of 8
PHYSICAL REVIEW B
110
, 205118 (2024)
Electronic topological transitions in cadmium under pressure studied via theoretical
and experimental x-ray absorption spectroscopy
Jasmine K. Hinton
,
1
,
2
,
3
,
*
,
Daniel Schacher
,
2
,
3
,
*
Wonseok Lee
,
4
,
*
G. Alexander Smith
,
5
,
6
,
*
Emily Siska,
2
Changyong Park
,
7
Paul B. Ellison
,
2
,
3
Scott K. Cushing
,
4
,
Craig P. Schwartz
,
2
,
§
Keith V. Lawler
,
2
,

and Ashkan Salamat
2
,
3
,
1
Neutron Scattering Division, Neutron Sciences Directorate,
Oak Ridge National Laboratory
, Oak Ridge, Tennessee 37830, USA
2
Nevada Extreme Conditions Laboratory,
University of Nevada, Las Vegas
, Las Vegas, Nevada 89154, USA
3
Department of Physics & Astronomy,
University of Nevada Las Vegas
, Las Vegas, Nevada 89154, USA
4
Division of Chemistry and Chemical Engineering,
California Institute of Technology
, Pasadena, California 91125, USA
5
National High Magnetic Field Laboratory,
Los Alamos National Laboratory
, Los Alamos, New Mexico 87545, USA
6
Department of Chemistry & Biochemistry,
University of Nevada Las Vegas
, Las Vegas, Nevada 89154, USA
7
HPCAT, X-ray Science Division,
Argonne National Laboratory
, Lemont, Illinois 60439, USA
(Received 29 June 2023; revised 16 August 2024; accepted 11 October 2024; published 7 November 2024)
An electronic topological transition (ETT) in cadmium below 1 GPa is investigated
in situ
with experimental
x-ray absorption spectroscopy and projecting calculated core-valence excitons onto the band structure. These
projections are a useful application of the Bethe-Salpeter equation approach that considers many-body effects.
The method described herein can be used for systems that are otherwise difficult to probe
in situ
; therefore,
it provides a generalizable approach to identifying and understanding ETTs under high pressure. Although
pressure-induced ETTs are often probed using indirect structural responses, our own x-ray diffraction and Raman
studies suggest a second-order structural transition around 3 GPa but are largely insensitive to or inconclusive
for the previously studied ETT in this region.
DOI:
10.1103/PhysRevB.110.205118
I. INTRODUCTION
The ground-state allotrope of elemental cadmium (Cd) re-
mains in the hexagonal close-packed (hcp) structural motif
under high-pressure compression up to 174 GPa [
1
]. Based
on geometric and energetic considerations, the ideal c/a ra-
tio for hcp is
8
/
3
1
.
633; however, Cd has an unusually
high ambient c/a ratio of about 1.86 [
1
]. Only zinc (Zn) and
mercury (Hg) have comparably high c/a ratios [
2
]. Nonideal
c/a ratios often indicate some nontrivial electronic structure,
and as such, materials with anomalous c/a ratios, like Cd, are
excellent test cases for new probes of electronic structure.
A discontinuity in the c/a ratio has historically been used
as preliminary evidence of electronic structural changes and
cause to further investigate such electronic properties via the
topology of the Fermi surface [
3
7
].
The topology of the Fermi surface indicates the relative
position in reciprocal space of the electrons at the Fermi
energy, and can be used to understand a variety of other
properties of a material in the metallic state [
8
]. With in-
creasing pressures and no structural phase transitions, the
Fermi surface will change continuously [
9
]. The presence
*
These authors contributed equally to this work.
Contact author: hintonjk@ornl.gov
Contact author: scushing@caltech.edu
§
Contact author: craig.schwartz@unlv.edu

