Thermal Transport Measurements in Strongly Correlated Electronic Systems

CHENG Shu-fan ,XU Hao ,BAO Song ,WEN Jin-sheng ,2

1.National Laboratory of Solid State Microstructures &Department of Physics,Nanjing University,Nanjing 210093,China

2.Collaborative Innovation Center of Advanced Microstructures,Nanjing University,Nanjing 210093,China

Abstract: When taking into account the electronic correlations such as the onsite Coulomb repulsion and coupling between electrons,spins and orbitals,many fascinating novel quantum states beyond the free-electron framework can emerge, e.g.,unconventional superconductivity and quantum spin liquids.The understanding of these new states not only will expand the existing territory of our knowledge,but also likely lead to revolution in quantum science and technology.Therefore,studying the strongly correlated physics is a cutting-edge theme in condensed matter physics.The parent state of cuprate high-temperature superconductors is a Mott insulator,an insulating state due to the strong electronic correlation,whereas the band theory predicts it to be metallic.Due to the Coulomb gap in Mott insulators,the charge degree of freedom is often frozen,which makes electrical transport measurements inapplicable.As a probe sensitive to the elementary excitations of quasiparticles not limited to electrons,but also including magnons,spinons,as well as phonons,thermal transport measurements play an important role in the study of strongly correlated electronic systems.In this paper,we review some of the recent interesting results on unconventional superconductors,heavy fermions and quantum spin liquids utilizing the longitudinal thermal transport measurements,complimentary to our recent review article on the progress of the transverse thermal conductivity measurements on the thermal Hall effect.

Key words: strongly correlated electronic systems;unconventional superconductivity;cuprates;heavy fermions;quantum spin liquids;geometrical frustration;thermal conductivity;Seebeck effect

I.INTRODUCTION

As quantum mechanics is recognized as the basic rule in the microscopic scale,researchers begin to study solid state physics with the theory derived from it,such as the band theory,which greatly advances condensed matter physics.Initially,the theory of energy band,which used single-electron approximation and neglected the correlation between electrons,successfully explained behaviors of most insulators,metals and semiconductors.The discovery of cuprates,a kind of unconventional superconductors which emerged from a strongly correlated electronic system known as Mott insulator,substantially boosted the research progress in strong correlation physics.Since a Mott insulator is a metal in the weak-coupling band theory,it is realized that the correlation effect between electrons cannot be ignored[1-2].

Correlation effects can bring many intriguing phenomena.Cuprate high-temperature superconductors are a typical example.They were first found by doping the parent compound La2CuO4,which was a Mott insulator,where the onsite Coulomb repulsionUopened a charge gap comparable toU,and made the otherwise conducting electron band fall well below the Fermi level,and thus the material became insulating.Injecting electrons or holes by doping will suppress the antiferromagnetic order in the Mott insulating state,and induce high-temperature superconductivity,which appears to be beyond the conventional BCS theory that describes superconductivity in the weak-coupling materials like metals well.Cuprate high-temperature superconductors provide plentiful phenomena like coherent superconducting peak,anisotropic superconducting gap,and pseudogap[3-4].Research on them is believed to contribute to deepen the understanding of quantum manybody physics and promote the development of applied science and industry greatly.Heavy fermion compounds are another classical correlated electronic system.It shows Fermi-liquid behavior but has huge effective electron mass caused by the hybridization of localized and itinerant electrons.Meanwhile,Kondo effect and Ruderman-Kittel-Kasuya-Yosida (RKKY)interaction will compete with each other and bring rich exotic phenomena.These systems attract sustained interest because they provide an ideal platform to delve into superconductivity,quantum magnetism,quantum phase transition,etc[5-7].Quantum spin liquid (QSL),a novel phase first proposed by Anderson,is also a Mott insulator and has been studied extensively recently[8].The spins maintain long-range correlation but do not exhibit any magnetic order even at absolute zero temperature,due to strong frustration induced by geometry(like in triangular or kagome lattice with antiferromagnetic interaction) or onsite anisotropic exchange coupling (like in honeycomb lattice with Kiteav interaction).The research of QSLs needs new theory of phase transition beyond Landau’s framework[9-11].It is expected to promote quantum communication and quantum computation because of the presence of long-range entanglement and exotic excitations like Majorana fermions that obey non-Abelian statistics.Because of the complexity,research on strongly correlated electronic systems calls for powerful experimental techniques.

