# Spin-tunable thermoelectric performance in monolayer chromium pnictides

Melania S. Muntini<EMAIL_ADDRESS>Department of Physics, Faculty
of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Surabaya
60111, Indonesia.  Edi Suprayoga<EMAIL_ADDRESS>Research Center for
Quantum Physics, National Research and Innovation Agency, Tangerang Selatan
15314, Indonesia.  Sasfan A. Wella  Research Center for Quantum Physics,
National Research and Innovation Agency, Tangerang Selatan 15314, Indonesia.
Iim Fatimah  Department of Physics, Faculty of Science and Data Analytics,
Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia.  Lila Yuwana
Department of Physics, Faculty of Science and Data Analytics, Institut
Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia.  Tosawat Seetawan
Center of Excellence on Alternative Energy, Research and Development
Institution, Sakon Nakhon Rajabhat University, Sakon Nakhon 47000, Thailand.
Program of Physics, Faculty of Science and Technology, Sakon Nakhon Rajabhat
University, Sakon Nakhon 47000, Thailand.  Adam B. Cahaya  Department of
Physics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia,
Depok 16424, Indonesia  Ahmad R. T. Nugraha  Research Center for Quantum
Physics, National Research and Innovation Agency, Tangerang Selatan 15314,
Indonesia.  Eddwi H. Hasdeo<EMAIL_ADDRESS>Research Center for
Quantum Physics, National Research and Innovation Agency, Tangerang Selatan
15314, Indonesia.  Department of Physics and Material Science, University of
Luxembourg, L-1511 Luxembourg.

###### Abstract

Historically, finding two-dimensional (2D) magnets is well known to be a
difficult task due to instability against thermal spin fluctuations. Metals
are also normally considered poor thermoelectric (TE) materials. Combining
intrinsic magnetism in two dimensions with conducting properties, one may
expect to get the worst for thermoelectrics. However, we will show this is not
always the case. Here, we investigate spin-dependent TE properties of
monolayer chromium pnictides (Cr$X$, where $X$ = P, As, Sb, and Bi) using
first-principles calculations of electron- and phonon-energy dispersion, along
with Boltzmann transport formalism under energy-dependent relaxation time
approximation. All the Cr$X$ monolayers are dynamically stable and they also
exhibit half metallicity with ferromagnetic ordering. Using the spin-valve
setup with antiparallel spin configuration, the half metallicity and
ferromagnetism in monolayer Cr$X$ enable manipulation of spin degrees of
freedom to tune the TE figure of merit ($ZT$). At optimized chemical potential
and operating temperature of $500$ K, the maximum $ZT$ values ($\approx 0.22$,
$0.12$, and $0.09$) with the antiparallel spin-valve setup in CrAs, CrSb, and
CrBi improve up to almost twice the original values ($ZT\approx 0.12$, $0.08$,
and $0.05$) without the spin-valve configuration. Only in CrP, which is the
lightest species and less spin-polarized among Cr$X$, the maximum $ZT$
($\approx 0.34$) without the spin-valve configuration is larger than that
($\approx 0.19$) with the spin-valve one. We also find that, at 500 K, all the
Cr$X$ monolayers possess exceptional TE power factors of about $0.02$–$0.08$
W/m.K2, which could be one of the best values among 2D conductors.

##  I  Introduction

Figure 1: (a) Geometrical structure of 2D chromium pnictides with honeycomb
(hexagonal) lattice. (b) Spin-valve device setup with antiparallel spin
configuration.

Thermoelectric (TE) materials convert heat directly into electricity so that
they could be beneficial for powering electronic devices in some situations
where heat loss is abundant [1], such as from automotive engines, high-
performance computers, or even human bodies. For most of the applications at
moderate temperature (around 500 K), the cost-effective power generation of
the TE materials requires a high power factor (PF) on the order of at least
$10^{-3}$ W/m.K2 [2]. Besides PF, there is also an efficiency-related quantity
known as the dimensionless figure of merit, $ZT$, which one can calculate by
the following formula [1]:

$ZT=\frac{\mathrm{PF}}{\kappa}T=\frac{S^{2}\sigma}{\kappa_{e}+\kappa_{ph}}T,$
(1)

where $S$ is the Seebeck coefficient, $\sigma$ is the electrical conductivity,
$T$ is the operating temperature, and $\kappa$ is the total thermal
conductivity (sum of the electronic contribution $\kappa_{e}$ and lattice (or
phonon) contribution $\kappa_{ph}$). Therefore, in conventional
thermoelectrics, to obtain as large $ZT$ as possible, we need simultaneously
high $S$, high $\sigma$, and low $\kappa$. This condition is, unfortunately,
difficult to achieve by nature [3, 4, 5]. For example, in normal metals,
$\sigma$ and electronic part of $\kappa$ (denoted by $\kappa_{e}$) depend on
each other through the Wiedemann-Franz law [6], which prohibits very high
$\sigma$ and very low $\kappa$ to emerge simultaneously. Furthermore, $S$
($\sigma$) in metals or other conducting materials is lower (higher) than $S$
($\sigma$) in semiconductors or insulators [7]. In this regard, much of
thermoelectrics research in the past decades has often focused on
semiconductors to find optimal values of the TE transport coefficients ($S$,
$\sigma$, and $\kappa$) that can give $ZT>1$ [8]. On the other hand, most
conductors usually only possess $ZT<0.1$ [9].

Recent advances in two-dimensional (2D) materials have also opened up an
opportunity for exploring high-performance thermoelectrics in low dimensions
[10, 11, 12]. For example, black phosphorus [13, 14, 15], transition metal
dichalcogenides [16, 17], and group-III chalcogenides [18, 19] in the family
of 2D semiconductors have been found to exhibit better TE performance than
their bulk counterparts. Pioneering studies by Dresselhaus’ group have pointed
out that the quantum confinement effect plays a crucial role in improving PF
or ZT in low dimensions [20, 21]. In particular, when the confinement length
is less than the thermal de Broglie wavelength of the material under
consideration, the PF enhancement is theoretically guaranteed to be achieved
[22]. However, the theory assumed the material does not have intrinsic
magnetism and is not spin-polarized. Therefore, to obtain high-performance
thermoelectrics in 2D materials with intrinsic magnetism, it seems that one
needs to consider another enhancement technique beyond quantum confinement and
find the materials that can utilize the mechanism.

Before looking for suitable 2D magnets for thermoelectrics, one should note
that the difficulty of obtaining long-range magnetic order in two dimensions
is a long-standing problem that prevents most 2D materials from possessing
intrinsic magnetism. According to the Mermin-Wagner theorem, the long-range
magnetic order in isotropic 2D magnets at finite temperatures is unstable
against thermal spin fluctuations [23]. However, by the presence of magnetic
anisotropy, some materials such as CrI3 and Cr2Ge2Te6 can emerge as 2D magnets
with the magnetic order depending on the number of layers [24, 25]. Analyzing
the structural simplicity of CrI3, one may expect the other 2D Cr-based
honeycomb structures [Fig. 1(a)] may also lead to intrinsic magnetism. Indeed,
recent studies on monolayer chromium pnictides (Cr$X$, where $X$ = P, As, Sb,
and Bi) suggested that monolayer Cr$X$ can be another 2D magnet, yet half-
metallic. While monolayer CrI3 as a _semiconductor_ is potentially a good TE
material [26, 27], we do not know whether monolayer Cr$X$ can perform
similarly to CrI3. Note that CrI3 also exhibit spin-dependent TE properties,
with the $ZT$ value, at 500 K, of about $0.26$, which is high enough for the
family of 2D materials (c.f. monolayer MoS2 with $ZT\sim 0.11$ [16] and InSe
with $ZT\sim 0.5$ [28] at the same temperature). However, the constant
relaxation time approximation used in Refs. [26] and [27] might not be
accurate enough for the $ZT$ calculation. Moreover, although the TE transport
coefficients in CrI3 are spin-dependent, there is no clear advantage of the
spin polarization for the $ZT$ enhancement in CrI3 because one cannot utilize
the spin degrees of freedom in semiconductors using the spin-valve TE device
[29]. By contrast, half metallicity in monolayer Cr$X$ may allow manipulation
of spin degrees of freedom to enhance the $ZT$.

In this work, we will show our simulation suggesting that the $ZT$ values in
monolayer Cr$X$ can increase using the spin-valve setup with antiparallel
configuration [Fig. 1(b)]. Since the majority carriers (spin-up states) are
metallic and the minority ones (spin-down states) are semiconducting, the PF
values in monolayer Cr$X$ are also exceptionally high for a 2D material
because the minority carriers contribute to a high Seebeck coefficient, while
the majority carriers contribute to high electrical conductivity. We perform
the calculations for TE transport coefficients within the linearized Boltzmann
transport theory and energy-dependent relaxation time approximation, with
electronic energy and phonon dispersion relations obtained from first-
principles density functional theory (DFT), as outlined in the next section.

##  II  Computational Methods

All optimized geometrical structures, electronic properties, and phonon
dispersion relations of Cr$X$ monolayers are calculated with DFT as
implemented in the Quantum ESPRESSO code [1]. We employ the optimized norm-
conserving Vanderbilt (ONCV) pseudopotentials [2, 3] to describe the
interaction between electrons and ions. We use the Perdew-Burke-Ernzerhof
functional [4] under generalized gradient approximation to describe exchange-
correlation energy and potential. The wave functions are expanded in plane-
wave basis sets with the cutoff energy as high as $\sim$820 eV. We sample the
Brillouin zone using dense $32\times 32\times 1$ and $64\times 64\times 1$
Monkhorst-Pack (MP) grids for calculations of optimized geometry and density
of states, respectively. All structures are relaxed until the maximum
Hellmann-Feynman force per atom is less than $0.26$ meV/Å. We set a vacuum
layer to 30 Å to avoid the interlayer interactions due to the lattice
periodicity. We then obtain the optimized lattice constants $a$ [see Fig.
1(a)] of $3.88$, $3.94$, $4.39$, and $4.58$ Å for CrP, CrAs, CrSb, and CrBi,
respectively, in good agreement with available data in the literature [34,
35]. With the optimized geometry data, we can calculate the ground-state
electronic structures of monolayer Cr$X$.

We use DFT+$U$ improvement of accuracy, especially for Cr, to include the
effects of strong electronic correlations. Following a prior theoretical study
[34], $U=3$ eV can be employed for Cr atoms with the spin polarization taken
into account for all the systems. The initial magnetic moment of each Cr$X$ is
set to three times Bohr magneton ($\mu_{B}$). After relaxation, the resulting
magnetic moments of CrP, CrAs, CrSb, and CrBi monolayers are $2.97\mu_{B}$,
$3.00\mu_{B}$, $3.00\mu_{B}$, and $3.00\mu_{B}$, respectively, which agree
well with the reported value of $3\mu_{B}$ in the literature [34, 35]. For the
spin-unpolarized band structures, readers can refer to Supplementary Material
Fig. S1 [36]. Although Cr$X$ monolayers here posssess non-zero magnetic
moments, the DFT+$U$ method is already sufficient and consistent with the
DFT+$U$+$J$ method because the $J$ parameter for these cases was found to be
small enough, around $10$ meV [34, 35]. The dynamical stability of Cr$X$ is
also confirmed by calculating the phonon dispersion [Fig. S2 in the
Supplementary Material] with the dynamical matrix evaluated on the $4\times
4\times 1$ MP grid of q-points [36]. Note that we do not consider van der
Waals interaction in this study because it will not change the band structure
significantly; it only gives a minor change in the lattice constants [34, 35].