Contact author: keith.lawler@unlv.edu
Contact author: ashkan.salamat@unlv.edu
of a noncontinuous deformation of the Fermi surface indi-
cates a nontrivial change in the electronic structure, and is
known as an electronic topological transition (ETT) or Lif-
shitz transition [
10
]. While such phase transitions can be
sampled by theoretical methods, characterizing these methods
experimentally, particularly under high-pressure conditions,
remains challenging. A variety of the probes used to charac-
terize phase transitions at high pressure, including diffraction,
vibrational spectroscopy, or using other characteristic thermo-
dynamic properties, are either largely insensitive to or do not
provide an atomic-level understanding of the underlying ETT
process [
11
]. When available, angle-resolved photoemission
spectroscopy (ARPES) is ideal for understanding electronic
behavior [
12
]; unfortunately, ARPES remains inapplicable for
high-pressure materials.
Several previous works have investigated the possible rela-
tionship between c/a anomalies and ETTs in Cd at varying
pressures [
6
,
13
15
]; however, c/a discontinuities can occur
for reasons other than ETTs [
16
,
17
]. Although the electronic
structure of Cd has been studied at low pressure [
18
,
19
], these
works do not provide direct quantifiable evidence of an ETT in
Cd. Vibrational Raman spectroscopy is often used to investi-
gate ETTs [
10
,
20
22
]; however, there are disagreements even
with the same system as to whether a mode should harden,
soften, or just change linewidth as the system experiences
an ETT. Additionally, some systems may not display optical
Raman modes due to their crystal structures [
9
]. The Fermi
surface of Cd was directly measured via the de Haas-van
Alphen effect under ambient conditions [
23
] and attempts
were made for measurements up to 2 GPa [
24
]; however, the
2469-9950/2024/110(20)/205118(8)
205118-1
©2024 American Physical Society
JASMINE K. HINTON
et al.
PHYSICAL REVIEW B
110
, 205118 (2024)
pressure calibration and hydrostaticity of the pressure trans-
mitting media (PTM) used in the latter work obfuscate the
certainty of an ETT. Resistance measurements [
25
], magne-
toresistance measurements [
26
], and structural measurements
using x-ray diffraction (XRD) [
1
,
7
,
27
] suffer from similar
hydrostaticity concerns and only infer the presence of an ETT.
Here, we observe a connection between the electronic be-
havior as observed spectroscopically and the projection of
the calculated core-valence exciton density onto the electronic
band structure, suggesting and describing an ETT in Cd below
1 GPa. This work further validates the technique of using
x-ray absorption spectroscopy (XAS) to explore ETTs at high
pressure [
28
] and, when paired with the computational re-
sults, gives new insights into the mechanism of ETTs under
pressure. Structural data as determined by XRD is shown and
compared with the spectroscopic data to separate electronic
and structural effects. No XAS discontinuities are observed
after the ETT, indicating that in the range studied, there are no
further ETTs.
II. METHODS
The absorption length of the Cd K-edge was calculated
in
Hephaestus
to be 23 μm. Cd (99.999%, Sigma-Aldrich)
samples were cut from wire in order to match the calculated
absorption length, as commercially available Cd powder sam-
ples with the appropriate mesh size could not be sourced at
the time of this work. All samples were handled in an inert
argon (Ar) environment prior to and during loading. Samples
were at no point exposed to ambient air conditions, and a
lack of reaction with air was verified with Raman for surface
effects and XRD for bulk effects. No oxidation effects were
seen by XAS, Raman, or XRD. Ruby fluorescence was used as
the primary pressure marker [
29
], with gold (Au) also loaded
for use as an XRD pressure marker [
30
]. Ruby and Au had
average dimensions of 15
×
15 μm (length
×
width), and were
arranged to be out of the absorption path of Cd. Using an in-
house-designed diamond anvil cell (DAC), samples were gas
loaded with helium (He) in a premachined beryllium (Be) gas-
ket, preindented to a thickness of about 23 μm. Samples were