Thermal transport measurement is one of such techniques..trictly speaking,thermal transport includes both longitudinal and transverse measurements.In our previous review paper,we have introduced the progress in the transverse thermal conductivity measurements,namely the thermal Hall effect[12].In this paper,we will focus on longitudinal transport properties,such as thermal conductivityκxxand Seebeck coefficientS,where the measured signals are parallel to the heat current.As a useful probe to detect excitations,thermal transport measurement can provide information about the ground and excited states as well as the interactions in the systems.Compared with electrical transport measurements,thermal transport measurements are harder to perform,because of the weaker signal over background ratio and the longer relaxation time,but are more useful under some conditions.For insulators,such as multiferroic materials and candidate materials of QSLs,where charge transport measurements are inapplicable,thermal transport research can be more effective and provide more information about the elementary excitations.For example,in a QSL,its longitudinal thermal conductivityκxxcan be written as:

where the linear term is contributed by spinons,which are charge-neutral fractional magnetic excitations carrying spin-1/2,and the cubic term is contributed by phonons.Plottingκ/T vs.T2,we can easily judge if spinons exist or not.Furthermore,using simple relation from Fourier heat equation,one can estimate the strength of spin correlations from the relation between thermal conductivityκand mean-free pathls:

whereCsis the heat capacity of spinons andvsis carrier velocity.Compared with heat capacity measurements,thermal transport measurements would not be disturbed by localized excitations and Schottky contributions,and thus provide more reliable details about itinerant excitations.Meanwhile,through pouring heat currents along different axes,thermal conductivity measurements can reflect the angular-dependence of a certain physical property,such as the four-fold symmetry in d-wave superconductors.

II.THERMAL TRANSPORT RESULTS IN CORRELATED SYSTEMS

A.Cuprates

The phenomenon of superconductivity was first discovered by Onnes in Hg monomers cooled in liquid helium,leading to a great deal of research on superconductivity[13].For conventional superconductors,BCS theory provides well-accepted explanations.Empirically,there is an upper limit of critical temperature,Tcof 30-40 K,which is known as the McMillan limit[14-15].However,with the discovery of unconventional superconductors,such as copper-and ironbased superconductors,this empirical limit has been broken[1,16-17].It is believed that the strength of the electron-phonon coupling is unable to pair up the strongly repulsive electrons,and sustain the highTcin the high-temperature superconductors.The mechanism of unconventional superconductivity is beyond the framework of BCS theory,but a commonly accepted alternative microscopic model is still unavailable,which deserves further exploration.

The parent compounds of cuprate superconductors are typical Mott insulators,like La2CuO4,which exhibits superconducting properties after doping with electrons or holes.Fig.1 shows the phase diagram of hole-and electron-doped cuprate superconductors[1].

Fig.1.Phase diagrams of cuprate high-temperature superconductors.Examples of electron-and hole-doped superconductors with Nd2-xCexCuO4 and La2-xSrxCuO4,respectively.AF and SC represent the antiferromagnetism and superconductivity,respectively[1].

One of the remarkable phases is the pseudogap phase that occurs in the hole-doped side.In the following section,we will introduce how thermal transport measurements play a role in the research of cuprates by showing a series of progress around the pseudogap region in La1.6-xNd0.4SrxCuO4.

In previous studies,the properties of some hole-doped superconductors,such as La2-xSrxCuO4(LSCO)and YBa2Cu3Oy(YBCO),will change remarkably around the critical doping concentrationp*,where it transitions from the pseudogap to the strange-metal state[18].For LSCO,its resistivity keeps linear temperature dependence until zero-temperature limit when the doping at and slightly larger thanp*.Otherwise,the resistivity turns up noteworthily below the pseudogap temperatureT*[19],corresponding to a crossover from a metal to an insulator[20].For LSCO and YBCO,the measurements of Hall coefficient reflect the abrupt drop of carrier densityn,such as the drop fromn=1+pforp ＞p*ton=pforp ＜p*in YBCO.This change of carrier density reveals that the pseudogap phase causes the transformation of Fermi surface[21-22].

La1.6-xNd0.4SrxCuO4(Nd-LSCO) is a kind of single-layer cuprates which is hole doped.Being similar with two materials discussed above,the resistivity and Hall number of Nd-LSCO change whenpcrosses over thep*=0.23.Below thep*,there is an upturn in the resistivity in the pseudogap phase as shown in Fig.2(a),which may be explained by the localization of excitations.Fig.2(b) shows the Hall coefficient near thep*: thepdecreases belowp*,there is a steep rise in the Hall coefficient,indicating the drop of carrier density and the transformation of Fermi surface.The loss of carrier density may also cause the upturn of the resistivity as shown in Fig.2(a).Therefor,the real mechanism(carrier localization or carrier density drop)that causes the abnormal behavior of the resistivity is worth investigating.

