# Travelling waves in an ensemble of excitable oscillators: the interplay of
memristive coupling and noise

Ivan A. Korneev  Institute of Physics, Saratov State University,
Astrakhanskaya str. 83, 410012 Saratov, Russia  Ibadulla R. Ramazanov
Institute of Physics, Saratov State University, Astrakhanskaya str. 83, 410012
Saratov, Russia  Andrei V. Slepnev  Institute of Physics, Saratov State
University, Astrakhanskaya str. 83, 410012 Saratov, Russia  Tatiana E.
Vadivasova  Institute of Physics, Saratov State University, Astrakhanskaya
str. 83, 410012 Saratov, Russia  Vladimir V. Semenov
<EMAIL_ADDRESS>Institute of Physics, Saratov State University,
Astrakhanskaya str. 83, 410012 Saratov, Russia

###### Abstract

Using methods of numerical simulation, we demonstrate the constructive role of
memristive coupling in the context of the travelling waves formation and
robustness in an ensemble of excitable oscillators described by the FitzHugh-
Nagumo neuron model. First, the revealed aspects of the memristive coupling
action are shown on an example of the deterministic model where the memristive
properties of the coupling elements provide for achieving travelling waves at
lower coupling strength as compared to non-adaptive diffusive coupling. In the
presence of noise, the positive role of memristive coupling is manifested as
significant increasing a noise intensity critical value corresponding to the
noise-induced destruction of travelling waves as compared to classical
diffusive interaction. In addition, we point out the second constructive
factor, the Lévy noise whose properties provide for inducing travelling waves.

Memristor; Adaptive coupling; Lévy noise; Travelling waves; Excitability

###### pacs:

05.10.-a, 05.45.-a, 05.40.Fb

††preprint: AIP/123-QED

> Travelling waves of different nature and properties are observed in many
> physical, biological and chemical media and networks. Models exhibiting such
> structures are widely used for description of a broad spectrum of regular
> and chaotic spatio-temporal evolutional processes from signal and energy
> transmission to climate change, spreads of epidemics, etc. To induce and
> suppress travelling waves and control their characteristics, one can vary
> the model parameters as well as apply regular or stochastic external
> forcing. One more approach for controlling travelling waves can be realized
> in networks of coupled oscillators: a control scheme based on tuning the
> coupling properties (for instance, the coupling strength, topology and
> adaptivity). In the current paper we address this issue in the context of
> interplay of adaptive coupling and noise perturbations. In addition, we
> extend a manifold of noise-induced effects observed in high-dimensional
> systems by Lévy-noise-induced travelling waves characterized by higher
> robustness in the presence of memristive coupling.

##  I Introduction

Travelling waves represent an interdisciplinary phenomenon uniting an
incredibly broad variety of dynamical processes in deterministic and
stochastic media Landa (1996); Ghazaryan, Lafortune, and Manukian (2023);
Smoller (1994); García-Ojalvo and Sancho (1999); Shepelev, Slepnev, and
Vadivasova (2016), networks Korneev  _et al._ (2021a, 2022); Semenov, Jalan,
and Zakharova (2023); Zakharova and Semenov (2023); Rybalova, Muni, and
Strelkova (2023) and delayed-feedback oscillators Giacomelli  _et al._ (2012);
Klinshov  _et al._ (2017); Semenov and Maistrenko (2018); Zakharova and
Semenov (2024). Exhibited as spatially-periodic structures, fronts, backs and
pulses, travelling waves are of a frequent occurrence in plasma physics Loarte
_et al._ (1999); Schwabe  _et al._ (2007), optics and electronics Ebeling
(1993); Akhmediev and Ankiewicz (1997); Semenov  _et al._ (2023); Liu (2016),
hydrodynamics Henry  _et al._ (2019); Debnath (1994); Johnson (1997); Dias and
Kharif (1999), chemistry Kuramoto (1984); Kapral and Showalter (1995); Epstein
and Pojman (1998), neurophysiology Hugles (1995); Muller  _et al._ (2018);
Mohan  _et al._ (2024), as well as on the edge of ecology, population biology
and epidemiology Sherratt and Smith (2008); Malchow, Petrovskii, and Venturino
(2008); Brauer and Castillo-Chavez (2012). Such diversity causes interest of
specialists in nonlinear dynamics and complex systems focused on revealing the
interdisciplinary, fundamental properties of travelling waves and controlling
their characteristics and stability.

In the current paper, travelling waves are studied in an ensemble of
stochastic excitable oscillators with local adaptive coupling. The features of
coupling are associated with the properties of the coupling element, the
memristor. The memristors have a wide range of practical applications, first
of all, neuromorphic computing and the implementation of new generation memory
elements Kozma, Pino, and Pazienza (2012); Chua, Tetzlaff, and Slavova (2022).
In the context of nonlinear dynamics, the memristor is interesting as an
element whose intrinsic properties can essentially change the dynamics of
oscillatory systems and are responsible for qualitatively new types of the
behavior from bifurcations without parameters in single memristor-based
oscillators with lines of equilibria Korneev, Vadivasova, and Semenov (2017);
Korneev and Semenov (2017); Korneev  _et al._ (2021b, 2023a, 2023b) to the
Turing patterns Buscarino  _et al._ (2016) and travelling waves in networks of
memristive elements Pham  _et al._ (2012). The presence of the memristor as a
coupling element provides for the observation of initial-condition-dependent
synchronization of regular Korneev, Semenov, and Vadivasova (2020) and chaotic
Korneev  _et al._ (2021c) self-oscillators and wave processes in single-layer
Korneev  _et al._ (2021a) and multilayer Korneev  _et al._ (2022) networks. In
the present study, we complement the results of paper Korneev  _et al._
(2021a) addressing the impact of memristive coupling on travelling waves by
the consideration of the stochastic dynamics.

