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ChinesePhysicsLettersVolume 33 Number 3 March 2016

CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter

Microscopic Theory of the Thermodynamic Properties of Sr3Ru2O7∗

Wei-Cheng Lee(李伟正)1,2, Congjun Wu(吴从军)1**1Department of Physics, University of California, San Diego, California, 92093, USA

2Department of Physics, Applied Physics, and Astronomy, Binghamton University-State University of New York,Binghamton, USA

(Received 12 February 2016)The thermodynamic properties of the bilayer ruthenate compound Sr3Ru2O7 at very low temperatures are inves-tigated by using a tight-binding model yielding the realistic band structure combined with the on-site interactionstreated at the mean-field level. We find that both the total density of states at the Fermi energy and the entropyexhibit a sudden increase near the critical magnetic field for the nematic phase, echoing the experimental find-ings. A new mechanism to explain the anisotropic transport properties is proposed based on scatterings at theanisotropic domain boundaries. Our results suggest that extra cares are necessary to isolate the contributionsdue to the quantum criticality from the band structure singularity in Sr3Ru2O7.

PACS: 72.80.Ga, 73.20.−r, 71.10.Fd DOI: 10.1088/0256-307X/33/3/037201

The bilayer ruthenate compound Sr3Ru2O7 hasaroused considerable attentions with various inter-esting properties. It was first considered exhibit-ing a field-tuned quantum criticality of the metam-agnetic transition.[1−3] Later, in the ultra-pure sin-gle crystal it has been found that the metamagneticquantum critical point is intervened by the emer-gence of an unconventional anisotropic (nematic) elec-tronic state,[4,5] stimulating considerable theoreticalefforts.[6−18] Sr3Ru2O7 is a metallic itinerant systemwith the active 𝑡2𝑔-orbitals of the Ru sites in the bi-layer RuO2 (𝑎𝑏) planes. At very low temperatures(∼ 1 K), it starts as a paramagnet at small mag-netic fields. Further increasing field strength leads totwo consecutive metamagnetic transitions at 7.8 and8.1 Tesla if the field is perpendicular to the 𝑎𝑏-plane.The nematic phase is observed between these two tran-sitions, identified by the observation of anisotropic re-sistivity without noticeable lattice distortions.

This nematic phase in Sr3Ru2O7 can be under-stood as the consequence of a Fermi surface Pomer-anchuk instability.[3] It is a mixture in both den-sity and spin channels with the 𝑑-wave symmetry,[19]though its microscopic origin remains controversial.Different microscopic theories have been proposedbased on the quasi-1D bands of 𝑑𝑥𝑧 and 𝑑𝑦𝑧,[14,15,18]and based on the 2d-band of 𝑑𝑥𝑦.[8,9,16,17] In thetheories of ours[14,18] and Raghu et al.,[15] the un-conventional (nematic) magnetic ordering was inter-preted as orbital ordering between the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. In particular, in Ref. [18] a realistic tight-binding model is constructed taking into account themulti-orbital features, which reproduces accuratelythe results of the angle-resolved photon emission spec-

troscopy (ARPES)[20] and the quasiparticle interfer-ence in the spectroscopic imaging scanning tunnelingmicroscopy (STM).[21]

The influence of quantum critical fluctuations inSr3Ru2O7 seems to be novel as well. Rost et al.[22,23]measured the entropy and specific heat in ultra-puresamples and found divergences near the metamagnetictransitions in both quantities. Although it is a com-mon feature in a quantum critical state that the spe-cific heat diverges as 𝐶/𝑇 ∼ [(𝐵 − 𝐵𝑐)/𝐵𝑐]

−𝛼 dueto quantum fluctuations, the exponent of 𝛼 is fittedto be 1 instead of 1/3 as predicted by the celebratedHertz–Millis theory.[24,25] The total density of states(DOS) measured by Iwaya et al.[26] using the STMshowed that the DOS at the Fermi energy (𝒟(𝜖

F)) in-

creases significantly under the magnetic field, but theDOS at higher and lower energy does not change ac-cordingly. This indicates that the Zeeman energy doesnot simply cause a relative chemical potential shift toelectrons with opposite spins (the DOS evolution withthe external magnetic field will be discussed in Sup-plemental Material III B, i.e., SM III B). These findingshave posted a challenge to understand the critical be-havior in this material.

In this Letter, we show that the realistic band fea-tures of Sr3Ru2O7 make this material very sensitiveto small energy scales. Parts of the Fermi surface areclose to the van Hove singularities, and Fermi surfacereconstructions in the external magnetic fields lead toa singular behavior in 𝒟(𝜖F). This results in the diver-gences observed in the experiments mentioned above.Because of the strong spin-orbit coupling and the un-quenched orbital moments, the Zeeman energy tendsto reconstruct the Fermi surfaces rather than just pro-

∗Supported by the NSF DMR-1410375 and AFOSR FA9550-14-1-0168, the President’s Research Catalyst Award (No CA-15-327861) from the University of California Office of the President, and the CAS/SAFEA International Partnership Program forCreative Research Teams.

**Corresponding author. Email: wucj@physics.ucsd.edu© 2016 Chinese Physical Society and IOP Publishing Ltd

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CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter

vides a relative chemical potential shift. Our resultssuggest that the influence of quantum critical fluctua-tions will be masked if 𝒟(𝜖

F) of the system exhibits a

non-monotonic behavior, implying that a more carefulanalysis is required in order to distinguish the role ofthe quantum criticality in the bilayer Sr3Ru2O7.

We employ the tight-binding model 𝐻0 for theband structure of Sr3Ru2O7 derived by two of usand Arovas in a previous work[18] as elaborated inSM I. It is featured by the 𝑡2𝑔-orbital structure (e.g.𝑑𝑥𝑧, 𝑑𝑦𝑧, 𝑑𝑥𝑦), the bilayer splitting, the staggered dis-tortion of the RuO octahedral, and spin-orbit couplingdescribed by a few parameters as follows. The intra-layer hoppings include the longitudinal (𝑡1) and trans-verse (𝑡2) hoppings for the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals, thenearest neighbor (𝑡3), the next nearest neighbor (𝑡4),the next next nearest neighbor (𝑡5) hoppings for the𝑑𝑥𝑦-orbital, and the next nearest neighbor (𝑡6) hop-ping between 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. Here 𝑡⊥ is thelongitudinal inter-layer hopping for the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. The rotations of the RuO octahedra induceadditional inra-(𝑡

INT) and inter-layer (𝑡⊥

INT) hoppings

between 𝑑𝑥𝑧 and 𝑑𝑦𝑧 orbitals. The onsite terms in-clude the spin-orbit coupling 𝜆𝜎 ·𝐿, the energy split-ting 𝑉𝑥𝑦 of the 𝑑𝑥𝑧 and 𝑑𝑦𝑧 orbitals relative to the𝑑𝑥𝑦-orbital, and the chemical potential 𝜇. A typicalFermi surface configuration is plotted in Fig. 1 withthe parameter values in the caption.

kx/π

ky/π

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5-1.0-1.0

Fig. 1. The Fermi surfaces using the bilayer tight-bindingmodel Eq. (1) in SM I with the parameters in units of 𝑡1 as:𝑡2 = 0.1𝑡1, 𝑡3 = 𝑡1, 𝑡4 = 0.2𝑡1, 𝑡5 = −0.06𝑡1, 𝑡6 = 0.1𝑡1,𝑡⊥ = 0.6𝑡1, 𝑡INT = 𝑡⊥

INT= 0.1𝑡1, 𝜆 = 0.2𝑡1, 𝑉𝑥𝑦 = 0.3𝑡1,

and 𝜇 = 0.94𝑡1. The thick dashed lines mark the bound-ary of the half Brillouin zone due to the unit cell doublinginduced by the rotation of RuO octahedra. The Fermi sur-faces of the bonding (𝑘𝑧 = 0) and the anti-bonding bands(𝑘𝑧 = 𝜋) are denoted by black solid and red dashed lines,respectively.

