $a$\- and $c$-axis are expanded,
indicating that CDW suppression can be realized through structural modulation
in various pathways.

The XPS shows that both ZrTe3 and (Cu,Ni)0.05ZrTe3 have mixed chemical states
of Zr4+ and Zr2+ as evidenced by two sets of binding energies of Zr $3d_{5/2}$
and Zr $3d_{3/2}$ at $\sim$ 183, 185.4 eV, and 180, 182.4 eV [Fig. 3(b)].
However, the atomic concentration ratio of Zr4+ to Zr2+ are dramatically
altered by the intercalation, which are 4.3, 0.32 and 0.23, for ZrTe3,
Cu0.05ZrTe3 and Ni0.05ZrTe3, respectively. The average valence Zr is $\sim$
+3.6 for ZrTe3, which decreases to $\sim$ +2.5 for Cu0.05ZrTe3 and +2.4 for
Ni0.05ZrTe3, respectively, in agreement with the trend in the XANES
measurement. The Zr-Te bonds around Zr are spatially anisotropic in bond type
and spatial distribution, causing spatially anisotropic charge orbital
environment around Zr and consequently inhomogeneous Zr valence. The XANES and
Raman results suggested that the lattice parameters of ZrTe3 are changed by
Cu,Ni intercalations, thus the spatial anisotropic degree of Zr-Te bonds
around Zr for (Cu,Ni)0.05ZrTe3 can be also expected to be changed, resulting
in the observed change in ratio of Zr4+ to Zr2+. Compared to Hf,Se-doping-
caused unit cell reduction accompanying by decreased Zr2+ to Zr4+ ratio
Gleason ; APL , Cu,Ni-intercalations-induced unit cell expansion is
accompanied by increased Zr2+ to Zr4+ ratio. Since 5% Ni induced a more
expanded unit cell than Cu, Zr2+ to Zr4+ ratio also increased more. The hole-
type 3D FS at the BZ center is mainly contributed by Zr $d$ orbital. ARPES
shows that the 3D FS is slightly reduced in ZrTe2.96Se0.04 when compared to
ZrTe3 4 . The reduced Zr2+ to Zr4+ ratio in ZrTe2.96Se0.04 APL  is coincident
with the reduction of hole-type carrier numbers in the 3D band as confirmed by
ARPES 4 , thus it is possible that Zr2+ contributed to the 3D FS though it is
not much involved in the formation of CDW. The quasi-1D FS responsible for CDW
and SC is mainly distributed on the BZ edge, in particular in the $D$-pocket 4
, is more sensitive to the change in BZ size modulated by lattice
expansion/reduction.

Figure 4: (Color online)Total and atomic contribution to density of states
(a-c) and calculated band structures near $\Gamma$, B, D, and Y high symmetry
points (d-f) for ZrTe3, Cu0.06ZrTe3 and Ni0.06ZrTe3.

For insight into the impact of Cu, Ni intercalation we further performed
comparative first-principles calculations of the density of states (DOS) [Fig.
4(a-c)] and band structure [Fig. 4(d-f)]. An intercalated atom in ZrTe3,
either Cu or Ni, resides in a pentacoordinated Wyckoff 2$e$ position within
the vdW space gap. This position is in agreement with the most favorable
interstitial position Ganose  as it is energetically favored when compared to
tetrahedrally coordinated 2$a$ and 2$b$ positions. Calculated DOS for ZrTe3
[Fig. 4(a)] is in agreement with previous KuboY . Upon intercalation, Ni
contribution to DOS at the Fermi level is somewhat higher when compared to Cu
[Fig. 4(b,c)], in agreement with the effective mass $m^{*}$ (Table I). The
effects of intercalation in vdW gap of ZrTe3 are not substantial at the Fermi
level. They are mainly visible in bands along $B$-$D$ high symmetry direction
as the increase in the spectral weight of the bands at energies (0.2-0.25) eV
above the Fermi level whereas quasi-1D band dispersion is unchanged between
$D$ and $Y$ indicating negligible charge transfer from intercalated atoms to
Te2-Te3 chains. On the other hand, we notice a shift near $\Gamma$ for
intercalated crystals [Fig. 4(e,f)] for the Zr 4$d$-derived band at about
-0.25 eV; the shift has been associated with Zr-Ni charge transfer in NixZrTe3
Ganose .