The repeatedly pointed out similarity between the memristor behaviour and the
functional peculiarities of neural cell synapses (for instance, see Refs. Jo
_et al._ (2010); Li  _et al._ (2013); Serb  _et al._ (2016); Williamson  _et
al._ (2013)) has inspired us to consider an ensemble of excitable neurons
coupled through the memristive coupling as a promising model for simulation
effects in biological neural networks. Noise sources with various
characteristics inevitably present in biological neural networks and
significantly affect the neuron dynamics Lindner  _et al._ (2004); Pisarchik
and Hramov (2023). One of the most common models to describe fluctuations in
such systems is white or coloured Gaussian noise. In certain cases (for
example, in the presence of abrupt stochastic impulses), stochastic processes
with the Lévy distribution can model the dynamics of real biological neurons
more accurately as compared to Gaussian noise Nurzaman  _et al._ (2011); Wu,
Xu, and Ma (2017). Motivated by the significance of Lévy processes in neural
systems, we consider a network of excitable elements subject to the Lévy white
noise. Thus, our research is focused on two factors affecting travelling
waves: the presence of memristive coupling and noise including occasional
high-amplitude impulses. This choice is dictated by an attempt to describe
effects that can be potentially observed in biological neural networks and to
reveal fundamental peculiarities of travelling waves in complex systems with
adaptive coupling.

##  II Model and methods

The model under study is schematically illustrated in Fig. 1 (a). It
represents an ensemble of the identical FitzHugh-Nagumo oscillators in the
excitable regime. The oscillators interact through local memristive coupling.
The model equations are:

$\left\\{\begin{array}[]{l}\dfrac{dx_{i}}{dt}=\dfrac{1}{\varepsilon}\left(x_{i}-y_{i}-x^{3}_{i}/3\right)\\\
+s\left[M(z_{i-1})(x_{i-1}-x_{i})+M(z_{i})(x_{i+1}-x_{i})\right],\\\
\dfrac{dy_{i}}{dt}=0.8x_{i}-y_{i}+0.2+\xi_{i}(t),\\\
\dfrac{dz_{i}}{dt}=x_{i}-x_{i+1}-\delta z_{i},\\\ \end{array}\right.$ (1)

where $x_{i}$ and $y_{i}$ are the fast and slow dynamic variables which define
the instantaneous state of the $i$-th oscillator ($i=1,2,\ldots,N$, where $N$
is the number of interacting oscillators). The parameter $\varepsilon$ is
usually assumed to be small, which corresponds to the relaxation behaviour of
the single oscillator. Variables $z_{i}$ determine the instantaneous states of
the memristive coupling elements, whose conductivity is given by the function
$M(z_{i})=a+bz_{i}^{2}$ (the cubic memristor model where $a$ and $b$ are
parameters). The memristor state equations $\dot{z}_{i}=x_{i}-x_{i+1}-\delta
z_{i}$ contain the parameter $\delta$ which characterizes the memristor
forgetting effect. The larger is the parameter $\delta$, the shorter is the
time range where the correlation between the initial and instantaneous states
persists Chang  _et al._ (2011); Chen  _et al._ (2013); Zhou, Fang, and Yang
(2019). To avoid the effects reported in paper Korneev  _et al._ (2021a) and
associated with a continuous dependence of the oscillatory dynamics
characteristics on the initial conditions, case $\delta=0$ (the ideal
memristor case) is excluded from the consideration.

In the absence of noise, model (1) was considered in Ref. Korneev  _et al._
(2021a) where the aspects of memristive coupling are described in more detail.
In the current paper, we study the coupled FitzHugh-Nagumo oscillators in the
excitable regime subject to Lévy noise. Statistically independent additive
Lévy noise sources $\xi_{i}(t)$ in Eqs. (1) are defined as the formal
derivatives of the Lévy stable motion. Lévy noise is characterized by four
parameters: a stability index $\alpha\in(0:2]$, a skewness (asymmetry)
parameter $\beta\in[-1:1]$, a parameter $\mu$ (is assumed to be zero) being a
mean value of the Lévy noise when $1\leq\alpha\leq 2$ and a scale parameter
$\sigma$. Parameter $D=\sigma^{\alpha}$ is introduced as the noise intensity.
The characteristic function of noise sources $\xi_{i}(t)$ takes the form
Janicki and Weron (1994); Dybiec and Gudowska-Nowak (2006); Dybiec, Gudowska-
Nowak, and Hänggi (2007):

Figure 1: (a) Schematic representation of an ensemble of the memristively
coupled (through elements $M_{i}$) FitzHugh-Nagumo neurons (see Eqs. (1)); (b)
Map of regimes on the plane ($b$,$s$) characterising the deterministic system,
$\xi_{i}(t)\equiv 0$. The only regime of the collective dynamics in area I is
the quiescent steady state regime (panel (c)). This regime coexists with
travelling waves (panel (d)) in area II; (c)-(d) Space-time plots $x_{i}(t)$
illustrating the ensemble dynamics. The upper insets show the ensemble state
at the last moment $t=10^{2}$. System parameters are $a=1$,
$\varepsilon=0.01$.

