# Enhancement of the Curie temperature in single crystalline ferromagnetic
LaCrGe3 by electron irradiation-induced disorder

E. H. Krenkel  Ames National Laboratory, Ames, Iowa 50011, USA  Department of
Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  M. A.
Tanatar<EMAIL_ADDRESS>Ames National Laboratory, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  M. Kończykowski  Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS, École
Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  R. Grasset  Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS, École
Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  Lin-Lin Wang  Ames National Laboratory, Ames, Iowa 50011, USA  S. L.
Bud’ko  Ames National Laboratory, Ames, Iowa 50011, USA  Department of Physics
and Astronomy, Iowa State University, Ames, Iowa 50011, USA  P. C. Canfield
Ames National Laboratory, Ames, Iowa 50011, USA  Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA  R. Prozorov
<EMAIL_ADDRESS>Ames National Laboratory, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA

(7 May 2024)

###### Abstract

LaCrGe3 has attracted attention as a potential candidate for studies of
quantum phase transitions in a ferromagnetic material. The application of
pressure avoids a quantum critical point by developing a new magnetic phase.
It was suggested that the disorder may provide an alternative route to a
quantum critical point. We used low-temperature 2.5 MeV electron irradiation
to induce relatively small amounts of point-like disorder in single crystals
of LaCrGe3. Irradiation leads to an increase of the resistivity at all
temperatures with some deviation from the Matthiessen rule. Hall effect
measurements show that electron irradiation does not cause any detectable
change in the carrier density. Unexpectedly, the Curie temperature,
$T_{\text{FM}}$, _increases_ with the increase of disorder from approximately
90 K in pristine samples up to nearly 100 K in the heavily irradiated sample,
with a tendency towards saturation at higher doses. Although the mechanism of
this effect is not entirely clear, we conclude that it cannot be caused by
effective “doping” or “pressure” due to electron irradiation. We suggest that
disorder-induced broadening of a sharp peak in the density of states, $D(E)$,
situated at $E_{p}=E_{F}-0.25$ eV below the Fermi energy, $E_{F}$, causes an
increase in $D(E_{F})$, leading to an enhancement of $T_{\text{FM}}$ in this
itinerant ferromagnet.

##  I Introduction

Monotonic suppression of the transition temperature of second-order phase
transitions to $T=0$ by a non-thermal tuning parameter (pressure, composition,
disorder, uniaxial strain, magnetic field) provides a fruitful playground for
the realization of novel exotic phases of matter [1, 2]. For example, quantum
fluctuations of the antiferromagnetic order parameter on approaching a quantum
critical point (QCP) lead to unconventional superconductivity in heavy
fermion, cuprate, organic, and iron-based superconductors [3, 4, 5].
Fluctuations of the charge density wave (CDW) order parameter lead to an
increase in superconducting $T_{c}$ in (Ca,Sr)3(Rh,Ir)4Sn13 compounds [6, 7].

The application of similar ideas to ferromagnetic compounds has a fundamental
problem. Quantum phase transitions in two-dimensional and three-dimensional
metallic systems from a paramagnetic (PM) phase to a homogeneous ferromagnetic
(FM) phase are generically first order, provided the material is sufficiently
clean [8]. This is determined by the interaction of the magnetization with the
electronic soft modes that exist in any metal, leading to a fluctuation-
induced first-order transition. The exceptions are non-centrosymmetric metals
with strong spin-orbit coupling [9]. As a result, when tuning material to QCP,
the continuous FM-PM transition either becomes an abrupt first-order
transition below a tricritical point or modulated magnetic phases appear; see
Refs.[10, 11] for reviews. However, it has also been shown that non-magnetic
quenched disorder suppresses the tricritical temperature and the transition
can remain second order down to zero temperature if the disorder strength
exceeds a certain critical value, which opens new opportunities in studies of
quantum criticality in small magnetic moment (fragile) ferromagnets [2, 8].
This suggests that disorder may play a pivotal role in quantum critical
behavior of ferromagnets.