Having complete information of electronic structures, we calculate the TE
transport coefficients using BoltzTraP2 code [6] within linearized Boltzmann
transport theory and energy-dependent relaxation time approximation. The
moment of the generalized transport coefficient with spin index $j=\uparrow$
and $\downarrow$ is given by,

$\mathcal{L}^{(\alpha)}_{j}=e^{2}\sum_{n,\textbf{k}}\tau_{n,\textbf{k}}v^{2}_{n,\textbf{k}}(\varepsilon_{n,\textbf{k}}-\mu)^{\alpha}\left(-\frac{\partial
f_{n,\textbf{k}}}{\partial\varepsilon_{n,\textbf{k}}}\right).$ (2)

This kernel is used to calculate the electrical conductivity, the Seebeck
coefficient, and the electron thermal conductivity as follows:

$\sigma_{j}=\mathcal{L}^{(0)},\quad
S_{j}=\frac{1}{eT}\frac{\mathcal{L}^{(1)}}{\mathcal{L}^{(0)}},\quad\kappa_{e,j}=\frac{1}{e^{2}T}\left[\mathcal{L}^{(2)}-\frac{\mathcal{L}^{(1)2}}{\mathcal{L}^{(0)}}\right],$
(3)

where $e$ is the unit electric charge, $\mu$ is the chemical potential,
$\varepsilon_{n,\textbf{k}}$ is the energy of $n$th band at wave vector k,
$\tau_{n,\textbf{k}}$ is the electronic relaxation time, $v_{n,\textbf{k}}$ is
the electronic group velocity, and $f_{n,\textbf{k}}$ is the Fermi-Dirac
distribution function. To fairly treat $\sigma$ and $\kappa_{e}$ of 2D
systems, we multiply the output $\sigma$ and $\kappa_{e}$ from BoltzTraP2 with
the thickness of the simulation box over the conventional width of the
monolayer. The total contributions from both spins read as:

$\sigma=\sigma_{\uparrow}+\sigma_{\downarrow},\quad
S=\frac{\sigma_{\uparrow}S_{\uparrow}+\sigma_{\downarrow}S_{\downarrow}}{\sigma},\quad\kappa_{e}=\kappa_{e,\uparrow}+\kappa_{e,\downarrow}.$
(4)

We consider the electron-phonon scattering as the most dominant scattering
contribution to the electronic relaxation time, which can be defined as
$\tau_{n,\textbf{k}}=\hbar/(2~{}\mathrm{Im}\sum_{n,\textbf{k}})$, where
$\hbar$ is the reduced Planck constant and $\mathrm{Im}\sum_{n,\textbf{k}}$ is
the imaginary part of the electronic self energy. Using the EPW code [5], we
compute the self energy using the electronic energy and the phonon dispersion
on relatively coarse $16\times 16\times 1$ k-point and $8\times 8\times 1$
q-point grids, respectively. To obtain a finer result, we fit
$1/\tau_{n,\textbf{k}}$ from EPW with an energy-dependent $\tau$ model:

$1/\tau(\varepsilon)=\mathcal{C}\cdot{\mathrm{DOS}}(\varepsilon)+1/\tau_{0},$
(5)

where $\mathcal{C}$ is a DOS-dependent fitting parameter and $\tau_{0}$ is the
relaxation time constant that is chosen to overcome the absence of
$\mathrm{DOS}$ in the band gap area in the spin-down band. The resulting
$1/\tau(\varepsilon)$ model for each Cr$X$ species is given in Fig. S3 [36].
Note that although the CrX family are ferromagnetic materials, we do not
consider contribution of magnetic spin excitations (known as magnons) to the
thermoelectric properties. The magnon effect in heterostructures [like the
system depicted in Fig. 1(b)] is suppressed because the magnons cannot escape
the ferromagnets [43].

Finally, to fully obtain $ZT$, we calculate the phonon thermal conductivity
$\kappa_{ph}$ from the linearized phonon Boltzmann transport equation for
phonons within the single-mode relaxation-time method, as implemented in the
Phono3py code [39, 40]. The second-order harmonic and third-order anharmonic
interatomic force constants are computed using a $2\times 2\times 1$ supercell
with the finite displacement method. To obtain a converged result, we use
$32\times 32\times 1$ q-point mesh to sample reciprocal space of the primitive
cells of all Cr$X$ monolayers.

Figure 2: Electronic structure (energy dispersion and density of states) of
chromium pnictides: (a) CrP, (b) CrAs, (c) CrSb, (d) CrBi. The blue dashed
(red solid) lines refer to spin-up (-down) states.

##  III Results and discussion

The Cr$X$ monolayers have a honeycomb structure without buckling along an axis
normal to the surface, similar to graphene [see Fig. 1]. Their electronic
structures, on the other hand, do not correspond to graphene mostly because of
the presence of $d$ orbitals in Cr. The $d$ orbitals play a significant role
to lift the spin degeneracy. In Fig. 2, we show the electronic structures of
(a) CrP, (b) CrAs, (c) CrSb, and (d) CrBi, which are all consistent with
previous first-principles results [34, 35]. In particular, one can compare the
good agreement of Fig. 2 in this study with Fig. 4 of Ref. [34]. The dashed
(solid) line refers to spin-up (down) states. Spin-up states are metallic
while spin-down states possess band gaps displaying the half-metallic phase.
At the Fermi level, chromium pnictides are ferromagnet. These electronic
structures are distinct from monolayer chromium iodide ($\mathrm{CrI_{3}}$
which shows ferromagnetic insulator [24]. As atomic number increases from CrP
to CrBi, both bandwidth and half-metallic band gap decrease. We will show
later how these behaviors will tune the TE transport.

Figure 3: Thermoelectric quantities of CrP: (a-c) Electrical conductivity,
(d-f) Seebeck coefficient, and (g-i) electron thermal conductivity of spin-up,
spin-down, and total, respectively.

In Fig. 3, we plot the TE transport coefficients as a function of chemical
potential for CrP as a representative of chromium pnictides at $T=500~{}\rm
K$. Meanwhile, the TE transport coefficients of CrAs, CrSb, and CrBi are given
in Figs. S4-S6 [36]. As expected, the spin-up conductivity dominates over the
spin-down conductivity as depicted in Figs. 3(a) and 3(b). Therefore, the
total conductivity $\sigma$ pretty much resembles the spin-up contribution
[Fig. 3(c)]. On the contrary, the Seebeck effect is dominated by the spin-down
states because of the presence of the band gap. According to the Mott formula,
the Seebeck coefficient is proportional to
$-\sigma^{\prime}(\mu)/\sigma(\mu)$; thus it will be large if the band gap
exists. For the spin-down states, the Seebeck coefficient reaches
$2000~{}\mathrm{\mu V/K}$. Normally, the Seebeck coefficient is proportional
to the band gap $\Delta$ in the non-degenerate limit ($k_{B}T\ll\Delta$). For
the spin-up states, the Seebeck coefficient is not entirely zero, typically on
the order of fundamental entropy per charge ($k_{B}/e=87~{}\mathrm{\mu V/K}$).

The total Seebeck coefficients $S$ are weighted by their conductivity so that
the results mostly follow the shape of the spin-up states that indicates
dominant contributions from majority carrier at the Fermi level. Despite the
small $S$ values, their oscillations around zero will play an important role
later when spin-dependent transport is considered. In Figs. 3(g) and 3(i), the
spin-up conductivity and total electron thermal conductivity $\kappa_{e}$
resemble the shape of $\sigma$, consistent with the Wiedemann-Franz law. In
fact, the Lorenz number $\kappa_{e}/\sigma T$ is $2.7\times 10^{-8}\ \rm
W.\Omega/K^{2}$, similar to those of ordinary metals.

Figure 4: Phonon thermal conductivity $\kappa_{ph}$ as a function of
temperature for different Cr$X$ species. Indicated by boxes are $\kappa_{ph}$
values at 500 K. Figure 5: (a-d) Power factors and (e-f) figures of merit of
CrP, CrAs, CrSb, and CrBi, respectively, as a function of chemical potential.
In the PF plots, we show the contribution of spin-up states, spin-down states,
and their total contribution to the PF values. In the ZT plots, with $T=500$
K, we show comparison of total $ZT^{(0)}$ [Eq. (6)] and $ZT^{\rm sv}$ that
includes the spin-valve effect.

We show phonon thermal conductivity $\kappa_{ph}$ of all Cr$X$ monolayers as a
function of $T$ in Fig. 4 with the assumption that the magnon effect is
neglected and that the materials are kept ferromagnetic. The Curie
temperatures of CrX monolayers are within $550$–$700$ K [34], above which the
ferromagnetic to paramagnetic phase transition and short-range magnetic
interaction may occur. Therefore, to avoid the issue of phase transition and
short-range magnetic interaction, we can focus on the practicality of our
system below $550$ K. For example, as indicated by the arrow and small box in
Fig. 4, at $T=500$ K, $\kappa_{ph}$ of CrP is smaller by almost two-order of
magnitudes than $\kappa_{e}$. This comparison generally holds for the other
chromium pnictides. This result suggests that sole electronic transport
calculation for the TE properties is sufficient as the phonon contribution to
$\kappa$ is minimum. As the atom gets heavier from CrP to CrBi, $\kappa_{ph}$
decreases following the Debye-Callaway model.

We turn our analysis on the PF and $ZT$ of Cr$X$ as shown in Fig. 5. The total
PF is dominated by the spin-up states as expected. With the increase of atomic
weight, maximum PF tends to decrease but PF at $\mu=0$ increases. CrP has the
highest PF because it has the largest Fermi velocity while CrBi has a high PF
at $\mu=0$ because it has an additional contribution from the spin-down state
at that energy. It is worth mentioning that the PF of Cr$X$ (about
$0.02$–$0.08$ W/m.K2) could be one of the best values among 2D conductors,
comparable to the extraordinary PF record in Mg3Bi2-based materials [41]. The
total figure of merit $ZT^{(0)}$, given by

$\displaystyle
ZT^{(0)}=\lim_{\sigma_{\uparrow}\gg\sigma_{\downarrow}}\frac{\left(\sigma_{\uparrow}S_{\uparrow}+\sigma_{\downarrow}S_{\downarrow}\right)^{2}T}{\left(\sigma_{\uparrow}+\sigma_{\downarrow}\right)\kappa}=\frac{\sigma_{\uparrow}S_{\uparrow}^{2}T}{\kappa}$
(6)

is also dominated by the spin-up states with a value of about $10^{-3}$ at
small $\mu$ and reaching maximum value of $\sim 0.1$ at a large hole doping
[see Figs. 5(e–h) gray area]. The figure of merit $ZT^{(0)}$ behaves similarly
with the PF trend, in which CrP possesses the largest one. The figure of merit
also decreases as the atomic weight increases.