$\phi(k)=\int\limits_{-\infty}^{+\infty}\exp(ikx)L_{\alpha,\beta}(\xi,\sigma,\mu)dx=\left\\{\begin{array}[]{l}\exp\left[-\sigma^{\alpha}|k|^{\alpha}\left(1-i\beta
sgn(k)\tan\dfrac{\pi\alpha}{2}\right)\right],\quad\text{for }\alpha\neq 1,\\\
\exp\left[-\sigma|k|\left(1+i\beta\dfrac{2}{\pi}sgn(k)\ln|k|\right)\right],\quad\text{for
}\alpha=1,\end{array}\right.$ (2)

where $L_{\alpha,\beta}(\xi,\sigma,\mu)$ is the probability density function.
To generate random sequences $\xi_{i}$ corresponding to characteristic
function (2), the Janicki-Weron algorithm is used Janicki and Weron (1994);
Weron and Weron (1995):

$\begin{array}[]{l}\xi_{i}=\sigma
S_{\alpha,\beta}\times\dfrac{\sin(\alpha(V+B_{\alpha,\beta}))}{(\cos(V))^{1/\alpha}}\times\left(\dfrac{\cos(V-\alpha(V+B_{\alpha,\beta}))}{W}\right)^{\dfrac{1-\alpha}{\alpha}},\quad\text{for
}\alpha\neq 1,\\\ \xi_{i}=\dfrac{2\sigma}{\pi}\left[\left(\dfrac{\pi}{2}+\beta
V\right)\tan(V)-\beta\ln\left(\dfrac{\dfrac{\pi}{2}W\cos(V)}{\dfrac{\pi}{2}+\beta
V}\right)\right],\quad\text{for }\alpha=1,\end{array}$ (3)

where
$B_{\alpha,\beta}=\left(\text{arctan}\left(\beta\tan\left(\dfrac{\pi\alpha}{2}\right)\right)\right)/\alpha$,
$S_{\alpha,\beta}=\left(1+\beta^{2}\tan^{2}\left(\dfrac{\pi\alpha}{2}\right)\right)^{1/2\alpha}$,
$V$ is a random variable uniformly distributed on
$\left(-\dfrac{\pi}{2}:\dfrac{\pi}{2}\right)$, $W$ is an exponential random
variable with mean 1 (variables $W$ and $V$ are statistically independent).
The same numerical procedure was used in Ref. Korneev, Zakharova, and Semenov
(2024) addressing the issue of Lévy noise-induced coherence resonance in the
single FitzHugh-Nagumo oscillator in the excitable regime. In case $\alpha=2$,
signals $\xi_{i}(t)$ represent independent sources of white Gaussian noise. In
case $\alpha<2$, the noise distribution is non-Gaussian and contains long
heavy tails associated with random impulses of high amplitude.

Numerical simulations were carried out by the integration of model equations
(1) using the Heun method Mannella (2002) with the time step $\Delta
t=10^{-3}$ or smaller. It is important to note that numerical modelling of
equations including $\alpha$-stable stochastic process with finite time step
implies the normalization of the noise term by $\Delta t^{1/\alpha}$ (see also
Xu  _et al._ (2016); Pavlyukevich  _et al._ (2010)). The boundary conditions
are chosen to be periodic: $x_{i\pm N}=x_{i}$, $y_{i\pm N}=y_{i}$ and $z_{i\pm
N}=z_{i}$. Sufficiently long time of transient processes are discarded to
observe the established natural and stationary dynamics. The exploration is
carried out by the analysis of plotted instantaneous spatial profiles and
spatio-temporal diagrams.

##  III Deterministic dynamics

First, model (1) is considered in the absence of noise, $\xi_{i}(t)\equiv 0$
to reveal the impact of memristive properties of coupling on travelling waves
in the deterministic system. In such a case, one can induce travelling waves
in the system by starting simulation from $x_{i}(0)=\sin(2\pi
i/N),~{}y_{i}(0)=\cos(2\pi i/N)$, if the coupling strength $s$ is sufficiently
high. Then continuous decreasing parameter $s$ allows to reveal the area of
travelling wave existence111We continuously vary the coupling strengths with
certain parameter step such that the last state at the previous iteration is
the initial condition for the next one.. Result of applying such method for
different values of the memristor parameter $b$ is depicted in Fig. 1 (b) as a
map of regimes on the plane ($b$,$s$) which contains two areas. The only kind
of collective dynamics in area I is a completely quiescent steady state regime
corresponding to the realization $x_{i}(t)\approx-1.076$ and
$y_{i}(t)\approx-0.661$ (coordinates of a stable steady state in the phase
space of the single oscillator). In contrast, the coexistence of the quiescent
regime and travelling waves is realized in area II. As can be seen in Fig. 1
(b), increasing the parameter $b$ being responsible for manifestation of
memristive properties allows to achieve travelling waves at lower values of
the coupling strength. Changing the parameter $\delta$ being responsible for
the memristor forgetting effect allows to shift the boundary between the areas
of the existence and the absence of travelling waves, but the revealed effect
persists: memristor-based adaptive coupling provides for achieving travelling
waves at lower coupling strength.

##  IV Impact of noise

###  IV.1 Gaussian noise

Figure 2: Transformation of travelling waves when increasing the intensity $D$
of white Gaussian noise in the absence ($b=0$, panel (a)) and in the presence
($b=1$, panel (b)) of the coupling element memristive properties. System
parameters are $s=0.7$, $a=1$, $\varepsilon=0.01$, $\delta=10^{-2}$. Noise
impact parameters are: $\alpha=2$, $\beta=0$, $\mu=0$ (white Gaussian noise),
the noise intensity is introduced as $D=\sigma^{\alpha}$.