Fig.2.(a)Temperature dependence of the resistivity of Nd-LSCO at μ0H=0 T (gray) and μ0H=33 T (red for p=0.22 and blue for p=0.24)[23].(b) Temperature dependence of Hall coefficient of Nd-LSCO at different doping concentrations[24].

Thermal transport measurements can provide more information about the excitations in these systems..ortunately,because of the relatively low upper critical field of Nd-LSCO,one can suppress the superconductivity with static magnetic fields instead of pulsed ones and then measure the thermal transport properties in the normal state with long enough relaxation time,which is unattainable for LSCO and YBCO due to their high upper critical fields.In Ref.[23],Michon et al.measured the ultralow-temperature thermal conductivityκof Nd-LSCO in a 15 T static magnetic field,and the results are shown in Fig.3.Combiningκwith the electrical resistivity measurements,they pointed out that whetherpwas smaller thanp*or not,the relation of thermal conductivity and electrical resistivity in metal,called Wiedemann-Franz law,was correct:

Here,κ(0)/Tis the residual electronic term of the thermal conductivity in the normal state,ρ(0) is electrical resistivity asT →0 K in the normal state andL0is the Lorenz number with Sommerfeld correction.

Fig.3.(a) Doping dependence of the thermal conductivity of Nd-LSCO plotted as κ0/T vs.p in the normal state(μ0H=15 T,red dots) and superconducting state (μ0H=0 T,red circles),where κ0 is the zero-temperature thermal conductivity and p is the doping concentration.Blue squares show the doping dependence of the resistivity at zero temperature ρ(0) in the normal state,plotted as L0/ρ(0) vs.p,where L0 is the Lorenz number with the Sommerfeld correction.(b) The ratio of κN/T to L0/ρ(0),where κN is the κ0 in normal state.The p* is the critical doping of pseudogap phase[23].

These results have several implications.First,the Wiedmann-Franz law holds acrossp*means no matter in or out of the pseudogap state,there are welldefined fermions,which are mobile and play roles in carrying energy and charge simultaneously.Second,althoughκ(0) andρ(0) follow the Wiedemann-Franz law whenpcrossesp*,their numbers abruptly change.This result indicates a metal-metal transition with the change of carrier density but not an insulator-metal crossover.Third,the sharp change of thermal conductivity acrossp*is observed at zero field as well,which means this transition is independent of the magnetic field and the sharp change in transport measurements may come from the loss of carrier density when the pseudogap phase appears.

Collignon et al.plotted the temperature-doping phase diagram of Nd-LSCO by measuring the Seebeck coefficientSwith different dopings[25].As shown in Fig.4,at low temperatures,pseudogap phase appears whenp ＜ p*and charge-density-wave (CDW) phase appears whenp ＜pCDW,the latter of which is argued to be due to the rebuilding of the Fermi surface.

Fig.4.Phase diagram of Nd-LSCO,PG and CDW represent pseudogap and charge-density-wave phases,respectively. Tc denotes the superconducting transition temperature. T*/TCDW and p*/pCDW refer to the temperature/doping of the borders of PG and CDW phases,respectively[25].

The data of the Seebeck coefficient is shown in Fig.5.The change inS/Tnear thep*provides further evidence that the increase in the Hall coefficientRHafter the appearance of the pseudogap is due to the loss of carrier density.The experimental results show that as the doping decreases belowp*,the Seebeck coefficient of Nd-LSCO in the pseudogap phase jumps to about five times of the value atp ＞p*.Eq.4 shows that the Seebeck coefficientSis controlled by the carrier densitynand heat capacity of electronsCe(whereTandeare temperature and electron charge,respectively).

In Ref.[26],it is shown that a change inpfrom 0.24 to 0.20 results in a decrease inCeof Nd-LSCO of about 50%.Therefore,the appearance of the pseudogap must be accompanied by a sharp decrease in carrier densityn,which suggests that the jumps in the Hall coeffi-cient and thermal conductivity at the critical dopingp*should be due to the change in carrier density.Note that although Eq.4 is simple,it has been proved to hold in strongly correlated systems such as heavy fermions and cuprates[27].

The above research successfully provides a finer phase diagram of Nd-LSCO,but it contradicts with some other works.For instance,Ref.[28]pointed out that there is an electronic pocket in the Fermi surface for the overdoped Nd-LSCO in theory,meaning that the Seebeck coefficient,which is positive in the previous experimental study[25],should be negative.

Essentially,Seebeck effect is a phenomenon induced by the difference between particles and holes that carry heat and electric charge.Considering the simplest case,if there is one particle and one hole near the Fermi surface and have the same dispersions and mean-free paths,the Seebeck coefficient should be zero.When the major carriers are electrons that have higher carrier density or longer mean-free path,a negative Seebeck coefficient will be observed.

