title: Incommensurate Spin Fluctuations in the Spin-triplet Superconductor Candidate UTe$_2$
authors: Chunruo Duan Kalyan Sasmal M. Brian Maple Andrey Podlesnyak Jian-Xin Zhu Qimiao Si Pengcheng Dai
text: 
  
  
  
  
  Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA
  
  
  Department of Physics, University of California, San Diego, CA 92093, USA
  
   Neutron Scattering Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
  
  Theoretical Division and Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
  
  
  pdai@rice.edu
  Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA
  
  
  
  
  Spin-triplet superconductors are of extensive current interest because 
  they can host topological
  state and Majorana ferimons important for quantum computation. 
  The uranium based heavy-fermion superconductor UTe_2 has been argued as a 
  spin-triplet superconductor similar to UGe_2, URhGe, and UCoGe, 
  where the superconducting phase is near (or coexists with) a ferromagnetic (FM) 
  instability and spin-triplet electron pairing is 
  driven by FM spin fluctuations. Here we use neutron scattering to show that 
  although UTe_2 exhibits no static magnetic order down to 0.3 K, its magnetism is dominated by incommensurate spin fluctuations near antiferromagnetic (AF) ordering wave vector and  
  extends to at least 2.6 meV. We are able to understand the dominant incommensurate spin fluctuations of UTe_2 in terms of its electronic structure calculated using 
  a combined density functional and dynamic mean field theory.
  
  
  
  Incommensurate Spin Fluctuations in the Spin-triplet Superconductor Candidate UTe_2
      Pengcheng Dai
      December 30, 2023
  ===================================================================================
  
  
  
  Superconductivity occurs in many metals when electrons form
  coherent Cooper pairs below the superconducting 
  transition temperature T_c <cit.>. In conventional and most unconventional superconductors, electron Cooper pairs in the superconducting state 
  form anti-parallel spin-singlets with the total spin S=0 <cit.>. However, electrons in the superconducting state can also form parallel spin-triplet Cooper pairs <cit.>, analogous to the equal spin pairing state in the superfluid ^3He <cit.>. The Pauli's exclusion principle
  can be fulfilled for both singlet and triplet Cooper pairs 
  by adjusting the symmetry of the orbital part of the wave function. For the 
  spin-singlet 
  pairing state, the orbital wave function
  has even parity (symmetric) with orbital 
  angular momentum L=0 (s-wave), 2 (d-wave), etc. For spin-triplet state, the orbital wave function has odd parity
   (antisymmetric) with orbital angular momentum L=1 (p-wave), 3 (f-wave), and etc <cit.>. While most unconventional superconductors have
  spin-singlet pairing associated with antiferromagnetic (AF) 
  spin fluctuations <cit.>, spin-triplet superconductors are rare, and the superconductivity is believed to be driven by longitudinal ferromagnetic (FM) spin fluctuations <cit.>. Since spin-triplet superconductors are intrinsically
  topological <cit.> and can host Majorana fermions
  important for quantum computation <cit.>, it is important
  to understand a candidate spin-triplet superconductor by determining the associated spin fluctuations.  
  
  
  In spin-triplet superconductors such as UGe_2 <cit.>, 
  URhGe <cit.>, and UCoGe <cit.>, superconductivity arises through
  suppression of the static FM order <cit.> or coexists 
  with static FM order <cit.>. Inelastic neutron 
  scattering (INS) experiments find 
  clear evidence of FM spin fluctuations in URhGe <cit.> and UCoGe <cit.>. For the spin-triplet candidate superconductor Sr_2RuO_4 <cit.>, 
  where the material is paramagnetic at all temperatures and 
  superconductivity does not coexist with static FM order,
  magnetism is dominated by incommensurate spin fluctuations
  arising from Fermi surface 
  nesting of itinerant electrons <cit.>, although weak FM spin fluctuations are also observed <cit.>. 
  Similarly, although considerable evidence exists for spin-triplet superconductivity in
  UPt_3 <cit.>, its superconductivity appears to be associated with AF order and spin fluctuations 
  instead of FM spin fluctuations <cit.>. 
  