We begin studying the impact of fluctuations from the case of white Gaussian
noise when fixing noise parameters $\alpha=2$, $\beta=0$, $\mu=0$ and varying
$\sigma$. In the absence of memristive properties (at $b=0$), increasing the
noise intensity results in suppression of travelling waves as demonstrated in
Fig. 2 (a1) and reduces the dynamics to the slightly fluctuating quiescent
regime. Further increasing the noise level induces spontaneous local spiking
activity which extends along the ensemble due to the action of coupling [Fig.
2 (a2)]. When the noise intensity growth continues, spiking activity becomes
more regular (the occurrence of coherence resonance) and spatially coherent
(the manifestation of synchronization of noise-induced oscillations) as
depicted in Fig. 2 (a3).

When the coupling elements express the memristive properties (at $b>0$),
travelling waves become much more robust. To illustrate this fact, panel (b1)
in Fig. 2 contains a travelling wave which persists at the noise intensity
$D=5.1\times 10^{-5}$. This value is 340 times higher than the critical value
of the noise intensity corresponding to the noise-induced travelling wave
destruction observed when the memristive properties of coupling are not
expressed (compare Fig. 2 (a1) and Fig. 2 (b1)). Moreover, further growth of
the noise intensity does not reduce travelling waves to the quiescent
dynamics, but transforms this regime to wandering wave motion including the
interaction of the initial wave with new ones spontaneously induced by noise
as depicted in Fig. 2 (b2). Noise of larger intensity induces synchronization
of the noise-induced oscillations [Fig. 2 (b3)]. Despite the space-time plot
in Fig. 2 (b3) is obtained at higher noise-intensity as compared to Fig. 2
(a3), the effect of synchronization is more pronounced which indicates the
constructive role of memristive coupling in the context of both travelling
waves and the phenomenon of synchronization.

###  IV.2 Lévy noise

In case $\alpha<2$, the noise impact is characterised by the appearance of
high-amplitude impulses [Fig. 3 (a)]. These impulses can both suppress and
induce travelling waves. If the coupling does not exhibit the memristive
properties ($b=0$) and the noise intensity is low, one can observe both
effects on the same spatio-temporal diagram (see points 1-5 in Fig. 3 (b)). In
particular, the initial wave process in Fig. 3 (b) is suppressed by noise in
point 1. However, two new waves travelling to the left and to the right are
induced by noise in point 2 and collide in point 3. Then a new pair of waves
is induced in point 4 and destroyed in points 5 due to the action of noise.

Figure 3: (a) Particular realization of Lévy noise signal $\xi_{i}(t)$ at
$\alpha=1.8$, $\beta=0$, $\mu=0$, $D=5\times 10^{-7}$. (b)-(d) Transformation
of travelling waves when increasing the intensity $D$ of Lévy noise in the
absence ($b=0$) of the coupling element memristive properties. System
parameters are $s=0.7$, $a=1$, $\varepsilon=0.01$, $\delta=10^{-2}$. Noise
impact parameters are $\alpha=1.8$, $\beta=0$, $\mu=0$.

Similarly to the case of Gaussian noise, travelling waves are much more robust
against the Lévy noise in the presence of memristive coupling. In particular,
high-amplitude noise impulses provide for appearance of new travelling waves
(see points $*$ in Fig. 4 (a),(b)), whereas spontaneous suppression of waves
is not observed even for high noise intensities (see Fig. 4 (c) where all the
noise-induced waves disappear through the collision). Further noise intensity
growth transforms the stochastic dynamics into the regime of synchronized
spiking activity [Fig. 4 (d)].

Figure 4: Transformation of travelling waves when increasing the intensity $D$
of Lévy noise in the presence ($b=1$) of the coupling element memristive
properties. System parameters are $s=0.7$, $a=1$, $\varepsilon=0.01$,
$\delta=10^{-2}$. Noise impact parameters are $\alpha=1.8$, $\beta=0$,
$\mu=0$.

##  V Conclusions

In the current paper, we have demonstrated a constructive role of memristive
coupling in the context of travelling waves observed in an ensemble of
deterministic excitable oscillators as well as in the presence of noise. In
the deterministic case, the memristive properties of the coupling elements
provide for realization of travelling waves at sufficiently lower coupling
strength as compared to the classical diffusive interaction of fixed
intensity. If travelling waves are exhibited by coupled stochastic
oscillators, memristive coupling becomes a factor supporting travelling waves
such that the noise-induced destruction of travelling waves is observed at
higher noise intensities in comparison with diffusive coupling not exhibiting
memristive properties.

The second intriguing effect consists in the action of noise. Depending on the
properties of the additive stochastic impact, it can both suppress and induce
travelling waves. Noise-induced travelling waves can be easily obtained if the
stochastic forcing includes high-amplitude impulses. We used the Lévy noise
model to visualize this fact.

For sufficiently high noise level, the regime of travelling waves is
transformed into the regime of synchronized noise-induced spiking activity
being a result of coherence resonance. In our paper, we illustrate this by
means of space-time diagrams. However, the revealed evolution into the regime
of synchronization requires additional detailed analysis which is an issue for
further study. In addition, the presented results can be extended in the
following by the consideration of another adaptive coupling models to
formulate a generalized conclusion on the impact of synaptic plasticity being
an intrinsic property of biological neural networks on effects associated with
wave propagation and syncronization in such networks.

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Acknowledgements

V.S. and I.K. acknowledge support by the Russian Science Foundation (project
No. 23-72-10040).