For doped ZrTe3, the electronic structure of quasi-1D FS i.e., the FS nesting
condition is modulated via atomic structural modulation that are not spatial
uniformly through out the entire ZrTe3 crystal. In other words, the lattice/BZ
modulations are more likely in short range order, consistent with the
observation of reduced intensity or linewidth of the Raman peak at 155 to 143
cm-1 [Fig. 3(a)]. Long range static CDW appears to compete with SC Ghi ; EH ;
TP , while short range CDW correlations are present over a substantial range
of carrier concentrations Ghi ; DL  that support the appearance of SC. This
can explain the higher $T_{c}$ for Se-doped or Cu,Ni-intercalated ZrTe3 than
that of pristine ZrTe3. For Hf-doped ZrTe3, because most of Te2-Te3 chains are
relaxed, thus are kept the original lattice periodicity, the long range static
CDW is preserved and is more stable at the cost of reduced density of states
of quasi-1D band at the Fermi level that are necessary for SC formation,
therefore, rendering suppressed SC. Compared with TiSe2 Morosan  and other CDW
materials, the SC can be substantially enhanced by Cu,Ni-intercalation in
ZrTe3 while the CDW is only slightly suppressed. Recently, the thermal
destruction of CDW in ZrTe3 studied by scanning tunneling microscopy (STM)
reveals a weak to strong impurity pinning across the CDW transition Jiaxin ,
calling for further STM study on the effects of Hf,Se-doping and Cu,Ni-
intercalation on the collective CDW and SC orders in ZrTe3.

##  IV CONCLUSIONS

In summary, 5% Cu,Ni intercalations in the vdW gap of ZrTe3 result in increase
in $T_{c}$ as well as electron correlation strength. The ratio of Zr2+ to Zr4+
atoms also increases, resulting in band shifts near the $\Gamma$ point in the
Brillouen zone. Intercalation results in a tensile lattice deformation of
Te2-Te3 atomic chains along the $a$-/$c$-axis, and consequent expansion of the
unit cell along these directions. This reduces the e-p coupling of quasi-1D FS
due to reduction of phonon energy but also affects its nesting condition via
atomic structural modulation, thus revealing the mechanism of CDW suppression.
It exposes the important role of mixed valence in the competition of
collective orders for the ground state at the Fermi surface.

## Acknowledgements

We thank Hengxin Tan and Binghai Yan for useful communication. Work at
Brookhaven National Laboratory (BNL) is supported by the Office of Basic
Energy Sciences, Materials Sciences and Engineering Division, U.S. Department
of Energy under Contract No. DE-SC0012704. This research used the 8-ID
beamline of the National Synchrotron Light Source II, and resources of the
Center for Functional Nanomaterials, which is a U.S. Department of Energy
Office of Science User Facility, at BNL under Contract No. DE-SC0012704. First
principle calculations were supported by the Ministry of Education, Science,
and Technological Development of the Republic of Serbia.

§Present address: Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA. ∗Present address: Anhui Province Key Laboratory of Condensed
Matter Physics at Extreme Conditions, High Magnetic Field Laboratory, Chinese
Academy of Sciences, Hefei 230031, China. †Present address: Department of
Physics and Beijing Key Laboratory of Opto-electronic Functional Materials
$\&$ Micro-nano Devices, Renmin University of China, Beijing 100872, China.
‡Present address: ALBA Synchrotron Light Source, Cerdanyola del Valles,
E-08290 Barcelona, Spain.