One of the potentially strong effects of point-like non-magnetic disorder is
its impact on the energy distribution of the density of states (DOS), $D(E)$.
The ferromagnetic instability in a metal occurs when the Stoner criterion
$ID(E_{F})>1$ is satisfied, here $I$ is the Stoner parameter of the strength
of the Coulomb interaction. Normally, random point-like disorder leads to
smearing of $D(E)$ causing a reduction of $D(E_{F})$, thus destabilizing
itinerant magnetism. However, if there is a peak in $D(E)$ not far from
$E_{F}$, this smearing may cause an increase in $D(E_{F})$ and enhance
$T_{\text{FM}}$.

Substitutional disorder is an unavoidable component of systems in which
composition variation is used as a tuning parameter, and disentangling its
role from chemical doping/steric effects is very hard and sometimes
impossible. Irradiation with energetic particles provides an alternative way
to control the disorder. The use of disorder as a non-thermal tuning
parameter, as, for example, in the cases of NbSe2 [12] and (Ba,K)Fe2As2 [13,
14], might provide a way to smear a first-order transition and enable a
ferromagnetic quantum critical point [10, 11].

LaCrGe3 is a typical example of avoided quantum criticality in a ferromagnetic
system. Upon the application of hydrostatic pressure, the Curie temperature
($T_{\text{FM}}$ ) is initially suppressed, but then the nature of the
transition changes to the first order and a new magnetic phase is stabilized
[15, 16]. When the magnetic transition becomes first-order at the tricritical
point, application of a magnetic field $H$ along the magnetization axis
reveals a wing structure phase diagram in the $T-p-H$ space, indicating the
possibility of a new type of field-induced quantum critical point [17].

Motivated by the idea of using disorder as a tuning parameter to reveal
quantum criticality in itinerant metallic ferromagnets, we present a study the
effects of a controlled disorder on magnetic ordering in LaCrGe3. Disorder was
introduced by low temperature electron irradiation, creating defects in form
of vacancies with densities on the order of 10-3 defects per atom (i.e.
comparable to the disorder associated with one out of 200 unit cells having a
defect). Contrary to theoretical predictions and general intuition, we found
that the transition temperature to a ferromagnetic state in LaCrGe3 is
enhanced by the disorder. The effect is most likely caused by the specifics of
the electronic band structure of this material, which shows a strong peak in
the density of states below $E_{F}$.

##  II Methods

###  II.0.1 Sample preparation

Unlike initial growths of this material [18], the single crystals of LaCrGe3
used in this study were grown in two steps [19] from melts of La18Cr12Ge70
[20, 21] using fritted Canfield Crucible Sets (CCS) [22, 23]. First
La18Cr12Ge70 was heated to 1150 ∘C and cooled to 950 ∘C over 50–100 h. At 950
∘C, a mixture of LaGe2-x plates and LaCrGe3 rods was separated from the
remaining liquid phase. Subsequently, this decanted liquid was resealed,
heated to 1000 ∘C to fully re-melt it, and then slowly cooled from 950 ∘C to
825 ∘C over roughly 100 h. At 825 ∘C the growth was decanted and the resulting
single phase LaCrGe3 crystalline solid phase was separated from excess liquid.

Samples from the same batches used in our electron irradiation study were
characterized by x-ray diffraction, resistance, and magnetization. The results
were consistent with previous reports [18, 16, 17] in terms of the
ferromagnetic Curie temperature ($T_{\text{FM}}$) and residual resistivity
ratio (RRR), see Fig. 1 below.

The rod-shaped single crystals with hexagonal cross sections were
characterized by electrical resistivity and Hall effect measurements, with the
electrical current along the long dimension of the sample corresponding to the
$c-$axis. Contacts to the samples were soldered in a standard four-probe
resistivity measurement configuration and 5-probe Hall effect configuration
with indium. Conducting Dupont 4929N silver paste was used to mechanically
stabilize the contacts [24]. The typical contact resistance was in the 10 to
100 m$\Omega$ range. Temperature-dependent resistivity and Hall effect
measurements were performed in a Quantum Design Physical Property Measurement
System (PPMS). Magnetic field for Hall effect measurements was applied along
the $a-$axis, transverse to the current flowing along the $c-$axis.
Resistivity measurements at 300 K were performed on 11 samples of LaCrGe3 with
current along the $c-$axis. Two samples were discarded as obvious outliers,
and from the remainder, the resistivity at room temperature was determined as
141 $\pm$ 12 $\mu\Omega\cdot$cm. In the remainder of this paper, we use this
value for all pristine $c-$axis transport samples, and normalize geometric
factors to match this value. For the irradiation study, we selected four
samples, shown in Fig.1, with highly reproducible temperature-dependent
resistivity curves that lay on top of each other and are indistinguishable to
the eye.