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# Temperature induced Lifshitz transition in WTe2

Yun Wu  Ames Laboratory, U.S. DOE and Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011, USA  Na Hyun Jo  Ames Laboratory,
U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA  Masayuki Ochi  RIKEN Center for Emergent Matter Science
(CEMS), Wako, Saitama 351-0198, Japan  JST ERATO Isobe Degenerate
$\pi$-Integration Project, Advanced Institute for Materials Research (AIMR),
Tohoku University, Sendai, Miyagi 980-8577, Japan  Lunan Huang  Ames
Laboratory, U.S. DOE and Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50011, USA  Daixiang Mou  Ames Laboratory, U.S. DOE and
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  Sergey L. Bud’ko  Ames Laboratory, U.S. DOE and Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA  P. C. Canfield
<EMAIL_ADDRESS>Ames Laboratory, U.S. DOE and Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA  Nandini Trivedi
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
Ryotaro Arita  RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama
351-0198, Japan  JST ERATO Isobe Degenerate $\pi$-Integration Project,
Advanced Institute for Materials Research (AIMR), Tohoku University, Sendai,
Miyagi 980-8577, Japan  Adam Kaminski<EMAIL_ADDRESS>Ames Laboratory,
U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA

###### Abstract

We use ultra-high resolution, tunable, VUV laser-based, angle-resolved
photoemission spectroscopy (ARPES) and temperature and field dependent
resistivity and thermoelectric power (TEP) measurements to study the
electronic properties of WTe2, a compound that manifests exceptionally large,
temperature dependent magnetoresistance. The temperature dependence of the TEP
shows a change of slope at T=175 K and the Kohler rule breaks down above
70-140 K range. The Fermi surface consists of two electron pockets and two
pairs of hole pockets along the X-$\Gamma$-X direction. Upon increase of
temperature from 40K, the hole pockets gradually sink below the chemical
potential. Like BaFe2As2, WTe2 has clear and substantial changes in its Fermi
surface driven by modest changes in temperature. In WTe2, this leads to a rare
example of temperature induced Lifshitz transition, associated with the
complete disappearance of the hole pockets. These dramatic changes of the
electronic structure naturally explain unusual features of the transport data.

The discovery of giant magnetoresistance (GMR) in Fe/Cr superlatticeBaibich
_et al._ (1988); Binasch  _et al._ (1989), opened a new era of applications in
magnetic field sensors, read heads in high density hard disks, random access
memories, and galvanic isolatorsDaughton (1999). In the quest of achieving
large MR in crystalline materials, a class of manganese oxides colossal
magnetoresistance (CMR) materials was discovered that exhibits a large change
in resistance with applied magnetic fields but only at low
temperaturesUrushibara  _et al._ (1995); Moritomo  _et al._ (1996); Ramirez
_et al._ (1997). Recently, extremely large MR has been observed in PtSn4Mun
_et al._ (2012), Cd3As2Liang  _et al._ (2015), NbSb2Wang  _et al._ (2014), and
WTe2Ali  _et al._ (2014). In both PtSn4 and WTe2, the MR shows no sign of
saturation and reaches an order of at least $10^{5}$% at low temperature.
Different mechanisms have been proposed to explain the large MR in these
materials, such as high electron/hole mobility and carrier density in PtSn4Mun
_et al._ (2012), protection mechanism from electron backscattering in
Cd3As2Liang  _et al._ (2015), electron-hole carrier compensation in WTe2Ali
_et al._ (2014); Pletikosic  _et al._ (2014); Alekseev  _et al._ (2015).
However, the exact origin of large MR in these materials remains an open
question.

The temperature dependent electrical resistivity, Hall coefficient and
thermoelectric power of tungsten-ditelluride have been known for several
decades Kabashima (1966) and a three-carrier semi-metal band model Brixner
(1962); Kabashima (1966) was proposed to explain the electrical resistivity.
Later on, density-functional based augmented spherical wave (ASW) electronic
structure calculations and relatively low resolution ARPES Augustin  _et al._
(2000) were used to study the electronic properties of WTe2 and further
supported the semimetallic nature of this material. However due to the low
resolution of these early experiments, no details about the Fermi surface or
band dispersion were obtained. Recent ARPES Pletikosic  _et al._ (2014) and
quantum oscillation Zhu  _et al._ (2015) results revealed presence of small
electron and hole pockets of roughly similar size. These findings were
consistent with carrier compensation mechanism as the primary source of the MR
effect Ali  _et al._ (2014); Alekseev  _et al._ (2015). Furthermore, this
study also reported a change of the size of the Fermi pockets between 20K and
100K. More recently, Jiang et al. Jiang  _et al._ (2015) claimed observation
of strong spin-orbital coupling effect and proposed that the backscattering
protection mechanism also may play a role in the large non-saturating MR of
WTe2.

Figure 1: (color online) (a) Magnetic field dependence of TEP measured at
T=2.2K with very clear quantum oscillations. Inset shows FFT of data after
subtraction of a smooth background. (b) Temperature dependence of TEP. Inset
shows first derivative with arrow marking the change of slope due to Lifshitz
transition. Blue line and red lines are calculated x component of TEP assuming
EF=0, and EF=20 meV. Green line shows calculated x component of TEP assuming
that EF changes from 0 meV at T=0 to 45 meV at T=300 K. (c) Generalized Kohler
plot. Arrow marks the point below which the Kohler rule is violated ($T<60$
K). Inset shows temperature dependence of the resistance measured for magnetic
field of 0 kOe, 50 kOe, 90 kOe and 140 kOe.

In this letter, we present the results from temperature dependent transport
and ultra-high resolution, tunable VUV laser based ARPESJiang  _et al._ (2014)
measurements. Our data show that there are two electron pockets and four hole
pockets along the X-$\Gamma$-X direction and a fully occupied light “hole”
band at the center of the Brillouin Zone (BZ) located just below the Fermi
energy. Systematic temperature dependence measurements revealed a Lifshitz
transition, i. e. change of the Fermi surface topology, close to 160 K above
which both pairs of the hole pockets vanish. This transition is likely
associated with change of slope observed in the derivative of the temperature
dependent thermoelectric power (TEP).