  
  Recently, UTe_2 has been identified as a new spin-triplet superconductor
  with T_c≈ 1.6 K <cit.>. 
  UTe_2 has an orthorhombic unit cell with space group Immm,
  where the U atoms form parallel ladders along the a-axis inside 
  trigonal prisms of Te atoms [Fig. 1(a)] <cit.>.
   The shortest U-U bond is along the rung of the ladder in the c-axis direction, while the easy-axis of the U spins is along the a-axis. 
  The symmetry operation that connects one ladder to its nearest neighbor is the body-center (1/2,1/2,1/2) translation. 
  A Curie-Weiss fit to the magnetic susceptibility data reveals an effective moment per U atom close to the 5f^2 or 5f^3 free ion value at high temperature <cit.>. No long-range magnetic order has been reported down to 0.25 K 
  <cit.>. Instead, a 
  sudden increase in the magnetic susceptibility below 10 K 
  in response to a magnetic field applied parallel to the a-axis resembles the quantum critical behavior of metallic ferromagnets, indicating strong FM spin fluctuations along the a-axis <cit.>. This suggests that UTe_2 sits at the paramagnetic end of a series of FM heavy fermion superconductors including UGe_2 <cit.>, URhGe <cit.>, 
  and UCoGe <cit.>. At the FM end, the compound UGe_2 is a pressure-induced superconductor with optimal T_c≈ 0.5 K at 1.2 GPa <cit.>. 
  Moving from UGe_2 to URhGe, superconductivity occurs at ambient pressure 
  below T_c≈ 0.25 K and coexists with static FM order below a Curie temperature T_C≈ 9.5 K <cit.>. Finally, UCoGe has coexisting superconductivity and
  FM order with increased T_c≈ 0.425 K and decreased T_C≈ 3 K, respectively <cit.>.
  
  
  The scenario that UTe_2 is a candidate
  spin-triplet superconductor <cit.> is supported 
  by a growing list of observations. These include the 
  upper critical fields H_C2 that exceed the Pauli limits 
  along all crystallographic directions <cit.>;
  temperature independent ^125Te Knight shift across T_c 
  in nuclear magnetic resonance measurements  <cit.>; 
  coexisting FM spin fluctuations and superconductivity <cit.>; 
  signatures of chiral superconductivity <cit.>; and breaking of time reversal symmetry expected for a spin-triplet superconductor <cit.>. There are also  theoretical <cit.> and experimental <cit.> efforts 
  to understand the underlying electronic structure of UTe_2. 
  
  
  In this Letter, we use INS to probe the wave vector and energy
  dependence of spin fluctuations in UTe_2. In addition to confirming 
  that UTe_2 exhibits no static magnetic order down to 0.3 K, 
  we discovered that the dominant spin fluctuations in UTe_2 are three-dimensional (3D)
  in reciprocal space, centered at the 
  incommensurate wave vector Q=(0,±(K+ 0.57),0) (K=0,1),
  and extend to energies of at least E= 2.6 meV. 
  FM spin fluctuations, if present, are much weaker than the incommensurate spin fluctuations. 
  Based on density functional theory (DFT) in the
  generalized gradient approximation (GGA) <cit.>, combined with dynamical 
  mean-field theory (DMFT) calculations <cit.>, we
  understand the dominant incommensurate 
  spin fluctuations by showing that the associated wave vector is approximately consistent 
  with the AF wave vector of the Ruderman-Kittel-Kasuya-Yosida (RKKY)
   interaction between the 5f moments.
  Therefore, in addition to a FM instability, 
  incommensurate (close to AF) spin fluctuations must also be considered to unveil the
  magnetic and superconducting properties of UTe_2.  
  
  
  
   
  
  Our INS experiments were carried out at the Cold Neutron Chopper Spectrometer (CNCS) at Oak Ridge National Laboratory. 
  The momentum transfer Q in 3D reciprocal 
  space is defined as
  Q = H a^* + K b^* + L c^*, where H, K and L are Miller indices and a^* = â2π/a, b^* = b̂2π/b, c^* = ĉ2π/c with a = 4.16 Å, b = 6.12 Å and c = 13.95 Å of the orthorhombic lattice <cit.>. 
  Single crystals of UTe_2 were prepared using the chemical vapor transport method with I_2 as the transport media <cit.>.
  The crystals are naturally cleaved along the ab-plane and form small flakes of about 0.5 to 1 mm thick. The typical dimension of the crystals in the ab-plane is from 1 to 2 mm and the mass is in the range of 10 to 30 mg. We co-aligned 61 pieces (total mass 0.7-g) of single crystals on oxygen-free Cu-plates using an X-ray Laue machine to check
  the orientation of each single crystal. The sample assembly is aligned in the
  [0,K,L] scattering plane as shown in the black frame of Fig. 1(b) and mounted inside a He^3 refrigerator. 
  Most of our measurements were carried out with E_i=3.37 meV on CNCS at different temperatures.
  