Figure 1:  Temperature dependent resistivity of the four selected samples of
LaCrGe3. The room-temperature resistivity of LaCrGe3 for the current along
$c-$axis was normalized to $141\mu\Omega\cdot$cm, a value obtained from an
array of 11 samples. The geometric factors of all samples were adjusted to
match this value. Insets zoom in on the transition temperature in resistivity
(top inset), and temperature derivative of the resistivity (bottom inset). The
two criteria used for the determination of the transition temperatures from
resistivity data, $T_{r}$, and from the onset of the derivative increase,
$T_{d}$, are shown.

###  II.0.2 Electron irradiation

Table 1: Sample irradiation summary  Sample | Thickness | Dose | Entry | Exit E | Exit $\sigma$ | Exit | Attenuation | $\rho(300\>K),(\%change)$
---|---|---|---|---|---|---|---|---
| ($\mu$ m) | $\;\text{C}/\text{cm}^{2}$ | $\;\text{dpa}\times 10^{-4}$ | MeV | barn | $\;\text{dpa}\times 10^{-4}$ | % | ($\mu\Omega cm$
A | 133 | 3 | 21.3 | 2.27 | 105 | 19.7 | 92 | 150.0 (6%)
B | 511 | 3 | 21.3 | 1.9 | 88 | 16.5 | 77 | 149 (5.6%)
C | 222 | 0.5 | 3.55 | 2.17 | 101 | 3.15 | 89 | 142.0 (0.3%)
D | 222 | 1 | 7.1 | 2.17 | 101 | 6.30 | 89 | 150.5 (7%)

Electron irradiation was performed at the “SIRIUS” Pelletron linear
accelerator operated by Laboratoire des Solides Irradiés at École
Polytechnique in Palaiseau, France [25]. The 2.5 MeV electrons impact the
sample kept in liquid hydrogen at 22 K to ensure efficient heat removal, and
to prevent clustering and recombination of created atomic defects, vacancy-
interstitial, called Frenkel pairs. Due to their small rest mass, the
relativistic energy transfer upon impact from electrons matches the energy of
ionic displacement barriers. Heavier particles and ions induce correlated
disorder from secondary collisions and cascades, such as extended clusters and
columnar defects. The typical threshold energy to displace the ions is between
10 and 100 eV [28, 29, 26, 27].

The interstitials usually have lower diffusion barriers compared to vacancies.
When the sample is returned to room temperature, some of the Frenkel pairs
recombine, and some interstitials leave the crystal through various sinks,
such as extended defects (dislocations/disclinations) and surfaces. The
remaining metastable population of vacancies and interstitials is the point
like disorder in our study [28, 29]. The exact rate of annealing is material-
dependent and we use resistivity increase to quantify the added disorder.

The samples for resistivity measurements during and after electron irradiation
were mounted on a thin mica plate, which was placed inside a hollow Kyocera
chip C-QFN (Ceramic Quad Flat Non-Leaded Packages) [30]. The Kyocera chip,
mounted on a special holder, was inserted into the irradiation chamber and
kept in liquid hydrogen during irradiation.

For the analysis of the defect density we used an average cross-section for
LaCrGe3 at 2.5 MeV, $\sigma=114\;\text{barn}$. This value gives $7.1\times
10^{-4}$ defects per atom per 1 C/cm2. While the incoming electron beam energy
is at 2.5 MeV, the energy is gradually lost and in a sample of LaCrGe3 it
should go to zero at the thickness of 1.3 mm (stopping distance). This limits
the practical acceptable thicknesses of the samples to 0.5 mm or less. In
Table 1 we summarize parameters for the first irradiation run involving all
the samples used in this study. For the thickest sample #B the energy of the
beam at the sample exit is calculated as 1.9 MeV. This energy is still
sufficient to knock out all species of ions in the lattice [28, 29]. Direct
evidence for this conclusion comes from the identical increase of
$T_{\text{FM}}$ and resistivity shift in samples #A (0.133 mm thick) and #B
(0.511 mm thick) after receiving an identical dose of irradiation; see Figures
2, 3 and Fig. 5 below.