<EMAIL_ADDRESS><EMAIL_ADDRESS>
## References

  * (1) G. Grüner, The dynamics of charge-density waves, Rev. Mod. Phys. 60, 1129 (1988).
  * (2) A. H. Castro Neto, Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides, Phys. Rev. Lett. 86, 4382 (2001).
  * (3) F. Schmitt, P. S. Kirchmann, U. Bovensiepen, R. G. Moore, L. Rettig, M. Krenz, J.-H. Chu, N. Ru, L. Perfetti, D. H. Lu, M. Wolf, I. R. Fisher, Z.-X. Shen, Transient electronic structure and melting of a charge density wave in TbTe3, Science 321, 1649 (2008).
  * (4) S. A. Kivelson, E. Fradkin and V. J. Emery, Electronic liquid-crystal phases of a doped Mott insulator, Nat. Phys. 393, 550 (1998).
  * (5) H. Barath, M. Kim, J. F. Karpus, S. L. Cooper, P. Abbamonte, E. Fradkin, E. Morosan, and R. J. Cava, Quantum and Classical Mode Softening Near the Charge-Density-Wave-Superconductor Transition of CuxTiSe2, Phys. Rev. Lett. 100, 106402 (2008).
  * (6) Y. I. Joe, X. M. Chen, P. Ghaemi, K. D. Finkelstein, G. A. de la Pena, Y. Gan, J. C. T. Lee, S. Yuan, J. Geck, G. J. MacDougall, T. C. Chiang, S. L. Cooper, E. Fradkin, and P. Abbamonte, Emergence of charge density wave domain walls above the superconducting dome in 1T-TiSe2, Nat. Phys. 10, 421 (2014).
  * (7) A. K. Geremew, S. Rumyantsev, F. Kargar, B. Debnath, A. Nosek, M. A. Bloodgood, M. Bockrath, T. T. Salguero, R. K. Lake, and A. A. Balandin, Bias-Voltage Driven Switching of the Charge-Density-Wave and Normal Metallic Phases in 1T-TaS2 Thin-Film Devices, ACS Nano 13, 7231 (2019).
  * (8) Z. Zhang, J. Niu, P. Yang, Y. Gong, Q. Ji, J. Shi, Q. Fang, S. Jiang, H. Li, X. Zhou, L. Gu, X. Wu, and Y. Zhang, Van der Waals Epitaxial Growth of 2D Metallic Vanadium Diselenide Single Crystals and their Extra-High Electrical Conductivity, Adv. Mater. 29, 1702359 (2017).
  * (9) C. B.-Constant, S. M.-Valero, R. Cordoba, and E. Coronado, Van der Waals Heterostructures Based on Atomically-Thin Superconductors, Adv. Electron. Mater. 7, 2000987 (2021).
  * (10) X. Zhou and E. E. Rodriguez, Tetrahedral Transition Metal Chalcogenides as Functional Inorganic Materials, Chem. Mater. 29, 5737 (2017).
  * (11) G. H. Han, D. L. Duong, D. H. Keum, S. J. Yun, and Y. H. Lee, Van der Waals Metallic Transition Metal Dichalcogenides, Chem. Rev. 118, 6927 (2018).
  * (12) L. Brattas and A. Kjekshus, On the properties of compounds with the ZrSe3 type structure, Acta Chem. Scand. 26, 3441 (1972).
  * (13) S. Furuseth, L. Brattas, and A. Kjekshus, On the crystal structures of TiS3, ZrS3, ZrSe3, ZrTe3, HfS3, and HfSe3, Acta Chem. Scand. Ser. A 29, 623 (1975).
  * (14) T. J. Wieting, A. Grisel, and F. Lévy, Lattice dynamical study of the low dimensional type B compounds TiS3, HfSe3, and ZrTe3, Physica 105B, 366 (1981).
  * (15) S. Takahashi, T. Sambongi, and S. Okada, Conduction properties of ZrTe3. J. de Physique 44, 1733 (1983).
  * (16) D. J. Eaglesham, Electron microscope study of superlattices in ZrTe3, J. Phys. C 17, L697 (1984).
  * (17) H. Nakajima, K. Nomura, and T. Sambongi, Anisotropic superconductiong transition in ZrTe3, Physica 143B, 240 (1986).
  * (18) K. Yamaya, Mixed bulk-filament nature in superconductivity of the charge-density-wave conductor ZrTe3. Phys. Rev. B 85, 184513 (2012).
  * (19) S. Tsuchiya, K. Matsubayashi, K. Yamaya, S. Takayanagi, S. Tanda, and Y. Uwatoko, Effects of pressure and magnetic field on superconductivity in ZrTe3: local pair-induced superconductivity, New J. Phy. 19, 063004 (2017).