  
  
  
  Figure 1(c) shows a map of reciprocal space in the [0,K,L] scattering plane 
  at 0.3 K and elastic position.
  We see nuclear Bragg peaks at 
  the expected (0,± 1,± 1), (0,± 2,0), and (0,±2,±2) positions.
  The spread of the Bragg peaks along the L direction indicates 
  a broad sample mosaic of ∼15 degrees. 
  To search for possible static FM or AF magnetic order, we show in Fig. 1(d) the temperature
  difference plot between 0.3 K and 2 K, and find no evidence of intensity gain anywhere 
  within the probed reciprocal space. We therefore conclude that UTe_2 does not exhibit
  static FM or AF order down to 0.3 K, consistent with earlier work <cit.>. 
  
  
  
  
  Despite the absence of long range magnetic order, INS experiments on UTe_2
  reveal clear evidence for excitations 
  at finite energy transfers. Figures 2(a-d) show 2D images of  
  constant-energy cuts in the [0,K,L] plane at different 
  energies below T_c (T=0.3 K).
  At E=0.4± 0.1 meV, we see clear excitations at incommensurate 
  wave vectors Q_ IC=(0,± 0.57,0) and a possible signal at FM wave vectors 
   Q_ FM=(0,± 1,± 1). On increasing energies to 
  E=1± 0.1, 1.3± 0.1, and 1.9± 0.1 meV, we see excitations at 
  Q_ IC=(0,±(K+0.57),0) with K=0,1 and no scattering 
  at Q_ FM [Figs. 2(b,c,d)].
  
  Figure 2(e) shows the energy dependence of excitations along the
  [0,K,0] direction, which reveals two dispersionless excitations
  at Q_ IC=(0,K+0.57,0) with K=0,1 and a dispersive excitation stemming
  from the (0,2,0) nuclear Bragg peak. The dispersionless excitations at
  incommensurate wave vectors Q_ IC starting from E=0.2 meV must be spin fluctuations since low-energy acoustic phonons must be dispersive and 
  originate from nuclear Bragg peak positions. To test if the dispersive excitation from
  (0,2,0) is indeed an acoustic phonon mode, we plot in Fig. 2(f) the 
  2D image of excitations
  along the [0,K,2] direction. While the Q_ IC excitations
  are no longer present, one can see a similar dispersive mode stemming from 
  the nuclear Bragg peak (0,2,2). By comparing the scattering 
  intensity of these modes with nuclear structure factors, we conclude that 
  the dispersive mode is the longitudinal acoustic phonon mode with a sound velocity of
  ∼1000 m/s comparable to the sound velocity of UTe
  measured by Brillouin light scattering <cit.>.
  Figures 2(g) and 2(h) show constant-energy cuts along the [0,K,0] direction
  at different energies marked in Fig. 2(e).
  At all energies probed, we see dispersionless incommensurate 
  spin excitations centered around Q_ IC=(0,0.57± 0.02,0)
  with the dynamic spin correlation length of ∼12 Å. This is close 
  to the commensurate AF wave vector of (0,0.5,0), thus indicating 
  that spin fluctuations in UTe_2 are predominately AF in nature.
  
   
  To see how incommensurate spin fluctuations
  around Q_ IC change across T_c and determine if there are strong FM spin fluctuations, we carried out  
  energy scans at wave vectors Q_1-Q_6 as marked in Fig. 2(a)
  at temperatures T=0.3, 2, and 12 K.
  Figures 3(a) and 3(b) summarize the key results at 
  the incommensurate wave vectors Q_1=(0,0.57,0) 
  and Q_2=(0,1.57,0), respectively. To accurately determine 
  the nuclear incoherent scattering backgrounds from the UTe_2 sample 
  and the Cu sample holder, we chose 
  Q_3=(0,1.1,2) [Fig. 3(c)] and Q_4=(0,2,1) [Fig. 3(d)] since 
  these positions are near the incommensurate and FM positions, respectively, 
  but are sufficiently away from the nuclear Bragg peak positions. For possible FM spin fluctuations, we consider nuclear Bragg
  peak positions Q_5=(0,1,1) [Fig. 3(e)] and 
  Q_6=(0,2,0) [Fig. 3(f)]. 
  