monitored via x-ray imaging to check for bridging between
the diamonds and found to not be bridging (Fig. S5 [
31
]).
XRD and XAS experiments were carried out at 16-BMD,
Sector 16 of the Advanced Photon Source [
32
]. XRD mea-
surements (
λ
=
0.4132 Å) used axial geometry, while XAS
measurements were taken in radial geometry through the x-
ray transparent Be gasket on the Cd K-edge (26.7112 keV).
The incoming x-ray beam size was around 4 μm at full width
at half maximum, and the full width at about 1% of maximum
is around 20 μm (Fig. S1 [
31
]). Due to focusing methods [
32
],
there are no practical effects of a higher harmonic on the
current measurement (see the Supplemental Material (SM)
[
31
] for further discussion).
Since the sample itself is pure elemental Cd, the lowest
pressure sample (loaded inertly at ambient pressure in an Ar
environment) is collected as the internally consistent reference
for energy calibration and referred to as the energy standard
loadings within this work. Energy scans were made repeatedly
to improve counting statistics. The scans were reproducible
within 0.1–0.5 eV deviation, depending on the number of
repetitions applied for each measurement. While the source of
deviation is, strictly speaking, unknown, it is suspected to be
due to heat accumulation on the monochromator motor and
a slight offset in the calibration parameter due to a thermal
expansion of mechanical parts. The energy calibration scheme
for 16-BMD, Sector 16 of the Advanced Photon Source can be
found in the work of Park
et al.
[
32
].
Raw XAS data, primarily x-ray absorption near edge struc-
ture (XANES), were reduced using the
Athena
software.
Extended x-ray absorption fine structure (EXAFS) data were
fit using the
Artemis
software package, and further analysis
details of EXAFS data can be found in the SM [
31
]. Ruby
spectra were fit using Fityk for the central wavelength po-
sition which was used to calculate pressure [
29
]. XRD data
were processed in Dioptas [
33
] and GSASII [
34
]. A La Bail
refinement was carried out on each data point to extract lattice
parameters. Au was fitted to the equation of state of Takemura
and Dewaele [
30
] to compare to Ruby fluorescence measure-
ments and was found to be in good agreement.
K-edge XAS calculations on Cd were performed with
density functional theory (DFT) using the Bethe-Salpeter
equation (BSE) as implemented in the OCEAN code [
35
37
]
as a function of pressure with optimized structures every
0.5 GPa between 0 and 5 GPa. The Perdew-Burke-Ernzerhof
(PBE) functional as implemented in the Quantum Espresso
package was used with an energy cutoff of 2450 eV (180 Ry)
for the DFT portion of the calculation [
38
40
]. The self-
consistent field (SCF) calculation was done on a 10
×
10
×
10
k
grid while the screening calculation was done on a 2
×
2
×
2
k
grid. The pseudopotentials have a valency of 4s
2
4p
6
4d
10
5s
2
[
41
]. The maximum energy for bands in the final state
wave function was set to 150 eV above the edge. Lorentzian
broadening was set to 7.32 eV to better match experimental
data [
42
]. This formalism allows for using the experimen-
tally determined crystallographic orientation of the sample
obtained from XRD to be applied to XAS calculations. The
same computational input parameters for the K-edge XAS
calculations are used with the generalized minimal residual
(GMRES) method [
43
] (rather than the Haydock recursion
method) to solve the BSE and calculate the real-space exci-
tonic wavefunction.
Ambient temperature Raman measurements as a function
of pressure are also presented. Steel and rhenium (Re) gaskets
are used with a lithium fluoride (LiF) PTM. Sm:SrB
4
O
7
is
used as a pressure marker for its temperature independence
[
44
,
45
]. A 532 nm excitation line is used with matching Opti-
grate filters. Further DAC specification followed the methods
of Hinton
et al.
[
46
].
III. RESULTS AND DISCUSSION
Previous calculations suggested an ETT in Cd at varying
pressures depending on the details of the method [
6
,
13
,
14
,
48
].
In our calculations probing for low pressure ETTs, distinct
changes are observed in the band structure and in the topology
of the Fermi surface between 0 and 0.5 GPa, particularly
along the