The flux of electrons was estimated by measuring the total electrical current
through a 5 mm diameter diaphragm using a Faraday cup positioned behind the
sample, so that only transmitted electrons were counted. The dose of electron
irradiation received by a sample is reported in $\text{C}/\text{cm}^{2}$. In
conventional units, $1~{}\text{C}/\text{cm}^{2}=6.24\times
10^{18}~{}\text{electrons}/\text{cm}^{2}$. Typically, the dose accumulated in
a session varies from 0.5 (overnight) to 3 $\text{C}/\text{cm}^{2}$ (weekend).
Larger doses are accumulated in several irradiation sessions, which can be
separated by 6 to 12 months with ex situ sample characterization in between.
Since the density of defects produced by irradiation is metastable, some
annealing occurs at room temperature between runs. Throughout the manuscript
we use “pristine” and “unirradiated” interchangeably to describe samples that
have not been exposed to electron irradiation. The doses shown in figures
involving sample B are cumulative - 3 C /cm2 from the first irradiation is
listed as 3 C/cm2. When the same sample receives a second irradiation of 3 C /
cm2, it is shown in the figure as 6 C/cm2 (for the total dose), and so on. The
initial doses for all samples are listed in column 2 of Table 1.

###  II.0.3 Electronic band structure calculations

Density functional theory (DFT) with local density approximation (LDA) as
exchange-correlation functional has been used to relax structures and
calculate the density of states (DOS) for LaCrGe3 without and with vacancies.
The DFT calculations have been done in Vienna Ab-initio Simulation Package
(VASP) using projected augmented wave method and a plane-wave basis set with a
kinetic energy cutoff of 227.2 eV. The primitive unit cell is fully relaxed on
a (7$\times$7$\times$8) Monkhorst-Pack k-point mesh including the $\Gamma$
point with a Gaussian smearing of 0.05 eV and an increased kinetic energy
cutoff of 1.25 times. The calculated lattice constants of 6.088 and 5.591 Å
agree with the previously calculated 6.078 and 5.587 Å[31], which are slightly
underestimated when compared to the experimental data of 6.165 and 5.748 Å.
For the accurate DOS calculation, a much denser (12$\times$12$\times$18)
k-point mesh with tetrahedron method is used. The (3$\times$3$\times$3)
supercell with vacancies is relaxed on a (2$\times$2$\times$3) k-point mesh
and the corresponding DOS calculated on a denser (4$\times$4$\times$6) k-mesh.
The absolute forces on ions are reduced to below 0.02 eV/Å during relaxation.
Previous DFT calculations have shown that LDA with the fully relaxed crystal
structure can give a total magnetic moment of 1.09-1.12 $\mu_{B}$ in a good
agreement to the experimental data of 1.22-1.25 $\mu_{B}$ [18, 19, 31]. Our
calculated 1.14 $\mu_{B}$ with LDA also agrees with these results.

##  III Results

We begin by comparing the impact of irradiation on different samples. Figure 1
shows the temperature-dependent resistivity of four crystals of LaCrGe3. The
inset zooms on the area close to $T_{\text{FM}}$ in resistivity (top) and the
temperature-dependent resistivity derivative (bottom). The selected samples
show good reproducibility of both $T_{\text{FM}}$ and RRR, as well as the
overall temperature dependence of the resistivity. Insets also show two ways
of the transition temperature determination. The crossing points of the linear
fits of the $\rho(T)$ curves above and below the transition was used to
determine $T_{r}$ (upper inset). Similar linear fits of the derivative curves
(lower inset) were used to define $T_{d}$.

Figure 2:  Temperature-dependent resistivity of sample #A before and after 3
C/cm2 of electron irradiation. Stars mark the ferromagnetic transition
temperature before and after irradiation using $T_{r}$ criterion. Inset shows
the derivative of resistivity with respect to temperature, with stars defining
the transition temperature using $T_{d}$ criterion. Blue line in the main
panel shows the difference between resistivity curves before and after
irradiation.