  
  
  
  
  
  Figures 3(c) and 3(d) show the nuclear incoherent scattering background
  at Q_3 and Q_4, respectively. As expected, the scattering is weakly
  wave vector and temperature dependent between 0.3 K and 12 K. 
  The green dashed lines in Figs. 3(a) and 3(b) are measured 
  incoherent background scattering. The incommensurate spin fluctuations are clearly above the background scattering and follow the Bose population factor on warming from 0.3 K to 12 K. Figures 3(g) and 3(h) show the temperature
  dependence of the imaginary part of the dynamic 
  susceptibility χ^''(E) at Q_1 and Q_2, obtained
  by subtracting the incoherent scattering backgrounds and correcting for the Bose population
  factor <cit.>.  χ^''(E) at both incommensurate 
  wave vectors increase with increasing energy, but show no dramatic  
  temperature dependence on warming from 0.3 K to 2 K across T_c, 
  and to 12 K.  This is reminiscent of the
   temperature dependent χ^''(E) in Sr_2RuO_4 <cit.>, but clearly different from spin-singlet unconventional
  heavy Fermion superconductors such as CeCoIn_5 <cit.>, CeCu_2Si_2 <cit.>, and etc., where there is a strong enhancement of χ^''(E), termed neutron spin resonance <cit.>, in the pink marked energy region below T_c.  
  
  Figures 3(e) and 3(f) summarize our attempt to extract FM spin fluctuations
  in UTe_2, where the green dashed lines are incoherent 
  scattering backgrounds measured at Q_3 and Q_4, respectively.
  Compared with Figs. 3(a) and 3(b) at Q_ IC, we see that 
  FM spin fluctuations, if present, are much smaller
  in magnitude and essentially vanish above ∼0.7 meV within the errors of
  our measurements. Although temperature dependence of the scattering suggests
  the presence of FM spin fluctuations, they do not dominate the spin fluctuations
  spectra and neutron polarization analysis <cit.> may 
  be necessary to conclusively identify FM spin fluctuations in UTe_2. 
  
  To understand these results, we have also performed the 
  electronic structure calculations of UTe_2 
  using the DFT+DMFT method <cit.>. 
  In a heavy fermion metal such as UTe_2, there are two potential origins for the wave vector of incommensurate spin fluctuations. One is the RKKY interaction 
  between the 5f moments, as appearing in a Kondo lattice 
  model, which is 
  determined by the electronic structure of the spd conduction electrons. 
  For this electronic structure, U-5f electrons  are treated
  as open core states (similarly via DFT to ThTe_2 <cit.>),  the calculated Fermi surface is shown 
  in Figs. 4(a,c,e) <cit.>. Noticeably, the electron momentum transfer across
  the two purple Fermi pockets is about 0.61  b^*, close to Q_ IC observed in the INS experiments. 
  Another potential source for the wave vector of incommensurate spin fluctuations is the Fermi-surface 
  nesting of the U-5f heavy bands, which we have determined 
  using the DFT+DMFT method <cit.>. The Fermi surface of the heavy bands
  is shown in Figs. 4(b,d,f). Specifically, at k_z=2π/c,  the Fermi surface exhibits a rectangular shape,
  and the electron momentum transfer across the short edges of the rectangular shape is about 0.72  b^*, which is slightly away from the observed Q_ IC.
  Therefore, the RKKY interaction of the 5f moments
  is likely driving the incommensurate spin fluctuations of UTe_2, although 
  the nesting of the strongly renormalized f-electron bands at the Fermi energy cannot
  be ruled out.
  
  In summary, we have discovered that the dominant spin fluctuations in UTe_2 are incommensurate near AF wave vector and  
  extend to at least 2.6 meV. These results are consistent with DFT+DMFT calculations,
  indicating that incommensurate spin fluctuations in UTe_2 may arise 
  from the Q-dependence of the RKKY interaction between the U-5f moments.
   We expect these incommensurate spin fluctuations to play an important role 
  in the development of the unconventional superconductivity.
  
  The INS work at Rice is supported by the U.S. DOE, BES 
  under grant no. DE-SC0012311 (P.D.). Part of
  the material characterization efforts at Rice is supported by the Robert A. Welch
  Foundation Grant Nos. C-1839 (P.D.). Research at UC San Diego was supported by the
  U.S. DOE, BES under grant no. DEFG02-04-ER46105 (single crystal growth) and U.S. NSF under Grant No. DMR-1810310 (characterization of
  physical properties). Work at Los Alamos was carried out under the auspices of the U.S. DOE National Nuclear Security Administration under Contract No. 89233218CNA000001, and was supported by the LANL LDRD Program.
  The theory work at Rice has been supported by the U.S. DOE, BES 
  under grant no. DE-SC0018197 and the Robert A. Welch Foundation Grant No. C-1411.
  A portion of this research
  used resources at the Spallation Neutron Source, a DOE Office of Science User
  Facility operated by ORNL. 
  
  
  
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