So far, we have seen Cr$X$ as ordinary metals with all TE properties are
dominated by the majority carrier (spin-up states). The spin-down states, on
the other hand, are gapped, and they possess a high Seebeck coefficient. It is
possible to incorporate the spin-down contribution in this system to enhance
the $ZT$ values using a special configuration [29, 42]. By inserting a non-
magnetic metal spacer between two ferromagnetic materials and shifting their
magnetization orientation by $\theta$, voltage generation can increase due to
finite spin accumulation [29, 43]

$\displaystyle\frac{S(\theta)}{S}=\frac{1+(1-P^{\prime}P)\zeta(\theta)}{1+(1-P^{2})\zeta(\theta)},$
(7)

where
$P=\left(\sigma_{\uparrow}-\sigma_{\downarrow}\right)/\left(\sigma_{\uparrow}+\sigma_{\downarrow}\right)$
is the spin polarization of the conductivity,
$P^{\prime}=\left(\sigma_{\uparrow}S_{\uparrow}-\sigma_{\downarrow}S_{\downarrow}\right)/\left(\sigma_{\uparrow}S_{\uparrow}+\sigma_{\downarrow}S_{\downarrow}\right)$
is the spin polarization of the Seebeck current and

$\displaystyle\zeta(\theta)\propto\frac{V_{s}(\theta)}{J_{s}(\theta)}\propto\frac{1-\cos\theta}{1+\cos\theta}=\tan^{2}\frac{\theta}{2}$
(8)

that depends on the ratio of spin accumulation $V_{s}(\theta)$ and spin
current $J_{s}(\theta)$ at the spacer. Note that since Cr$X$ are half metals,
their value of $P$ is around $1$ across the half-metallic gap while
$P^{\prime}$ can be large when the sign of $S_{\uparrow}$ and $S_{\downarrow}$
are opposite to each other. This $P^{\prime}$ value can be large when
$S_{\uparrow}$ oscillates around zero as shown in Fig. 3(d). However, the
large value of $P^{\prime}$ also means that the total Seebeck coefficient is
small [see Eq. (4)].

Cahaya et al. introduced antiparallel spin configuration ($\theta=\pi$) as
shown in Fig. 1(b) that maximizes voltage and temperature gradients [29]. This
configuration utilizes a $p$-$n$ junction (a junction with the opposite value
of Seebeck coefficient at two ends) and maximizes spin accumulation at its
spacer. A nonmagnetic metal that connects $p$\- and $n$-types Cr$X$ materials
acts as a spin accumulator that ensures spin current is minimized. Using the
antiparallel configuration, we show spin-valve effect on the figure of merit
$ZT^{\rm sv}$ by using $S(\pi)$ in Eq. (7) to Eq. (6). Solid green lines in
Figs. 5(e–h) show that $ZT^{(0)}$ can be enhanced by a factor of two with this
configuration.

We summarize the maximum $ZT$ ($ZT_{\rm max}$) values along with their optimum
chemical potential for all the Cr$X$ monolayers in Fig. 6. At the optimized
chemical potential $\mu_{\rm opt}$ and operating temperature of $T=500$ K, the
$ZT_{\rm max}$ values ($\approx 0.22$, $0.12$, and $0.09$) with the
antiparallel spin-valve setup in CrAs, CrSb, and CrBi improve up to almost
twice the original values ($ZT^{(0)}\approx 0.12$, $0.08$, and $0.05$) without
the spin-valve configuration. Only in CrP, which is the lightest species and
less spin-polarized among Cr$X$, the $ZT_{\rm max}$ value ($\approx 0.34$)
without the spin-valve configuration is larger than that ($\approx 0.19$) with
the spin-valve one. Optimized $\mu$ occurs at large hole doping around $-1$ eV
to $-2$ eV with only exception for CrBi, whose $ZT^{(0)}_{\rm max}$ occurs
near $\mu=0$. Note that both $ZT^{(0)}$ and $ZT^{\rm sv}$ have weak
temperature dependence because of strong resemblance to the Wiedemann-Franz
law.

One may question that the requirement of large doping to obtain the $ZT$ value
($<0.4$ for the Cr$X$ monolayers) in this work is still disappointing for
practical interest of using Cr$X$ in real TE devices. Nevertheless, we have
shown that, by using first-principles calculations with almost the best level
of approximation in thermoelectrics theory, the enhancement of $ZT$ due to the
half metallicity and spin polarization is truly possible. As for the practical
interest, in addition to $ZT$, we can also talk about the PF, which is useful
when the heat input (source) is abundant. For example, the PF value of CrBi in
this work can reach $0.03$ W/m.K2, which is an excellent value for a 2D
material, at chemical potential near zero, thus it does not require large
doping. Further design of the metallic 2D magnets to obtain larger $ZT$ at low
chemical potential shall be an interesting problem of further studies.

Figure 6: Maximum $ZT^{(0)}$ and $ZT^{\rm sv}$ (left axis) and the
corresponding optimal chemical potential $\mu_{\rm opt}$ (right axis) for each
Cr$X$ species with $T=500$ K.

##  IV Conclusions

Using monolayer Cr$X$ as a prototype, we have proven that the TE performance
of half-metallic 2D ferromagnets can improve due to the spin-dependent TE
transport coefficients. The synergistic effects of different spin degrees of
freedom are possible through the antiparallel spin-valve configuration, which
leads to the $ZT$ enhancement up to almost twice the original $ZT$ values. The
origin of the $ZT$ enhancement is that the majority carriers with spin-up
states contribute to high electrical conductivity. On the other hand, the
minority carriers with spin-down states contribute to a high Seebeck
coefficient. The same principle also leads to the exceptional PF in monolayer
Cr$X$ within $0.02$–$0.08$ W/m.K2. This work paves the way toward selecting
suitable 2D ferromagnets for TE applications.

###### Acknowledgements.

M.S.M. acknowledges financial support from the research and community service
center, Institut Teknologi Sepuluh Nopember, Indonesia. E.H.H. acknowledges
financial support by Fonds National de la Recherche Luxembourg, Core 14764976
and 15752388. E.S., S.A.W., A.R.T.N, and E.H.H. performed numerical
calculations with Mahameru High-Performance Computing (HPC) facilities
provided by BRIN and HPC facilities of the University of Luxembourg (see
http://hpc.uni.lu). This work was supported by e-ASIA JRP.

## References

  * Goldsmid [2010] H. J. Goldsmid,  _Introduction to Thermoelectricity_ (Springer-Verlag, Berlin Heidelberg, 2010).
  * Liu  _et al._ [2016] W. Liu, H. S. Kim, Q. Jie, and Z. Ren, Scripta Mater. 111, 3 (2016).
  * Minnich  _et al._ [2009] A. J. Minnich, M. S. Dresselhaus, Z. F. Ren, and G. Chen, Energy Environ. Sci. 2, 466 (2009).
  * Zebarjadi  _et al._ [2012] M. Zebarjadi, K. Esfarjani, M. S. Dresselhaus, Z. F. Ren, and G. Chen, Energy Environ. Sci. 5, 5147 (2012).
  * Heremans  _et al._ [2013] J. P. Heremans, M. S. Dresselhaus, L. E. Bell, and D. T. Morelli, Nat. Nanotechnol. 8, 471 (2013).
  * Franz and Wiedemann [1853] R. Franz and G. Wiedemann, Ann. Phys. 165, 497 (1853).
  * Shakouri [2011] A. Shakouri, Annu. Rev. Mater. Res. 41, 399 (2011).
  * Dresselhaus  _et al._ [2007] M. Dresselhaus, G. Chen, M. Tang, R. Yang, H. Lee, D. Wang, Z. Ren, J.-P. Fleurial, and P. Gogna, Adv. Mater. 19, 1043 (2007).
  * Markov  _et al._ [2018] M. Markov, X. Hu, H.-C. Liu, N. Liu, S. J. Poon, K. Esfarjani, and M. Zebarjadi, Sci. Rep. 8, 1 (2018).
  * Markov and Zebarjadi [2019] M. Markov and M. Zebarjadi, Nanoscale Microscale Thermophys. Eng. 23, 117 (2019).
  * Kanahashi  _et al._ [2020] K. Kanahashi, J. Pu, and T. Takenobu, Adv. Energy Mater. 10, 1902842 (2020).
  * Li  _et al._ [2020] D. Li, Y. Gong, Y. Chen, J. Lin, Q. Khan, Y. Zhang, Y. Li, H. Zhang, and H. Xie, Nano-Micro Lett. 12, 1 (2020).
  * Fei  _et al._ [2014] R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, Nano Lett. 14, 6393 (2014).
  * Liao  _et al._ [2015] B. Liao, J. Zhou, B. Qiu, M. S. Dresselhaus, and G. Chen, Phys. Rev. B 91, 235419 (2015).
  * Saito  _et al._ [2016] Y. Saito, T. Iizuka, T. Koretsune, R. Arita, S. Shimizu, and Y. Iwasa, Nano Lett. 16, 4819 (2016).
  * Jin  _et al._ [2015] Z. Jin, Q. Liao, H. Fang, Z. Liu, W. Liu, Z. Ding, T. Luo, and N. Yang, Sci. Rep. 5, 1 (2015).
  * Yoshida  _et al._ [2016] M. Yoshida, T. Iizuka, Y. Saito, M. Onga, R. Suzuki, Y. Zhang, Y. Iwasa, and S. Shimizu, Nano Lett. 16, 2061 (2016).
  * Wickramaratne  _et al._ [2015] D. Wickramaratne, F. Zahid, and R. K. Lake, J. Appl. Phys. 118, 075101 (2015).
  * Zeng  _et al._ [2018] J. Zeng, X. He, S.-J. Liang, E. Liu, Y. Sun, C. Pan, Y. Wang, T. Cao, X. Liu, C. Wang, L. Zhang, S. Yan, G. Su, Z. Wang, K. Watanabe, T. Taniguchi, D. J. Singh, L. Zhang, and F. Miao, Nano Lett. 18, 7538 (2018).
  * Hicks and Dresselhaus [1993] L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B 47, 12727 (1993).
  * Hicks  _et al._ [1996] L. D. Hicks, T. C. Harman, X. Sun, and M. S. Dresselhaus, Phys. Rev. B 53, R10493 (1996).
  * Hung  _et al._ [2016] N. T. Hung, E. H. Hasdeo, A. R. T. Nugraha, M. S. Dresselhaus, and R. Saito, Phys. Rev. Lett. 117, 036602 (2016).
  * Mermin and Wagner [1966] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  * Gong  _et al._ [2017] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, _et al._ , Nature 546, 265 (2017).
  * Huang  _et al._ [2017] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, _et al._ , Nature 546, 270 (2017).
  * Gao  _et al._ [2020] B. Gao, T. Zhou, A.-J. Hong, F. Liang, G. Song, Q.-Q. Xu, J. Zhang, G.-N. Li, Y. Wang, and C. Dang, Appl. Phys. Express 13, 045001 (2020).
  * Sheng  _et al._ [2020] H. Sheng, Y. Zhu, D. Bai, X. Wu, and J. Wang, Nanotechnology 31, 315713 (2020).
  * Hung  _et al._ [2017] N. T. Hung, R. T. Nugraha, A, and R. Saito, App. Phys. Lett. 111, 092107 (2017).
  * Cahaya  _et al._ [2015] A. B. Cahaya, O. A. Tretiakov, and G. E. W. Bauer, IEEE Trans. Magn. 51, 1 (2015).
  * Giannozzi  _et al._ [2017] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, J. Phys. Condens. Matter 29, 465901 (2017).
  * Hamann [2013] D. R. Hamann, Phys. Rev. B 88, 085117 (2013).
  * Schlipf and Gygi [2015] M. Schlipf and F. Gygi, Comput. Phys. Commun. 196, 36 (2015).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Mogulkoc  _et al._ [2020] A. Mogulkoc, M. Modarresi, and A. N. Rudenko, Phys. Rev. B 102, 024441 (2020).
  * [35] A. Mogulkoc, M. Modarresi, and A. Rudenko, Phys. Rev. Applied 15, 064053 (2021).
  * [36] See supplementary material at “URL”. We give additional data for, (1) electronic band structures of all Cr$X$ without spin polarization, (2) phonon dispersion relations of all Cr$X$, (3) relaxation time model, and (4) thermoelectric transport coefficients of CrAs, CrSb, and CrBi.
  * Madsen  _et al._ [2018] G. K. Madsen, J. Carrete, and M. J. Verstraete, Comput. Phys. Commun. 231, 140 (2018).
  * Poncé  _et al._ [2016] S. Poncé, E. Margine, C. Verdi, and F. Giustino, Comput. Phys. Commun. 209, 116 (2016).
  * Togo  _et al._ [2015] A. Togo, L. Chaput, and I. Tanaka, Phys. Rev. B 91, 094306 (2015).
  * Mizokami  _et al._ [2018] K. Mizokami, A. Togo, and I. Tanaka, Phys. Rev. B 97, 224306 (2018).
  * Mao  _et al._ [2019] J. Mao, H. Zhu, Z. Ding, Z. Liu, G. A. Gamage, G. Chen, and Z. Ren, Science 365, 495 (2019).
  * Hatami  _et al._ [2007] M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly, Phys. Rev. Lett. 99, 066603 (2007).
  * Hatami  _et al._ [2009] M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly, Phys. Rev. B 79, 174426 (2009).