The Hubbard model contains the on-site intra andinter orbital interactions as

𝐻int = 𝑈∑𝑖,𝑎,𝛼

��𝛼𝑖𝑎↑��

𝛼𝑖𝑎↓ +

𝑉

2

∑𝑖,𝑎,𝛼=𝛽

��𝛼𝑖𝑎 ��

𝛽𝑖𝑎, (1)

where the Greek index 𝛼 refers to the orbitals 𝑥𝑧, 𝑦𝑧

and 𝑥𝑦; the Latin index 𝑎 refers to the upper andlower layers. The other two possible terms in themulti-band Hubbard interaction are the Hund rulecoupling and pairing hopping terms, which do notchange the qualitative physics and are neglected. Weassume the external 𝐵-field lying in the 𝑥𝑧-planewith an angle 𝜃 tilted from the 𝑧-axis. The occu-pation and spin in each orbital and layer are de-fined as follows: 𝑛𝛼

𝑎 ≡∑

𝑠⟨𝑑𝛼 †𝑠,𝑎(𝑖)𝑑

𝛼𝑠,𝑎(𝑖)⟩, 𝑆𝛼

𝑧 𝑎 ≡12

∑𝑠 𝑠 ⟨𝑑𝛼 †

𝑠,𝑎(𝑖)𝑑𝛼𝑠,𝑎(𝑖)⟩, 𝑆𝛼

𝑥𝑎 ≡ 12

∑𝑠⟨𝑑𝛼 †

𝑠,𝑎(𝑖)𝑑𝛼𝑠,𝑎(𝑖)⟩,

where 𝑠 refers to spin index. The detailed mean-fieldtheory solution for the Hamiltonian 𝐻0 +𝐻int is pre-sented in SM II, and order parameters are computedself-consistently. It was pointed out in Refs. [14,15]that the nematic phase can be identified as the or-bital ordering between the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. Thenematic (𝒩 ) and the magnetization (ℳ) order pa-rameters are defined as

𝒩 =∑𝑎

(𝑛𝑦𝑧𝑎 − 𝑛𝑥𝑧

𝑎 ), ℳ =∑𝑎𝛼

𝑆𝛼𝑎 . (2)

Throughout this study, the interaction parameter val-ues are taken as 𝑈/𝑡1 = 𝑉/𝑡1 = 3.6, such that nospontaneous magnetization occurs in the absence ofthe external magnetic field.

0.25

0.20

0.15

0.10

0.05

0.000.001 0.002 0.003 0.004 0.005 0.006

Ord

er

para

mete

rs

µBB/t1

M

N

Fig. 2. The order parameters as a function of 𝜇B𝐵 for𝜃 = 0. The nematic phase (in the green area) is boundedby two magnetization increases which correspond to meta-magnetic transitions.

We present the mean-field results of low tempera-ture thermodynamic properties at 𝐵 ‖ 𝑐, i.e., 𝜃 = 0.Many experimentally observed results can be repro-duced and understood by the singular behavior of𝒟(𝜖

F) under the magnetic field. The order param-

eters |ℳ| and 𝒩 as functions of 𝜇B𝐵 are shown in

Fig. 2. There are three rapid increases in the mag-netization, consistent with the experiment measure-ments of the real part of the very low frequencyAC magnetic susceptibility at 7.5 T, 7.8 T and 8.1T,respectively.[4] Experimentally, only the last two ex-hibit dissipative peaks in the imaginary part of theAC susceptibility, which characterize the first ordermetamagnetic transitions. The first jump measured isconsidered as a crossover. The nematic ordering devel-ops in the area bounded by the last two magnetizationjumps, reproducing the well-known phase diagram ofthe Sr3Ru2O7.[4] In particular, if we adopt the resultsfrom LDA calculations[27,28] that 𝑡1 ≈ 300meV, we

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CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter

can find that three jumps in the magnetization appearat 𝐵 ≈ 0.0032𝑡1/𝜇B , 0.0053𝑡1/𝜇B , and 0.0059𝑡1/𝜇B ∼15.7T, 26T, and 29 T, which are within the sameorder to the experimental values. This is an im-provement compared to the results in previous theorycalculations.[8−11,15,16] in which the nematic orderingdevelops at much higher field strength 𝜇B𝐵/𝑡1 ≈ 0.02.

The sensitivity of the nematic phase to the smallenergy scale like the Zeeman energy is because a partof the Fermi surfaces, mostly composed of quasi-1Dbands, is approaching the van Hove singularities at(𝜋, 0) and (0, 𝜋). The evolution of the Fermi sur-face structures as increasing the 𝐵-field across thenematic phase boundaries is presented in detail inSM IIIA. When the system is in the nematic phase,the Fermi surfaces only have 2-fold symmetry as ex-pected. Particularly, the nematic distortion is mostprominent near (±𝜋, 0) and (0,±𝜋) whose Fermi sur-faces are dominated by the quasi-1D bands, support-ing the mechanism of orbital ordering in quasi-1Dbands driven by the van Hove singularities.

4.4

4.0

3.6

3.20.005 0.006 0.007

µBB/t1

S↼B↽/T

Fig. 3. (a) The entropy landscape represented by thequantity 𝑆(𝐵)/𝑇 in units of 𝑘2𝐵/𝑡1 within the range of0.0045 ≤ 𝜇B𝐵/𝑡1 ≤ 0.007. A sudden increase near thenematic region (green area) is clearly seen.