  * (20) K. Stöwe and F. R. Wagner, Crystal structure and calculated electronic band structure of ZrTe3, J. Solid State Chem. 138, 160 (1998).
  * (21) C. Fesler, E. W. Finckh, H. Kleinke, F. Rocker, and W. Tremel, Electronic properties of ZrTe3, J. Mater. Chem. 8, 1787 (1998).
  * (22) P. Starowicz, C. Battaglia, F. Clerc, L. Despont, A. Prodan, H. J. P. van Midden, U. Szerer, A. Szytuta, M. G. Garnier, and P. I. Aebi, Electronic structure of ZrTe3, J. Alloys Compd. 442, 268 (2007).
  * (23) M. Hoesch, X. Cui, K. Shimada, C. Battaglia, S. Fujimori, and H. Berger, Splitting in the Fermi surface of ZrTe3: A surface charge density wave system, Phys. Rev. B 80, 075423 (2009).
  * (24) S. Lyu, L. Yu, J. Huang, C. Lin, Q. Gao, J. Liu, G. Liu, L. Zhao, J. Yuan, C. Chen, Z. Xu, and X. Zhou, Detailed electronic structure of three-dimensional Fermi surface and its sensitivity to charge density wave transition in ZrTe3 revealed by high resolution laser-based angle-resolved photoemission spectrosopy, Chin. Phys. B 27, 087503 (2018).
  * (25) A. Perucchi, L. Degiorgi, and H. Berger, Infrared signature of the charge-density wave gap in ZrTe3, Eur. Phys. J. B 48, 489 (2005).
  * (26) T. Yokoya, T. Kiss, A. Chainani, S. Shin, and K. Yamaya, Role of charge-density-wave fluctuations on the spectral function in a metallic charge-density-wave system, Phys. Rev. B 71, 140504(R) (2005).
  * (27) M. Chinotti, J. Ethiraj, C. Mirri, X. Zhu, L. Li, C. Petrovic, and L. Degiorgi, Impact of the charge density wave state in the electrodynamic response of ZrTe3-xSex: Optical evidence for a pseudogap phase, Phys. Rev. B 97, 045117 (2018).
  * (28) M. Hoesch, A. Bosak, D. Chernyshov, H. Berger, and M. Krisch, Giant Kohn anomaly and the phase transition in charge density wave ZrTe3, Phys. Rev. Lett. 102, 086402 (2009).
  * (29) Y. Hu, F. Zheng, X. Ren, J. Feng, and Y. Li, Charge density waves and phonon-electron coupling in ZrTe3, Phys. Rev. B 91, 144502 (2015).
  * (30) S. L. Gleason, Y. Gim, T. Byrum, A. Kogar, P. Abbamonte, E. Fradkin, G. J. MacDougall, D. J. Van Harlingen, X. Zhu, C. Petrovic, and S. L. Cooper, Structural contributions to the pressure-tuned charge-density-wave to superconductor transition in ZrTe3: Raman scattering studies, Phys. Rev. B 91 155124 (2015).
  * (31) X. Yu, X. Wen, W. Zhang, L. Yang, H. WU, X. Lou, Z. Xie, Y. Liu, and H. Chang, CrystEngComm 21, 5586 (2019).
  * (32) J. B. Goodenough, Chem. Mater. 26, 820 (2014).
  * (33) C. M. Varma, Rev. Mod. Phys. 48, 219 (1976).
  * (34) T. M. Rice and L. Sneddon, Phys. Rev. Lett. 47, 689 (1981).
  * (35) M. Kim, S. Klenner, G. M. McNally, J. Nuss, A. Yaresko, U. Wedig, R. K. Kremer, R. Pöttgen, and H. Takagi. Chem. Mater. 33, 6787 (2021).
  * (36) R. Yomo, Pressure effect on competition between charge density wave and superconductivity in ZrTe3: Appearance of pressure induced reentrant superconductivity. Phys. Rev. B 71, 132508 (2005).
  * (37) M. Hoesch, G. Garbarino, C. Battaglia, P. Aebi, and H. Berger, Evolution of the charge density wave superstructure in ZrTe3 under pressure, Phys. Rev. B 93, 125102 (2016).
  * (38) K. Gu, R. A. Susilo, F. ke, W. Deng, Y. Wang, L. Zhang, H. Xiao, and B. Chen, Pressure-induced enhancement in the superconductivity of ZrTe3, J. Phys.: Condens. Matter 30, 385701 (2018).
  * (39) H. Lei, X. Zhu, and C. Petrovic, Raising $T_{c}$ in charge density wave superconductor ZrTe3 by Ni intercalation, Europhys. Lett. 95, 17011 (2011).