Fig.5.(a)-(c) Seebeck coefficients of Nd-LSCO in different magnetic fields and temperatures for p=0.24 (a),0.22 (b) and 0.17 (c).(d) Seebeck coefficient of Nd-LSCO in the normal state (red squares for μ0H=16 T and blue dots for μ0H=33 T) at T=5 K[25].

Fig.6.In-plane and out-of-plane Seebeck coefficients (Sa and Sc) of Nd-LSCO in the strange-metal phase (p=0.24 and μ0H=16 T)[29].

Considering the asymmetry of the energy dependence in the scattering rate,Gourgout et al.described the variation of the Seebeck coefficient with temperature of Nd-LSCO in the strange-metal phase successfully[29].Temperature dependence ofS/Tis shown in Fig.6 and there are some notable features: 1),Sakeeps negative at about 300 K and then turns into positive at lower temperatures.2),Scis always positive in the experiment and has a shoulder at about 60 K.At lower temperatures,Scgrows rapidly as the temperature decreases.Considering the metal nature of Nd-LSCO[23],the authors chose Mott formula that has momentum-dependent elastic scattering rate to describe the behavior of the Seebeck coefficient at low temperatures.The results whose scattering rate is independent of the energy is shown with the dashed line in Fig.7(b) and (d).Compared with the experimental data shown in Fig.7(a) and (c),this description gives negative values in the whole temperature range forSaand gives a wrong magnitude forSc.After considering the correction which has asymmetric energy dependence,a better way to describe the Seebeck coefficient of Nd-LSCO is brought forward,shown as the solid lines in Fig.7(b) and (d).The correction of the scattering rate is shown in Eq.5:

The scattering rate term 1/τSMFLcomes from“skewed”marginal Fermi liquid (SMFL) model and “skewed”means the energy coefficienta+(a+for? ＞0 anda-for? ＜0).Energy-independent scattering rate 1/τ0(→k) and temperature coefficientαare model parameters whose values came from angle-dependent magnetoresistance measurements[30].The reason why the authors chose this model was the good linear temperature dependence of resistivity in both directions for Nd-LSCO,which agreed with the marginal Fermi liquid ansatz[31-32].

Fig.7.(a) and (b) Comparison between the experimental data and calculated values of the in-plane Seebeck coefficients Sa/T of Nd-LSCO.Dashed lines and solid lines are calculated with scattering rate 1/τ0(→k) and 1/τ0(→k)+1/τSMFL(?,T)respectively,SMFL means skewed marginal Fermi liquid.(c) and (d) Comparison of the out-of-plane Seebeck coefficients Sc/T similar to (a) and (b)[29].

The scattering rate caused by the SMFL mechanism,1/τSMFL,is shown in Fig.8.It is obviously that carriers with negative energies have smaller scattering rates than the positive ones.Although electronic pockets exist in the Fermi surface,holes play more important roles in the energy transportation and thus causes a positive Seebeck coefficient.Note that as the temperature increases,the effect of the particle-hole asymmetry diminishes,so the electronic and hole pockets on the Fermi surface will almost compensate with each other,leading to an almost zero Seebeck coefficient.This is manifested in Fig.6,where the positiveSaalmost vanishes at high temperature.At 300 K,contributions from the electronic pockets slightly win over those from hole pockets,giving rise to a small negative Seebeck coefficient corresponding to the electronic carriers.

B.Heavy fermions

Fig.8.Energy dependence of scattering rate 1/τSMFL comes from skewed marginal Fermi liquid model at different temperature 2 K (blue),25 K (green) and 50 K (red)[29].

Most of heavy Fermion systems are compounds containing rare-earth elements that exhibit 4f electronic properties predominantly and usually reflecting the behavior of Landau Fermi liquids which are typical strongly correlated electronic systems.The terminology “heavy” is reflected in the fact that,according to Landau Fermi liquid theory,the effective mass of electrons is usually thousands of times of that of a typical Fermi liquid.Unlike cuprates whose parent materials exhibit insulator-like properties and the localization of electrons,the main properties of heavy fermions are contributed by the itinerant 4f electrons,which are also of great interest for research in superconductivity,topology,quantum phase transitions and other directions.