Whereas most of the previous measurements have been carried out on WTe2
crystals grown via chemical vapor transport using halogens as transport
agentsBrown (1966); Ali  _et al._ (2014), we have grown WTe2 single crystals
from a Te-rich binary melt. High purity, elemental W and Te were placed in
alumina crucibles in W1Te99 and W2Te98 ratios. The crucibles were sealed in
amorphous silica tubes and the ampoules were heated to 1000∘C over 5 hours,
held at 1000∘C for 10 hours, and then slowly cooled to 460∘C over 100 hours
and finally decanted using a centrifugeCanfield and Fisk (1992). The resulting
crystals were blade or ribbon like in morphology with typical dimensions of 3
$\times$ 0.5 $\times$ 0.01 mm with the crystallographic $c$ axis being
perpendicular to the larger crystal surface; the crystals are readily cleaved
along this crystal surface.

Temperature and field dependent transport measurements were performed in
Quantum Design Physical Property Measurement system for 1.8 K $\leq$ T $\leq$
350 K and $|$H$|$ $\leq$ 140 kOe. The thermoelectric power (TEP) measurements
were performed by a dc, alternating temperature gradient techniqueMun  _et
al._ (2010). Temperature and field dependent resistivity measurements made on
our solution grown samples (see Figs SI xx-zz) demonstrate exceedingly high
values of the residual resistivity ratio (RRR) in excess of 900 and MR values
at 1.8 K and 90 kOe up to 6x10${}^{5}~{}$%.

Samples were cleaved in situ at 40 K in ultrahigh vacuum (UHV). The data were
acquired using a tunable VUV laser ARPES system, consisting of a Scienta R8000
electron analyzer, picosecond Ti:Sapphire oscillator and fourth harmonic
generatorJiang  _et al._ (2014). Data were collected with a tunable photon
energies ranging from 5.3 eV to 6.7 eV. Momentum and energy resolution were
set at $\sim$ 0.005 Å-1 and 2 meV. The size of the photon beam on the sample
was $\sim$30 $\mu$m.

For first-principles band structure calculations, we used the Perdew-Burke-
Ernzerhof parametrization of the generalized gradient approximation Perdew
_et al._ (1996) and the full-potential (linearized) augmented plane-wave plus
local orbitals (FP-(L)APW+lo) method including the spin-orbit coupling as
implemented in the wien2k code Blaha  _et al._ (2001). Experimental crystal
structure taken from Ref. [Mar  _et al._ , 1992] was used. The muffin-tin
radii for W and Te atoms, $r_{\rm W}$ and $r_{\rm Te}$, were set to 2.4 and
2.38 a.u., respectively. The maximum modulus for the reciprocal lattice
vectors $K_{\rm max}$ was chosen so that $r_{\rm Te}K_{\rm max}$ = 9.00. TEP
was calculated using a 52$\times$29$\times$13 $k$-point mesh with the
BoltzTrap code Madsen and Singh (2006).

The field dependence of the TEP at 2.2 K is shown in Fig.1(a). As was the case
for PtSn4Mun  _et al._ (2012), very clear quantum oscillations can be seen in
the TEP of WTe2. FFT analysis (Fig.1(a) inset) gives F1 = 0.93 MOe, F2 = 1.31
MOe, F3 = 1.47 MOe and F4 = 1.70 MOe in excellent agreement with the values
found from Shubnikov-de Hass data (Fig. S2 in Supplemental Material). The peak
labeled F5 is thought to be a result magnetic breakdown of F1 and F4 low field
orbitsZhu  _et al._ (2015). The temperature dependence of the TEP is plotted
in Fig.1(b). On warming TEP first increases with temperature reaching local
maximum at $\sim$30K, then decreases, crossing zero near 50K and its absolute
value continues to increase rapidly up to $\sim$160K. For higher temperatures
the TEP changes at lower rate reaching value of - 30 $\mu$V/K at room
temperature. The change of the rate at $\sim$160K is illustrated in the inset
of 1(b), where first derivative of TEP displays clear kink marked by an arrow.
The temperature and field dependence of the extraordinarily large MR of WTe2
are shown in 1(c). The generalized Kohlers plot shows that there is fairly
good scaling of the data with an exponent of $\sim$1.98 at lower temperatures.
As temperature is increased Kohlers law scaling breaks down. Given that
temperature is an implicit variable, the temperature values associated with
the vertical arrow in 1 c range from 70 K to 140 K.

Figure 2: (color online) Fermi surface plot and band dispersion. a) Fermi
surface plot - ARPES intensity integrated within 10 meV about the chemical
potential. b) Constant energy plot at binding energy of 25 meV. c) Band
dispersion along cut #1. Dashed lines mark the two left branches of the two
hole bands. d) Band dispersion at the center of the electron pocket along cut
#2. e) Band dispersion at the center of the left hole pocket along cut #3. f)
Band dispersion at the center of the zone along cut #4. g) Band dispersion at
the center of the right hole pocket along cut #3.  Figure 3: (color online)
(a) Calculated band structure along $\Gamma$-X symmetry direction. Blue and
red dashed lines mark the values of chemical potential used for calculating
TEP. b) Fermi surface plot - ARPES intensity integrated within 10 meV about
the chemical potential measured at T=40K. c) same as (a) but for T=160K d-k)
temperature dependence of the band dispersion at the hole pocket (i. e. along
cut #1 in Fig. 1a) divided by the Fermi function. l) EDCs divided by Fermi
function at the center of the hole pocket for several temperatures. Black line
marks the energy of the peak. m) Temperature dependence of the area of the
hole pocket and energy of the top of the hole band. Purple arrows in (a, l and
k) point to a band located above the hole band that is identified in
calculation, band dispersion and Fermi function divided EDCs.

