Strain control of a bandwidth-driven spin reorientation in Ca 3Ru2O7 C. D. Dashwood1A. H. Walker1M. P. Kwasigroch2 3L. S. I. Veiga1 4Q. Faure1 5 J. G. Vale1D. G. Porter4P. Manuel6D. D. Khalyavin6F. Orlandi6C. V. Colin7

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Strain control of a bandwidth-driven spin reorientation in Ca3Ru2O7

C. D. Dashwood,1, ∗A. H. Walker,1, †M. P. Kwasigroch,2, 3 L. S. I. Veiga,1, 4 Q. Faure,1, 5

J. G. Vale,1D. G. Porter,4P. Manuel,6D. D. Khalyavin,6F. Orlandi,6C. V. Colin,7

O. Fabelo,8F. Kr¨uger,1, 6 R. S. Perry,1R. D. Johnson,9A. G. Green,1and D. F. McMorrow1

1London Centre for Nanotechnology and Department of Physics and Astronomy,

University College London, London, WC1E 6BT, United Kingdom

2Department of Mathematics, University College London, London, WC1H 0AY, United Kingdom

3Trinity College, Cambridge, CB2 1TQ, United Kingdom

4Diamond Light Source, Harwell Science and Innovation Campus,

Didcot, Oxfordshire, OX11 0DE, United Kingdom

5Laboratoire Le´on Brillouin, CEA, CNRS, Universit´e Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France

6ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory,

Didcot, Oxfordshire, OX11 0QX, United Kingdom

7Universit´e Grenoble Alpes, CNRS, Institut N´eel, 38000 Grenoble, France

8Institut Laue-Langevin, 71 Avenue des Martyrs, CS 20156, 38042 Grenoble, France

9Department of Physics and Astronomy, University College London, London, WC1E 6BT, United Kingdom

The layered-ruthenate family of materials possess an intricate interplay of structural, electronic

and magnetic degrees of freedom that yields a plethora of delicately balanced ground states. This is

exempliﬁed by Ca3Ru2O7, which hosts a coupled transition in which the lattice parameters jump, the

Fermi surface partially gaps and the spins undergo a 90◦in-plane reorientation. Here, we show how

the transition is driven by a lattice strain that tunes the electronic bandwidth. We apply uniaxial

stress to single crystals of Ca3Ru2O7, using neutron and resonant x-ray scattering to simultaneously

probe the structural and magnetic responses. These measurements demonstrate that the transition

can be driven by externally induced strain, stimulating the development of a theoretical model in

which an internal strain is generated self-consistently to lower the electronic energy. We understand

the strain to act by modifying tilts and rotations of the RuO6octahedra, which directly inﬂuences

the nearest-neighbour hopping. Our results oﬀer a blueprint for uncovering the driving force behind

coupled phase transitions, as well as a route to controlling them.

INTRODUCTION

The coupling between structural and electronic degrees

of freedom in quantum materials generates a variety of

ground states and drives transitions between them. Text-

book examples include the Peierls transition in 1D ma-

terials, in which a periodic lattice deformation leads to a

metal-insulator transition and the formation of a charge-

density wave [1,2]. Similarly, the cooperative Jahn-Teller

eﬀect describes the spontaneous distortion of a crystalline

lattice to lower the electronic degeneracy and give rise

to orbital ordering [3]. Recently, there has been consid-

erable interest in twisted bilayer materials, which host

a spectrum of electronic phases—from Mott insulators

[4] to unconventional superconductors [5]—as the band-

width is tuned by the twist angle. The key role of the

lattice in many quantum materials oﬀers a powerful set

of control parameters with which to tune their phases,

but also presents a considerable challenge in developing

a comprehensive understanding of the interactions that

give rise to these phases.

In this context, the application of stress has arisen

as a powerful method to tune the electronic properties

∗cameron.dashwood.17@ucl.ac.uk; These authors contributed

equally to this work.

†a.walker.17@ucl.ac.uk; These authors contributed equally to this

work.

of quantum materials, including superconducting [6,7],

charge/spin-density wave [8–12], nematic [13–16] and

topological [17,18] phases. Here, we use uniaxial stress

to drive a coupled spin reorientation and Fermi surface

reconstruction in Ca3Ru2O7, uncovering the central role

of the lattice and facilitating a microscopic understand-

ing of the transition.