One of the intriguing puzzles experimentally ob-served is the critical exponent of the divergence inentropy (as well as specific heat) when approachingthe nematic region, or the quantum critical point. Asmentioned in the introduction, the exponent does notmatch the Hertz-Millis theory based on the assump-tion of a constant DOS 𝒟(𝜖

F). However, Sr2Ru3O7

has a complicated Fermi surface evolutions under the𝐵-field, and is in the Fermi liquid state at low temper-atures inferred from the temperature dependences ofthe resistivity. It is then worthy of studying first thecontribution from the band structure to the entropybefore considering the quantum fluctuations. The en-tropy per Ru atom can be evaluated by

𝑆(𝐵) = − 𝑘B

𝑁

′∑𝑘

∑𝑗

[𝑓(𝐸𝑗(𝑘)) ln 𝑓(𝐸𝑗(𝑘))

+ (1− 𝑓(𝐸𝑗(𝑘))) ln(1− 𝑓(𝐸𝑗(𝑘)))], (3)

where 𝑓 is the Fermi distribution function, and 𝐸𝑗(𝑘)is the mean-field energy spectra of the 𝑗th band.In Fig. 3, we plot 𝑆(𝐵)/𝑇 at a low temperature of1/(𝛽𝑡1) = 0.002 for the 𝐵-fields in the vicinity of

the nematic region. 𝑆(𝐵)/𝑇 increases first, beingsuppressed, and then decreases, which is consistentwith the experiment.[22] Since the nematic transitionis driven by the sudden increase of 𝒟(𝜖

F), the entropy

should also be enhanced from outside towards the ne-matic region.

While it is generally expected that the quantumfluctuations near the critical point contribute addi-tional entropy, our results demonstrate the singularbehavior of 𝒟(𝜖

F) already produces diverging behavior

in entropy under magnetic field at constant temper-ature, although the critical exponent 𝛼 is difficult beextracted from the current theory. Similar argumenthas been proposed in a previous study,[23] in whichthe effect of a rigid band shift away from van Hovesingularities in a perfect 1D band is discussed.

ρ

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5-0.050 0.000 0.050

ω/t

AllQuasi-1Dbands

µBB/t1

kBT/t 1

0.0051

0.009

0.006

0.003

0.0057 0.0063

0.0000.0050.0100.0150.020

(a) (b)

Fig. 4. (a) The 𝑇–𝐵 phase diagram. Magnitudes of 𝒩 arerepresented by the color scales. The areas with light colorshave large 𝒩 , defining the region for the nematic order.The re-entry of the nematic order at higher temperatureis seen at fields between 0.0058 < 𝜇B𝐵/𝑡1 < 0.0063. (b)The DOSs of the all bands (solid line) and quasi-1D bands(dashed line) at 𝜇B𝐵/𝑡1 = 0.006. The yellow areas referto the energy window bounded by ±𝑘B𝑇/𝑡1 with temper-ature 𝑘B𝑇/𝑡1 = 0.003. It can be seen that this thermalenergy window covers a region in which the DOS increasesabruptly, driving the nematic phase at finite temperature.

An intriguing experimental observation is the“muffin”-shaped phase boundary of the nematic phasein the 𝑇–𝐵 phase diagram.[22] At field strengthsslightly below 7.8T and above 8.1T the nematic phaseappears at finite temperature but vanishes at zerotemperature, to which we term as the “re-entry” be-havior. It means that the entropy is actually higherinside the nematic phase than the adjacent normalphases. By inspecting Fig. 3 closer, it can be seen thatthe entropy drops as entering the nematic phase fromthe lower-field boundary but raises as entering fromthe upper-field boundary, thus the present theory hasthe “re-entry” at the upper-field boundary but not atthe lower-field boundary, which is further confirmedby the temperature dependence of 𝒩 as a function of𝐵-field shown in Fig. 4(a).

The re-entry behavior can be understood as fol-lows. The mechanism for nematic ordering inSr3Ru2O7 based on van Hove singularities is all aboutincreasing 𝒟(𝜖

F) abruptly by driving the system closer

to the van Hove singularities with the magnetic field.At the field strength slightly above the upper critical

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CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter

field, 𝒟(𝜖F) is large but still not enough for the occur-rence of the nematic ordering. If the thermal energyis large enough to cover enough DOS within the ther-mal energy window 𝜖

F−𝑘

B𝑇 < 𝜖

F< 𝜖

F+𝑘

B𝑇 but still

low enough so that the thermal fluctuations are small,the re-entry of the nematic phase at higher temper-ature is possible. As an illustration, Fig. 4(b) plotsthe DOS for 𝜇B𝐵/𝑡1 = 0.006 at which the nematicordering first occurs at 𝑘

B𝑇/𝑡1 = 0.003 in our calcula-

tion. It can be seen that the thermal energy windowfor 𝑘

B𝑇/𝑡1 = 0.003 (yellow areas) covers a region in

which the DOS increases abruptly, consistent with themechanism for the re-entry behavior discussed above.

Now let us consider the case of titled magneticfield. Figure 5 presents the nematic order parame-ter 𝒩 as functions of 𝜇

B𝐵 and 𝜃. Since the in-plane

component of the orbital Zeeman energy explicitlybreaks the 𝐶4 symmetry down to the 𝐶2 symmetry,𝒩 is non-zero as long as 𝜃 = 0. Nevertheless, theexperimentally observable nematic phase can still beidentified by the jumps in 𝒩 . Our results show thatthe nematicity is strongly enhanced with increasing𝜃, which is due to the orbital Zeeman energy. Theanisotropic in-plane component of the orbital Zeemanenergy term −𝜇B𝐵

∑𝑖,𝑎 𝐿𝑥,𝑖𝑎 sin 𝜃 is clearly propor-

tional to 𝐵 and largest at 𝜃 = 90∘ (i.e. �� ‖ ��), whichinduces the anisotropy of Fermi surface as explicitlyshown in Fig. 3 in SM IV. Although such anisotropy inthe band structure is not important at low field, it canbe amplified by the effect of the interactions, drivingthe system more susceptible to the nematic phase asthe critical points are approached. As a result, theportion of the nematic phase in the phase diagram isenlarged as 𝜃 increases from 0∘ to 90∘ as seen in ourcalculations.

θ (

deg)

µBB/t1

90

60

30

00.002 0.003 0.004 0.005 0.006 0.007

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

Fig. 5. The nematic order parameter 𝒩 as functions of𝜇B𝐵 and 𝜃. The magnitudes of 𝒩 are represented by thecolor scales.

However, experimentally the resistive anisotropydisappears quickly as the magnetic field is tilted awayfrom the 𝑐-axis,[3,5,15] suggesting that the nematic or-dering vanishes with the increase of the field angle𝜃. This observation is seemingly in contradiction withour theory, but this disagreement can be reconciled

as follows. It has been argued that the resistivitymeasurement may not be a good indicator for the ne-matic phase in Ref. [15]: The nematic phase is mostlyassociated with states near the van Hove singular-ity where Fermi velocities are too small to contributesignificantly to transport properties. The observedanisotropic resistivity is mostly likely due to the scat-terings on nematic domains. The tilt of the magneticfield aligns domains, which makes the anisotropy ex-plicit. However, if the domains are fully aligned, theresistivity measurement will become insensitive to thenematic phase due to the diminished scatterings be-tween nematic domains even though the nematic ordercould be larger.