Figure 2 shows the temperature-dependent resistivity of sample #A before
(black curve) and after 3 C/cm2 of irradiation (red curve). The $\rho(T)$
curve demonstrates a nearly parallel shift with irradiation. The difference,
$\Delta\rho(T)=\rho(T,3\>\text{C/cm}^{2})-\rho(T,\text{pristine})$ (blue line,
main panel), shows minor violation of Matthiessen’s rule, increasing from 9.5
$\mu\Omega cm$ at 300 K to 12.5 $\mu\Omega cm$ at 95 K, just above $T_{N}$.
Below the transition, $\Delta\rho$ drops to 10.5 $\mu\Omega cm$ and increases
on further cooling to 10 K to 12 $\mu\Omega cm$, saturating at this value.

Importantly, $T_{\text{FM}}$ increases with irradiation, from 89.9 K to 92.9 K
using $T_{r}$ criterion and from 90.6 K to 94.9 K using $T_{d}$ criterion, as
shown in inset of Fig. 2). Because of the transition temperature increase,
there is a notable feature in the difference plot at $T_{\text{FM}}$.

Figure 3:  Evolution of the temperature dependent longitudinal resistivity of
sample #B with sequential electron irradiations. Inset shows the derivative of
the resistivity with respect to temperature. Resistivity shows nearly parallel
up-shift with irradiation dose.

Figure 3 shows the longitudinal resistivity of the sample #B in zero magnetic
field (also used for Hall measurements) in the pristine state (black curve)
and after multiple irradiations with cumulative doses of 3 C/cm2 (red), 6
C/cm2 (green, second irradiation of 3C/cm2 sample), 9 C/cm2 (cyan, added 3
C/cm2 to 6 C/cm2) and 11 C/cm2 (blue, added 2 C/cm2 to 9 C/cm2 sample). The
irradiation for the 9 and 11 C/cm2 doses was applied approximately 6 months
after the previous doses, while the 6 C/cm2 dose was applied one year after
the sample initially received 3 C/cm2. This explains the visibly smaller shift
between the 3 and 6 C/cm2 curves compared to the other doses, due to longer
time for partial annealing of the sample at room temperature [26, 27]. Samples
#C and #D were subjected to significantly smaller doses of irradiation (0.5
C/cm2 and 1 C/cm2 respectively) and are shown in Fig. 4. Sample #C followed
the same trend in the transition temperature and resistivity with irradiation
as samples #A and #B. Sample #D showed a larger increase in both resistivity
and the transition temperature per dose than the rest.

The temperature-dependent shift, $\Delta\rho$, for all samples is compared in
Fig. 5. It has the same shape for samples #A, #B and #C, showing slight
increase on cooling except for in the transition area.

Figure 4: Temperature-dependent resistivity of samples #C (left panel) and #D
(right panel) before (black curves) and after (red curves) electron
irradiation. Stars mark the ferromagnetic transition temperature before and
after irradiation using $T_{r}$ criterion. Insets show the derivative of
resistivity with respect to temperature, with stars defining the transition
temperatures using $T_{d}$ criterion. Blue lines show the difference,
$\Delta\rho=\rho(\text{irradiated})-\rho\text{(pristine)}$.  Figure 5:  Left
panel: Shift of the temperature dependent resistivity,
$\Delta\rho(T,\text{Dose})=\rho(T,\text{Dose})-\rho(T,0\text{C}/\text{cm}^{2}$).
Sample #A, after a dose of 3 C/cm2 (black); sample #B after multiple
irradiations with cumulative doses of 3 C/cm2 (red), 6 C/cm2 (light green), 9
C/cm2 (cyan) and 11 C/cm2 (dark blue); sample #C after 0.5 C/cm2 (yellow); and
sample #D after 1 C/cm2 (dark green) irradiation. The right panel shows the
relative shifts in resistivity with respect to their values at room
temperature, $\Delta\rho(T)-\Delta\rho(300K)$.  Figure 6:  Top panel: Dose
dependence of ferromagnetic Curie temperature $T_{\text{FM}}$ determined using
$T_{r}$ (closed symbols) and $T_{d}$ (open symbols) criteria. Bottom panel:
Dose dependence of the shift of the room temperature resistivity
$\Delta\rho(300\>\text{K},\text{Dose})=\rho(300\>\text{K},\text{Dose})-\rho(300\>\text{K},0)$.