–M, H–K, and H–L high symmetry paths in the
first Brillouin zone. Between

and M, the peak of one band
initially below the Fermi energy rises above it as the pressure
is increased from 0 to 0.5 GPa [Fig.
1(a)
]. Across the same
205118-2
ELECTRONIC TOPOLOGICAL TRANSITIONS IN CADMIUM ...
PHYSICAL REVIEW B
110
, 205118 (2024)
FIG. 1. Computational evidence for ETT between 0 and 1 GPa.
(a) Electronic band structure of Cd for 0 GPa (solid blue) and 0.5 GPa
(dashed orange). (b), (c) The Fermi surface of Cd in the first Brillouin
zone, with high symmetry points marked in green: (b) 0 GPa, and
(c) 0.5 GPa. Note the qualitative change in the Fermi surface between
0 and 0.5 GPa. Plotted with Fermisurfer software [
47
].
symmetry M–K path, a connection is formed in the sixfold
symmetric corner features of the Fermi surface [Figs.
1(b)
and
1(c)
]. Along H–K, a surface that is connected at 0 GPa
becomes disconnected at 0.5 GPa [Figs.
1(b)
and
1(c)
], and
we observe a band crossing the Fermi energy along this same
symmetry path [Fig.
1(a)
]. Along H–L, a new surface appears
in the Fermi surface [Figs.
1(b)
and
1(c)
], and this corresponds
to another band rising above the Fermi energy in Fig.
1(a)
.
The changes described lead to the formation of a new
hole in the manifold, which represents a discontinuous change
in the topology between 0 and 0.5 GPa. A new feature in
the Fermi surface is also found with increasing pressure that
occurs between the A and L points. These effects continue to
increase in magnitude with increasing pressure and the mani-
fold connection grows throughout the Brillouin zone parallel
to M–K as the pressure rises from 0.5 to 5 GPa. Despite
these electronic changes at the Fermi energy, the total and
partial densities of states show minimal changes as the ETT
is crossed (Figs. S19 and S20 [
31
]). These computational
observations suggest an ETT between 0 and 1 GPa, and no
other ETT is predicted to occur up to 10 GPa.
To experimentally explore the predicted ETT, we first turn
to XRD. Diffraction is widely used for observing phase tran-
sitions; however, it is largely insensitive to ETTs that do not
correspond with significant first or second-order atomic struc-
ture changes. Here, the high-pressure diffraction is performed
using a He PTM, the highest degree of hydrostaticity possible
to compare to previous studies. Strong preferred orientation
leading to poor powder averaging is observed across multiple
samples (Figs. S6–8 [
31
]). Structural exploration revealed that
previously observed anomalies in the c/a ratio with volume
FIG. 2. Structural information for Cd obtained via XRD from
0to25GPa.(a)c/avsV
/
V
0
plot with experimental (closed pink
circles) and computational data (open orange circles) from this work
plotted over literature data of Pratesi
et al.
[
27
] (black open circles),
Takemura [
1
] (black closed circles and black open diamonds), and
Godwal
et al.
[
13
] (black closed squares). Errors for the experimental
points are less than the diameter of the points shown. (b) Normalized
pressure (F) vs Eulerian strain (
f
E
) plot from XRD where
F
=
P
[3
f
E
(1
+
2
f
E
)
5
/
2
]
1
and
f
E
=
(1
/
2)[(
V
0
/
V
)
2
/
3
1]. Linear fits of
the data are shown for above and below 2.95 GPa, and for the full
data set.
at about 12 GPa (or around V
/
V
0
=
0
.
85) [
1
,
7
,
27
] are not
observed when a hydrostatic PTM, He, is used [Fig.
2(a)
].
No further structural discontinuities are observed in the c/a
ratio experimental data. Although calculated volumes predict
discontinuities in c/a between 0–1 and 2–3 GPa, no fur-
ther structural discontinuities are observed in the c/a ratio
experimental data.
A change in slope in the pressure derivative of the bulk
modulus is observed in the F-f plot [Fig.
2(b)
] correspond-
ing to approximately 3 GPa. The change in slope suggests a
second-order phase transition. No discontinuities to the Fermi
surface or electronic band structure are observed in correlation
with this second-order structural transition. While a change in
slope cannot be ruled out between 0–1 GPa in the F-f plot, the
lack of experimental data in this range and the fact that F-f
plots cannot use points at 0 GPa prevent a definitive conclu-
sion; therefore, we cannot assign the lowest data point alone
as a slope change, since this being an outlier is a possibility.
Our limited EXAFS data also shows no structural anomaly,
supporting no structural phase transition below 1 GPa
(Fig. S12 [
31
]).
The room temperature responses of the Raman active E
2
g
phonon mode of Cd frequency, linewidth, and intensity are
shown in Figs. S26–35 and Table SIII [
31
] for pressures
from 0–10 GPa. Data points were collected from compression,
decompression, as well as for samples that were previously
205118-3
JASMINE K. HINTON
et al.
PHYSICAL REVIEW B
110
, 205118 (2024)
heated to above the melting temperature. The calculated zone-
centered Raman frequencies, electron-phonon couplings, and
phonon linewidths across the predicted 0.5 GPa ETT are given
in Table SIV [
31
]. Further details and values of experimental
frequency versus pressure and zone center phonon calcula-
tions are available in the SM [
31
]. Our experimental Raman
data do not show clear discontinuities about the ETTs in the
aggregate for frequency, linewidth, or intensity. Additionally,
our calculated values do not indicate discontinuities in the
frequency, linewidth, or electron-phonon couplings. Taken to-
gether, this indicates that neither structural probe, diffraction,