Ca3Ru2O7is a bilayer member of the Ruddlesden-

Popper ruthenates, An+1RunO3n+1, across which vary-

ing structural distortions lead to a diversity of ground

states. In the monolayer compounds these range from

superconductivity in Sr2RuO4[6,20] to a Mott insulat-

ing state in Ca2RuO4[21,22], while bilayer Sr3Ru2O7

displays an electronic nematic phase with spin-density

wave order [23–26]. Ca3Ru2O7crystallises in the Bb21m

space group (a≈5.3˚

A, b≈5.5˚

A, c≈19.5˚

A), in

which a combination of octahedral tilts around b(X−

a−a−c0in Glazer notation) and rotations around c(X+

a0a0c+) combine to unlock polar lattice displacements

[27]. It undergoes a coupled structural, electronic and

magnetic transition, where the lack of inversion symme-

try causes a magnetic cycloid to form and mediate a

spin-reorientation transition (SRT). Below TN≈60 K,

the spins align along a, coupled ferromagnetically within

the bilayers and antiferromangetically between them, in

the AFMaphase (see Fig. 1b) [19]. On cooling through

49 K, an incommensurate cycloid (ICC) develops with

q= (δ, 0,1) (δ≈0.023) and the spins rotating in the

arXiv:2210.12555v2 [cond-mat.str-el] 26 Sep 2023

Fig. 1.Neutron scattering under stress. a Schematic of the strain cell used in the neutron scattering experiment. The

bridges of the cell and sample plates are shown in grey, and the sample is shown with a blue-to-yellow colourmap to indicate

the strain gradient induced along its length. Neutrons are scattered in transmission through the central strained region of the

sample. bMagnetic structure of Ca3Ru2O7in the AFMaphase [19]. cIntegrated intensity of the commensurate (0,0,1) and

incommensurate (±δ, 0,1) peaks as a function of applied strain, ∆L/L. The background signal comes from scattering from

the strain cell, and errors are standard deviations. The insets show detector images at zero and maximum compressive/tensile

applied strain. dMagnetic structure of Ca3Ru2O7in the AFMbphase [19].

a–bplane [28,29]. The envelope of the cycloid evolves

from elongated along a, to circular, to elongated along b.

The SRT concludes at 46 K with the cycloid collapsing

into the collinear AFMbphase, where the spins are glob-

ally rotated by 90◦from AFMa(Fig. 1d). The SRT is

accompanied by a rapid change in the lattice parameters

[30] and a partial gapping of the Fermi surface [31–34].

Although there have been various proposals for the

mechanism behind the SRT, including recent work that

attributes it to the energy gain associated with a Rashba-

based hybridisation of bands at the Fermi level [34], these

proposals have generally neglected the structural degrees

of freedom. We use piezoelectric cells to apply continu-

ously tuneable uniaxial stresses to Ca3Ru2O7single crys-

tals. A combination of neutron and resonant x-ray scat-

tering enables us to directly probe the magnetic structure

and oﬀer insight into the response of the lattice to ap-

plied stress. Our measurements reveal that the SRT can

be fully driven by strain at ﬁxed temperature, and allow

the construction of temperature-strain phase diagrams

for orthogonal in-plane stresses. This motivates the de-

velopment of a minimal theoretical model in which strain

tunes the electronic hopping. As well as reproducing our

strain data, a self-consistent solution of the model (with

realistic values for the electronic parameters) reveals that

the transition with temperature is driven by an internal

strain that arises to lower the electronic energy. We in-

terpret this strain as acting via the tilts and rotations

of the RuO6octahedra, as corroborated by temperature-

dependent diﬀraction measurements. Our results there-

fore oﬀer a promising mechanism to drive electronic and

magnetic phase transitions in correlated perovskite ma-

terials.