The shift in the resistivity at 300 K is a reasonable proxy for defect
concentration. The bottom panel of Fig.6 presents variation of resistivity at
$\rho(300\>\text{K})$ as a function of irradiation dose. The top panel
presents dose dependence of the ferromagnetic transition temperatures
determined using $T_{r}$ (closed symbols) and $T_{d}$ (open symbols) as
criteria. The dependence of the ferromagnetic transition temperature on
$\rho(300\>\text{K})$ is shown in Fig. 7. It is clear, that despite some
variation of $T_{\text{FM}}$ in the pristine state, all samples show a
consistent trend of $T_{\text{FM}}$ increase with increased disorder. When
sample #B was subjected to higher doses of irradiation, it revealed a trend
toward $T_{\text{FM}}(\rho)$ saturation.

Figure 7:  Change in the ferromagnetic transition temperature in single
crystals of LaCrGe3 as a function of the room temperature resistivity, which
is used here as a proxy for the amount of disorder.  Figure 8:  A product of
the transverse Hall resistance $R_{xy}$ of sample #B and its thickness $t$
before and after irradiation as a function of magnetic field. Lines show the
linear fit used for finding the Hall coefficient at 300 K. At this
temperature, the difference in carrier density with irradiation is
insignificant to within error ($9.0\pm 0.2\cdot 10^{-4}$ cm3 / C vs. $8.7\pm
0.2\cdot 10^{-4}$ cm3 / C), showing that irradiation does not dope the system.

Finally, Fig. 8 presents field-dependent Hall resistance $R_{xy}$ multiplied
by sample thickness in sample #B at room temperature, in the paramagnetic
state far above the long range ordering at $T_{\text{FM}}$. The data in the
pristine state are shown with solid symbols, and after 3 C/cm2 electron
irradiation with open symbols. Unfortunately, the contact for the transverse
(Hall) resistance measurement deteriorated after 6 C/cm2 of irradiation and
mechanically detached after 9 C/cm2. For both data sets the Hall resistance is
linear in magnetic field, and have identical linear fit slopes (grey and
dashed lines) to within error ($9.0\pm 0.2\times 10^{-4}$ cm${}^{3}/C$ to
$8.7\pm 0.2\times 10^{-4}$ cm${}^{3}/C$). This suggests that the change
(increase) of the carrier density is less than 3%, if any. This should be
compared to the 6% change of resistivity for the same doses. The sign of the
Hall effect is consistent with the sign of reported thermopower [34].

##  IV Discussion

Summarizing our observations, we see three effects of controlled disorder on
the properties of LaCrGe3 crystals. (1) The resistivity increases with
disorder roughly independently of temperature and reveals only minor
deviations from the Matthiessen’s rule both above and below $T_{\text{FM}}$.
These deviations diminish further with dose increase. (2) The resistivity
increase is not accompanied by any sizable change of the Hall effect in the
paramagnetic phase well above $T_{\text{FM}}$. This suggests that irradiation
primarily changes the elastic scattering rate and is not changing the Fermi
surface. (3) The ferromagnetic transition temperature, T${}_{\text{FM}}$, is
increased nearly linearly as a function of dose for small amounts of
irradiation in all samples studied. At high doses there is a tendency toward
saturation.

Figure 9: Top panel: the supercell structure of LaCrGe3 with La, Cr and Ge
ions shown by green, blue and grey spheres, respectively. One formula unit
(i.e. one La, one Cr and three Ge atoms labeled with yellow crosses) are
removed to create vacancies of 1.8% to represent disorder. Bottom panel: the
density of states, $D(E)$, of ferromagnetic LaCrGe3 in pristine state (black
line) and in a (3$\times$3$\times$3) supercell with 1.8% vacancies, as shown
in the top panel. Red line is calculation with the rigid structure, blue line
is calculation after ionic relaxation. The spin majority (minority) DOS is
plotted as positive (negative) value. The sharp DOS peak in the pristine state
near $E_{F}-0.25$ eV is broadened by vacancies and leads to a notable increase
in the DOS at $E_{F}$.

