Line-Graph Approach to Spiral Spin Liquids Shang Gao1 2yGanesh Pokharel2 3Andrew F. May2Joseph A. M. Paddison2Chris Pasco2Yaohua Liu1Keith M. Taddei1Stuart Calder1David G. Mandrus2 3 4Matthew B. Stone1and Andrew D. Christianson2

2025-05-03 0 0 3.06MB 20 页 10玖币

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Line-Graph Approach to Spiral Spin Liquids∗

Shang Gao,1, 2, †Ganesh Pokharel,2, 3 Andrew F. May,2Joseph A. M. Paddison,2Chris Pasco,2Yaohua Liu,1Keith

M. Taddei,1Stuart Calder,1David G. Mandrus,2, 3, 4 Matthew B. Stone,1and Andrew D. Christianson2

1Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

2Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

3Department of Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA

4Department of Material Science & Engineering,

University of Tennessee, Knoxville, Tennessee 37996, USA

(Dated: October 24, 2022)

Competition among exchange interactions is able to induce novel spin correlations on a bipar-

tite lattice without geometrical frustration. A prototype example is the spiral spin liquid, which

is a correlated paramagnetic state characterized by sub-dimensional degenerate propagation vec-

tors. Here, using spectral graph theory, we show that spiral spin liquids on a bipartite lattice can

be approximated by a further-neighbor model on the corresponding line-graph lattice that is non-

bipartite, thus broadening the space of candidate materials that may support the spiral spin liquid

phases. As illustrations, we examine neutron scattering experiments performed on two spinel com-

pounds, ZnCr2Se4and CuInCr4Se8, to demonstrate the feasibility of this new approach and expose

its possible limitations in experimental realizations.

Introduction.— A spiral spin liquid (SSL) is an ex-

otic correlated paramagnetic state of sub-dimensional

degeneracy, meaning that the propagation vectors qof

the ground states form a continuous manifold, or spiral

surface, in a dimension that is reduced from the orig-

inal system [1–14]. Similar to geometrically frustrated

magnets [15, 16], a SSL may host topological spin tex-

tures [17–19] and quantum spin liquid states [3–5, 20–22].

What diﬀerentiates a SSL from a conventional frustrated

magnet is the sub-dimensional degeneracy, which induces

highly distinctive dynamics since the spins are conﬁned

to ﬂuctuate collectively as nonlocal spirals [23]. Recent

calculations on a square lattice reveal that the low-energy

ﬂuctuations in a SSL may behave as topological vortices

in momentum space [23], leading to an eﬀective tensor

gauge theory with unconventional fracton quadrupole ex-

citations that are deeply connected to theories of quan-

tum information, elasticity, and gravity [24–29].

To date, bipartite lattices have been the primary av-

enue through which SSLs are studied. This is because

the ground state degeneracy on a bipartite lattice can

be exact, so that all spin spirals with qover the spi-

ral surface have exactly the same energy [5]. Although

this degeneracy stabilizes the SSL down to very low tem-

∗This manuscript has been authored by UT-Battelle, LLC un-

der Contract No. DE-AC05-00OR22725 with the U.S. Depart-

ment of Energy. The United States Government retains and the

publisher, by accepting the article for publication, acknowledges

that the United States Government retains a non-exclusive, paid-

up, irrevocable, world-wide license to publish or reproduce the

published form of this manuscript, or allow others to do so, for

United States Government purposes. The Department of En-

ergy will provide public access to these results of federally spon-

sored research in accordance with the DOE Public Access Plan

(http://energy.gov/downloads/doe-public-access-plan).

†sgao.physics@gmail.com

peratures [1, 3], it also imposes a strong constraint on

real materials because most of the known bipartite-lattice

compounds are dominated by the nearest-neighbor inter-

actions J1[30–44]. Even for the established model com-

pounds where the second-neighbor interactions J2are rel-

atively strong [17, 45, 46], the degeneracy over the spiral

surface is only approximate due to the existence of fur-

ther perturbations [14]. This degeneracy lifting results

in an approximate SSL state at elevated temperatures

where thermal ﬂuctuations overcome the slight energy

diﬀerence among the spirals.