Ce-based compound is a classical heavy Fermion system and CeMIn5(M=Co,Rh,Ir) is one of the most heavily studied systems.Being similar with cuprates,CeCoIn5was believed to have a d-wave superconducting gap with adx2-y2symmetry,exhibiting a four-fold rotation invariability aboutc-axis in both heat capacity and thermal transport measurements with magnetic field rotated ina-bplane[33-34].Its cousin material,CeIrIn5,was pointed out to be similar to CeCoIn5in thermal and spin-lattice relaxation measurements[35-37],although itsTcof 0.4 K was much lower than that of 2.3 K in CeCoIn5[34].Meanwhile,Ref.[38]pointed out that there were two separated superconducting phases in the phase diagram of Ce(Rh1-xIrx)In5,which suggests these two materials may hold different superconducting mechanisms.These experimental results call for further investigations into the pairing mode of electrons in CeIrIn5.As a probe sensitive in detecting quasiparticle excitations,thermal conductivity measurements not only reflect the presence or absence of superconducting energy gap,but also provide a basis for determining the fine structure such as the symmetry of gap or the location of node.Thus,careful thermal conductivity measurements have been studied in CeIrIn5.

As shown in Fig.9,the heat capacity of CeIrIn5does seem to have the four-fold rotation symmetry aboutc-axis like CeCoIn5which holdsdx2-y2symmetry[33-36].However,Shakeripour et al.challenged this conclusion by measuring in-plane and out-of-plane thermal conductivity for CeIrIn5[39].They pointed out that their results were not consistent with the line nodes indx2-y2symmetry but instead might be consistent with a hybrid gap structure with a nodal loop in the equator and two nodal points at the north and south poles as shown in Fig.10.

Before showing the results from Ref.[39],it is necessary to illustrate how the energy gap structure influences thermal conductivity in the superconducting state.For the simplest one,s-wave,the energy gap is isotropic in the reciprocal space,meaning that there are no Fermi-like excitations in the zero-temperature limit and the linear term coefficient of thermal conductivity,κ0/T,should be zero in this case.If there are line or point nodes existing in the gap,they will contribute to gapless excitations which will causeκ0/0 in the zero-temperature limit[40].Fig.11 showsκ/Tfor two materials NbSe2and Tl2Ba2CuO6+δ,which are typical s-and d-wave superconductors,respectively.Note that the latter has a non-zero linear term contributed by zero-energy excitations from the line nodes.

Now let us come to the results from Ref.[39]as shown in Fig.12.As the temperature decreases and crossesTc,thermal conductivity of the normal state(induced by a 0.5 T magnetic field) and the superconducting state separates for current flowing either along theaorc-axis,as shown in Fig.12(a).The suppression ofκin the superconducting state indicates the presence of energy gap and decrease of carriers.For the in-plane thermal conductivity,there is a non-zero intercept when extrapolating the data toT=0 K,which calls for a line node structure in thekx-kyplane.For the current flowing alongc-axis,no non-zero term exists when temperature goes to zero,which strongly contradicts with adx2-y2symmetry that has four vertical line nodes alongc-axis in the Fermi surface.Combing these results with the symmetry of the point group,the authors suggested that the (1,i) state of theEgrepresentation,which had a hybrid gap structure as shown in Fig.10 was the only candidate gap structure for CeIrIn5.

It is worth noting that doping can affect the validity of the above criteria in identifying gap structures.In Ref.[40],it is pointed out that for materials with gapped excitations,doping may introduce some gapless excitations and causeIn order to study the universal behaviors of the energy gap,Shakeripour et al.continued their thermal transport measurements in CeIrIn5with La doping in Ref.[41].

Fig.9.Heat capacity C and thermal conductivity κ of CeCoIn5 (upper row) and CeIrIn5 (lower row) with magnetic field H rotated in the a-b plane[33-36].

Fig.10.Superconducting energy gap for d-wave with a dx2-y2 symmetry (left) or hybrid structure (right).Red lines and red points show the line nodes and point nodes respectively[39].

Let us focus on the changes after La doping,as shown in Fig.13.The in-plane thermal conductivity of Ce0.999La0.001IrIn5becomes smaller than the pure one,which may be caused by more scattering from La atoms,as shown in Fig.13(a)[41].However,in the zerotemperature limit,the intercept does not change obviously.Reference[40]discussed this case.In the clean limit,a universal heat conduction from line node atT →0 has a remarkable property: doping will not influenceκ/T|T=0because the increase in the zeroenergy density of states caused by the impurity is offset by the decrease of the mean-free path.Thus,the thermal transport measurements in thea-bplane after doping further proves the existence of a horizontal line node in the basal plane.Forκcin Fig.13(b),Ce0.999La0.001IrIn5has a non-zero linear term at low temperatures,suggesting that impurity induces gapless excitations alongc-axis.Limited by the measurement error,they could not infer the gap structure in the Brillouin zone withkz0 yet.

Fig.11.Thermal conductivity κ of s-wave (NbSe2,blue)and d-wave (Tl2Ba2CuO6+δ,red) superconductor plotted as κ/T vs.T2[40].