Theoretically, for a general case of an itinerant system, an increase in
disorder suppresses the ferromagnetic transition temperature, $T_{\text{FM}}$
due to the smearing of DOS leading to the reduction of $D(E_{F})$. Is it
possible that similar smearing could instead result in an increase of
$D(E_{F})$? In Fig. 9 we present DOS in LaCrGe3 calculated for the pristine
structure and for the structure with 1.8% vacancies, the lowest concentration
of vacancies suitable for a (3$\times$3$\times$3) supercell, as shown in the
top panel. The average distance among these vacancy sites is about 8 Å. The
vacancy sites are separated among themselves by at least a full unit cell in
all directions. The characteristic feature of band structure of LaCrGe3 is a
sharp peak in $D(E)$ located near $E_{F}-0.25$ eV (black line) [31]
originating from the almost flat Cr 3$d$ bands ($d_{xz}$ and $d_{yz}$
orbitals) in the spin majority along the $\Gamma$-M-K-$\Gamma$ direction.
Noticeably DOS right at $E_{F}$ for spin majority is very small and even
smaller than that of the spin minority. Vacancies induce disorder by
disrupting chemical bonds. As shown in the bottom panel of Fig. 9, 1.8%
vacancies notably reduce and broaden the peak near $E_{F}-0.25$ eV (red line).
However, as the consequence of the broadening, the DOS at $E_{F}$ shows a
sizable increase for the majority spin, while it slightly decreases for the
minority spin. Such opposite changes in the DOS at $E_{F}$ vs. $E_{F}-0.25$ eV
are also true with ionic relaxation (blue line). The overall DOS at $E_{F}$
increases by 0.5 states/f.u. or 25% accounting for both spin channels. The
broadening of almost-flat bands at EF-0.25 eV leads to an increase of the DOS
at EF.

The calculated significant increase of DOS for a sample with 1.8% vacancies,
should be scaled down to match the significantly lower densities of the
vacancies introduced by irradiation in our experiments. As we have shown, the
actual density is about one order of magnitude smaller for the highest doses
achieved. Although an increase in $D(E_{F})$ provides a plausible mechanism
for $T_{c}$ increase, it should lead to a slope change of resistivity curve
due to carrier density variation, violation of Matthiessen’s rule and a change
in the Hall effect. Neither is obvious in our experiment, which may suggest
that the states from the broadened flat bands remain localized.

Suppression of the ordering transition temperature with disorder is a general
trend for materials with different types of order. Examples include a
nematic/antiferromagnetic transition in the parent iron-based superconductor
BaFe2As2, isoelectron substituted BaFe2(As1-xPx)2 [33], and in hole-doped
(Ba1-xKx)Fe2As2 [35]. Similar suppression of the charge density wave
transition temperature with disorder is found in NbSe2 [36, 12], TaSe2 and
TaS2 [36, 37]. Disorder suppresses the transition temperature in nodal
superconductors [38], but the $T_{c}$ is practically insensitive to disorder
in conventional $s-$wave superconductors (known as Anderson theorem) [39],
with the exceptions coming from the effect of strong disorder on the density
of states as observed in aluminum [40].

The exceptions to this common trend of ordering temperature suppression are
mostly found in systems with competing orders. For example, in NbSe2 [36, 12],
TaSe2 [36], TaS2 [36, 37], Lu5Ir4Si20 [41], cuprates [42] and some Remeika
3-4-13 compounds [32] superconducting $T_{c}$ increases on suppression of CDW.
In all these materials two electronic orders are competing for the density of
states at the Fermi surface, and disorder tips the balance in favor of
subdominant order (superconductivity) by suppressing dominant (higher
transition temperature) order [43]. Similar phenomena are observed in FeSe, in
which superconductivity is competing with nematic order [44]. The only
exception to this rule is observed in (Ba1-xKx)Fe2As2 where dominant $C_{2}$
stripe antiferromagnetic order is stabilized when the subdominant (lower
transition temperature) $C_{4}$ phase is suppressed by disorder [13]. The
possibility of competing phases in LaCrGe3 is suggested by the observation of
several types of ordering in the pressure phase diagram [15]. Another
ferromagnetic phase was suggested to be stabilized in LaCrGe3 at ambient
pressure below $\sim$70 K [19, 45].