We now proceed to elucidate the electronic origin of the strange transport
behavior i. e. change of the slope of the TEP and violation of the Kohler’s
rule. The details of the Fermi surface and band dispersion along key
directions in the Brillouin zone are shown in Fig.2. In Fig.2(a) we plot the
ARPES intensity integrated within 10 meV about the chemical potential, where
high intensity contours mark the location of the Fermi surface sheets. The
Fermi surface consists of a pair of hole pockets and an electron pocket on
each side of the $\Gamma$ point. The electron pocket on the right side is not
visible as it is outside of the scan range. The intensity at binding energy of
25 meV is shown in Fig.2(b). The size of the contours for the electron pockets
decreases and size of the contour of the hole pockets increases demonstrating
the correct assignment of the character of the carriers. The band dispersion
along the main symmetry axis is plotted in Fig.2(c). Due to matrix elements,
only the right branch of the electron band is clearly visible. This is
followed by two crossings of the left sides of the hole bands (marked by
dashed lines), and then a single crossing of the right branch of both hole
bands. At the center of the BZ, the top of hole band is located just below EF,
thus there is no hole pocket at the center of the zone. Detailed band
dispersion along vertical cuts are shown in Fig.2(d-g). Fig.2(d) shows that
the bottom of the electron band joins with the top of a lower band and appears
just like the structure of a Dirac stateGeim and Novoselov (2007)
approximately 70 meV below EF. This is different from calculations shown in
Fig. 3a, where bottom of the electron pocket is separated from the band below
by a 200 meV gap. Cut #3 and Cut #5 show the dispersion of the hole pockets.
At the cut location, the bands that belong to this pair of hole pockets seem
already merged within our resolution. At the $\Gamma$ point (Cut #4), two
bands are visible with the top one located just below EF.

We now proceed to describe one of more intriguing electronic property of this
material that leads to unusual transport properties. Fig.3 a shows the
calculated band structure along $\Gamma$-X direction. The band calculation
predicts pair of hole pockets and an electron pocket between $\Gamma$ and X,
in agreement with ARPES data presented above and previous calculationsAli  _et
al._ (2014). In Figs.3b, c we show the Fermi surface map measured at 40K and
160K. The hole pockets shrink from two circles to a spot of intensity and
electron pocket expands with increasing temperature. This is consistent with
previous comparison of low and high temperature FS size Pletikosic  _et al._
(2014). We detail this behavior by plotting ARPES intensity divided by the
Fermi function along the vertical cut at the center of the hole pocket in
Fig.3(d-k). A clearly visible hole band moves down in energy and by 160 K its
top touches the chemical potential and at temperature of 280 K, the top of
this band has sunk below the chemical potential. To quantify this effect we
plotted the EDCs divided by the Fermi function at the center of the hole
pocket for several temperatures in Fig.3(l). We extract the energy position of
the peak by fitting a gaussian and plot such extracted energy location of the
top of the hole band in Fig.3(m). The top of the hole band moves down in
energy upon increasing temperature from 18 meV at 120 K to -7 meV at 280 K. We
also extracted the area of the hole pocket by measuring the separation between
MDC peaks as a function of temperature and plot this quantity in Fig.3(m). The
energy position of the top of the hole band moves below the chemical potential
and thus the area of the hole pocket must vanish above $\sim$160 K. These data
provide an archetypical example of temperature induced Lifshitz transition
since they demonstrate a change of the Fermi surface topology upon heating.
Chemical substitution induced Lifshitz transitions are quite common and were
previously observed in Ba(Fe1-xCox)2As2 at Co concentrations of 3.8% 11% and
20% Liu  _et al._ (2010, 2011). On the other hand, a temperature induced
Lifshitz transition in the absence of a structure or magnetic phase transition
is extremely rare. (Interestingly enough in Ba(Fe1-xRux)2As2, where the Fermi
surface consists of hole and electron pockets similar, significant shifts of
the chemical potential and resulting changes in size of both pockets were
observedDhaka  _et al._ (2013).) The temperature dependent TEP, in particular
the change of slope of $\partial S/\partial T$ at $\sim 160$ K (Fig. 1, inset)
is consistent with the suggested temperature induced Lifshitz transition, as
TEP is expected to be very sensitive to the changes in the Fermi surface
topology Varlamov  _et al._ (1989).

Whereas the Lifshitz transition itself can be inferred from both the TEP and
ARPES data as outlined above, the dramatic, temperature dependent change in
relative size of the electron and hole pockets should manifest itself in other
ways over wider temperature ranges. Blue and red curves in 1 b show calculated
x-component of TEP using two fixed values of EF of 0 meV and 20 meV
respectively. Neither of these two curves matches the experimental data even
on qualitative level. The EF=0 meV scenario (blue line) has positive peak at
low temperature, but also positive values in high temperature limit unlike
seen in the experimental TEP data. The EF=20 meV curve has the correct high
temperature limit, but lacks positive peak at low temperature. Clearly,
calculation with fixed chemical potential does not reflect experimental data
even on a qualitative level. The temperature induced shift of the chemical
potential discussed above naturally solves this problem. In conventional
metals, the chemical potential does not change with temperature appreciably at
room temperature when the kBT$\ll$EF. In WTe2 and other semimetals, where the
top of the hole band and bottom of there electron band is in close proximity
(few tens of meV) of EF one can expect significant changes of the chemical
potential with temperature (also see calculated density of states presented in
Supplemental Material). To demonstrate this we calculated the shift of the
chemical potential with temperature by imposing requirement that the total
number of electrons in all bands is conserved (see Supplemental Material).
This leads to a change of the chemical potential of 14 meV between T=0 K and
T=300 K. We repeated calculation of TEP using such obtained temperature
dependent value of the chemical potential scaled so that $E_{F}=$45 meV at
T=300 K to account for possible renormalization effects. The high and low
temperature values of the chemical potential used in calculation of TEP are
illustrated by dashed lines in Fig.3 a. The result of the calculation is
plotted as green line in Fig. 1 b. Although this is an approximation and does
not take into account the thermal expansion and other effects such as phonon
drag Barnard (1972) it does reproduce key features of the TEP data on
qualitative level including positive peak at low temperature and correct trend
in high temperature limit. The MR data presented in Fig. 1 c also indicates
that a simple Kohlers rule model breaks down as the sample is heated. Such a
breakdown is likely associated with a changing ratio of electron and hole
carriers implied by the data in 3 m and caused by the temperature induced
shift of the chemical potential.