Supplementary Material for:
Spin-tunable thermoelectric performance in monolayer chromium pnictides

Melania S. Muntini,1,∗ Edi Suprayoga,2,† Sasfan A. Wella,2 Iim Fatimah,1 Lila
Yuwana,1 Tosawat Seetawan,3,4 Adam B. Cahaya,5 Ahmad R. T. Nugraha,2 Eddwi H.
Hasdeo2,6,‡

1Department of Physics, Faculty of Science and Data Analytics, Institut
Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia
2Research Center for Physics, National Research and Innovation Agency (BRIN),
South Tangerang, 15314, Indonesia
3Center of Excellence on Alternative Energy, Research and Development
Institution, Sakon Nakhon Rajabhat University, Sakon Nakhon 47000, Thailand
4Program of Physics, Faculty of Science and Technology, Sakon Nakhon Rajabhat
University, Sakon Nakhon 47000, Thailand
5Department of Physics, Faculty of Mathematics and Natural Sciences,
Universitas Indonesia, Depok 16424, Indonesia
6Department of Physics and Materials Science, University of Luxembourg,
Luxembourg 1511, Luxembourg ∗Electronic address<EMAIL_ADDRESS>
†Electronic address<EMAIL_ADDRESS>
‡Electronic address<EMAIL_ADDRESS>
(Dated: )

##  S1 Electronic band structure without spin polarization

In Figs. S1(a)–(d), we show the electronic band structure for non-spin
polarized Cr$X$, calculated by the Quantum ESPRESSO package [1] with the
following DFT details. We set a vacuum layer to 30 Å to avoid the interlayer
interactions due to the lattice periodicity. We employ the optimized norm-
conserving Vanderbilt (ONCV) pseudopotentials [2, 3] to describe the
interaction between electrons and ions. The Perdew-Burke-Ernzerhof (PBE)
functional [4] under generalized gradient approximation (GGA) is used to
describe exchange-correlation energy and potential. We sample the Brillouin
zone using a dense $32\times 32\times 1$ Monkhorst-Pack grid for scf
calculation. The plane-wave basis set is used to expand wave functions with
cutoff energy of $\sim$820 eV.

Figure S1: Electronic band structure for spin-unpolarized Cr$X$: (a) CrP, (b)
CrAs, (c) CrSb, and (d) CrBi.

##  S2 Phonon dispersion

In Figs. S2(a)–(d), we show the phonon dispersion relations for CrX from the
ph.x calculations in the Quantum ESPRESSO package with the dynamical matrix
integral of $4\times 4\times 1$ mesh of q-points. We found no negative and no
imaginary frequency of phonons for all species of Cr$X$, thus indicating the
dynamical stability of the calculated structures. Unlike their electronic
structures, phonon band structures of chromium pnictides resemble that of
graphene. There are six phonon modes comes from two atoms in a unit cell with
three polarization directions. Heterogeneous atoms in a unit cell splits the
in-plane optical modes located at the top branches i.e., longitudinal optic
and in-plane tangential optic modes. These two modes are degenerate in
graphene at $\Gamma$ point. The highest optical branch is, interestingly, flat
in CrAs, CrSb, and CrBi. As atomic number increases from CrP to CrBi, phonon
bandwidth and sound velocity decrease as expected from harmonic oscillator
model. This feature will play roles in controlling thermal conductivity of
chromium pnictides.

Figure S2: Phonon dispersion relations of spin-polarized Cr$X$: (a) CrP, (b)
CrAs, (c) CrSb), and (d) CrBi.

##  S3 Relaxation time model

We include the electron-phonon scattering rate in the relaxation time, that
can be defined as
$\tau_{n,\textbf{k}}=\hbar/(2~{}\mathrm{Im}\sum_{n,\textbf{k}})$, where
$\hbar$ is the reduced Planck constant and $\mathrm{Im}\sum_{n,\textbf{k}}$ is
the imaginary part of the electronic self-energy, as implemented in the EPW
code [5]. We compute the electron energy (and the phonon dispersion) on a
relatively coarse $16\times 16\times 1$ ($8\times 8\times 1$) k-point
(q-point) grid. To obtain a finer result, we fit $1/\tau_{n,\textbf{k}}$ from
EPW with an energy-dependent relaxation time:
$1/\tau(\varepsilon)=\mathcal{C}\cdot{\mathrm{DOS}}(\varepsilon)+1/\tau_{0}$,
where $\mathcal{C}$ is a DOS-dependent fitting parameter and $\tau_{0}$ is the
relaxation time constant that is chosen to overcome the absence of
$\mathrm{DOS}$ in the band gap area in the spin-down band. For all Cr$X$
monolayers, the fitting parameter is set to $\mathcal{C}=10^{12}$ eV/s and
$\tau_{0}=10^{-13}$ s. The inverse relaxation time as a function of energy for
Cr$X$ are shown in Figs. S3(a)–(d).

Figure S3: Inverse relaxation time or scattering rate model as a function of
energy for each Cr$X$: (a) CrP, (b) CrAs, (c) CrSb, and (d) CrBi.

##  S4 Thermoelectric transport coefficients

In Fig. S4–S6, we plot the thermoelectric transport coefficients as a function
of chemical potential of CrAs, CrSb, and CrBi at $T=500~{}\rm K$. The
thermoelectric properties are calculated using BoltzTraP2 code [6] within
Boltzmann transport equation under the energy-dependent relaxation time
approximation.

Figure S4: Thermoelectric transport coefficients of CrAs. Figure S5:
Thermoelectric transport coefficients of CrSb. Figure S6: Thermoelectric
transport coefficients of CrBi.

## References

  * Giannozzi  _et al._ [2017] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, J. Phys. Condens. Matter 29, 465901 (2017).
  * Hamann [2013] D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Phys. Rev. B 88, 085117 (2013).
  * Schlipf and Gygi [2015] M. Schlipf and F. Gygi, Optimization algorithm for the generation of ONCV pseudopotentials, Comput. Phys. Commun. 196, 36 (2015).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Poncé  _et al._ [2016] S. Poncé, E. Margine, C. Verdi, and F. Giustino, EPW: Electron–phonon coupling, transport and superconducting properties using maximally localized wannier functions, Comput. Phys. Commun. 209, 116 (2016).
  * Madsen  _et al._ [2018] G. K. Madsen, J. Carrete, and M. J. Verstraete, BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients, Comput. Phys. Commun. 231, 140 (2018).