Inspired by recent density functional theory (DFT)

calculations for the breathing pyrochlore lattice com-

pounds [47], here we seek the realization of an approx-

imate SSL, i.e. a SSL with an approximate degener-

acy, beyond the bipartite lattices. According to the

Luttinger-Tisza theory [48, 49], the degeneracy of a SSL

model is encoded in the minimum manifold of the in-

teraction matrix. Using graph theory [50, 51], we show

that the J1-J2model on a bipartite lattice shares the

same minimum manifold with a J1-J3model on the cor-

responding line-graph lattice, where J3denotes the third-

neighbor interaction. Thus an approximate SSL state is

achieved in the latter case when J3is suﬃciently strong,

which greatly expands the range of materials that may

support a SSL state. This line-graph approach to SSL

is vetted through neutron scattering experiments per-

formed on two Cr-based chalcogenide spinels ZnCr2Se4

and CuInCr4Se8.

Line-graph approach.— Our starting point is a Heisen-

berg model on a l-regular lattice, where lcounts the num-

ber of the nearest-neighbor (NN) sites. In the presence of

a uniform NN exchange interaction J1, the coupled spins

form a undirected graph G= (V, D), with Vdenoting the

set of vertices (i.e. the spin sites) and Ddenoting the set

of edges (i.e. the NN bonds). Two vertices iand jare

arXiv:2210.11781v1 [cond-mat.str-el] 21 Oct 2022

called adjacent if the graph contains an edge e={i, j},

and the adjacency matrix is deﬁned as A(G)ij = 1 for

{i, j} ∈ Dand 0 otherwise. Following this deﬁnition, the

spin Hamiltonian, H=J1PhijiSi·Sjwith hijidenoting

the NN bonds, can be expressed through the adjacency

matrix as H=1

2J1A(G)ij Si·Sj. Therefore, according

to the Luttinger-Tisza theory [1, 5], the classical ground

state of Hcan be determined from the eigensolution of

A(G), of which the eigenvalues are deﬁned as the spec-

trum of the graph, denoted as σ(G).

Algebraic graph theory deﬁnes that the related graphs

should have related spectra [50, 51]. Of interest here is

the line graph LG[52, 53], whose vertices correspond to

the edges of the root graph Gand are adjacent if the

original edges share a vertex: LG= (D, {{e, e0} | e∪e06=

∅, e 6=e0}). For a regular bipartite graph G, a conve-

nient way to deﬁne its line graph is to select the vertices

at the midpoints of the original edges. As illustrated in

Fig. 1, for the honeycomb (l= 3) and diamond (l= 4)

lattices that are the prototype hosts of the SSL, their line

graphs form the kagome and pyrochlore lattices, respec-

tively. According to graph theory [50, 51], the spectra of

Gand LGare related by

σ(LG) = (−2)m−n∪σ(G) + l−2 , (1)

where m(n) is the total number of edges (vertices) of the

root graph G. In reciprocal space, Eq. (1) indicates the

existence of ﬂat eigenbands on LGwith a degeneracy of

l−2, which has been the focus of many recent studies [51,

54–56]. More importantly, it reveals that the non-ﬂat

eigenbands of σ(LG) and σ(G) share the same dispersion

up to a constant of l−2, and their eigenvectors are related

by the incident matrix as discussed in Ref. [57].

Such a spectrum correspondence can be immediately

veriﬁed for Heisenberg models. On a regular lattice of g

sublattices and Nprimitive cells, the interaction matrix

of a J1-only model

Jαβ

1(q) = J1

i∈α,j∈β

ij∈hiji

exp [−iq·(ri−rj)] (2)

is a g×ghollow matrix with zero diagonal elements.

The eigenbands ν(q) for J1<0 are shown as solid lines

in Fig. 1(b) for the honeycomb and kagome lattices, and

in Fig. 1(d) for the diamond and pyrochlore lattices. The

minimum of the eigenbands νmin has been subtracted for

comparison. Analytical expressions for the eigenbands

are presented in the Supplemental Material [58], which

includes additional Refs. [59–69]. Aside from the top ﬂat

bands in blue color, the two dispersive bands ν±(q)−νmin

on the kagome (pyrochlore) lattice overlap exactly with

those on the honeycomb (diamond) lattice, which is a

direct consequence of spectral graph theory.