In summary, we used ultra-high resolution tunable laser ARPES to investigate
the electronic structure of WTe2. Fermi surface consists of two electron
pockets and two pairs of hole pockets. This is consistent with the band
calculation reported earlierAli  _et al._ (2014). Upon increasing temperature,
both electron and hole bands move to higher binding energy, or in other words
the chemical potential is increasing. Eventually this leads to vanishing of
the hole pocket for T$>$160K and thus a Lifshitz transition occurs, above
which the Fermi surface consists of only two electron pockets. Around this
temperature we observe discontinuity of the slope of the derivative of TEP,
confirming that the Lifshitz transition affects the transport and
thermodynamics of this material. This effect is also naturally explains
features of temperature dependence of the TEP and breakdown of Kohler rule. We
speculate that the temperature induced shift of the chemical potential and
resulting change of the relative size of electron and hole pockets causes the
large magnetoresistance effect to vanish at elevated temperatures. Reported
here change of the chemical potential upon increase of the temperature is not
expected for simple materials, but was observed previously in iron arsenic
high temperature superconductorsDhaka  _et al._ (2013), where both electron
and hole pockets are located in close proximity of EF.

We would like to thank Mohit Randeria for very useful discussions. Research
was supported by the US Department of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering. Ames Laboratory is operated
for the US Department of Energy by the Iowa State University under Contract
No. DE-AC02-07CH11358. Na Hyun Jo is supported by the Gordon and Betty Moore
Foundation EPiQS Initiative (Grant No. GBMF4411).

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Supplementary material for ”Temperature induced Lifshitz transition in WTe2”

##  I Transport measurements

Resistance of the samples as a function of temperature and magnetic field
(Fig. S4) was measured in a standard, linear four probe configuration using
the ACT option of a Quantum Design Physical Property Measurement System (PPMS)
instrument. Clear Shubnikov - de Haas oscillations are seen in the
magnetoresistance at low temperatures. For analysis, the oscillatory part of
magnetoresistance (after subtraction of a monotonic part approximated by a
second order polynomial) is plotted as a function of $1/H$ (Fig. S5) and fast
Fourier transform algorithm is used.

Hall resistivity, $\rho_{H}$, was measured using a four probe method, in
positive ($\rho_{+H}$) and negative ($\rho_{-H}$) magnetic field and was
calculated as $\rho_{H}=(\rho_{+H}-\rho_{-H})/2$ to separate odd in field
$\rho_{H}$ from possible contamination of the even in field magnetoresistance.
The results are presented in Fig. S6.

Figure 4: (color online) Resistivity data with residual resistivity ratio
of$\sim 907$ and magnetoresistance of $\sim 6.5\times 10^{5}\%$ at 1.8 K and
90 kOe (H$\parallel$c). Figure 5: (color online) Shubnikov-de Haas
oscillations for $H\parallel c$ at selected temperatures. Upper inset shows
fast Fourier transform (FFT) spectra. It represents five different
frequencies: $F^{1}=0.92~{}MOe$, $F^{2}=1.32~{}MOe$, $F^{3}=1.52~{}MOe$,
$F^{4}=1.70~{}MOe$, $F^{5}=1.73~{}MOe$. The frequency, $F^{5}$, seems to be
due to the magnetic breakdown of $F^{1}$ and $F^{4}$ orbits. The slope of the
lower inset demonstrates effective mass of each frequencies based on Lifshitz-
Kosevich theory, $m^{*}_{F^{1}}=0.27m_{e}$, $m^{*}_{F^{2}}=0.24m_{e}$,
$m^{*}_{F^{3}}=0.27m_{e}$, $m^{*}_{F^{4}}=0.30m_{e}$. Figure 6: (color online)
Hall coefficient, $R_{H}=\rho_{H}/H$, at H=90 kOe. Inset shows Hall
resistivity of WTe2 at T=1.8, 10, 25, 50, 100, 200, 300 and 330 K. Quantum
oscillations in Hall resistivity are clearly seen at 1.8K

##  II Theory Calculations

The technical details of the band structure calculation are provided in main
text. Fig. S4a shows calculated band structure along main symmetry directions.
In panel Fig. S4b we plot the calculated density of states (DOS). The negative
slope of the calculated DOS (i. e. DOS decreasing when increasing the energy)
implies that the chemical potential will increase with increasing temperature
and DOS in this temperature range is dominated by the hole pockets.

The changes of the value of the chemical potential with temperature are shown
in Fig. S5 and were calculated by fixing the total number of electrons in all
bands while applying a Fermi distribution for set temperature.

Figure 7: (color online) Calculated band structure (left) and corresponding
density of states (right). The upper dashed line marks energy of the top of
the hole band that corresponds to minimum in DOS, the lower dashed line marks
$E_{F}$. Figure 8: (color online) Calculated changes of the value of the
chemical potential with temperature.