Our results have posted a possibility that the ne-matic order could occur in a larger range of the mag-netic field for 𝐵 ‖ �� than for 𝐵 ‖ 𝑧. Detection meth-ods other than resistivity would be desirable. One fea-sible way is to measure the quasiparticle interference(QPI) in the spectroscopic imaging STM, which hasbeen examined in detail in our previous work.[18] It hasbeen predicted by us that if there is a nematic order,QPI spectra will manifest patterns breaking rotationalsymmetry. Another possible experiment is the nuclearquadruple resonance (NQR) measurement, which hasbeen widely used to reveal ordered states in high-𝑇𝑐 cuprates[29−31] and recently the iron-pnictides.[32]This technique utilizes the feature that a nucleus witha nuclear spin 𝐼 ≥ 1 has a non-zero electric quadruplemoment. Because the electric quadruple moment cre-ates energy splittings in the nuclear states as a electricfield gradient is present, a phase transition could be in-ferred if substantial changes in the resonance peak areobserved in the NQR measurement. In addition, sincethis is a local probe at the atomic level, it is highlysensitive to the local electronic change. Given that Ruatom has a nuclear spin of 𝐼 = 5/2[33] and the orbitalordering in the quasi-1D bands significantly changesthe charge distribution around the nuclei, a system-atic NQR measurement as functions of magnetic fieldand field angle will reveal more conclusive informationabout nematicity.

One remaining puzzle on the transport anisotropyin the nematic phase is why the easy axis for the cur-rent flow is perpendicular to the in-plane componentof the 𝐵-field.[5] In the following, we provide a nat-ural explanation based on the anisotropic spatial ex-tension of domain boundaries. Assuming the 𝐵-fieldlying in the 𝑥𝑧-plane, the in-plane (𝑥𝑦) orbital Zee-man energy reads 𝐻in−plane = −𝜇

B𝐵 sin 𝜃

∑𝑖,𝑎 𝐿𝑥,𝑖𝑎,

which couples the 𝑑𝑥𝑦 and 𝑑𝑥𝑧-orbitals and breaks thedegeneracy between the 𝑑𝑥𝑧 and 𝑑𝑦𝑧-orbitals. Sincethe 𝑑𝑥𝑦-orbital has lower on-site energy due to thecrystal field splitting than that of 𝑑𝑥𝑧, the 𝑑𝑥𝑧-orbitalbands are pushed to higher energy than 𝑑𝑦𝑧-orbitalbands by this extra coupling. As a result, the ne-matic state with preferred 𝑑𝑦𝑧-orbitals (i.e., 𝒩 > 0)has lower energy in the homogeneous system. At small

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CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201 Express Letter

angles of 𝜃, domains with preferred 𝑑𝑥𝑧-orbitals (i.e.𝒩 ≤ 0) could form as depicted in Fig. 6(a) as meta-stable states, which occupy less volume than the ma-jority domain of 𝒩 > 0.

(b)

y

x

(a)

Fig. 6. Illustration of the energetically-favored domainstructures as the in-plane magnetic field is along the �� axis.Orbital ordered phase with 𝒩 > 0 are dominant (shadedareas) and high-energy domains with 𝒩 ≤ 0 (green ovals)coexist. (b) The domain walls extend longer in the 𝑦 di-rection because it costs less energies if less 𝑦-than-�� bondsare broken. The white oval represents the wavefunction of𝑑𝑦𝑧 orbital on each site.

Let us consider the shape of the domain bound-aries. Because of the quasi-1D features of the 𝑑𝑥𝑧and 𝑑𝑦𝑧-orbitals, the horizontal (vertical) domain wallbreaks the bonds of the 𝑑𝑦𝑧(𝑑𝑥𝑧)-orbital as depictedin Fig. 6(b), respectively. Since the 𝑑𝑦𝑧-orbital is pre-ferred by 𝐻in−plane, the horizontal domain wall costsmore energy. Consequently, the domain structure il-lustrated in Fig. 6(a) with longer vertical walls thanthe horizontal walls is energetically favored. Since theelectrons suffer less domain scatterings hopping alongthe 𝑦-axis in this domain structure, it becomes theeasy axis for the current flow. At large values of 𝜃,higher energy domains are suppressed and eventuallyvanish, and thus the resistivity measurement becomesinsensitive to the nematic phase because of vanishingof the domain scatterings.

In summary, we have shown that many impor-tant properties observed in the Sr3Ru2O7 could bequalitatively consistent with a realistic tight-bindingmodel together with on-site interactions treated atmean-field level. The band structure is complicatedby multibands, bilayer splitting, rotations of RuO oc-tahedra, and the spin-orbit coupling, collectively lead-ing to the high sensitivity to the small energy scales,which is the main cause of the singular behavior in theevolution of the Fermi surfaces under magnetic field.For the case of magnetic field parallel to the 𝑐-axis, thenematic order, which is interpreted as the orbital or-dering in quasi-1D 𝑑𝑦𝑧 and 𝑑𝑥𝑧 bands, appears. Thesingular behavior in 𝒟(𝜖

F) also results in the diver-

gences in the entropy and specific heat landscapes.As the magnetic field is tilted away from the 𝑐 axis(𝜃 = 0), we find that the nematic region expands in-stead of shrinking as the resistivity measurement hasindicated. To explain this discrepancy, we adopt thedomain scattering argument.[15] Furthermore, we havegiven an explanation for another experimental puzzlethat the easy axis for the current flow is always per-pendicular to the in-plane magnetic field. Measure-ments like quasiparticle interference in the spectro-scopic imaging STM and NQR which could detect theorbital ordering directly have been proposed.

We thank J. E. Hirsch and A. Mackenzie for valu-able discussions.

References[1] Grigera S A et al 2001 Science 294 329[2] Perry R S et al 2001 Phys. Rev. Lett. 86 2661[3] Grigera S A et al 2003 Phys. Rev. B 67 214427[4] Grigera S A et al 2004 Science 306 1154[5] Borzi R A et al 2007 Science 315 214[6] Millis A J, Schofield A J, Lonzarich G G and Grigera S A

2002 Phys. Rev. Lett. 88 217204[7] Green et al A G 2005 Phys. Rev. Lett. 95 086402[8] Kee H Y and Kim Y B 2005 Phys. Rev. B 71 184402[9] Yamase H and Katanin A 2007 J. Phys. Soc. Jpn. 76

073706[10] Yamase H 2007 Phys. Rev. B 76 155117[11] Puetter C, Doh H and Kee H Y 2007 Phys. Rev. B 76

235112[12] Berridge A M, Green A G, Grigera S A and Simons B D

2009 Phys. Rev. Lett. 102 136404[13] Berridge A M, Grigera S A, Simons B D and Green A G

2010 Phys. Rev. B 81 054429[14] Lee W C and Wu C 2009 Phys. Rev. B 80 104438[15] Raghu S et al 2009 Phys. Rev. B 79 214402[16] Puetter C M, Rau J G and Kee H Y 2010 Phys. Rev. B 81

081105[17] Fischer M H and Sigrist M 2010 Phys. Rev. B 81 064435[18] Lee W C, Arovas D P and Wu C 2010 Phys. Rev. B 81

184403[19] Wu C, Bergman D, Balents L and Das Sarma S 2007 Phys.