This eigenband correspondence is maintained un-

der the addition of certain further-neighbor interac-

tions. For the J1-J2model on a bipartite lattice,

LΓX W K Γ

E/|J|

J3b

J3a

× 2

× 1

(a)

honeycomb - kagome

(b)

(d)

honeycomb - kagome

(c)

J3b

J3a

KΓM K

E/|J|

diamond - pyrochlore

FIG. 1. (a) The line graph of a bipartite honeycomb lat-

tice (large spheres linked by orange bonds) is a kagome lat-

tice (small spheres linked by gray bonds). J1is the nearest-

neighbor exchange coupling. Black dashed (red solid) arrows

indicate the second-neighbor coupling J2(third-neighbor cou-

plings J3aand J3b) over the honeycomb (kagome) lattice. (b)

Eigenbands of the interaction matrix of Heisenberg models on

the honeycomb and kagome lattices. Orange solid (dashed)

lines are the two eigenbands for the J1model (J1-J2model

with J2/J1=−0.25) on the honeycomb lattice, which overlap

with the two lower eigenbands of the J1model (J1-J3model

with J3/J1=−0.25) on the kagome lattice shown in blue

solid (dashed) lines. (c-d) Similar correspondence exists be-

tween the diamond and pyrochlore lattices. The interaction

matrices for the J1model (J1-J2model with J2/J1=−0.3)

on the diamond lattice and the J1model (J1-J3model with

J3/J1=−0.3) on the pyrochlore lattice share two same

eigenbands shown by the overlapping orange and blue solid

(dashed) lines.

J2couples the spins of the same sublattices. There-

fore, its contribution to J(q) is a diagonal matrix

J2(q) that commutes with J1(q), leading to a q-

dependent shift γG(q) of the eigenbands with γG(q) =

(J2/Ng)Pij∈hhijii exp [−iq·(ri−rj)], where hhijii are

the second-neighbor bonds [58]. Similar conclusions can

be drawn for the J1-J3model on the line-graph lattices,

as J3(including both J3aand J3b) also couples spins of

the same sublattices [58]. Since J2(G) on the root-graph

lattice and J3(LG) on the line-graph lattice share the

same exchange paths as compared in Fig. 1(a,c), we ex-

pect, in the case of J2(G) = J3(LG), the same dispersive

eigenbands up to a constant on the root- and line-graph

lattices. This correspondence is illustrated by the dashed

lines in Fig. 1(b,d).

Approximate SSL.— An approximate SSL can be re-

alized on the line-graph lattices through the eigenband

correspondence. Following the results of the J1-J2model

on the bipartite lattices [1, 5], it is clear that for the J1-

J3model on a line-graph lattice the eigenband minima of

J(q) will form a degenerate manifold in reciprocal space

at |J3/J1|>1/(2l), where lis the number of the NN sites

on the root-graph lattice. Here J1and J3should be FM

and AF, respectively, as the additional ﬂat bands on line

graphs remove the FM-AF duality of a bipartite lattice.

Since the equal moment constraint over the eigenvectors

of J(q) is not always satisﬁed for qover the minimum

manifold [58], the SSL realized through the line-graph ap-

proach is approximate and needs to be stabilized by ther-

mal ﬂuctuations. As discussed in the Supplemental Ma-

terial [58], the degeneracy breaking over the spiral surface

is relatively weak, which contrasts the strong modulation

in the previously studied half-moon patterns [57, 70–72].

This weak degeneracy breaking on the line-graph lattices

leads to a stable SSL state in a wide temperature regime

that is comparable to that on the bipartite lattices.