* Chen et al. (2018) Yutian Chen, Aja Huang, Ziyu Wang, Ioannis Antonoglou, Julian Schrittwieser, David Silver, and Nando de Freitas.  Bayesian optimization in alphago.  _arXiv preprint arXiv:1812.06855_ , 2018.
  * Chevalier and Ginsbourger (2013) Clément Chevalier and David Ginsbourger.  Fast computation of the multi-points expected improvement with applications in batch selection.  In _International Conference on Learning and Intelligent Optimization_ , 2013.
  * Contal et al. (2013) Emile Contal, David Buffoni, Alexandre Robicquet, and Nicolas Vayatis.  Parallel Gaussian process optimization with upper confidence bound and pure exploration.  In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , 2013.
  * Cortes and Vapnik (1995) Corinna Cortes and Vladimir Vapnik.  Support-vector networks.  _Machine Learning_ , 1995.
  * Cover and Thomas (2012) Thomas M Cover and Joy A Thomas.  _Elements of information theory_.  John Wiley & Sons, 2012.
  * Cutajar et al. (2019) Kurt Cutajar, Mark Pullin, Andreas Damianou, Neil Lawrence, and Javier González.  Deep Gaussian processes for multi-fidelity modeling.  _arXiv preprint arXiv:1903.07320_ , 2019.
  * Deshwal et al. (2020) Aryan Deshwal, Syrine Belakaria, and Janardhan Rao Doppa.  Mercer features for efficient combinatorial Bayesian optimization.  _arXiv preprint arXiv:2012.07762_ , 2020.
  * Dodge et al. (2017) Jesse Dodge, Kevin Jamieson, and Noah A Smith.  Open loop hyperparameter optimization and determinantal point processes.  _arXiv preprint arXiv:1706.01566_ , 2017.
  * Frazier et al. (2008) Peter I Frazier, Warren B Powell, and Savas Dayanik.  A knowledge-gradient policy for sequential information collection.  _SIAM Journal on Control and Optimization_ , 2008.
  * Gillenwater et al. (2012) Jennifer Gillenwater, Alex Kulesza, and Ben Taskar.  Near-optimal map inference for determinantal point processes.  In _Advances in Neural Information Processing Systems_ , 2012.
  * Gómez-Bombarelli et al. (2018) Rafael Gómez-Bombarelli, Jennifer N Wei, David Duvenaud, José Miguel Hernández-Lobato, Benjamín Sánchez-Lengeling, Dennis Sheberla, Jorge Aguilera-Iparraguirre, Timothy D Hirzel, Ryan P Adams, and Alán Aspuru-Guzik.  Automatic chemical design using a data-driven continuous representation of molecules.  _ACS Central Science_ , 2018.
  * Gong et al. (2019) Chengyue Gong, Jian Peng, and Qiang Liu.  Quantile stein variational gradient descent for batch Bayesian optimization.  In _International Conference on Machine Learning_ , 2019.
  * González et al. (2014) Javier González, Joseph Longworth, David C James, and Neil D Lawrence.  Bayesian optimization for synthetic gene design.  _Advances in Neural Information Processing Systems: Workshop in Bayesian Optimization_ , 2014.
  * González et al. (2016a) Javier González, Zhenwen Dai, Philipp Hennig, and Neil Lawrence.  Batch Bayesian optimization via local penalization.  In _Artificial intelligence and statistics_ , 2016a.
  * González et al. (2016b) Javier González, Michael Osborne, and Neil Lawrence.  Glasses: Relieving the myopia of Bayesian optimisation.  In _Artificial Intelligence and Statistics_ , 2016b.
  * Griffiths and Hernández-Lobato (2020) Ryan-Rhys Griffiths and José Miguel Hernández-Lobato.  Constrained Bayesian optimization for automatic chemical design.  _Chemical Sciences_ , 2020.
  * Hennig and Schuler (2012) Philipp Hennig and Christian J Schuler.  Entropy search for information-efficient global optimization.  _Journal of Machine Learning Research_ , 2012.
  * Hensman et al. (2013) James Hensman, Neil D Lawrence, and Magnus Rattray.  Hierarchical Bayesian modelling of gene expression time series across irregularly sampled replicates and clusters.  _BMC Bioinformatics_ , 2013.
  * Hernández-Lobato et al. (2016) Daniel Hernández-Lobato, Jose Hernandez-Lobato, Amar Shah, and Ryan Adams.  Predictive entropy search for multi-objective bayesian optimization.  In _International Conference on Machine Learning_ , pages 1492–1501, 2016.
  * Hernández-Lobato et al. (2014) José Miguel Hernández-Lobato, Matthew W Hoffman, and Zoubin Ghahramani.  Predictive entropy search for efficient global optimization of black-box functions.  In _Neural Information Processing Systems_ , 2014.
  * Hernández-Lobato et al. (2017) José Miguel Hernández-Lobato, James Requeima, Edward O Pyzer-Knapp, and Alán Aspuru-Guzik.  Parallel and distributed Thompson sampling for large-scale accelerated exploration of chemical space.  In _International Conference on Machine Learning_ , 2017.
  * Hoffman and Ghahramani (2015) Matthew W Hoffman and Zoubin Ghahramani.  Output-space predictive entropy search for flexible global optimization.  In _Advances in Neural Information Processing Systems: Workshop on Bayesian Optimization_ , 2015.
  * Huang et al. (2006) Deng Huang, Theodore T Allen, William I Notz, and Ning Zeng.  Global optimization of stochastic black-box systems via sequential kriging meta-models.  _Journal of Global Optimization_ , 2006.
  * Jiang et al. (2020) Shali Jiang, Henry Chai, Javier Gonzalez, and Roman Garnett.  Binoculars for efficient, nonmyopic sequential experimental design.  In _International Conference on Machine Learning_ , 2020.
  * Jones et al. (1993) Donald R Jones, Cary D Perttunen, and Bruce E Stuckman.  Lipschitzian optimization without the Lipschitz constant.  _Journal of optimization Theory and Applications_ , 1993.
  * Jones et al. (1998) Donald R Jones, Matthias Schonlau, and William J Welch.  Efficient global optimization of expensive black-box functions.  _Journal of Global optimization_ , 1998.
  * Junge et al. (2020) Kai Junge, Josie Hughes, Thomas George Thuruthel, and Fumiya Iida.  Improving robotic cooking using batch Bayesian optimization.  _IEEE Robotics and Automation Letters_ , 2020.
  * Jurafsky and Martin (2014) Dan Jurafsky and James H Martin.  _Speech and Language Processing_.  2014\.
  * Kandasamy et al. (2016) Kirthevasan Kandasamy, Gautam Dasarathy, Junier B Oliva, Jeff Schneider, and Barnabás Póczos.  Gaussian process bandit optimisation with multi-fidelity evaluations.  In _Advances in Neural Information Processing Systems_ , 2016.
  * Kandasamy et al. (2017) Kirthevasan Kandasamy, Gautam Dasarathy, Jeff Schneider, and Barnabás Póczos.  Multi-fidelity Bayesian optimisation with continuous approximations.  _International Conference on Machine Learning_ , 2017.
  * Kandasamy et al. (2018a) Kirthevasan Kandasamy, Akshay Krishnamurthy, Jeff Schneider, and Barnabás Póczos.  Parallelised Bayesian optimisation via Thompson sampling.  In _International Conference on Artificial Intelligence and Statistics_ , 2018a.
  * Kandasamy et al. (2018b) Kirthevasan Kandasamy, Willie Neiswanger, Jeff Schneider, Barnabas Poczos, and Eric P Xing.  Neural architecture search with Bayesian optimisation and optimal transport.  In _Advances in Neural Information Processing Systems_ , 2018b.
  * Kathuria et al. (2016) Tarun Kathuria, Amit Deshpande, and Pushmeet Kohli.  Batched Gaussian process bandit optimization via determinantal point processes.  In _Advances in Neural Information Processing Systems_ , 2016.
  * Kennedy and O’Hagan (2000) Marc C Kennedy and Anthony O’Hagan.  Predicting the output from a complex computer code when fast approximations are available.  _Biometrika_ , 2000.
  * Kim et al. (2015) Sujin Kim, Raghu Pasupathy, and Shane G Henderson.  A guide to sample average approximation.  In _Handbook of simulation optimization_. 2015.
  * Klein et al. (2017a) Aaron Klein, Stefan Falkner, Simon Bartels, Philipp Hennig, and Frank Hutter.  Fast Bayesian optimization of machine learning hyperparameters on large datasets.  _International Conference on Artificial Intelligence and Statistics_ , 2017a.
  * Klein et al. (2017b) Aaron Klein, Stefan Falkner, Numair Mansur, and Frank Hutter.  RoBo: A flexible and robust Bayesian optimization framework in Python.  In _Neural Information Processing Systems: Workshop on Bayesian Optimization_ , 2017b.
  * Kleywegt et al. (2002) Anton J Kleywegt, Alexander Shapiro, and Tito Homem-de Mello.  The sample average approximation method for stochastic discrete optimization.  _SIAM Journal on Optimization_ , 2002.
  * Ko et al. (1995) Chun-Wa Ko, Jon Lee, and Maurice Queyranne.  An exact algorithm for maximum entropy sampling.  _Operations Research_ , 1995.
  * Kohavi (1995) Ron Kohavi.  A study of cross-validation and bootstrap for accuracy estimation and model selection.  In _International Joint Conference on Artificial Intelligence_ , 1995\.
  * Kulesza et al. (2012) Alex Kulesza, Ben Taskar, et al.  Determinantal point processes for machine learning.  _Foundations and Trends in Machine Learning_ , 2012.
  * Kusner et al. (2017) Matt J Kusner, Brooks Paige, and José Miguel Hernández-Lobato.  Grammar variational autoencoder.  _arXiv preprint arXiv:1703.01925_ , 2017.
  * Le Gratiet and Garnier (2014) Loic Le Gratiet and Josselin Garnier.  Recursive co-kriging model for design of computer experiments with multiple levels of fidelity.  _International Journal for Uncertainty Quantification_ , 2014.
  * Maas et al. (2011) Andrew L Maas, Raymond E Daly, Peter T Pham, Dan Huang, Andrew Y Ng, and Christopher Potts.  Learning word vectors for sentiment analysis.  In _Association for Computational Linguistics_ , 2011.
  * Marmin et al. (2015) Sébastien Marmin, Clément Chevalier, and David Ginsbourger.  Differentiating the multipoint expected improvement for optimal batch design.  In _International Workshop on Machine Learning, Optimization and Big Data_ , 2015.
  * Matérn (1960) Bertil Matérn.  Spatial variation, volume 36 of.  _Lecture Notes in Statistics_ , 1960.
  * Mockus et al. (1978) Jonas Mockus, Vytautas Tiesis, and Antanas Zilinskas.  The application of Bayesian methods for seeking the extremum.  _Towards global optimization_ , 1978.
  * Moss et al. (2018) Henry Moss, David Leslie, and Paul Rayson.  Using J-K-fold cross validation to reduce variance when tuning nlp models.  In _International Conference on Computational Linguistics_ , 2018\.
  * Moss and Griffiths (2020) Henry B Moss and Ryan-Rhys Griffiths.  Gaussian process molecule property prediction with flowmo.  _Advances in Neural Information Processing Systems: Workshop on Machine Learning for Molecules_ , 2020.
  * Moss et al. (2019) Henry B Moss, Andrew Moore, David S Leslie, and Paul Rayson.  Fiesta: Fast identification of state-of-the-art models using adaptive bandit algorithms.  _Association for Computational Linguistics_ , 2019.
  * Moss et al. (2020a) Henry B Moss, Vatsal Aggarwal, Nishant Prateek, Javier González, and Roberto Barra-Chicote.  Boffin tts: few-shot speaker adaptation by Bayesian optimization.  In _International Conference on Acoustics, Speech and Signal Processing_ , 2020a.
  * Moss et al. (2020b) Henry B. Moss, Daniel Beck, Javier Gonzalez, David L. Leslie, and Paul. Rayson.  Boss: Bayesian optimisation over string spaces.  In _Advances in neural information processing systems_ , 2020b.
  * Moss et al. (2020c) Henry B Moss, David S Leslie, and Paul Rayson.  Bosh: Bayesian optimization by sampling hierarchically.  _International Conference on Machine Learning: Workshop on Active Learning and Experimental Design_ , 2020c.
  * Moss et al. (2020d) Henry B. Moss, David S. Leslie, and Paul Rayson.  MUMBO: Multi-task max-value Bayesian optimization.  In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , 2020d.
  * O’Hagan (1998) Anthony O’Hagan.  A Markov property for covariance structures.  _Statistics Research Report_ , 1998.
  * Paleyes et al. (2019) Andrei Paleyes, Mark Pullin, Maren Mahsereci, Neil Lawrence, and Javier González.  Emulation of physical processes with emukit.  In _Advances in Neural Infromation Processing Systems: Workshop on Machine Learning and the Physical Sciences_ , 2019.
  * Pearce et al. (2019) Michael Pearce, Matthias Poloczek, and Juergen Branke.  Bayesian optimization allowing for common random numbers.  _Winter Simulation Conference_ , 2019.
  * Perdikaris et al. (2017) Paris Perdikaris, Maziar Raissi, Andreas Damianou, Neil D Lawrence, and George Em Karniadakis.  Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling.  _Royal Society A: Mathematical, Physical and Engineering Sciences_ , 2017.
  * Picheny et al. (2010) Victor Picheny, David Ginsbourger, and Yann Richet.  Noisy expected improvement and on-line computation time allocation for the optimization of simulators with tunable fidelity.  _International Conference on Engineering Optimisation_ , 2010.
  * Prieß et al. (2011) Malte Prieß, Slawomir Koziel, and Thomas Slawig.  Surrogate-based optimization of climate model parameters using response correction.  _Journal of Computational Science_ , 2011.
  * Rahimi and Recht (2008) Ali Rahimi and Benjamin Recht.  Random features for large-scale kernel machines.  In _Advances in neural information processing systems_ , 2008.
  * Raileanu and Stoffel (2004) Laura Elena Raileanu and Kilian Stoffel.  Theoretical comparison between the gini index and information gain criteria.  _Annals of Mathematics and Artificial Intelligence_ , 2004.
  * Rasmussen (2004) Carl Edward Rasmussen.  Gaussian processes in machine learning.  In _Advanced Lectures on Machine Learning_. 2004.
  * Ru et al. (2018) Binxin Ru, Michael A Osborne, Mark McLeod, and Diego Granziol.  Fast information-theoretic Bayesian optimisation.  In _International Conference on Machine Learning_ , pages 4384–4392, 2018.
  * Ru et al. (2020) Binxin Ru, Ahsan S Alvi, Vu Nguyen, Michael A Osborne, and Stephen J Roberts.  Bayesian optimisation over multiple continuous and categorical inputs.  _International Conference on Machine Learning_ , 2020.
  * Salton and Buckley (1988) Gerard Salton and Christopher Buckley.  Term-weighting approaches in automatic text retrieval.  _Information Processing an Management_ , 1988.
  * Shah and Ghahramani (2015) Amar Shah and Zoubin Ghahramani.  Parallel predictive entropy search for batch global optimization of expensive objective functions.  In _Advances in Neural Information Processing Systems_ , pages 3330–3338, 2015.
  * Snoek et al. (2012) Jasper Snoek, Hugo Larochelle, and Ryan P Adams.  Practical Bayesian optimization of machine learning algorithms.  In _Neural Information Processing Systems_ , 2012.
  * Srinivas et al. (2010) Niranjan Srinivas, Andreas Krause, Sham M Kakade, and Matthias Seeger.  Gaussian process optimization in the bandit setting: No regret and experimental design.  _International Conference on Machine Learning_ , 2010.
  * Swersky et al. (2013) Kevin Swersky, Jasper Snoek, and Ryan P Adams.  Multi-task Bayesian optimization.  In _Neural Information Processing Systems_ , 2013.
  * Swersky et al. (2020) Kevin Swersky, Yulia Rubanova, David Dohan, and Kevin Murphy.  Amortized Bayesian optimization over discrete spaces.  In _Conference on Uncertainty in Artificial Intelligence_ , 2020.
  * Takeno et al. (2020) Shion Takeno, Hitoshi Fukuoka, Yuhki Tsukada, Toshiyuki Koyama, Motoki Shiga, Ichiro Takeuchi, and Masayuki Karasuyama.  Multi-fidelity Bayesian optimization with max-value entropy search.  _International Conference on Machine Learning_ , 2020.
  * Tanaka and Iwata (2018) Ryokei Tanaka and Hiroyoshi Iwata.  Bayesian optimization for genomic selection: a method for discovering the best genotype among a large number of candidates.  _Theoretical and Applied Genetics_ , 2018.
  * Vakili et al. (2020) Sattar Vakili, Victor Picheny, and Artem Artemev.  Scalable Thompson sampling using sparse Gaussian process models.  _arXiv preprint arXiv:2006.05356_ , 2020.
  * Wang and Jegelka (2017) Zi Wang and Stefanie Jegelka.  Max-value entropy search for efficient Bayesian optimization.  In _International Conference in Machine Learning_ , 2017.
  * Wang et al. (2017) Zi Wang, Chengtao Li, Stefanie Jegelka, and Pushmeet Kohli.  Batched high-dimensional Bayesian optimization via structural kernel learning.  In _International Conference on Machine Learning_ , 2017.
  * Wu and Frazier (2016) Jian Wu and Peter Frazier.  The parallel knowledge gradient method for batch Bayesian optimization.  In _Advances in Neural Information Processing Systems_ , 2016.
  * Wu and Frazier (2017) Jian Wu and Peter I Frazier.  Continuous-fidelity Bayesian optimization with knowledge gradient.  _Advances in Neural Information Processing Systems: Workshop in Bayesian Optimization_ , 2017.
  * Yang and Pedersen (1997) Yiming Yang and Jan O Pedersen.  A comparative study on feature selection in text categorization.  In _International Conference on Machine Learning_ , 1997.
  * Zhang et al. (2017) Yehong Zhang, Trong Nghia Hoang, Bryan Kian Hsiang Low, and Mohan Kankanhalli.  Information-based multi-fidelity Bayesian optimization.  In _Advances in Neural Information Processing Systems: Workshop on Bayesian Optimization_ , 2017.