  * (40) X. Zhu, H. Lei, and C. Petrovic, Coexistence of bulk superconductivity and charge density wave in CuxZrTe3, Phys. Rev. Lett. 106, 246404 (2011).
  * (41) C. S. Yadav and P. L. Paulose, Superconductivity at 5.2 K in ZrTe3 polycrystals and the effect of Cu and Ag intercalation, J. Phys.: Condens. Matter 24, 235702 (2012).
  * (42) X. Zhu, B. Lv, F. Wei, Y. Xue, B. Lorenz, L. Deng, Y. Sun, and C. Chu, Disorder-induced bulk superconductivity in ZrTe3 single crystals via growth control, Phys. Rev. B 87, 024508 (2013).
  * (43) X. Zhu, W. Ning, L. Li, L. Ling, R. Zhang, J. Zhang, K. Wang, Y. Liu, L. Pi, Y. Ma, H. Du, M. Tian, Y. Sun, C. Petrovic, and Y. Zhang, Superconductivity and charge density wave in ZrTe3-xSex, Sci. Rep. 6, 26974 (2016).
  * (44) S. Cui, L. He, X. Hong, X. Zhu, C. Petrovic, and S. Li, Multiband nodeless superconductivity near the charge-density-wave quantum critical point in ZrTe3-xSex, Chin. Phys. B 25, 077403 (2016).
  * (45) A. M. Ganose, L. Gannon, F. Fabrizi, H. Nowell, S. A. Barnett, H. Lei, X. Zhu, C. Petrovic, D. O. Scanlon, and M. Hoesch, Local corrugation and persistent charge density wave in ZrTe3 with Ni intercalation, Phys. Rev. B 97, 155103 (2018).
  * (46) H. Yan, N. Shirato, X. Zhu, D. Rosenmann, X. Tong, W. Xu, C. Petrovic, V. Rose, and E. Nazaretski, X-ray assisted scanning tunneling microscopy and its application for materials science: the first results on Cu doped ZrTe3, Crystals 9, 588 (2019).
  * (47) L. Yue, S. Xue, J. Li, W. Hu, A. Barbour, F. Zheng, L. Wang, J. Feng, S. B. Wilkins, C. Mazzoli, R. Comin, and Y. Li, Distinction between pristine and disorder-perturbed charge density waves in ZrTe3, Nat. Commun. 11, 98 (2020).
  * (48) C. Mirri, A. Dusza, X. Zhu, H. Lei, H. Ryu, L. Degiorgi, and C. Petrovic, Excitation spectrum in Ni- and Cu-doped ZrTe3, Phys. Rev. B 89, 035144 (2014).
  * (49) C. Collignon, A. Ataei, A. Gourgout, S. Badoux, M. Lizaire, A. Legros, S. Licciardello, S. Wiedmann, J. Yan, J. Zhou, Q. Ma, B. D. Gaulin, N. Doiron-Leyraud, and L. Tillefer, Thermopower across the phase diagram of the cuprate La1.6-xNd0.4CuO4: Signatures of the pesudogap and charge density wave phases, Phys. Rev. B 103, 155102(2021).
  * (50) K. Wang, H. Lei, and C. Petrovic, Evolution of correlation strength in KxFe2-ySe2 superconductor doped with S, Phys. Rev. B 84, 054526 (2011).
  * (51) K. Behnia, D. Jaccard and J. Flouquet, On the thermoelectricity of correlated electrons in the zero-temperature limit, J. Phys.: Condens. Matter. 16, 5187 (2004).
  * (52) Y. Liu, Z. Hu, M. O. Ogunbunmi, E. Stavitski, K. Attenkofer, S. Bobev, and C. Petrovic, Giant thermoelectric power factor anisotropy in PtSb1.4Sn0.6, Inorg. Chem. 61, 13586 (2022).
  * (53) A. Pourret, L. Malone, A. B. Antunes, C. S. Yadav, P. L. Paulose, B. Fauque, and K. Behnia, Strong correlation and low carrier density in Fe1+yTe0.6Se0.4 as seen from its thermoelectric response, Phys. Rev. B 83, 020504(R) (2011).
  * (54) M. Demeter, M. Neumann, and W. Reichelt, Mixed-valence vanadium oxides studied by XPS, Surf. Sci. 454-456, 41 (2000).
  * (55) G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  * (56) J. P. Perdew, A. Ruzsinszky, C. Gábor, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces, Phys. Rev. Lett. 100, 136406 (2008).
  * (57) V. Wang, N. Xu, J. Liu, G. Tang, and W. Geng, VASPKIT: A user-friendly interface facilitating high-throughput computing and analysis using VASP code. Computer Physics Communications 267, 108033 (2021).