Rev. Lett. 99 70401[20] Tamai A et al 2008 Phys. Rev. Lett. 101 026407[21] Lee J et al 2009 Nat. Phys. 5 800[22] Rost A W et al 2009 Science 325 1360[23] Rost A W et al 2010 Phys. Status Solidi B 247 513[24] Hertz J A 1976 Phys. Rev. B 14 1165[25] Millis A J 1993 Phys. Rev. B 48 7183[26] Iwaya K et al 2007 Phys. Rev. Lett. 99 057208[27] Liebsch A and Lichtenstein A 2000 Phys. Rev. Lett. 84 1591[28] Eremin I, Manske D and Bennemann K 2002 Phys. Rev. B

65 220502[29] Hammel P C et al 1998 Phys. Rev. B 57 R712[30] Teitel’baum G B, Büchner B and de Gronckel H 2000 Phys.

Rev. Lett. 84 2949[31] Singer P M, Hunt A W and Imai T 2002 Phys. Rev. Lett.

88 047602[32] Lang G et al 2010 Phys. Rev. Lett. 104 097001[33] Ishida K et al 1997 Phys. Rev. B 56 R505

037201-5

Supplemental Material for

“Microscopic Theory of the Thermodynamic Properties of Sr3Ru2O7”

(See CHIN.PHYS. LETT. Vol. 33, No. 3 (2016) 037201)

Wei-Cheng Lee1, 2, ∗ and Congjun Wu1, †

1Department of Physics, University of California, San Diego, California, 92093, USA2Department of Physics, Applied Physics, and Astronomy,

Binghamton University-State University of New York, Binghamton, USA

I. THE TIGHT-BINDING MODEL

In this section, we explain the detailed band structureSr3Ru2O7, which is complicated by the t2g-orbital struc-ture (e.g. dxz, dyz, dxy), the bilayer splitting, the stag-gered distortion of the RuO octahedra, and spin-orbitcoupling. We have constructed a detailed tight-bindingHamiltonian which gives rise to band structures in agree-ment with the ARPES data in a previous work1. Wefound that a difference of the on-site potential betweenthe two adjacent RuO layers, Vbias, should be added1 inorder to fit the shape of the Fermi surfaces observed inthe ARPES experiments2. This term appears becauseARPES is a surface probe and this bilayer symmetrybreaking effect is important near the surface. Since wefocus on the thermodynamic properties which are all bulkproperties, Vbias is set to be zero in this paper. Below wewill start from this model and refer readers to Ref. [1]for more detailed information.The tight-binding band Hamiltonian H0 can be re-

duced to block forms classified by kz = 0, π correspond-ing to bonding and anti-bonding bands with respect tolayers as:

H0 = h0(kz = 0) + h0(kz = π), (1)

with h0(kz) defined as

h0(kz)=∑k

′Φ†

k,kz,s

(h0s(k, kz) g†(k, kz)

g(k, kz) h0s(k + Q, kz)

)Φk,kz,s

,

(2)

where the spinor Φ†k,kz,s

operator is defined as

Φ†k,kz,s

=(dyz †k,s,kz

, dxz †k,s,kz

, dxy †k,−s,kz

,

dyz †k+Q,s,kz

, dxz †k+Q,s,kz

, dxy †k+Q,−s,kz

); (3)

dαs,kz(k) annihilates an electron with orbital α and spin

polarization s at momentum (k, kz); Q = (π, π) is thenesting wavevector corresponding to unit cell doublinginduced by the rotations of RuO octahedra;

∑k′means

that only half of the Brillouin zone is summed. Pleasenote the opposite spin configurations s and -s for thedxz, dyz and dxy-orbitals in Eq. 3, which is convenientfor adding spin-orbit coupling later.

The diagonal matrix kernels h0s in Eq. 2 are definedas

h0s(k, kz) = As(k) + B1 cos kz − µI, (4)

where

As(k) =

ϵyzk

ϵoffk

+ isλ − sλ

ϵoffk

− isλ ϵxzk

iλ

−sλ −iλ ϵxyk

, (5)

and

B1 =

−t⊥ 0 00 −t⊥ 00 0 0

; (6)

where t⊥ is the longitudinal inter-layer hopping for thedxz and dyz orbitals. λ is the spin-orbit coupling strengthwhich comes from the on-site spin-orbit coupling term

as Hso = λ∑

i Li · Si; µ is the chemical potential; thedispersions for the dyz, dxz, and dxy bands in Eq. 5 aredefined as

ϵyzk

= −2t2 cos kx − 2t1 cos ky,

ϵxzk

= −2t1 cos kx − 2t2 cos ky,

ϵxyk

= −2t3(cos kx + cos ky

)− 4t4 cos kx cos ky

−2t5(cos 2kx + cos 2ky

)− Vxy

ϵoffk

= −4t6 sin kx sin ky, (7)

which includes longitudinal (t1) and transverse (t2) hop-ping for the the dxz and dyz orbitals, respectively, as wellas are nearest neighbor (t3), next-nearest neighbor (t4),and next-next-nearest neighbor (t5) hopping for the dxyorbital. Following the previous LDA calculations3, Vxy isintroduced to account for the splitting of the dyz and dxzstates relative to the dxy states. While symmetry forbidsnearest-neighbor hopping between different t2g orbitalsin a perfect square lattice without the rotation of Ru oc-tahedra, a term describing hopping between dxz and dyzorbitals on next-nearest neighbor sites (t6) is allowed andput into the tight-binding model.

The off-diagonal matrix kernel g(k, kz) in Eq. 2 reads

g(k, kz) = G(k)− 2B2 cos kz, (8)

where

2

B2 =

0 t⊥INT 0−t⊥INT 0 00 0 0

, (9)

and

G(k) =

0 −2tINT γ(k) 0

2tINT γ(k) 0 00 0 0

, (10)

with γ(k) = cos kx + cos ky. tINT and t⊥INT describe theintra- and inter-layer hopping between dxz and dyz in-duced by the rotations of RuO octahedra, providing the

coupling between k and k + Q.When describing the Zeeman energy, we can choose

the magnetic field B to lie on the xz plane and define

θ as the angle between B and the c-axis of the samplewithout loss of the generality. Consequently, the Zeemanterm becomes:

HZeeman = HorbitalZeeman +Hspin

Zeeman,

HorbitalZeeman = −µB B

∑i,a

(Lz,ia cos θ + Lx,ia sin θ

),

HspinZeeman = −2µB B

∑i,a,α

(Sαz ia cos θ + Sα

x ia sin θ),

(11)

where a is the layer index, B = |B|, and the matricesLx,z can be found in Ref. [1].For θ = 0, the extra Zeeman terms from x-component

of L and S couple Φ†k,kz,↑

and Φ†k,kz,↓

. Defining ϕ†k,kz

≡(Φ†

k,kz,↑,Φ†

k,kz,↓

), the Zeeman term can be written in the

matrix form as:

HZeeman = µBB∑k

′ ∑kz

ϕ†k,kz

HZ(θ)ϕk,kz

,

(12)

where

HZ(θ)=

HD

Z (θ,+) 0 HO †Z (θ) 0

0 HDZ (θ,+) 0 HO †

Z (θ)

HOZ (θ) 0 HD

Z (θ,−) 0

0 HOZ (θ) 0 HD

Z (θ,−)

,

(13)

HDZ (θ, s) = cos θ ×

−s −i 0i −s 00 0 s

, (14)

with s = ±1 and

HOZ (θ) = sin θ ×

−1 0 00 −1 −i0 i −1

. (15)

In realistic band structures measured by ARPES2,there exists an additional δ-band arising from the dx2−y2-orbital which is not covered by the current model. Theparticle filling in the t2g-orbitals is not fixed. For theconvenience of calculation, we fix the chemical potentialµ = 0.94t1 instead of fixing particle filling in the t2g-orbitals while changing magnetic fields and orientations.The corresponding fillings inside the t2g-orbitals per Ruatom varies within the range between 4.05 and 4.06 inFigs. 2, 4, 5 in the main text. This treatment does notchange any essential qualitative physics.

II. THE MEAN-FIELD THEORY

In this part, we present the process of mean-field the-ory solution.

The standard mean-field decomposition of Hint leadsto

HMFint =

∑i,a,α

∑s

Wαs d

α †s,a(i)d

αs,a(i)− USα

x dα †s,a(i)d

αs,a(i) ,

(16)where

Wαs = U

(12n

α − s Sαz

)+ V

∑β =α

nβ , (17)

with the assumptions of nαa = nα, Sα

x,z a = Sαx,z.

Since the order parameters {nαa} are non-zero even

without magnetic field, we require that the renormalizedFermi surface at zero field to be the one given in Fig. 1 inthe main text. As a result, in addition to the optimizedparameters obtained in our previous work1, we need tosubtract the following term from Eq. 16:

Hshift =∑i,a,α

∑s

Wαs (0)d

α †s,a(i)d

αs,a(i), (18)

where

Wαs (0) = 1

2Unα(0) + V∑β =α

nβ(0), (19)

and nα(0) is the occupation number in orbital α corre-sponding to the Fermi surfaces shown in Fig. 1 in themain text. This is an effect of the renormalization of thechemical potential µ and Vxy due to interactions. Afterputting all the pieces together, we finally arrive at themean-field Hamiltonian as:

HMF = H0 +HMFint −Hshift +HZeeman

≡∑k

′ ∑kz

ϕ†k,kz

HMF (k)ϕk,kz

. (20)

The order parameters are computed self-consistently.

3

FIG. 1: Fermi surface evolution for the case of magnetic field parallel to c axis (a) before (µBB = 0.0048t1) (b) inside(µBB = 0.00544t1), and (c)after (µBB = 0.006t1) the nematic phase. Significant changes in the Fermi surface topology underthe magnetic field can be seen (see Fig. 1 in the main text for the Fermi surfaces at zero field). The nematic distortion is mostobvious in the Fermi surfaces near (±π, 0) and (0,±π) as indicated by the yellow areas in (b). These parts of Fermi surfacesare composed mostly of quasi-1D bands, supporting the intimacy of nematic phase to the orbital ordering.

III. THE CASE OF THE PERPENDICULARMAGNETIC FIELD (θ = 0)

In this section we present supplemental informationfor thermodynamic properties at low temperatures arecalculated within the mean-field theory for the case of

B ∥ c, i.e., θ = 0. They can be reasonably reproducedand understood by the singular behavior of D(ϵF ) underthe magnetic field.

A. Evolution of Fermi surface

We plot the part of Fermi surfaces mostly composedof quasi-1D bands as shown in the yellow areas in Fig. 1(b), which is close to the van Hove singularities at (π, 0)and (0, π). The evolution of the Fermi surface structuresas increasing the B-field across the nematic phase bound-aries is presented in Fig. 1 a (before), b (inside), and c(after) at µBB/t1 = 0.0048, 0.00544, 0.006, respectively.Before and after the nematic phases, the Fermi surfaceshave the 4-fold rotational symmetry as exhibited in Fig.1 (a) and (c), while the 4-fold symmetry is broken into2-fold in the nematic phase.It is worthy of mentioning that the onsite spin-orbit

coupling Hso = λ∑

i Li · Si has important effects onthe Fermi surface evolutions. Hso hybridizes the oppo-site spins between quasi-1D bands dyz,xz and the 2-Dband dxy. As the spin Zeeman energy is present, thespin majority (minority) bands of dyz,xz couples to thespin minority (majority) band of dxy. Moreover, theorbital Zeeman energy provides more hybridizations be-tween quasi-1D dxz,yz bands. Combined with the abovetwo effects, the addition of the spin and orbit Zeeman en-ergies causes reconstruction of the Fermi surfaces ratherthan just chemical potential shifts. These results show

that the complexity and the sensitivity of the Sr3Ru2O7

band structure can be captured very well by our tight-binding model with a reasonable quantitative accuracy.In the following, the same model will be used to fur-ther investigate some novel physical properties observedin experiments.

B. Total density of states

Iwaya et al. has measured the STM tunneling differen-tial conductance dI

dV in the B-field for Sr3Ru2O7, whichcorresponds to measurement of the DOS. It has been ob-served that while DOS at higher and lower energy doesnot change, the DOS at the Fermi energy (D(ϵF )) in-crease significantly under the application of the magneticfield, demonstrating the violation of the rigid band pic-ture.

In our model the total DOS can be evaluated using:

ρtot(ω) = 1πN

∑k

′TrIm

[GMF (k, ω)

]GMF (k, ω) ≡

(ω + iη −HMF (k)

)−1(21)

where HMF (k) is given in Eq. 20 with the self-consistentorder parameters and N is the total number of sites inthe bilayer square lattices.

The profiles of the total DOS at several differentmagnetic field strength are plotted in Fig. 2(a), andclearly a rigid band picture does not apply here. D(ϵF )(ρtot(ω = 0) in the plot) increases significantly as thenematic phase is approached. This feature can also bedirectly understood by the picture of Fermi surface recon-struction, since the changes in the Fermi surface topol-ogy inevitably lead to the non-monotonic behavior in

4

FIG. 2: (a) The total DOS ρtot(ω) as a function of µBB.The broadening factor η is set to be η = 0.002t1. ρtot(ω)does not follow the rigid band picture and ρtot(ω = 0) hasa sudden increase near the nematic region. Inset: the totalDOS at zero field for a wider range of |ω|/t1 ≤ 0.4. The peakscorresponding to van Hove singularities are near ω/t1 ≈ −0.2and ω/t1 ≈ 0.35. (b) The DOS of the quasi-1D bands. Inset:the DOS of quasi-1D bands at zero field for a wider rangeof |ω|/t1 ≤ 0.4. Only the peak around ω/t1 ≈ 0.35 remains,meaning that this peak is due to the van Hove singularitiesin quasi-1D bands.