RESULTS

Driving the spin reorientation with strain

As a well-established bulk probe of magnetic order,

neutron scattering is a natural choice for our experi-

ments. The large sample size generally needed is, how-

ever, at odds with the inverse scaling of achievable strain

on sample length – an issue that is compounded by the

increased background scattering from the strain cell. To

overcome these issues, we used the latest generation of

piezoelectric strain cells from Razorbill Instruments [35],

which can apply large strains to samples measured in

a transmission scattering geometry with minimal back-

ground from the cell. Combined with the high count-rate

of the WISH instrument at the ISIS Neutron and Muon

Source [36], this enabled us to achieve suﬃcient signal at

strains of up to 0.5%. A schematic of the strain cell is

shown in Fig. 1a, with a Ca3Ru2O7sample spanning a

distance Lbetween the sample plates. Applying a volt-

age across the piezoelectric stacks changes the distance

between the plates by ∆L, producing an applied strain

∆L/L along the a-axis of the sample. Further experi-

mental details can be found in the Methods.

Figure 1c shows the result of a strain sweep at a

ﬁxed temperature in the centre of the ICC phase. We

tracked the intensities of the commensurate (0,0,1) mag-

netic peak, which arises from both the AFMaand AFMb

structures, and satellite (±δ, 0,1) peaks that arise from

the mediating cycloidal phase. At ∆L/L = 0, we see

intense satellite peaks and a small remnant commensu-

rate peak due to temperature and strain gradients in

the sample. Under both tensile (∆L/L > 0) and com-

pressive (∆L/L < 0) strain the satellite peaks are sup-

pressed and the commensurate peak is simultaneously

enhanced. The commensurate intensity is lower under

compression, consistent with entering the AFMaphase,

while the higher commensurate intensity under tension

is consistent with the AFMbphase [28]. This assignment

is conﬁrmed by strain dependences at temperatures just

above and below the SRT (see the Supplementary Infor-

mation). The slight change in incommensurate intensity

at low strains is consistent with the variation of moment

size through the ICC phase. Our neutron data therefore

provide strong evidence that we can continuously drive

the cycloid-mediated SRT with strain at ﬁxed tempera-

ture.

Despite this, our neutron setup has a number of draw-

backs. The transitions between the magnetic phases can

be seen to occur over a ﬁnite width of around 0.3% due

to the neutron beam illuminating a large region of the

sample across which there is a signiﬁcant strain gradient

(see Fig. 1a). Further, despite the large displacements

that can be generated by the cell, the maximum strains

of |∆L/L| ≈ 0.5% are not large enough to fully enter

the collinear phases. Finally, and most signiﬁcantly, the

scattering geometry necessitated by the strain cell blocks

access to any nuclear Bragg peaks with a ﬁnite compo-

nent along the stress direction. This precludes a determi-

nation of the lattice parameters, and therefore the true

strain generated in the sample, forcing us to rely on the

displacement of the sample plates as a measure of the ap-

plied strain. As we will see in the following section, the

fact that the epoxy is signiﬁcantly softer than the sam-

ple leads to only a fraction of the applied strain being

transmitted to the sample.

Response of the lattice to applied stress

A full characterisation of the response of the lattice

to applied stress is crucial to understanding the mag-

netic response, as highlighted by previous contradictory

reports of the electronic and magnetic changes under uni-

axial pressure in Ca3Ru2O7[37,38]. To achieve this, we

turned to synchrotron x-ray scattering at beamline I16

of the Diamond Light Source. Size limitations of the

closed-cycle cryostat required use of a smaller strain cell

than the neutron measurements. The lower stresses are

mitigated, however, by the high ﬂux and small focus of

the x-ray beam, allowing the use of smaller samples to

achieve applied strains over 2%. To maximise beam ac-

cess we mounted the samples on top of raised sample

plates, as shown in Fig. 2a (further details can be found

in the Methods). Two samples were measured, one with

stress applied along aand the other along b.