nor vibrational Raman, were effective probes for identifying
this ETT in Cd. Although it is still possible (and common)
to investigate ETTs by both structural methods, our dataset
for Cd makes a good case study for systems where these
methods will not work based on the fundamental properties
of the material. For example, it is possible for a material to
have no Raman modes allowed, or experience an isostructural
electronic change; in this case neither probe would be able to
indicate a change.
Although these structural probes did not provide conclu-
sive evidence correlating to the ETT observed in calculations,
XAS can also be used to investigate the presence of an ETT
as it relates directly to the electronic behavior. XAS can be
broken into two regions: the XANES region and EXAFS re-
gion. EXAFS uses high energy scattering states to probe local
atomic structure (or short-range order) through scattering. The
near edge is an indicator of electronic behavior—XAS at
the K-edge excites the 1
s
core electron to states above the
Fermi energy. By projection of the calculated core-valence
excitonic wavefunction onto the electron band structure, we
can infer the
p
-orbital character of the band. Since the XAS
process involves forcing a core electron into an unoccupied
state above the Fermi surface, examining the XANES regions
provides information pertaining to the electronic behaviors of
the predicted ETT.
Figure
3
gives the experimental and calculated XANES
from 0 to 5 GPa, with the experiment represented in dashed
lines vertically offset above the calculations in solid lines. The
spectra at 0 GPa, obtained from the energy standard DAC, are
black and the higher pressures are in color as indicated by the
legend. The experimental XAS show a discontinuous change
between 0 and 0.8 GPa indicated by the shift between 0 GPa
and the higher pressure lines 0.8–4.5 GPa. Above 0.8 GPa,
the changes are more gradual. The spectra shown in Fig.
3
are
from one Cd sample (called sample 2 in the SM). We observe a
similar shift in another sample (sample 3 in the SM). Sample
3 had been pressure-cycled and thus, some hysteresis in the
energy shift is observed. The behavior of sample 3 suggests
that the ETT observed in sample 2 is reproducible. Further
data and details are available in the SM [
31
].
Theoretically calculated spectra are presented from 0.5–
5.0 GPa in 0.5 GPa resolution. In addition to the energy offset
applied to the calculated spectra, a scaling factor of about
16% is applied to the energy to account for the well-known
DFT error of underestimating the band gap [
49
]. The same
background correction used in the experiment is applied to
the calculated spectra to account for the gradually dropping
oscillator strength inherent to the calculation. Both of the
above changes are for ease of visual comparison. See the SM
FIG. 3. Calculated and experimental XANES spectra collected
for Cd between 0 and 5 GPa (dashed lines show experimental ob-
servations). The 0 GPa point is from the energy standard, and the
subsequent points are from sample 2. The large shift between the
black lines at 0 GPa and the color lines at pressure is consistent with
an electronic change occurring. Solid lines are calculated XANES
spectra using OCEAN; the same energetic shift and scaling factor are
applied to each calculated spectra. Points have 0.5 GPa resolution.
The same large peaks at approximately 26720, 26740, and 26775 eV,
as well as the blue shift as a function of pressure are seen in both
experiment and theory. See text and the SM [
31
] for normalization
and alignment details.
[
31
] for a comparison between experimental and calculated
spectra that have not had scaling or background subtractions.
Our simulations show a large discontinuous change be-
tween 0 and 0.5 GPa, followed by a gradual change above
0.5 GPa. Theoretical calculations here consider both the ori-
entation of the sample and the polarization of incident light
orientation. Since Cd is hexagonal, the contributions from the
a
and
b
planes will be equal to each other while the
c
plane is
different. This can be accounted for in the crystal orientation
tensor, and is most visible in the third XANES peak. The third
peak will be asymmetric if the contribution from only one
plane is considered, and the asymmetry shifts depending on
which plane is considered. The asymmetry observed in the
third peak is more obvious in the experiment than in theory.
Figure
3
shows that the third XANES peak of the experi-
mental spectra is asymmetrical to the right, while the third
XANES peak of the theoretically calculated spectra seems
to show more of a doublet. This difference is likely due to
the sample’s intense preferred orientation, which is common
for a sample sourced from wire. XRD patterns demonstrating
different samples’ preferred orientation are in the SM [
31
].
Both experiment and theory see the peak at
26740 eV move
higher in energy with increasing pressure, even when consid-
ering preferred orientation, suggesting the electronic structure
is well captured by the theory.
To explore the electronic nature of this change in the Cd
XANES data further, the x-ray core-valence exciton distri-
butions are projected onto the electronic band structure. To
accomplish this, the core-valence exciton wavefunction at en-
ergy
ω
is defined using the two-particle Green’s function and
photon operator
ˆ
T
acting on the ground state
|

0

:
|

(
ω
)
=
1
ω
H
BSE
+
i
η
ˆ
T
|

0

.
(1)
205118-4