ρtot(ω = 0). In particular, comparing Fig. 1 in themain text and Fig. 1 in this Supplemental Material, it isstraightforward to see that more Fermi surfaces appearnear the (±π, 0) and (0,±π) as the magnetic field in-creases. Since there are van Hove singularities near these

four k points, ρtot(ω = 0) is expected to increase. Asthe magnetic field is increased further so that the vanHove singularities are all covered below the Fermi sur-faces, ρtot(ω = 0) starts to drop (not shown here).

At the first glance, the entropy measurement and ourresults of the total DOS seem to contradict with the STMmeasurement. While both the entropy measurement andour results develop a maximum around the nematic re-gion in D(ϵF ), the STM measurement showed insteadthat D(ϵF ) keeps increasing even after the nematic re-gion is passed. To resolve this discrepancy, several realis-

tic features need to be considered before comparing ourcalculations with the STM results. Since the STM is asurface probe and the surface of the material is usuallycleaved to have the oxygen atoms in the outermost layer,there is an oxygen atom lying above each uppermost Ruatom. Consequently, the tunneling matrix element willbe mostly determined by the wavefunction overlaps be-tween the p-orbitals of the oxygen atom and the d-orbitalsof the Ru atom, resulting in a much smaller matrix ele-ment for dxy orbital compared to dxz,yz orbitals1. Theminimal model to take this effect into account is to ex-tract the DOS only from the quasi-1D orbitals, which isplotted in Fig. 2(b). Although the overall profile in Fig.2(b) is not exactly the same as that in Ref. 4, which is at-tributed to more complicated momentum dependence oftunneling matrix elements5–7 not considered here, it cap-tures the increasing DOS with the magnetic field whichis more consistent with Ref. 4.

The insets in Fig. 2 plot the total and quasi-1D bandDOSs at zero field within a wider range of |ω|/t1 ≤ 0.4.The peaks corresponding to the van Hove singularitiesreside at ω/t1 ≈ −0.2 and ω/t1 ≈ 0.35, far away from theFermi energy. The reason why the small energy scale likeZeeman energy (∼ 0.003t1) can push the system to getcloser to the van Hove singularities at energies far awayfrom the Fermi energy is the help of the metamagnetism.In the mean-field theory, the magnetization produces aneffective chemical potential shift as sUSα

z for electrons atorbital α and spin s. As a result, under the magnetic fieldthe jump in the magnetization gives Sα

z ∼ 0.05 withinthe range of experimental interests. This leads to theeffective chemical potential shift about ±0.18t1 for U =3.6t1, which is large enough to push the system closer tothe van Hove singularities. This renormalization of thechemical potential by the interaction is also part of thecause for the violation of the rigid band picture.

IV. THE CASE OF THE TILTED MAGNETICFIELDS (θ = 0)

We present the Fermi surface configuration in the pres-ence of tilted magnetic field. The Fermi surfaces without

any interaction for B ∥ x with strength µB |B| = 0.1t1is plotted in Fig. 3, and an anisotropy can be seen.Although such anisotropy in the band structure is notimportant at low field, it can be amplified by the effectof the interactions, driving the system more susceptibleto the nematic phase as the critical points is approached.As a result, the portion of the nematic phase in the phasediagram is enlarged as θ increases from 0◦ to 90◦ as seenin our calculations.

V. SUMMARY AND MORE DISCUSSIONS

The failure of a rigid band picture is essentially a con-sequence of the interplay between spin-orbit coupling and

5

FIG. 3: The non-interacting Fermi surfaces for B ∥ x andµBB = 0.1t1. The anisotropy in Fermi surfaces is alreadyvisible, especially for the areas near (±π, 0) and (0,±π).

the Zeeman energy, despite the strong correlation effectcould also result in the violation of the rigid band shiftupon doping8. Because the spin-orbit coupling hybridizesthe quasi-1D bands and 2-D bands with opposite spins,the Zeeman energy naturally induces the reconstructionof the Fermi surfaces instead of just rigid chemical poten-tial shifts. This singular behavior in D(ϵF ) also resultsin the divergences in the entropy and specific heat land-scapes, since at very low temperature both quantities areapproximately proportional to D(ϵF ). Because the diver-gence observed by Rost el. al.9 start approximately at6-7 Tesla which is not very close to the quantum criti-cal point residing about 8 Tesla, a direct application ofthe quantum critical scaling seems to be inappropriate.The explanation of the critical exponent associated withthis divergence could not be complete without taking theband structure singularity into account in this particularmaterial10.As the magnetic field is tilted away from the c axis

(θ = 0), we find that nematic region expands instead ofshrinking as the resistivity measurement has indicated.From the theoretical viewpoints, the tilt of the magneticfield induces an extra in-plane component of the orbitalZeeman energy which explicitly breaks the C4 symmetrydown to the C2 symmetry. As argued above that thissystem is very sensitive to small energy scale, the effectof this extra Zeeman energy is not important at low fieldbut could amplify the effect of interaction to drive thesystem toward nematicity as the quantum critical pointis approached. As a result, the nematic phase is more fa-vored and stable in the presence of the in-plane magneticfield and it requires another Fermi surface reconstruc-

tion at even higher magnetic field in order to weaken thenematic phase by reduced D(ϵF ).

To explain this discrepancy between our theory and theresistivity measurement, we adopt the domain scatteringargument proposed by Raghu et. al.11. Furthermore,we have given an explanation for another experimentalpuzzle that the easy axis for the current flow is alwaysperpendicular to the in-plane magnetic field. Measure-ments like quasiparticle interference in the spectroscopicimaging STM and NQR which could detect the orbitalordering directly have been proposed to be more reliableprobes for the nematicity in this material than the resis-tivity measurement.

Finally we would like to comment on limitation of thepresent theory. Although we have found the ’re-entry’ be-havior of the nematic phase, i.e., the appearance of thenematic phase only at the finite temperature but not atthe zero temperature, near the upper-field boundary, theexperiments showed this behavior near both upper- andlower- field boundary. In our calculations, the re-entrybehavior is due to the increase of the density of stateswithin the narrow energy window around the Fermi en-ergy opened by thermal energy, but we do not reject otherschemes for the re-entry behavior. One possible schemeis an analogue of ferromagnetism without exchange split-ting proposed by Hirsch12. He showed that the nearestneighbor interactions could result in a spin-dependentrenormalization on the bandwidth (equivalently, the ef-fective mass). As a result, the filling for different spinbands can be different because of the unequal effectivemasses, leading to the ferromagnetism even without theexchange splitting as in the Stoner model.

In Hirsch’s original paper, the re-entry of the ferromag-netism at higher temperature was found. Since we onlyconsidered the on-site interactions in our model, suchan effect is beyond the scope of the current theory. Itis possible that after including the nearest neighbor in-teraction, the renormalizations of the bandwidths havenovel temperature-dependences, leading to a phase dia-gram better consistent with the experiments. If this isthe correct scheme, the re-entry of the nematic statesshould be accompanied by a change in the kinetic energydue to the effective mass renormalization, which could beexamined by the sum rules for the optical properties12–16.