# Scale-invariant magnetoresistance in a cuprate superconductor

P. Giraldo-Gallo  J. A. Galvis  National High Magnetic Field Laboratory,
Florida State University, Tallahassee, FL 32310, USA  Z. Stegen  National High
Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310,
USA  K. A. Modic  Max-Planck-Institute for Chemical Physics of Solids,
Noethnitzer Strasse 40, D-01187, Dresden, Germany  F. F Balakirev  J. B. Betts
Los Alamos National Laboratory, Los Alamos, NM 87545, USA  X. Lian  C. Moir
S. C. Riggs  National High Magnetic Field Laboratory, Florida State
University, Tallahassee, FL 32310, USA  J. Wu  A. T. Bollinger  Brookhaven
National Laboratory, Upton, New York 11973-5000, USA.  X. He  I. Božović
Brookhaven National Laboratory, Upton, New York 11973-5000, USA.  Applied
Physics Department, Yale University, New Haven, Connecticut 06520, USA.  B. J.
Ramshaw  Laboratory for Atomic and Solid State Physics, Cornell University,
Ithaca, NY 14853, USA  Los Alamos National Laboratory, Los Alamos, NM 87545,
USA  R. D. McDonald  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
G. S. Boebinger  National High Magnetic Field Laboratory, Florida State
University, Tallahassee, FL 32310, USA  Department of Physics, Florida State
University, Tallahassee, FL 32310, USA  A. Shekhter<EMAIL_ADDRESS>National High Magnetic Field Laboratory, Florida State University,
Tallahassee, FL 32310, USA

###### Abstract

The anomalous metallic state in high-temperature superconducting cuprates is
masked by the onset of superconductivity near a quantum critical point. Use of
high magnetic fields to suppress superconductivity has enabled a detailed
study of the ground state in these systems. Yet, the direct effect of strong
magnetic fields on the metallic behavior at low temperatures is poorly
understood, especially near critical doping, $x=0.19$. Here we report a high-
field magnetoresistance study of thin films of
$\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ cuprates in close vicinity to
critical doping, $0.161\leq x\leq 0.190$. We find that the metallic state
exposed by suppressing superconductivity is characterized by a
magnetoresistance that is linear in magnetic field up to the highest measured
fields of $80$T. The slope of the linear-in-field resistivity is temperature-
independent at very high fields. It mirrors the magnitude and doping evolution
of the linear-in-temperature resistivity that has been ascribed to Planckian
dissipation near a quantum critical point. This establishes true scale-
invariant conductivity as the signature of the strange metal state in high-
temperature superconducting cuprates.

Near optimal doping high-temperature superconductivity in cuprates is born
directly out of a “strange” metallic state that is characterized by a linear-
in-temperature resistivity up to the highest attainable temperatures.linear-
BiSCCO ; Anderson-science ; Hussey-pheno ; Zaanen-review  Such transport
behavior indicates scale-invariant dynamics, i.e., lack of an intrinsic energy
scale. This scale invariance is further corroborated by the frequency
dependence of the infrared reflectivityBasov-infrared  and the energy
dependence of the width of the quasiparticle peak in ARPES measurements.Valla-
ARPES  In conventional metals, for current to be carried by electron-like
quasiparticles near the Fermi surface, the scattering length cannot be
significantly shorter than the de Broglie wavelength of electrons.IoffeRegel ;
Mott1972 ; bad-metals ; Hussey2004-MIR  In contrast, the resistivity in the
strange metal phase of cuprates does not saturate or exhibit a crossover at
the temperature where the inferred quasiparticle scattering length is
comparable to the electronic wavelengths, calling into question the very
existence of quasiparticles in this state. This behavior is referred to as
“Planckian dissipation”, suggesting that the transport relaxation rate, $ℏ/τ$,
is instead limited by the thermal energy scale $k_{B}T$.Varma1989 ;
Mackenzie2013 ; Zaanen-planckian ; Zaanen-review ; AjiVarma ; Zaanen-string
In the two classes of unconventional superconductors with the highest
transition temperatures, the cuprates and the pnictides,Hussey2009 ;
Fisher2014  superconductivity emerges from a metallic state that exhibits
Planckian dissipation, a striking fact that is yet to be fully understood.

Because magnetic fields couple directly to charge carriers, their use has been
essential in the study of conventional and correlated metals. Previous studies
of high-temperature cuprates have utilized magnetic fields as a way to
suppress superconductivity to reveal the normal ground state.LSCO-logT ;
Boebinger1996 ; Boebinger-BLSCO-nature-2003 ; Boebinger2009 ; Taillefer-
lsco-2016 ; Taillefer-lsco-ybco-2016  Other studies have extrapolated
magnetotransport behavior in an attempt to infer the carrier dynamics in the
zero field limit.Hussey2009 ; Proust2016  What has been missing is a
comprehensive study of the direct effects of magnetic field on the electron
dynamics in the strange metal phase of cuprates over a broad range of magnetic
fields, an experimental challenge due to the optimal transition temperature
and high critical magnetic fields. To this end, we study electrical transport
in extremely high magnetic fields in
$\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ for a range of compositions near
critical doping. We find a scale-invariant response to magnetic field that is
distinct from the well-understood response of charged quasiparticles to the
Lorentz force as they move on the Fermi surface.Abrikosov  Strikingly, linear-
in-field resistivity, together with linear-in-temperature resistivity, emerges
as an intrinsic characteristic of the strange metal phase in a cuprate
superconductor near critical doping.

Scale-invariant transport is associated with metallic quantum criticality
where a characteristic energy scale is continuously tuned by an external
parameter and is vanishing when the tuning parameter crosses a critical
value.Mackenzie2013 ; Zaanen-review  In the underdoped regime, $p<0.19$, the
Hall effect in $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ Ando-hall  and
quantum oscillations in $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$
Taillefer-QO ; Ramshaw-mass-Science2015  evidence a small electron pocket,
believed to be associated with a charge density wave.CDW-Xray-Hayden-2012 ;
CDW-Xray-Damascelli-2014 ; CDW-LSCO-Hayden  By contrast, on the overdoped
side, $p>0.19$, quantum oscillations in
$\text{TlBa}_{2}\text{Cu}\text{O}_{6+\delta}$ evidence a large Fermi
surfaceVignolle-over-2008 , in line with band structure calculations.OKA
Measurements of Hall resistivity,Boebinger2009 ; Boebinger-BLSCO-nature-2003 ;
Taillefer-lsco-ybco-2016  critical magnetic fieldTaillefer-Hc2  and
quasiparticle massRamshaw-mass-Science2015  as well as the zero temperature
collapse of a line of phase transitionsBourges ; Kerr ; ultrasound ; dichroism
; secondharmonic  suggest a quantum critical point near $p=0.19$. At this
doping the linear-in-temperature resistivity extends to the lowest
temperaturesHussey2009 ; Boebinger1996 ; Zaanen-review  and therefore here one
might expect to access the anomalous behavior in the strange metal phase in
the broadest range of magnetic fields.

Figure 1 shows the in-plane resistivity in a
$\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ cuprate sampleBozovic2009 ;
Bozovic2016  at $x=0.190$Liang-doping  (SI) in magnetic fields up to $80$T for
temperatures in the range $4$K to $180$K. Linear-in-temperature resistivity
down to the superconducting temperature, $T_{c}=38.6$K (Figure 1b) indicates
close proximity to critical doping. Figure 1a shows that magnetoresistance
below 40K is linear in magnetic field over the entire normal-state field range
up to the highest measured field of $80$T. To quantify this observation we
define the field-slope, $β(B,T)=dρ(B,T)/dB$, and the temperature-slope,
$α(T)=dρ(T)/dT$. We find that the field-slope saturates at both high magnetic
fields and low temperatures. As field is increased, the field-slope $β(B,T)$
saturates after superconductivity is suppressed (SI), and as temperature is
decreased at a fixed field of $\sim 70$T, the slope saturates below about
$25$K (Figure 1d). This low-temperature, high-field saturation value of
$β(B,T)$ defines the field- and temperature-independent field-slope $β$, which
in natural energy units ($μ_{B}B$ for magnetic field) is
$β/μ_{B}=5.2μΩ\text{cm/meV}$. This value is comparable in magnitude to the
temperature-slope, $α/k_{B}=11.8μΩ\text{cm/meV}$ in energy units ($k_{B}T$ for
temperature).