Fig.12.(a)Normal state(circle)and superconducting state(dot)thermal conductivity of CeIrIn5 with heat current flowing in the a-b plane (top) and along the c-axis (bottom).(b) Anisotropy ratio κc/κa of CeIrIn5 in the normal (circle) and superconducting state (orange dot).Inset: Anisotropy ratio for d-wave (blue) and s-wave (red) superconducting gap from theory[39].

Fig.13.Thermal conductivity κ of pure (black) and doped (blue) CeIrIn5，plotted as κ/T.Open symbols for the normal state (μ0H=0.5 T) and filled symbols for the superconducting state (μ0H=0 T)[41].Top and bottom panels are the results for current flowing along a-and c-axis,respectively.

The interplane thermal conductivity of CeIrIn5,κc,with in-plane and out-of-plane magnetic field have also been measured recently,reflecting more details in the gap structure and revealing some subphases in the superconducting state[42].

Thermal conductivityκcof CeIrIn5with magnetic field applied alongc-axis is shown in Fig.14.Extrapolating the data toT=0 K with the linear term,fitted asκc/T=κ0/T+κ1/T ×T,gives a negative intercept,which is nonphysical.However,using square termκ2that is expected in the superconductors with point nodes to replace the linear termκ1,as

Fig.14.Interplane thermal conductivity κc of CeIrIn5 with magnetic field applied along c-axis.Solid and dashed lines are fits with linear term and square term,respectively[42].

gives a more reasonableκ0/Twhich is positive.Theoretically,the square term comes from the contribution of point nodes.This result provides evidence for the hybrid gap structure which has two point nodes at poles,supporting the conclusion drawn in Ref.[39].

Magnetic field changes the structure of the energy gap in CeIrIn5.When the magnetic field is applied perpendicular to the heat current that is alongc-axis,a kink inκcappears at about 0.07 K,as illustrated with black arrows in Fig.15(a).This kink may indicate a phase transition caused by the magnetic field at low temperatures.Furthermore,let us examine the results about the residual thermal conductivity termκ0that reflects the density of state for gapless excitations,as shown in Fig.15(b) with red dots.Theoretically,for the gap structure without line nodes alongkcdirection,κ0(H) grows with magnetic field in the exponential form,until the superconductivity is fully suppressed byHand the system turns into the normal state.The relation ofκ0and the thermal conductivity in the normal stateκNis:κ0(H)～κN(eH/Hc-1),as shown in Fig.15(b) with the dashed line.For CeIrIn5,the residual thermal conductivity termκ0normalized to the normal stateκN(μ0Hc2=1 T)is measured and shown in Fig.15(b).Theκ0of CeIrIn5grows faster than the exponential form when magnetic fieldHis in the range of 0.05Hc2＜H ＜0.4Hc2.The onset of this abnormal behavior in the thermal conductivity and the characteristic magnetic fieldHkinkin the heat capacity measurements pointed out with the black arrow in Fig.15(b)are the same.Both of them indicate a reconstruction of the energy gap from the hybrid structure to the one with line nodes alongkcas shown in the bottom of the phase diagram in Fig.16.

C.Quantum spin liquids

Quantum spin liquid(QSL)is a novel phase in condensed matter physics.It represents a state where the spins are highly entangled but do not establish longrange magnetic order even down to absolute zero temperature,due to the existence of strong frustrations and quantum fluctuations[9-11].The ground state of a QSL cannot be described with any local order parameter and so is beyond the framework of phase and phase transition from Landau.Although it has rich physics and is believed to be related to quantum communication and computation,difficulties in identifying a QSL hinder the development.Long-range quantum entanglement and presence of fractional magnetic excitations such as spinons are the two key characteristics of a QSL.It is challenging to probe the former one experimentally at present.Observations of the continuum in the inelastic neutron scattering (INS) measurements provide useful clue to pin down a QSL as spinons will give rise to a continuum[43-48].As a probe highly sensitive to lowenergy elementary excitations such spinons in QSLs,thermal conductivity measurement is getting more and more attention.Especially,since typical QSL candidates are Mott insulators,which electrical transport measurements cannot get reasonable response,thermal transport measurements become even more important.Examining the existence of the linear term in the thermal conductivity as inκ(T)=aT+bT3provides a powerful way to hunt spinons.