Before making comparisons to the experimental data,

we further generalize the SSL model by incorporating

a breathing distortion on the line-graph lattice so that

the NN interactions are modulated alternately as J1and

1[47, 73, 74]. For the breathing pyrochlore lattice shown

in Fig. 2(a), the eigenbands of J(q) can be solved as [58]:

ν1,2=J3κ(q)±q4(J1−J0

1)2+J1J0

1|η(q)|2+ (J1+J0

1),

ν3,4=J3κ(q)−(J1+J0

1) ,

where η(q) = P4

n=1 exp(−iq·dn) with dndenoting

the 4 bonding vectors around each spin site [58] and

κ(q) = |η(q)|2−4. Assuming J1< J0

1<0 and J3>0,

an approximate SSL state is realized for 1

J1J0

|J1+J0

1|< J3<

J1J0

|J1−J0

1|. The corresponding phase diagram is presented

in Fig. 2(b). Representative spiral surfaces in the approx-

imate SSL phase at |J3/J1|= 0.2 are shown in Fig. 2(c-e)

for J0

1/J1= 0.8, 0.6, and 0.5, respectively. The surfaces

are identical to those on the diamond lattice [1]. Similar

conclusions on the breathing kagome lattice are presented

in the Supplemental Material [58].

ZnCr2Se4with a regular pyrochlore lattice.— The Cr-

based chalcogenide spinels present nearly ideal model

compounds to demonstrate the proposed line-graph ap-

proach to SSLs. In these systems, J1is FM due to the

90◦superexchange path, while J2∼0 due to negligible

orbital overlap [75, 76]. As the ﬁrst example, we study

the short-range spin correlations in ZnCr2Se4, where the

Cr3+ (S= 3/2) ions form a regular pyrochlore lattice.

Single crystals of ZnCr2Se4were grown using the chem-

ical vapor transport method [58]. Figure 3 summa-

rizes the diﬀuse neutron scattering results measured on

CORELLI at the Spallation Neutron Source (SNS), Oak

Ridge National Laboratory (ORNL) [58]. With the sta-

tistical chopper, the elastic channel of our CORELLI

data has an average energy resolution of about 0.8 meV

for an incident neutron energy range of 13 to 33 meV. At

20 K, below the N´eel transition temperature TN∼22 K,

0 0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

1D degeneracy

= 0

(a)

J1/J1

|J3 /J1|

J3a

J3b

(b)

Approx. SSL

e d

ZCS

FIG. 2. (a) The breathing pyrochlore lattice is composed of

corner-sharing tetrahedra of two diﬀerent sizes, which results

in alternating J1and J0

1couplings over the gray and orange

tetrahedra, respectively. The exchange paths of the J2,J3a,

J3b, and J4interactions are also indicated. (b) Phase dia-

gram on the breathing pyrochlore lattice with ferromagnetic

J1and J0

1interactions together with antiferromagnetic J3in-

teractions assuming J3a=J3b. The shaded area indicates

the region where an approximate SSL phase can be stabilized

by thermal ﬂuctuations. Contour lines for constant |η(q)|are

shown in the approximate SSL phase. Along these lines, the

spiral surface stay the same. Location of ZnCr2Se4(ZCS)

with J1=J0

1and |J3/J1|= 0.15 is indicated by the yel-

low dot. Inset shows the degenerate manifold (blue lines)

for J3>1

J1J0

|J1−J0

1|out of the SSL regime. (c-e) Characteris-

tic spiral surfaces with quasi-degenerate energies in the ﬁrst

Brillouin zone at |J3/J1|= 0.2 and J0

1/J1= 0.8 (c), 0.6 (d),

and 0.5 (e). These points are indicated on the phase diagram

in (b) using the panel labels.

magnetic Bragg peaks indexed by q= (0,0,0.47) are ob-

served (see Fig. 3(a)). This is consistent with the helical

ground state reported previously [77–79], with the weak

ring-like scattering mainly arising from the low-energy

magnon excitations [76, 80]. At 30 K, above TN, mag-

netic Bragg peaks are replaced by broad diﬀuse scattering

with a spherical shape (see Fig. 3(b-d)), evidencing the

emergence of an approximate SSL state where gapped

excitations are replaced by quasielastic ﬂuctuations [58].

Assuming a Heisenberg model with exchange interac-

tions up to the fourth neighbors (J4), we ﬁt the diﬀuse

scattering data using the self consistent Gaussian approx-

imation (SCGA) method [58]. The calculated slices in

Fig. 3(b-d) reproduce the experimental data. The cou-

pling strengths are ﬁtted as J1=−2.86(8), J2= 0.00(1),

J3= 0.48(1), and J4=−0.057(1) meV, which is very

close to the values determined by inelastic neutron scat-

tering (INS) [76, 80] and indicates marginal changes in

the coupling strengths across the phase transition. The

weak strengths of the J2and J4interactions allow a direct

veriﬁcation of the line-graph approach that is based on

2.4

1.6 2

(h,0,2)

2.4

1.6 22.4

1.6 2

(h,0,2) (h,0,2)

(a) (hk2), 20 K (hhl), 30 K

(111), 30 K

calc.