We tracked the positions of multiple structural Bragg

Fig. 2.X-ray scattering under stress. a Schematic of

the strain cell used for x-ray scattering. The principles of

operation are the same as the neutron setup, except that the

sample is mounted on top of raised sample plates to give a

large sphere of access for the incident and scattered beams.

b2θscans of the structural (1,0,7) Bragg peak at various

applied strains. cTrue strain, ∆b/b, as a function of applied

strain, ∆L/L, at a range of temperatures through the SRT.

dTemperature dependences of the Poisson ratios determined

from the relative strains along orthogonal axes. All errors are

standard deviations.

peaks while applying compressive stresses. Figure 2b

shows representative 2θscans of the (1,0,7) peak for

stress applied along b. The shifting of the peak arises

from the changing lattice parameters, from which we

can determine the true strain, ∆b/b. This is shown in

Fig. 2c, where we see a linear, temperature-insensitive

dependence on ∆L/L, with around 40% of the applied

strain transmitted to the sample. We were unable to ap-

ply tensile strains due to the asymmetric sample mount-

ing, which also induced a slight bending of the sample

under compression (see the Supplementary Information).

The bending did not induce signiﬁcant strain gradients

through the probed region, however, as evidenced by the

minimal broadening of the peaks in Fig. 2b.

Our x-ray measurements also enable us to quantify

the strains along the other crystalline axes, εx= ∆x/x,

that arise from the uniaxial stresses applied by the cell,

and thereby calculate the Poisson ratios νxy =−εy/εx.

The results are plotted as a function of temperature in

Fig. 2d. The anisotropy of the in-plane (νab ≈νba ≈0.5)

and out-of-plane (νac ≈νbc ≈0.2) Poisson ratios is as

expected for a layered material like Ca3Ru2O7.

We can also use x-ray scattering to probe the mag-

netic phases by exploiting the resonant scattering en-

hancement on tuning the incident x-ray energy to the Ru

L2edge (2.967 keV). Alongside the positions of the struc-

tural peaks, we monitored the intensities of the (δ, 0,5)

and (0,0,5) magnetic peaks at each strain value, the lat-

-0.15 -0.1 -0.05 0

a/a (%)

Temperature (K)

AFMa

ICC

AFMb

a b c

-0.6 -0.4 -0.2 0

b/b (%)

AFMa

ICC

AFMb

-0.04 -0.02 0

app (%)

0.92

0.94

0.96

0.98

TSapp) / TS(0)

AFMa

AFMb

Fig. 3.Temperature-strain phase diagrams. a, b Experimental phase diagrams for stress applied along the a- and b-axes

respectively. The points with errors (standard deviations) are transition temperatures extracted from the strain measurements,

and the squares indicate the zero-strain transition temperatures from ref. [28]. The dashed lines are guides to the eye. The

insets indicate the tensile strains induced along the orthogonal directions according to the Poisson ratios. cTheoretical phase

diagram under applied strain εapp, calculated by self-consistently minimising the free energy in Eq. (2) using the parameter

values U= 8, λ= 0.5, θ= 15◦,µ= 8.5, κ= 400, ν= 12, t0= 1.12 and ε0= 0.066.

ter at two orthogonal azimuthal angles to diﬀerentiate

between the AFMaand AFMbphases [28]. Repeating

these strain sweeps at a range of temperatures enabled

us to construct the phase diagrams shown in Fig. 3a, b

for stress applied along aand brespectively (further de-

tails of how the phase diagrams were constructed can be

found in the Supplementary Information).

The phase diagrams show the transition temperatures

varying linearly with compressive strain along both aand

b. The data quality is worse for the sample with stress

applied along a, most likely due to the poorer crystalline

quality of the sample causing a heterogeneous distribu-

tion of strain (see the Supplementary Information). De-

spite this, we can still determine a rate of change of the

transition temperatures of approximately 40 K per per-

cent strain along a. The transition temperatures change

at a slower rate for the sample stressed along b, at around

7 K per percent strain along b. Although we could not

access the tensile sides of the phase diagrams in our x-ray

experiments, our neutron results suggest that the trends

should continue unchanged. It is interesting to note that

the phase boundaries move in the same direction in both

phase diagrams, despite the opposite in-plane deforma-

tions of the lattice (as depicted in the insets of Fig. 3a,

b).

A strain-coupled electronic model

To understand the role of strain in the SRT, we de-

veloped a phenomenological model in which strain tunes

the electronic hopping between nearest-neighbour Ru or-

bitals. In the spirit of developing a minimal model

that captures the physics under study, we included only

the Ru dxz and dyz orbitals in a perovskite monolayer.