Another possible scheme for the re-entry behavior isthe quantum critical fluctuations. It is well-known thatthe influences of the critical fluctuations extend from thequantum critical point to finite temperature in a V -shaperegion in the phase diagram. Moreover, the critical fluc-tuations in this material contain not only the ferromag-netic but also the nematic ones. As a result, it is notsurprising that the competition between these two typesof critical fluctuations leads to a intriguing phase dia-gram at the finite temperature, and the study towardthis direction is currently in progress.

6

∗ Electronic address: leewc@physics.ucsd.edu† Electronic address: wucj@physics.ucsd.edu1 W.-C. Lee, D. P. Arovas, and C. Wu, Phys. Rev. B 81,184403 (2010).

2 A. Tamai et al., Phys. Rev. Lett. 101, 026407 (2008).3 D. Singh and I. Mazin, Phys. Rev. B 63, 165101 (2001).4 K. Iwaya et al., Phys. Rev. Lett. 99, 057208 (2007).5 J. Tersoff and D. Hamann, Phys. Rev. Lett. 50, 1998(1983).

6 Y. Zhang et al., Nat. Phys. 4, 627 (2008).7 W.-C. Lee and C. Wu, Phys. Rev. Lett. 103, 176101(2009).

8 J. Farrell et al., Phys. Rev. B 78, 180409 (2008).9 A. W. Rost et al., Science 325, 1360 (2009).

10 J. Lee et al., Nat. Phys. 5, 800 (2009).11 S. Raghu et al., Phys. Rev. B 79, 214402 (2009).12 J. E. Hirsch, Phys. Rev. B 59, 6256 (1999).13 Y. Okimoto et al., Phys. Rev. Lett. 75, 109 (1995).14 Y. Okimoto et al., Phys. Rev. B 55, 4206 (1997).15 D. N. Basov and T. Timusk, Rev. Mod. Phys. 77, 721

(2005).16 A. D. LaForge et al., Phys. Rev. Lett. 101, 097008 (2008).

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CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMALPROPERTIES

036101 Bubble Generation in Germanate Glass Induced by Femtosecond LaserJue-Chen Wang, Qiang-Bing Guo, Xiao-Feng Liu, Ye Dai, Zhi-Yu Wang, Jian-Rong Qiu

036201 Spall Strength of Resistance Spot Weld for QP SteelChun-Lei Fan, Bo-Han Ma, Da-Nian Chen, Huan-Ran Wang, Dong-Fang Ma

036202 Ground-State Structure and Physical Properties of NB2 Predicted from First PrinciplesJing-He Wu, Chang-Xin Liu

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL,MAGNETIC, AND OPTICAL PROPERTIES

037101 Splitting Phenomenon Induced by Magnetic Field in Metallic Carbon NanotubesGui-Li Yu, Yong-Lei Jia, Gang Tang

037201 Microscopic Theory of the Thermodynamic Properties of Sr3Ru2O7

Wei-Cheng Lee, Congjun Wu

037301 Light-Emitting Diodes Based on All-Quantum-Dot Multilayer Films and the Influence ofVarious Hole-Transporting Layers on the PerformanceHui-Li Yin, Su-Ling Zhao, Zheng Xu, Li-Zhi Sun

037302 Tunneling Negative Magnetoresistance via δ Doping in a Graphene-Based Magnetic TunnelJunctionJian-Hui Yuan, Ni Chen, Hua Mo, Yan Zhang, Zhi-Hai Zhang

037303 Spin Caloritronic Transport of 1,3,5-Triphenylverdazyl RadicalQiu-Hua Wu, Peng Zhao, De-Sheng Liu

037401 Single Crystal Growth and Physical Property Characterization of Non-centrosymmetricSuperconductor PbTaSe2

Yu-Jia Long, Ling-Xiao Zhao, Pei-Pei Wang, Huai-Xin Yang, Jian-Qi Li, Hai Zi, Zhi-An Ren, Cong Ren,Gen-Fu Chen

037501 Zero-Magnetic-Field Oscillation of Spin Transfer Nano-Oscillator with aSecond-Order-Perpendicular-Anisotropy Free LayerYuan-Yuan Guo, Fei-Fei Zhao, Hai-Bin Xue, Zhe-Jie Liu

037502 A Single-Crystal Neutron Diffraction Study on Magnetic Structure of theQuasi-One-Dimensional Antiferromagnet SrCo2V2O8

Juan-Juan Liu, Jin-Chen Wang, Wei Luo, Jie-Ming Sheng, Zhang-Zhen He, S. A. Danilkin, Wei Bao

037801 Improvement of the Conductivity of Silver Nanowire Film by Adding Silver Nano-ParticlesYi Shen, Ruo-He Yao

CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE ANDTECHNOLOGY

038101 Characterization of Elastic Modulus of Granular Materials in a New Designed UniaxialOedometric SystemQin-Wei Ma, Yahya Sandali, Rui-Nan Zhang, Fang-Yuan Ma, Hong-Tao Wang, Shao-Peng Ma,Qing-Fan Shi

038201 Comparisons of Criteria for Analyzing the Dynamical Association of Solutes in AqueousSolutionsLiang Zhao, Yu-Song Tu, Chun-Lei Wang, Hai-Ping Fang

038501 Highly Efficient Greenish-Yellow Phosphorescent Organic Light-Emitting Diodes Based on aNovel 2,3-Diphenylimidazo[1,2-a]Pyridine Iridium(III) Complex

Jun Sun, Min Xi, Zi-Sheng Su, Hai-Xiao He, Mi Tian, Hong-Yan Li, Hong-Ke Zhang, Tao Mao,Yu-Xiang Zhang

038502 Electrical Instability of Amorphous-Indium-Gallium-Zinc-Oxide Thin-Film Transistors underUltraviolet IlluminationLan-Feng Tang, Hai Lu, Fang-Fang Ren, Dong Zhou, Rong Zhang, You-Dou Zheng, Xiao-Ming Huang,

038503 Current Controlled Relaxation Oscillations in Ge2Sb2Te5-Based Phase Change MemoryDevicesYao-Yao Lu, Dao-Lin Cai, Yi-Feng Chen, Yue-Qing Wang, Hong-Yang Wei, Ru-Ru Huo, Zhi-Tang Song

038801 Comprehensive Study of SF6/O2 Plasma Etching for Mc-Silicon Solar Cells

Tao Li, Chun-Lan Zhou, Wen-Jing Wang

038901 Fractal Analysis of Mobile Social NetworksWei Zheng, Qian Pan, Chen Sun, Yu-Fan Deng, Xiao-Kang Zhao, Zhao Kang

GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS

039801 Consistency Conditions and Constraints on Generalized f(R) Gravity with ArbitraryGeometry-Matter CouplingSi-Yu Wu, Ya-Bo Wu, Yue-Yue Zhao, Xue Zhang, Cheng-Yuan Zhang, Bo-Hai Chen