1.EtMe3Sb[Pd(dmit)2]2

EtMe3Sb[Pd(dmit)2]2(dmit-131) is a layered organic dimer-based Mott insulator,which has a nearly regular-triangular lattice ofdimers[49].13C NMR measurements showed that the spins in dmit-131 kept liquid-like state down to 19.4 mK,which was smaller than 0.01% of exchange interactionJ[49].Meanwhile,the heat capacity of dmit-131 was larger than its non-magnetic cousin material Et2Me2Sb[Pd(dmit)2]2(dmit-221)[50].The excess contribution is believed to come from spinons,after excluding contributions from electrons,magnons and phonons..nother organic QSL candidateκ-(BEDT-TTF)2Cu2(CN)3had similar behaviors with dmit-131,showing no magnetic order and no spin freezing at ultralow temperatures[51].Compared with the antiferromagnetκ-(BEDT-TTF)2[N(CN)2]Cl,magnetically-disordered materialκ-(BEDT-TTF)2Cu2(CN)3even shows larger heat capacity[52].This implies that there are some excitations existing in the latter one,providing more heat capacity than magnons in the former.The excitations in the QSL candidateκ-(BEDT-TTF)2Cu2(CN)3are argued to be gapless spinons.However,in this material,there is no linear term observed in the thermal conductivity measurement,contradicting the result of heat capacity which indicates the presence of gapless spinons[53].So it is haunted that whether a gapless QSL can show a residual linear term in the thermal conductivity or not.

Fig.15.Thermal conductivity of CeIrIn5,which was measured with heat current flowing along c-axis and magnetic field along a-axis.(a) The temperature dependence of the thermal conductivity along c-axis,plotted as κc/T vs.T for different magnetic fields.Dashed lines show the extrapolation with the equation κc/T=κ0/T+κ2/T×T2.(b)Thermal conductivity κc normalized to the normal-state thermal conductivity κN,plotted as [κc/T]/[κN/T] vs.H/Hc2.The data at temperature T →0 are the interceptions of the dashed lines on the y-axis in (a) and are plotted with red dots.Experimental data at T=57 mK and 100 mK are shown with open circles and green triangles,respectively.Black squares are the heat capacity of CeIrIn5 normalized to the normal state plotted on the right y-axis and the kink field of heat capacity is marked by the vertical arrow[42].

Fig.16.Phase diagram for CeIrIn5 with magnetic field perpendicular to the c-axis. Tc donates the superconducting temperature and Tkink denotes the temperature of the kink feature of CeIrIn5 in the thermal conductivity. Eg(1,i) and Eg(1,0) states appear at the bottom of the diagram with different magnetic fields.The energy gap structures of the two states are shown with red lines and dots[42].

In 2010,Yamashita et al.reported that the residual linear term of the thermal conductivity existed in the zero-temperature limit[54],as shown in Fig.17(a).It is the first time that the linear term is observed in QSL candidates,which indicates the gapless spinons,as low-lying elementary excitations,are present in dmit-131.This result is consistent with the thermodynamic measurement as well[50].At the same time,the value of the residual linear term bears immediate implications on the strength of spin correlations.Considering spinons in dmit-131 as a two-dimensional Fermionic system,one can estimate the thermal conductivityκas:

wheredis interlayer distance andais nearest-neighbor spin distance.From Eq.7,one can get the mean-free path of spinonsls～1μm,which is about 1 000 times of the nearest-neighbor spin distance[50].This result shows the strong spin correlations and the high mobility of spinons in dmit-131.

Fig.17.(a) Temperature dependence of κxx/T as a function of T2 for dmit-131 (pink),dmit-221 (green) and κ-(BEDT-TTF)2Cu2(CN)3 (black)[53].(b) Magnetic response of κ at untralow temperatures[54].

In 2019,Ni et al.[55]and Bourgeois-Hope et al.[56]separately reported their results of thermal conductivity measurements in dmit-131,both of them did not observe the residual linear term in the zero-temperature limit,even their samples were provided by the same group as in Ref.[54].As shown in Fig.18,the question is not only whether the linear term exists or not,but also that the values of the thermal conductivity are 10 times smaller than those in the previous report[54].These new results are consistent with NMR measurements in dmit-131 which pointed out that the dynamical spin susceptibility disappeared in the zerotemperature limit,meaning there are no gapless spin excitations in dmit-131[57].

Because these results strongly question the previous conclusion,Yamashita et al.responded in two papers[58-59].In the first paper[58],they argued that because of the obvious sample dependence,the difference might be caused by i) phonons were scattered by spin excitations;ii) structural domain formation;iii)microcracks.In the second paper[59],they argued that dmit-131 was a kind of organic compound,so the results might be strongly affected by the cooling rate (their data is shown in Fig.19.