In conventional metals magnetoresistance originates in the motion of electron
quasiparticles about the Fermi surface under the action of the Lorentz
force.Pippard  Magnetoresistance can be strong at high magnetic fields when
the cyclotron frequency $ω_{c}$ is comparable to the quasiparticle momentum
relaxation rate, $1/τ_{0}$, i.e., when $ω_{c}τ_{0}$ is near unity. The
resistivity at $p=0.190$ in Figures 1a,c decreases by nearly a factor of two
between $25$K and $4$K at $80$T, comparable to the expected resistivity change
associated with linear-in-temperature resistivity at zero-field for the same
temperature interval (Figure 1b,c).ybco-comment  In this temperature range the
field-slope of magnetoresistance $β(B,T)$ is large and finite, yet independent
of magnetic field and temperature (Figure 1a,d). The inferred quasiparticle
relaxation rate, $1/τ_{0}$, would change by about a factor of two in this
temperature range (Figure 1a,b,c). Such robustness of magneto-transport to
variations in the inferred magnitude of $1/\tau_{0}$ at very high fields
naturally conforms to scale-invariant transport behavior: at very high
magnetic fields the transport relaxation rate is set directly by magnetic
field $\hbar/\tau=bμ_{B}B$ where $b$ is a constant of order of unity. Such
behavior is not easy to reconcile with conventional motion of quasiparticles
about a Fermi surface.

The absence of a resistive anomaly associated with the significant change in
Fermi surface between underdoped and overdoped cuprates,Taillefer-QO ;
Vignolle-over-2008  as well as the smooth evolution of the temperature-slope
$α(x)$ across critical dopingHussey2009 , is an indication of a lack of well-
formed quasiparticles in the strange metal phase at high temperatures (the red
region of Figure 3) res-anomaly . The doping evolution of the high-field
magnetoresistance provides further insight into the character of charge
transport in the strange metal phase (the blue region of Figure 3). We
measured ab-plane resistivity in magnetic fields up to $55$T in
$\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ over the range of dopings
$x=0.161$ to $x=0.184$ (Figure 2). All samples in this doping range exhibit
linear-in-temperature resistivity at high temperatures and a small
extrapolated zero-temperature intercept (Figure 2b). The limiting low-
temperature, high-field value of field-slope $β(x)$ is shown in Figure 2c
along with the temperature-slope $α(x)$ (see SI for details). Both the
temperature-slope $α(x)$, and field-slope $β(x)$, decrease monotonically with
doping. Furthermore, when expressed in natural energy units, $β(x)/μ_{B}$ and
$α(x)/k_{B}$ evolve at a similar rate in this doping range. This weak doping
dependence of the slopes at high fields and temperatures is in apparent
contrast with the step in Hall coefficientTaillefer-lsco-2016 ; Boebinger2009
; Boebinger-BLSCO-nature-2003  and the divergence of the effective mass
Ramshaw-comment  as the critical doping is traversed at low temperature and
moderate magnetic fields (the white regions of Figure 3).away-comment  This
indicates that, despite the observation of quantum oscillations at low
temperatures (in $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ and
$\text{HgBa}_{2}\text{CuO}_{4+\delta}$ ), the dominant contribution to the
high-field magnetoresistance does not originate from the motion of charge
quasiparticles about the Fermi surface.

Figure 2a suggests that the field range of linear magnetoresistance extends to
progressively lower fields as we approach critical doping, represented by the
wedge-shaped boundary of the blue region in the field-doping plane (Figure 3).
The transport relaxation rate is linear in temperature in the temperature-
doping plane (marked in red in Figure 3), $\hbar/τ=a(x)k_{B}T$, and linear in
field in the magnetic field-doping plane (marked in blue),
$\hbar/τ=b(x)μ_{B}B$. When both temperature and field are finite, the
transport relaxation rate is set by the largest energy scale,
$\hbar/τ=\max\\{ak_{B}T,bμ_{B}B\\}$ (Figure 1), extending the region of scale-
invariant behavior into the temperature-field-doping phase space, as
illustrated schematically by dashed lines in Figure 3. This behavior has been
phenomenologically captured in pnictides by adding the thermal and magnetic
field scales in quadratureHayes2016 . The temperature-field competition that
we observe in cuprates is subtly different. Unlike the saturation of the
magnetoresistance at high temperatures, the temperature dependence of
resistivity at very high fields does not saturate at low temperatures (Figure
1d).

The linear-in-temperature resistivity has been instrumental in the development
of several theoretical approaches to understanding the strange metal state in
the cuprates.VarmaRep ; Zaanen-LinearT ; Zaanen-Book ; Kivelson2017  A
striking universality in the behavior of relaxation rates in the strange metal
state revealed in this study through observation of the linear-in-field
resistivity at very high magnetic fields is a new challenge for theory because
microscopic effects of magnetic field are highly constrained.

Acknowledgments: The high-field resistivity measurements in the 60T long-pulse
and 100T magnet systems were performed at the NHMFL’s Pulsed Field Facility
which is supported by the NSF through DMR-1157490 and the DOE/BES “Science at
100T” grant. All MBE synthesis, lithography, and characterization of the
samples (by electron and X-ray diffraction, transport and magnetic
measurements) were done at BNL and were supported by the U.S. Department of
Energy, Basic Energy Sciences, Materials Sciences and Engineering Division.
X.H. was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative
through Grant GBMF4410. AS acknowledges the hospitality of the Aspen Center
for Physics, where part of the analysis was done. Aspen Center for Physics is
supported by National Science Foundation grant PHY-1066293. JAG’s present
address is Departamento de Ciencias Naturales, Facultad de Ingeniería y
Ciencias Básicas, Universidad Central, 110311 Bogotá, Colombia.

## References

  * (1) S. Martin, A. T. Fiory, R. M. Fleming, L. F. Schneemeyer, J. V. Waszczak, Normal-state transport properties of $\text{Bi}_{2}\text{Sr}_{2}\text{CaCu}_{2}\text{O}_{8+\delta}$ crystals, _Phys. Rev. B_ 41, 846(R) (1990).
doi:10.1103/PhysRevB.41.846

  * (2) P. W. Anderson, Experimental Constraints on the Theory of High-Tc Superconductivity, _Science_ 256, 1526-1531 (1992).
doi:10.1126/science.256.5063.1526

  * (3) N. E. Hussey, Phenomenology of the normal state in-plane transport properties of high-Tc cuprates, _J. Phys. Condens. Matter_ 20, 123201 (2008).
doi:10.1088/0953-8984/20/12/123201

  * (4) B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, _Nature_ 518, 179–186 (2015).
doi:10.1038/nature14165

  * (5) A. V. Puchkov, D. N. Basov, T. Timusk, The pseudogap state in high-Tc superconductors: an infrared study, _J. Phys.: Condens. Matter_ 8, 10049 (1996).
doi:10.1088/0953-8984/8/48/023

  * (6) T. Valla, A. V. Fedorov, P. D. Johnson, Q. Li, G. D. Gu, N. Koshizuka, Temperature dependent scattering rates at the Fermi surface of optimally doped $\text{Bi}_{2}\text{Sr}_{2}\text{CaCu}_{2}\text{O}_{8+\delta}$ , _Phys. Rev. Lett._ 85, 828 (2000).
doi:10.1103/PhysRevLett.85.828

  * (7) A. F. Ioffe, and A.R. Regel, _Prog. Semicond._ 4, 237 (1960).
  * (8) N. F. Mott, Conduction in non-crystalline systems IX. the minimum metallic conductivity, _Phil. Mag._ 26, 1015-1026 ( 1972).
doi:10.1080/14786437208226973

  * (9) V. J. Emery, S. A. Kivelson, Superconductivity in Bad Metals, _Phys. Rev. Lett._ 74, 3253 (1995).
doi:10.1103/PhysRevLett.74.3253

  * (10) N. E. Hussey, K. Takenaka, H. Takagi, Universality of the Mott-Ioffe-Regel limit of metals, _Phil. Mag._ 84, 2847 (2004).
doi:10.1080/14786430410001716944

  * (11) C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, A. Ruckenstein, Phenomenology of the normal state of Cu-O high-temperature superconductors, _Phys. Rev. Lett._ 63, 1996 (1989).
doi:10.1103/PhysRevLett.63.1996

  * (12) J. A. N. Bruin, H. Sakai, R. S. Perry, A. P. Mackenzie, A universal scaling relation in high-temperature superconductors, _Science_ 339, 804 (2013).
doi:10.1126/science.1227612

  * (13) Jan Zaanen, Why the temperature is high, _Nature_ 430, 512 (2004).
doi:10.1038/430512a

  * (14) V. Aji, C. M. Varma, Theory of the Quantum Critical Fluctuations in Cuprate Superconductors, _Phys. Rev. Lett_ 99, 067003 (2007).
doi:10.1103/PhysRevLett.99.067003

  * (15) J. Zaanen, Y. W. Sun, Y. Liu, K. Schalm, Holographic duality for condensed matter physics, (Cambridge Univ. Press, 2015).
  * (16) R. A. Cooper, Y. Wang, B. Vignolle, O. J. Lipscombe, S. M. Hayden, Y. Tanabe, T. Adachi, Y. Koike, M. Nohara, H. Takagi, C. Proust, N. E. Hussey, Anomalous Criticality in the Electrical Resistivity of $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ , _Science_ 323, 603 (2009).
doi:10.1126/science.1165015

  * (17) J. G. Analytis, H-H. Kuo, R. D. McDonald, M. Wartenbe, P. M. C. Rourke, N. E. Hussey, I. R. Fisher, Transport near a quantum critical point in $\text{BaFe}_{2}(\text{As}_{x}\text{P}_{1-x})_{2}$ , _Nature Physics_ 10, 194 (2014).
doi:10.1038/nphys2869

  * (18) Yoichi Ando, G.S. Boebinger, A. Passner, Tsuyoshi Kimura, Kohji Kishio, Logarithmic divergence of both in-plane and out-of-plane normal-state resistivities of superconducting $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ in the zero-temperature limit, _Phys. Rev. Lett._ 75, 4662 (1995).
doi:10.1103/PhysRevLett.75.4662

  * (19) G. S. Boebinger, Yoichi Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, S. Uchida, Insulator-to-metal crossover in the normal state of $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ near optimum doping, _Phys. Rev. Lett._ 77, 5417 (1996).
doi:10.1103/PhysRevLett.77.5417

  * (20) G. S. Boebinger, Yoichi Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, S. Uchida, Signature of optimal doping in Hall-effect measurements on a high-temperature superconductor, _Nature_ 424, 912-915 (2003).
doi:10.1038/nature01890

  * (21) F. F. Balakirev, J. B. Betts, A. Migliori, I. Tsukada, Yoichi Ando, G. S. Boebinger, Quantum phase transition in the magnetic-field-induced normal state of optimum-doped high-$T_{c}$ cuprate superconductors at low temperatures, _Phys. Rev. Lett._ 102, 017004 (2009).
doi:10.1103/PhysRevLett.102.017004