  * (58) Z. Muhammad, B. Zhang, H. Lv, H. Shan, Z. Rehman, S. Chen, Z. Sun, X. Wu, A. Zhao, and L. Song, Transition from semimetal to semiconductor in ZrTe2 induced by Se substitution, ACS Nano 14, 835 (2020).
  * (59) T. E. Westre, P. Kennepohl, J. G. DeWitt, B. Hedman, K. O. Hodgson, and E. I. Solomon, A Multiplet Analysis of Fe K-Edge 1$s$ $\rightarrow$ 3$d$ Pre-Edge Features of Iron Complexes, J. Am. Chem. Soc. 119, 6297 (1997).
  * (60) N. Barišić, I. Smiljanić, P. Popčević, A. Bilušić, E. Tutiš, A. Smontara, H. Berge, J. Jaćimović, O. Yuli, and L. Forró, High-pressure study of transport properties in Co0.33NbS2, Phys. Rev. B 84, 075157 (2011).
  * (61) R. D. Barnard, Thermoelectricity in Metals and Alloys (Taylor & Francis, London, 1972).
  * (62) K. Miyake and H. Kohno, Theory of quasi-universal ratio of Seebeck coefficient to specific heat in zero-temperature limit in correlated metals, J. Phys. Soc. Jpn. 74, 254 (2005).
  * (63) Y. Liu, Z. Hu, X. Tong, D. Leshchev, X. Zhu, H. Lei, E. Stavitski, K. Attenkofer, and C. Petrovic, Thermal transport and mixed valence in ZrTe3 doped with Hf and Se, Appl. Phys. Lett. 120, 022601 (2022).
  * (64) S. Yang, H. Cai, B. Chen, C. Ko, V. O. Özçelik, D. F. Ogletree, C. E. White, Y. Shen and S. Tongay, Environmental stability of 2D anisotropic tellurium containing nanomaterials: anisotropic to isotropic transition, Nanoscale 9, 12288 (2017).
  * (65) M. Hoesch, L. Gannon, K. Shimada, B. J. Parrett, M. D. Watson, T. K. Kim, X. Zhu, and C. Petrovic, Disorder quenching of the charge density wave in ZrTe3, Phys. Rev. Lett. 122, 017601 (2019).
  * (66) Y. Kubo, Y. Sakurao, M. Ito, A. Koizumi, K. Yamaya, and Y. Uwatoko, Electron Momentum Density in the Low-Dimensional Layered System ZrTe3, J. Phys. Soc. Jpn. 76, 064711 (2007).
  * (67) G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca, A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala, D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer, and L. Braicovich, Long-rang incommensurate charge fluctuations in (Y,Nd)Ba2Cu3O6+x, Science 337, 821 (2012).
  * (68) E. H. da Silva Neto, P. Aynajian, A. Frano, R. Comin, E. Schierle, E. Weschke, A. Gyenis, J. Wen, J. Schneeloch, Z. Xu, S. Ono, G. Gu, M. Le Tacon, and A. Yazdani, Ubiquitous interplay between charge ordering and high-temperature superconductivity in cuprates, Science 343, 393 (2014).
  * (69) T. P. Croft, C. Lester, M. S. Senn, A. Bombardi, and S. M. Hayden, Charge density wave fluctuations in La2-xSrxCuO4 and their competition with superconductivity, Phys. Rev. B 89, 224513 (2014).
  * (70) D. LeBoeuf, N. Doiron-Leyraud, B. Vignolle, M. Sutherland, B. J. Ramshaw, J. Levallois, R. Daou, F. Laliberté, O. Cyr-Choinière, J. Chang, Y. J. Yo, L. Balicas, R. Liang, D. A. Bonn, W. N. Hardy, C. Proust, and L. Taillefer, Lifshitz critical point in the cuprate superconductor YBa2Cu3Oy from high-field Hall effect measurements, Phys. Rev. B 83, 054506 (2011).
  * (71) F. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G. Bos, Y. Onose, T. Klimczuk, A. P. Ramirez, N. P. Ong, and R. J. Cava, Superconductivity in CuxTiSe2, Nat. Phys. 2, 544 (2006).
  * (72) L. Liu, C. Zhu, Z. Y. Liu, H. Deng, X. B. Zhou, Y. Li, Y. Sun, X. Huang, S. Li, X. Du, Z. Wang, T. Guan, H. Mao, Y. Su, R. Wu, J.-X. Yin, J.-G. Cheng, and S. H. Pan, Thermal dynamic of charge density wave pinning in ZrTe3, Phys. Rev. Lett. 126, 256401 (2021).