Such a two-band model is consistent with ARPES stud-

ies that show that the dxy orbital does not contribute

to the Fermi surface [33]. We consider a Hamiltonian

H=ˆ

Ht+ˆ

HU+ˆ

Hλ−µˆ

N.ˆ

Htis a tight-binding Hamil-

tonian ﬁtted to ARPES data [33] (see the Methods).

HU=UPiτα ˆniτα↑ˆniτα↓is the on-site intra-orbital Hub-

bard interaction, where the lattice site is labelled i, the

orbital is labelled α∈ {xz, yz}, and τ∈ {A, B}is

a sublattice index that is introduced to allow for the

staggered octahedral tilting. The interaction is treated

in the Hartree-Fock mean-ﬁeld approximation, and all

other local electron-electron interactions are neglected for

simplicity. The Hamiltonian is then a function of site-

averaged charge, ρτα, and magnetisation, Mτα, ﬁelds for

each orbital and sublattice. ˆ

Hλis the spin-orbit interac-

tion [39],

Hλ=−iλ

i,τ

c†

iτ,xzστ

zciτ,yz + h.c.,(1)

where c†

iτα = (c†

iτα↑, c†

iτα↓). This is an on-site term

that couples the two orbitals. Octahedral tilting aﬀects

the spin-orbit term by rotating the spin quantisation

axes on the sublattice sites by ±θabout the b-axis, as

σA(B)

z= cos(θ)σz±sin(θ)(σx−σy)/√2. The octahedral

rotation (as well as the polar distortion that leads to

Rashba-like spin-orbit eﬀects) does not enter the Hamil-

tonian explicitly, although it does inﬂuence the hopping

as described in the next section.

Strain enters the model via a phenomenological ﬁeld, ε,

that couples to the nearest-neighbour hopping, t, of the

tight-binding model (see the Methods for further details).

We take a minimal coupling of t=t0+νε, where t0and

νare parameters, which is valid when the strain magni-

tude is small. The coupling of strain to further-neighbour

hopping is neglected. We also introduce a strain cost set

by a parameter, κ, in the free energy

F(M, ρ, ε, εapp) = −TX

n,k

ln 1 + e−(ϵnk−µ)/T 

+UX

τ,α M2

τα −ρ2

τα+1

2κ(ε0+ε−εapp)2,(2)

where ϵnkare the eigenvalues of the electronic Hamil-

tonian, µis the chemical potential, and ε0and εapp are

parameters. The last term is the contribution of the non-

electronic degrees of freedom to quadratic order. Within

this parameterisation, εis the strain measured relative to

the unstrained system which we deﬁne, via our choice of

value of ε0and without loss of generality, as the system

at the N´eel temperature. This is a natural reference point

for studying the magnetic phase diagram. As well as the

internal strain ε, we include an externally applied strain

εapp that moves the minimum of the strain cost function

away from zero. The electronic state is determined at a

particular temperature and externally applied strain by

minimising the free energy with respect to the electronic

degrees of freedom, Mτ α and ρτ α, and the strain ﬁeld, ε,

in a self-consistent manner.

In the absence of strain, κ=ν= 0, the solution of this

model exhibits easy a- and easy b-axis ferromagnetism

that corresponds to the AFMaand AFMbphases in the

full structure. This results from the interplay of the on-

site interaction and spin-orbit coupling, which generate

the magnetic anisotropy, and the octahedral tilting that

breaks its symmetry in the a–bplane. The anisotropy

is strongly dependent on the nearest-neighbour hopping,

and an a→breorientation of the ferromagnetic order

is achieved by increasing tand therefore the electronic

bandwidth. Although our minimal model cannot repro-

duce the cycloid that mediates the SRT, we would expect

this to appear naturally when the full bilayer is consid-

ered, due to a uniform Dzyaloshinskii-Moriya interaction

that becomes dominant close to the reorientation transi-

tion [28].