  * (22) F. Laliberté, W. Tabis, S. Badoux, B. Vignolle, D. Destraz, N. Momono, T. Kurosawa, K. Yamada, H. Takagi, N. Doiron-Leyraud, C. Proust, Louis Taillefer, unpublished,
arxiv:1606.04491

  * (23) S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud, Louis Taillefer, Cyril Proust, Change of carrier density at the pseudogap critical point of a cuprate superconductor, _Nature_ 531, 210 (2016).
doi:10.1038/nature16983

  * (24) Cyril Proust, B. Vignolle, J. Levallois, S. Adachi, N. E. Hussey, Fermi liquid behavior of the in-plane resistivity in the pseudogap state of $\text{YBa}_{2}\text{Cu}_{4}\text{O}_{8}$, _PNAS_ 113, 13654-13659 (2016).
doi:10.1073/pnas.1602709113

  * (25) A.A. Abrikosov, Fundamentals of the Theory of Metals, (Amsterdam: North Holland, 1988).
  * (26) Yoichi Ando, Y. Kurita, Seiki Komiya, S. Ono, Kouji Segawa, Evolution of the Hall coefficient and the peculiar electronic structure of the cuprate superconductors, _Phys. Rev. Lett._ 92, 197001 (2004).
doi:10.1103/PhysRevLett.92.197001

  * (27) Nicolas Doiron-Leyraud, Cyril Proust, David LeBoeuf, Julien Levallois, Jean-Baptiste Bonnemaison, Ruixing Liang, D. A. Bonn, W. N. Hardy, Louis Taillefer, Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor, _Nature_ 447, 565 (2007).
doi:10.1038/nature05872

  * (28) B. J. Ramshaw, S. E. Sebastian, R. D. McDonald, J. Day, B. Tan, Z. Zhu, J. B. Betts, R. Liang, D. A. Bonn, W. N. Hardy, N. Harrison, Quasiparticle mass enhancement approaching optimal doping in a high-$T_{c}$ superconductor, _Science_ 348, 317 (2014).
doi:10.1126/science.aaa4990

  * (29) J. Chang, E. Blackburn, A. T. Holmes, N. B. Christensen, J. Larsen, J. Mesot, Ruixing Liang, D. A. Bonn, W. N. Hardy, A. Watenphul, M. v. Zimmermann, E. M. Forgan, S. M. Hayden, Direct observation of competition between superconductivity and charge density wave order in $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6.67}$, _Nature Physics_ 8, 871 (2012).
doi:10.1038/nphys2456

  * (30) R. Comin, A. Frano, M. M. Yee, Y. Yoshida, H. Eisaki, E. Schierle, E. Weschke, R. Sutarto, F. He, A. Soumyanarayanan, Yang He, M. Le Tacon, I. S. Elfimov, Jennifer E. Hoffman, G. A. Sawatzky, B. Keimer, A. Damascelli, Charge order driven by Fermi-arc instability in $\text{Bi}_{2}\text{Sr}_{2−x}\text{La}_{x}\text{CuO}_{6+δ}$, _Science_ 343, 390 (2014).
doi:10.1126/science.1242996

  * (31) T. P. Croft, C. Lester, M. S. Senn, A. Bombardi, S. M. Hayden, Charge density wave fluctuations in $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ and their competition with superconductivity, _Phys. Rev. B_ 89, 224513 (2014).
doi:10.1103/PhysRevB.89.224513

  * (32) B. Vignolle, A. Carrington, R. A. Cooper, M. M. J. French, A. P. Mackenzie, C. Jaudet, D. Vignolles, Cyril Proust, N. E. Hussey, Quantum oscillations in an overdoped high-Tc superconductor, _Nature_ 455, 952 (2008).
doi:10.1038/nature07323

  * (33) O.K. Andersen, A.I. Liechtenstein, O. Jepsen, F. Paulsen, LDA energy bands, low-energy hamiltonians, $t′$, $t″$, $t_{⊥}(k)$, and $J_{⊥}$, _J. Phys. Chem. Solids_ 56, 1573-1591 (1995).
doi:10.1016/0022-3697(95)00269-3

  * (34) G. Grissonnanche, O. Cyr-Choiniere, F. Laliberte, S. Renede Cotret, A. Juneau-Fecteau, S. Dufour-Beausejour, M.-E. Delage, D. LeBoeuf, J. Chang, B.J. Ramshaw, D.A. Bonn, W.N. Hardy, R. Liang, S. Adachi, N.E. Hussey, B. Vignolle, C. Proust, M. Sutherland, S. Kramer, J.-H. Park, D. Graf, N. Doiron-Leyraud, Louis Taillefer, Direct measurement of the upper critical field in cuprate superconductors, _Nat. Comm._ 5, 3280 (2014).
doi:10.1038/ncomms4280

  * (35) B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhès, C. T. Lin, X. Chaud, P. Bourges, Magnetic order in the pseudogap phase of high-$T_{c}$ superconductors, _Phys. Rev. Lett._ 96, 197001 (2006).
doi:10.1103/PhysRevLett.96.197001

  * (36) J. Xia, E. Schemm, G. Deutscher, S. A. Kivelson, D. A. Bonn, W. N. Hardy, R. Liang, W. Siemons, G. Koster, M. M. Fejer, A. Kapitulnik, Polar Kerr-effect measurements of the high-temperature $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ superconductor: Evidence for broken symmetry near the pseudogap temperature, _Phys. Rev. Lett._ 100, 127002 (2008).
doi:10.1103/PhysRevLett.100.127002

  * (37) Arkady Shekhter, B. J. Ramshaw, Ruixing Liang, W. N. Hardy, D. A. Bonn, Fedor F. Balakirev, Ross D. McDonald, Jon B. Betts, Scott C. Riggs, Albert Migliori, Bounding the pseudogap with a line of phase transitions in $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ , _Nature_ 498, 75 (2013).
doi:10.1038/nature12165

  * (38) Y. Lubashevsky, L. Pan, T. Kirzhner, G. Koren, N. P. Armitage, Optical birefringence and dichroism of cuprate superconductors in the THz regime, _Phys. Rev. Lett._ 112, 147001 (2014).
doi:10.1103/PhysRevLett.112.147001

  * (39) L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage, D. Hsieh, A global inversion-symmetry-broken phase inside the pseudogap region of $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ , _Nature Physics_ online, doi:10.1038/nphys3962 (2016).
doi:10.1038/nphys3962

  * (40) G. Logvenov, A. Gozar, I. Božović, High-temperaturesuperconductivity in a single copper-oxygen plane, _Science_ 326, 699 (2009).
doi:10.1126/science.1178863

  * (41) I. Božović, X. He, J. Wu, A. T. Bollinger, Dependence of the critical temperature in overdoped copper oxides on superfluid density, _Nature_ 536 , 309 (2016).
doi:10.1038/nature19061

  * (42) Ruixing Liang, D. A. Bonn, W. N. Hardy, Evaluation of $\text{CuO}_{2}$ plane hole doping in $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ single crystals, _Phys. Rev. B_ 73, 180505(R) (2006).
doi:10.1103/PhysRevB.73.180505

  * (43) A.B. Pippard, Magnetoresistance in metals, (Cambridge University press, 1989).
  * (44) Measurements of longitudinal and Hall resistivity in $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ result in a similar estimate for the change in the inferred value of $ω_{c}τ_{0}$ in this temperature range.Taillefer-lsco-ybco-2016
  * (45) At low temperatures and low magnetic fields the resistive anomaly is masked by superconductivity.
  * (46) In $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ cuprates the mass increases by nearly a factor of three in the doping range of $p=0.116$ to $p=0.152$ and is expected to diverge at $p=0.19$Ramshaw-mass-Science2015 .
  * (47) Away from critical doping higher magnetic fields are necessary to access the strange metal behavior compared with the fields required for suppressing superconductivity at low temperature.
  * (48) I. M. Hayes, R. D. McDonald, N. P. Breznay, T. Helm, P. J. W. Moll, M. Wartenbe, Arkady Shekhter, J. G. Analytis, Scaling between magnetic field and temperature in the high-temperature superconductor $\text{BaFe}_{2}(\text{As}_{x}\text{P}_{1-x})_{2}$ , _Nature Physics_ 12, 916 (2016).
doi:10.1038/nphys3773

  * (49) C.M. Varma, Quantum-critical fluctuations in 2D metals: strange metals and superconductivity in antiferromagnets and in cuprates, _Rep. Prog. Phys._ 79, 082501 (2016).
doi:10.1088/0034-4885/79/8/082501

  * (50) Richard A. Davison, Koenraad Schalm, Jan Zaanen, Holographic duality and the resistivity of strange metals, _Phys. Rev. B_ 89, 245116 (2014).
doi:10.1103/PhysRevB.89.245116

  * (51) Jan Zaanen, Koenraad Schalm, Ya-Wen Sun, Yan Liu, Holographic Duality in Condensed Matter Physics, (Cambridge University press, 2015).
  * (52) Samuel Lederera, Yoni Schattnerb, Erez Bergb, and Steven A. Kivelson, Superconductivity and non-Fermi liquid behavior near a nematic quantum critical point, _Proc. Nat. Acad. of Sci._ 114, 4905 (2017).
doi:10.1073/pnas.1620651114

Figure 1. Magnetoresistance up to $80$T of the thin-film
$\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ at $x=0.190$. a Field scans up to
$80$T for a set of temperatures up to $180$K. The vertical ticks on the right
indicate the resistivity in temperature units, $ρ/α$, obtained using linear
fit in panel b. The aspect ratio is such that $1$inch either horizontally or
vertically represent the same value in natural energy units, $μ_{B}B$ and
$k_{B}T$ ($80$T corresponds approximately to $53.7$K). b Zero-field
resistivity in a broad temperature range up to room temperature. The gray line
indicates a linear-fit for resistivity above superconducting transition
temperature, $T_{c}$, $ρ=a+αT$, where the intercept $a≈1.5(\pm
1.5)\text{μΩcm}$ and the temperature-slope $α≈1.02(\pm 0.01)\text{μΩcm/K}$.
The uncertainty in $a,α$ reflects variation in a running slope analysis over
broad temperature range. c Temperature dependence of resistivity at fixed
field (indicated by color legend). Gray points indicate the zero-field
resistivity from panel a. The vertical ticks on the right indicate the
resistivity in units of temperature, $ρ/α$, same as in panel a. The solid line
through a set of 80T points at low temperatures is a guide for the eye. d
Temperature dependence of field-slope of resistivity at fixed field
(calculated as linear regression for $65T<B<77T$ field range). The slope
saturates below about $25$K.

Figure 2. Magnetoresistance up to $55$T of
$\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ thin-film samples in the range of
dopings $x=0.161-0.184$ (SI). a Field scans for a set of temperatures in the
range of dopings $x=0.161-0.184$. b Zero-field resistivity vs temperature for
the same set of dopings (the dopings are color-coded in the same way in a and
b). c Doping evolution of the temperature-slope (red) and field-slope (blue)
in the whole measured doping range, $x=0.161-0.190$. The values of the slopes
$α/k_{B}$ and $β/μ_{B}$ in energy units are indicated on the left axis. The
right axis on panel indicates the value of $α$.

Figure 3. Sketch of the x-B-T phase diagram in the vicinity of critical doping
$x_{\text{crit}}$ as suggested by our measurement (superconducting transition
is not shown). Solid-lines indicate the edge of the region of linear-T
behavior (marked in red) and linear-B behavior (marked in blue). Dashed-lines
indicate the region of the x-B-T phase space where temperature and magnetic
field compete.