Institute of Biotechnology’s Imaging Facility is supported by
NYSTEM (C29155) and the NIH (S10OD018516)Author
contributions:A.E.R., E.C., M.S., and S.H. wrote the manuscript.
A.E.R. performed immunolocalizations and in situ hybridizations.
A.E.R., J.C., and E.C. developed the models. R.J. performed in situ
hybridizations. R.K. helped with the modeling. B.C. performed
confocal imaging of theArabidopsisleaf primordia. A.E.R., A.B.R.,
R.J., M.S., S.H., and E.C. developed the project. S.H., M.S., E.C., A.R., and
H.K. provided funding.Competing interests:The authors declare no
competing interests.Data and materials availability:Microscopy data
are available through the Edinburgh University DataShare (https://
datashare.ed.ac.uk/handle/10283/3987) in The Plant Shape Lab
collection. All materials are available on request from the corresponding
authors. The growing polarized tissue framework (GFtbox) software
is freely available at: https://coensoft.jic.ac.uk. The computational
model codes are available through GitHub ( 37 ) and https://
coensoft.jic.ac.uk. All other data provided are in the main paper or
the supplementary materials.
SUPPLEMENTARY MATERIALS
science.org/doi/10.1126/science.abf9407
Materials and MethodsSupplementary Text: Model Descriptions
Figs. S1 to S10
Tables S1 to S4
References ( 38 – 44 )
Movies S1 to S7
MDAR Reproducibility Checklist1 December 2020; accepted 8 October 2021
10.1126/science.abf9407SUPERCONDUCTIVITY
Discovery of segmented Fermi surface induced
by Cooper pair momentum
Zhen Zhu^1 †, MichałPapaj^2 †, Xiao-Ang Nie^1 , Hao-Ke Xu^1 , Yi-Sheng Gu^1 , Xu Yang^1 , Dandan Guan^1 ,
Shiyong Wang^1 , Yaoyi Li^1 , Canhua Liu^1 , Jianlin Luo^3 , Zhu-An Xu^4 , Hao Zheng^1 ,
Liang Fu^2 , Jin-Feng Jia^1 *
A sufficiently large supercurrent can close the energy gap in a superconductor and create gapless
quasiparticles through the Doppler shift of quasiparticle energy caused by finite Cooper pair momentum.
In this gapless superconducting state, zero-energy quasiparticles reside on a segment of the
normal-state Fermi surface, whereas the remaining Fermi surface is still gapped. We use quasiparticle
interference to image the field-controlled Fermi surface of bismuth telluride (Bi 2 Te 3 ) thin films under
proximity effect from the superconductor niobium diselenide (NbSe 2 ). A small applied in-plane magnetic
field induces a screening supercurrent, which leads to finite-momentum pairing on the topological surface
states of Bi 2 Te 3. We identify distinct interference patterns that indicate a gapless superconducting
state with a segmented Fermi surface. Our results reveal the strong impact of finite Cooper pair momentum
on the quasiparticle spectrum.
I
n the presence of supercurrent flow, Cooper
pairs in a superconductor acquire finite
momentum ( 1 , 2 ). This Cooper pair mo-
mentumqresults in a Doppler shift of the
energy of the Bogoliubov quasiparticle
excitations (Fig. 1A)
EðÞ¼kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðÞDkm^2 þD^2q
þ1
2vk q ð^1 Þwhich is characterized by the Doppler shift
energy scaleED≈vkq=2, wherevk¼@Dk=@k
is the velocity at momentumk,Dkis the
normal-state dispersion,mis the chemical
potential, andDis the superconducting gap
in the absence of supercurrent. This Doppler
shift changes the energy of the quasiparticles
that move parallel or antiparallel to the super-
current but does not affect the quasipar-
ticles that move in the perpendicular direction,
resulting in an anisotropic quasiparticle energy
dispersion.