(hk2), 30 K

(b)

(c)

(d)

0 8 12

Int. (a.u.)

2.4

1.6 2

(h,h,0)

2.4

1.6 2

(h,h,0)

2.4

1.6 2

(h,h,0)

2.4

1.6 2

(h,h,0)

2.4

1.6

(0,k,2)

2.4

1.6

(0,k,2)

2.4

1.6

(0,0,l )

0.2

0.1

-0.1

-0.2

(l,-l,2l+2)

calc.

FIG. 3. Quasistatic spin correlations in ZnCr2Se4measured

on CORELLI at (a) 20 K and (b-d) 30 K. Slices are along

the (hk2) plane in panels (a,b), (hhl) plane in panel (c), and

(111) plane in panel (d). The right sub-panel in (a) shows the

complete spiral surface for the J1-J3model on a pyrochlore

lattice with J3/J1=−0.15. The ﬂat planes correspond to

the slice directions as indicated by the colored plaquette at

the top left corner in each panel. The right sub-panels in (b-

d) are the calculated diﬀuse scattering patterns for the ﬁtted

J1-J2-J3-J4model. Solid line is the spiral surface predicted

by the J1-J3model with J3/J1=−0.15. The same linear

intensity scale is employed in all panels.

aJ1-J3model. The solid circles on top of the calculated

patterns in Fig. 3(b-d) are the contours of the spiral sur-

face predicted by the J1-J3model with J3/J1=−0.15.

The contours capture the shapes of the diﬀuse scatter-

ing patterns. As compared in the Supplemental Mate-

rial [58], the existence of the FM J4interaction slightly

reduces the radius of the spiral surface while strongly

modulates the scattering intensities.

CuInCr4Se8with a breathing pyrochlore lattice.— Until

now, we have assumed a uniform J3interaction. In real

materials, the J3exchange paths on a line-graph lattice

can be diﬀerent as indicated in Fig. 1. This may lead

to diﬀerent coupling strengths and destabilize the SSL

state. As the second example, we study the spin cor-

relations in the breathing pyrochlore lattice compound

CuInCr4Se8[81], which has been proposed as a SSL can-

didate from DFT calculations [47].

A polycrystalline sample of CuInCr4Se8was synthe-

sized through the solid state reaction method [58]. INS

experiments were performed on SEQUOIA at the SNS,

ORNL. Neutron diﬀraction experiments were performed

on HB-2A at the high ﬂux isotope reactor (HFIR),

ORNL. As is consistent with the previous report [81],

magnetic susceptibility shown in Fig. 4(a) suggests a spin

glass-like transition at Tf∼15 K with a clear frequency

dependence. This is conﬁrmed in the neutron diﬀraction

results presented in Fig. 4(b), where only broad mag-

netic features are observed down to 0.25 K. The weak

features at Q∼0.41 and 0.72 ˚

A−1can be indexed by

q= (0.48,0.48,0), while their intensities exhibit a his-

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Q (Å-1)

100

150

Int. (a.u.)

0 100 200 300

T (K)

0.2

0.4

0.6

0.8

1.2

(emu/mol-Cr Oe)

10 Hz

10 kHz

(a)

10 15 20

T(K)

0.5

'(emu/mol-Cr Oe)

0.5 1 1.5 2 2.5

Int. (a.u.)

