PNAS
2024 Vol. 121 No. 7 e2315787121
https://doi.org/10.1073/pnas.2315787121
1 of 8
Quantum interference in superposed lattices
Yejun Feng
a,1
, Yishu Wang
b,c
, T. F. Rosenbaum
d
, P. B. Littlewood
e,f
, and Hua Chen
g
Edited by J.C. Davis, University of Oxford, Oxford, United Kingdom; received September 11, 2023; accepted January 4, 2024
RESEARCH ARTICLE
|
PHYSICS
Charge transport in solids at low temperature reveals a material’s mesoscopic prop-
erties and structure. Under a magnetic field, Shubnikov–de Haas (SdH) oscillations
inform complex quantum transport phenomena that are not limited by the ground state
characteristics and have facilitated extensive explorations of quantum and topological
interest in two
-
and three
-
dimensional materials. Here, in elemental metal Cr with two
incommensurately superposed lattices of ions and a spin
-
density
-
wave ground state, we
reveal that the phases of several low
-
frequency SdH oscillations in
휎
xx
(
휌
xx
)
and
휎
yy
(
휌
yy
)
are no longer identical but opposite. These relationships contrast with the SdH oscil-
lations from normal cyclotron orbits that maintain identical phases between
휎
xx
(
휌
xx
)
and
휎
yy
(
휌
yy
)
. We trace the origin of the low
-
frequency SdH oscillations to quantum
interference effects arising from the incommensurate orbits of Cr’s superposed reciprocal
lattices and explain the observed
휋
-
phase shift by the reconnection of anisotropic joint
open and closed orbits.
π-
phase shift | quantum oscillations | incommensurate reciprocal lattices | fermiology
Functional materials with interesting and useful electronic, magnetic, and optical responses
can be created through the engineering of their electronic structure (1–3). While conven-
tional crystalline materials still hold many hidden degrees of freedom for unconventional
quasiparticles of topological nature (1), an alternative route to access complex electronic
behavior is to artificially superpose lattices upon each other to avoid the confinement of
the three
-
dimensional space group symmetry (2, 3). Examples include the control of the
termination layer at the molecular
-
beam
-
epitaxy growth interface of two insulating oxides
(2) and the Moiré construction of two graphene sheets twisted at small angles (3). New
properties peculiar to the composite lattice can emerge, such as superconductivity (4, 5).
These examples suggest the power of exploring degrees of freedom beyond the examples
cited above to construct unique types of composite lattices. One natural but largely unex-
plored example exists in incommensurately modulated crystalline materials (6). For con-
ventional crystals, the electronic structures are dictated by their three
-
dimensional space
groups (1). The symmetry property of an incommensurate structure, on the other hand,
is mathematically constructed from space groups of a higher dimensional space, before
being sectioned into three dimensions (7). Many incommensurate structures exist in
metals, such as helical magnets and spin and charge density waves (6, 8, 9). The incom-
mensurate superstructure involving either charge or spin introduces a second set of recip-
rocal lattices which interacts with the underlying ions. While historically treated as simply
opening a gap at the Femi surface of the first set of reciprocal lattices (10–12), multiple
applications of the superposed incommensurate reciprocal lattice vectors can provide for
more complex possibilities and properties. As we detail below, sophisticated galvanomag-
netic behavior emerges from such composite electronic structures because of the incom-
mensurate nature of their superposition.
We explore unconventional electronic characteristics in the archetypical spin
-
density
-
wave
(SDW) system, Cr. Cr possesses a simple body
-
centered cubic Bravais lattice and a
one
-
atom basis, which allows a high
-
fidelity theoretical understanding of its paramagnetic
band structure. The paramagnetic Fermi surface is composed of only closed forms with
no open sheets and is isomorphic to those of W and Mo (Fig.
1
A
) (8, 13). Below
T
N
=
311.5 K, long
-
range SDW order with an incommensurate wavevector
Q
= (0.952, 0, 0)
develops in Cr as the result of a nesting instability at the Fermi surface (Fig.
1) (8, 12).
Because of different sizes of the hole and electron octahedra, the paramagnetic Fermi
surface is imperfectly gapped with residual fragments of both carrier types (Fig. 1).
We find that the superposed reciprocal lattices of the ionic lattice and the SDW lead to
two low
-
frequency SdH oscillations of 36 T and 40 T, with different galvanomagnetic
behavior in field and temperature dependences, anisotropy, and existence of harmonics.
Despite their differences, the SdH oscillations at each frequency reliably demonstrate oppo
-
site phases (or equivalently, a
π-
phase shift) between two configurations of electrical current
I
flowing either parallel or perpendicular to
Q
. In each case, the external magnetic field
H
Significance
Understanding the Fermi surface
is of fundamental importance to
metals. Here, we explore a
situation where two sets of
incommensurate lattices are
superposed together in the
elemental metal Cr. These two
reciprocal lattices are connected
through many
-
body electron
correlations that have distinct
experimental signatures. They
combine to build a Fermi surface
with many small de Haas–van
Alphen orbits and quantum
interference paths that are not
detectable by standard
photoemission techniques. In the
spin
-
and charge
-
density
-
wave
state of Cr, this construction
leads to Shubnikov–de Haas
oscillations with opposite phases
between two orthogonal
channels of transverse
magnetoresistance. This
phenomenon represents a
natural three
-
dimensional
analogue to topological and
Moiré systems.
Author contributions: Y.F. designed research; Y.F., Y.W.,
T.F.R., P.B.L., and H.C. performed research; Y.F., Y.W.,
T.F.R., P.B.L., and H.C. analyzed data; and Y.F. wrote the
paper.
The authors declare no competing interest.
This article is a PNAS Direct Submission.
Copyright © 2024 the Author(s). Published by PNAS.
This open access article is distributed under
Creative
Commons Attribution License 4.0 (CC BY)
.
1
To whom correspondence may be addressed. Email:
yejun@oist.jp.
This article contains supporting information online at
https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.
2315787121/-
/DCSupplemental
.
Published February 5, 2024.
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