When the Doppler energy shiftEDexceeds
D, the superconducting gap closes, and a seg-
mented Fermi surface appears with zero-
energy quasiparticles that are a superposition
of electrons and holes ( 2 – 4 ). A schematic ex-
ample of this behavior for a system with a
circular Fermi surface in the normal state is
shown in Fig. 1B. The supercurrent decreases
the energy of quasiparticles moving parallel to
it. With sufficiently large current, zero-energy
quasiparticles first appear on a segment of
the normal-state Fermi surface in the direc-
tion of the current. As the current increases,
the momentum-space contour of these zero-
energy quasiparticles grows, forming a banana-
like shape and contributing to finite density
of states (DOS) atE= 0. The solid and dashed
lines in Fig. 1B indicate whether the excita-
tion is more electron- or hole-like, respec-
tively, in a given part of the contour. At the
same time, the rest of the normal-state Fermi
surface remains fully gapped because the
Doppler shift energy has not overcome the
superconducting gap there. Although the ex-
istence of supercurrent-induced gapless qua-
siparticles, known as the Volovik effect ( 5 ),
has been established by tunneling ( 6 , 7 ) and
specific heat measurements ( 8 , 9 )ond-ands-wave superconductors, the Fermi surface
of a current-carrying superconductor has not
been observed directly. Detection of the seg-
mented Fermi surface of Bogoliubov quasi-
particles requires spectroscopic techniques
with both energy and momentum resolu-
tion. Furthermore, the supercurrent that is
necessary to close the superconducting gap
is often larger than the depairing current at
which the superconductor switches to a re-
sistive state ( 6 ).
In this work, we overcome the aforemen-
tioned difficulty with an alternative approach
and material platform. Instead of passing a
transport current directly through a super-
conductor, we apply a small external magnetic
fieldBextto produce a screening current near
the surface by means of the Meissner effect.
Then, the superconducting order parameter
on the surface develops a spatially varying
phaseDðÞ¼r DexpðÞiq rat positionr, with
qin the direction of the screening current that
is perpendicular to the fieldBext. Our platform
of choice (Fig. 1C) consists of a thin film of
Bi 2 Te 3 , the quintessential topological insulator
( 10 , 11 ), grown by molecular beam epitaxy
on top of bulk crystal NbSe 2 , an s-wave super-
conductor. The superconducting proximity
effect from NbSe 2 induces a hard gap in Bi 2 Te 3
at zero field, as demonstrated in previous
scanning tunneling microscopy (STM) meas-
urements ( 12 ). This material combination pro-
vides ideal synergy for creating and detecting
the gapless superconducting state. NbSe 2 is
a clean s-wave superconductor with a large
superconducting gap and long London pene-
tration depthlLthat exceeds 100 nm ( 13 ).
Notably, this means that a small in-plane mag-
netic field can create a large screening current
and Cooper pair momentum on the surface,
without introducing vortices in the measure-
ment region.
Our platform is ideal to observe formation
of the segmented Fermi surface, owing to a
combination of two factors. A Bi 2 Te 3 film
under the proximity effect has a simple Fermi
pocket surrounding theGpoint ( 14 ) that orig-
inates from the topological surface states.
The proximity-induced gap in the top layer
of Bi 2 Te 3 is reduced compared with the bulk
gap of NbSe 2 , and the topological surface
state has a substantially higher Fermi velocity
(~2.6 eV·Å) ( 14 ) than the states of the parentSCIENCEscience.org 10 DECEMBER 2021•VOL 374 ISSUE 6573 1381
(^1) School of Physics and Astronomy, Key Laboratory of
Artificial Structures and Quantum Control (Ministry of
Education), Shenyang National Laboratory for Materials
Science, Tsung-Dao Lee Institute, Shanghai Jiao Tong
University, Shanghai 200240, China.^2 Department of
Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA.^3 Beijing National Laboratory
for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China.
(^4) Department of Physics, Zhejiang University, Hangzhou
310027, Zhejiang, China.
*Corresponding author. Email: [email protected] (H.Z.);
[email protected] (L.F.); [email protected] (J.-F.J.)
†These authors contributed equally to this work.
RESEARCH | REPORTS