Another mechanism that could potentially explain the observed change in
$T_{\text{FM}}$ would be an effective change in the average volume of the unit
cell due to induced atomic defects. Experiments on neutron-irradiated Cu [46],
electron-irradiated Cu[47] and electron-irradiated Al [48] showed that the
lattice expands at low temperature, which is equivalent to a “negative
effective pressure”. When measurements were performed as a function of
temperature, it was found that at room temperature about 16% of the effective
lattice parameter change at low temperatures, $\Delta a/a$, remained in copper
and no resolvable leftover strain was observed in aluminum. The annealing
process has several stages and is quite complicated. In this scenario, the
effect of irradiation would result in an effective shrinkage of the lattice,
exerting ”positive effective pressure”. Therefore, the sign of the effect
depends on a material and conditions of irradiation and thermal history of the
sample.

Regardless the sign, let us estimate the volume change due to defects produced
by irradiation, which will allow comparison with studies of LaCrGe3 under
pressure. One way is to estimate the total volume of all vacancies, assuming
that no interstitials are left. Irradiation with the dose of 1 C/cm2 induces
about $3.5\times 10^{-3}$ vacancies per conventional unit cell, which is
approximately 0.005 Å3. With a unit cell volume of 218.5 Å3, this gives
$\Delta V/V\approx 2.2\times 10^{-5}$ change. Our highest irradiation dose was
11 C/cm2, see Fig.3, which results in $\Delta V/V\approx 2.5\times 10^{-4}$
total volume change. On the opposite side of the estimate, we can assume that
all defects stay and result in a rate of change reported for Cu and Al in
Ref.47. Remarkably, it was found that at low temperatures, the rate of change,
$\eta=(\Delta V/V)/\Delta\rho$, does not vary much for a wide range of
irradiation conditions [46, 47, 48]. For example, for copper irradiated with
2.6 MeV electrons at 6 K (quite similar to the parameters of our work),
$\eta=1.75\;(\mu\Omega\cdot\text{cm})^{-1}$ was reported [47]. As shown in
Fig.3, 11 C/cm2 results in a resistivity change of $\Delta\rho\approx
38\;\mu\Omega\cdot\text{cm}$. This gives $\Delta V/V\approx 4.4\times 10^{-4}$
total volume change, which is of the same order as the first estimate.
However, the reported change of resistivity in the LaCrGe3 was measured after
the irradiated sample was brought up to the room temperature. According to
Ref. [47] only about 16% of the effect remains for the linear change of the
lattice parameter $\Delta a/a$. Therefore, the estimate becomes $\Delta
V/V\approx 3.5\times 10^{-6}$ for the total volume change, which is much
smaller than the first estimate.

In our experiments, the enhancement of $T_{\text{FM}}$ per 1 C/cm2 is about
0.6 K. From the pressure dependence studies, this change is achieved at
approximately 0.04 GPa. With a bulk modulus of 88 GPa (Ref. 45 in Ref.[49]),
this corresponds to $\Delta V/V\approx 4.5\times 10^{-4}$, which is more than
20 times larger than our largest estimate for the effect of irradiation.
Therefore, steric effects are unlikely to be the reason for the disorder-
induced enhancement of the transition temperature.

##  V Conclusion

In conclusion, we observed an enhancement of the ferromagnetic transition
temperature in LaCrGe3 with the increasing concentration of point-like
defects. This contradicts theoretical predictions of the suppression of
itinerant ferromagnetism by disorder [10, 11]. We suggest that the flat band
peak in $D(E)$ 0.25 eV below the $E_{F}$ broadens with disorder, causing an
apparent increase in $D(E_{F})$ which leads to an increase of $T_{\text{FM}}$
in this itinerant system.

##  VI Acknowledgements

The research was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, Division of Materials Sciences and Engineering. Ames National
Laboratory is operated for the U.S. Department of Energy by Iowa State
University under Contract No. DE-AC02-07CH11358. Electron irradiation was
performed on SIRIUS platform supported by French National network of
accelerators for irradiation and analysis of molecules and materials EMIR&A
under Project No. 21-3700.

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