With a ﬁnite κand ν, we ﬁnd an internal strain being

generated self-consistently even with zero applied strain,

εapp = 0, as shown in Fig. 4. We see εincrease as the

temperature is reduced through the AFMaphase, low-

ering the energy of the electronic system at the expense

of an elastic energy cost. When the hopping reaches a

threshold value the SRT is triggered. This transition is

ﬁrst order and results in a small positive jump in the

strain (see zoomed region in Fig. 4) as well as a jump in

the magnetisation. We also capture some of the changes

in the Fermi surface that are seen in ARPES studies.

We ﬁnd a continuous Fermi surface reconstruction (insets

in Fig. 4) that is not directly driven by the transition,

but instead by the increase in magnetisation (and Stoner

gap) as temperature is reduced. We emphasise that in

Temperature

-1

-0.5

0.5

1.5

1.05

1.1

1.15

1.2

1.25

1.3

1.35

AFMa

AFMb

Fig. 4.Self-consistent internal strain. Strain ﬁeld, ε,

as a function of temperature, with the magnetic phase tran-

sitions indicated by vertical dashed lines. The corresponding

change in the eﬀective nearest-neighbour hopping parameter,

t, is measured on the right-hand axis. The zoomed region

shows a small discontinuity in εacross the SRT of magnitude

|∆ε| ∼ 0.005%. The insets show Fermi surfaces calculated in

the two magnetic phases. The parameter values are the same

as in Fig. 3c. The calculated value of the hopping parameter

at the transition corresponds to U/t ∼6.6, which is consis-

tent with values of t[33] and U[40] previously ﬁtted to the

observed electronic structure of Ca3Ru2O7.

this self-consistent calculation, the SRT and Fermi sur-

face reconstruction are achieved as a function of temper-

ature without varying the band ﬁlling. The model does

not capture the partial gapping of the Fermi surface that

is also seen by ARPES [31,33].

To make contact with our strain experiments, we also

solve the model with a negative applied strain, εapp <0.

This reduces the self-consistent strain, ε, and thus the

nearest-neighbour hopping, resulting in a reduction in

the critical temperature. As shown in Fig. 3c, the linear

suppression of the transition temperature matches the

experimental phase diagrams for compressive stress ap-

plied along aand bin Fig. 3a, b. As well as reproducing

the main sequence of magnetic phases with temperature,

our minimal model is therefore able to capture the qual-

itative features of our strain measurements.

Microscopic understanding of the transition

Our theoretical model is able to reproduce the phe-

nomenology of the SRT by introducing a strain ﬁeld

that couples to the nearest-neighbour hopping. For a

full understanding of the mechanism behind the tran-

sition, however, we need to connect this strain ﬁeld to

microscopic distortions of the Ca3Ru2O7lattice. Given

their strong inﬂuence on the ground states across the

Ruddlesden-Popper series of ruthenates, the octahedral

tilt and rotation degrees of freedom (depicted in Fig. 5a,

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摘要：

Straincontrolofabandwidth-drivenspinreorientationinCa3Ru2O7C.D.Dashwood,1,∗A.H.Walker,1,†M.P.Kwasigroch,2,3L.S.I.Veiga,1,4Q.Faure,1,5J.G.Vale,1D.G.Porter,4P.Manuel,6D.D.Khalyavin,6F.Orlandi,6C.V.Colin,7O.Fabelo,8F.Kr¨uger,1,6R.S.Perry,1R.D.Johnson,9A.G.Green,1andD.F.McMorrow11LondonCentreforNanotech...

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Strain control of a bandwidth-driven spin reorientation in Ca 3Ru2O7 C. D. Dashwood1A. H. Walker1M. P. Kwasigroch2 3L. S. I. Veiga1 4Q. Faure1 5 J. G. Vale1D. G. Porter4P. Manuel6D. D. Khalyavin6F. Orlandi6C. V. Colin7.pdf

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Strain control of a bandwidth-driven spin reorientation in Ca 3Ru2O7 C. D. Dashwood1A. H. Walker1M. P. Kwasigroch2 3L. S. I. Veiga1 4Q. Faure1 5 J. G. Vale1D. G. Porter4P. Manuel6D. D. Khalyavin6F. Orlandi6C. V. Colin7

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