0.5 1 1.5 2 2.5

-1 0 1

(h,0,0)

-1 0 1

(h,0,0)

-1

(0,k,0)

-1

(0,k,0)

0.5 1 1.5 2 2.5

-1 0 1

(h,0,0)

-1

(0,k,0)

-1 0 1

(h,0,0)

-1

(0,k,0)

(0.48,0.48,0) (0.48, 0.48,1)(0.3, 0.3, 0)

T = 20 K0.25 K − 25 K

(b)

T = 100 KT = 40 K T = 200 K

× 1.5 × 2.5 × 3.5

(d)(c) (e)

Q (Å-1)

Q (Å-1)Q (Å-1)

FIG. 4. (a) DC magnetic susceptibility of CuInCr4Se8mea-

sured in a 1 ×104Oe ﬁeld (circles). Red line shows the

Curie Weiss ﬁt at temperatures between 200 and 380 K.

The ﬁtted magnetic moment and Weiss temperature are

µeﬀ = 4.6µB/Cr and 97(2) K, respectively. The inset shows

the frequency dependence of the real part of the ac suscep-

tibility measured in a 10 Oe ﬁeld. Imaginary part of the ac

susceptibility is presented in the Supplemental Material [58].

(b) Diﬀerence between the neutron diﬀraction intensity at

0.25 and 25 K measured on HB-2A. The dashed line is a

guide to the eyes. Positions of the characteristic wave vec-

tors are indicated by arrows. The full width at half maxi-

mum (FHWM) of the closest nuclear reﬂection is indicated

by the horizontal bar. Inset shows the calculated neutron

diﬀuse scattering pattern in the (h, k, 0) plane at T= 20 K

for J3a= 0.07 meV (see text). (c-e) Circles show the Q-

dependence of the equal-time spin correlations at T= 40 (c),

100 (d), and 200 K (e). Results of the SCGA ﬁts using the

J1-J0

1-J3a-J3bmodel are shown by the red solid lines. Insets

are the calculated neutron diﬀuse scattering patterns in the

(h, k, 0) plane for J3a= 0.07 meV at the corresponding tem-

peratures (see text).

tory dependence as expected for a spin glass state.

Figures 4(c-e) present the equal-time spin correla-

tions obtained by integrating the INS spectra from [−20,

20] meV [58]. Assuming a J1-J0

1-J3a-J3bHeisenberg

spin model, we ﬁt the diﬀuse scattering data by the

SCGA method [58, 82]. The ﬁtted results are shown in

Fig. 4(c-d) as solid lines. The ﬁtted coupling strengths

are J1=−1.6(2), J0

1=−5.4(3), J3a= 0.1(1), and

J3b= 0.6(1) meV, where the diﬀerent strengths for J3a

and J3bare necessary for a satisfactory ﬁt. As shown

in the inset of Fig. 4(b-e), the spin correlations in the

(hk0) plane does not follow a circular shape, implying

the absence of a SSL state due to the unequal J3aand

J3binteractions.

The ﬁtted parameter set provides an explanation for

the glassy ground state in CuInCr4Se8.J3a, being the

weakest interaction in the J1-J0

1-J3a-J3bmodel, is how-

ever the most crucial parameter in determining the exact

length of the LRO q= (q, q, 0). Within the standard de-

viation of J3a,qvaries from 0 at J3a= 0 up to ∼0.4 at

J3a= 0.2 meV. Therefore, the broad diﬀuse scattering at

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Line-GraphApproachtoSpiralSpinLiquidsShangGao,1,2,yGaneshPokharel,2,3AndrewF.May,2JosephA.M.Paddison,2ChrisPasco,2YaohuaLiu,1KeithM.Taddei,1StuartCalder,1DavidG.Mandrus,2,3,4MatthewB.Stone,1andAndrewD.Christianson21NeutronScatteringDivision,OakRidgeNationalLaboratory,OakRidge,Tennessee37831,USA2Mat...

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Line-Graph Approach to Spiral Spin Liquids Shang Gao1 2yGanesh Pokharel2 3Andrew F. May2Joseph A. M. Paddison2Chris Pasco2Yaohua Liu1Keith M. Taddei1Stuart Calder1David G. Mandrus2 3 4Matthew B. Stone1and Andrew D. Christianson2.pdf

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Line-Graph Approach to Spiral Spin Liquids Shang Gao1 2yGanesh Pokharel2 3Andrew F. May2Joseph A. M. Paddison2Chris Pasco2Yaohua Liu1Keith M. Taddei1Stuart Calder1David G. Mandrus2 3 4Matthew B. Stone1and Andrew D. Christianson2

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