Fig.17(b) shows strong magnetic field response inκfrom Ref.[54].The authors pointed out that the field dependence was governed byκspinbecauseκphononwas insensitive to magnetic field.From the magnetic field dependence,they also argued that there were also spin-gap-like excitations in addition to the gapless excitations.Theoretically,the coexistence of gapless and gapped excitations is predictable in kagome lattice with strong frustration[60]but unexpected in triangular lattice.However,Fig.20 shows different field dependence of dmit-131.References[55-56]both show thatκhardly responds to magnetic field.By contrast,it has a 20%-30% increase in high fields in Ref.[54].

Fig.18.Thermal conductivity measurements in dmit-131 from Ref.[55] (a) and Ref.[56] (b),in comparison to those from Ref.[54] (purple ones).

Fig.19.Cooling rate dependence of κxx/T in dmit-131. Yaxis is displayed in logarithmic scale[59].

Fig.20.Magnetic field dependence of κ/T at low temperatures from Ref.[55] (a) and Ref.[56] (b).The inset in (b)shows the relative change in κ(T)caused by a magnetic field of μ0H=10 T.

2.YbMgGaO4

YbMgGaO4,a kind of rare-earth compounds with two-dimensional triangular lattice as shown in Fig.21,has antiferromagnetic interactions and effective spin-1/2 local moments.This material was first synthesized by Li et al.[61].Later on,many experiments have provided evidence that there is no magnetic order but persistent spin dynamics down to mili-Kelvin,and especially that inelastic neutron scattering measurements have observed the continuum which is expected for a QSL,all suggesting that YbMgGaO4is an ideal candidate material to realize the QSL state on a triangular lattice[61-64].

In 2016,Xu et al.presented their ultralowtemperature specific heat and thermal conductivity measurements on single crystals of YbMgGaO4[65].Magnetic heat capacity measurements showed there were two different types of spin excitations with different fields.In zero field,the temperature dependence of magnetic heat capacity showed a power law,Cm=cT0.74,which is consistent with Ref.[61]and indicates that YbMgGaO4is a gapless QSL with U(1)gauge fluctuations[63].Under high magnetic fields (6 and 9 T),Cmcould be fitted by an exponential form:Cm=de-Δ/kBT,which was attribute to the gapped magnons from fully-polarized state.

Fig.21.Schematic crystal structure of YbMgGaO4[61].Magnetic ions Yb3+ form two-dimensional triangular lattice as shown in (b).

Theoretically,with gapless excitations,a non-zero linear term should be observed in thermal transport measurement of YbMgGaO4,like shown in Fig.17(a)for dmit-131[54].However,in Ref.[65],the authors reportedκ(T) with negligible linear term at ultralow temperatures (Fig.22),suggesting that only phonons carry heat in this system.In Fig.22(a) and (b),compared YbMgGaO4with the nonmagnetic counterpart LuMgGaO4,one can find that the magnitude of thermal conductivity for the former is much less than the later.This means that the heat carriers,phonons,are strongly scattered by magnetic excitations.This conclusion was proved by the measurements under magnetic fields as well,as shown in Fig.22(b) and (c).Firstly,κ(H) decreases slowly with higher magnetic field below 2 T,then has a fast increase due to the opening of a spin gap by the magnetic field,and finally keeps steady after 5 T,where the spins turn into a fully-polarized state.In Ref.[47],similar results on the thermal conductivity have also been obtained in a sister compound YbZnGaO4.

In Ref.[65],the authors provided some possible reasons to explain why the thermal transport measurements contradicted with the expectation that YbMgGaO4was a QSL with gapless spin excitations: i)the gapless spinons do exist in YbMgGaO4but do not contribute to thermal transport,which may be caused by the strong scattering from phonons,magnetic domains and impurities;ii) the gapless spinons do not exist in YbMgGaO4,and the results of magnetic heat capacity measurement and inelastic neutron scattering INS spectrum are not originating from QSLs but have other origins,such as spin-glass state.Later on,the study in YbZnGaO4shows that the spin-glass state can simulate spin-liquid behaviors such as the power-law behavior in the heat capacity,and continuum in the INS measurements[47].

More recently,Rao et al.reported their ultralowtemperature thermal conductivity and magnetic torque measurements which strongly suggested the existence of QSL state in YbMgGaO4[66].They observedκ(T)to be well fitted byκ(T)=aT+bT2at two different temperature regions,and both could give a finite linear term indicating the presence of the spinons (Fig.23).Using the same way shown in Ref.[54],they estimated the mean-free path of the itinerant spin excitations,spinons,which was 7.84 nm at 200 mK＜T ＜600 mK (about 23 times of the inter-spin distance)and 2.16 nm below 200 mK(about 6 times of the inter-spin distance).Both of these distances are much shorter than that of dmit-131[54]and 1T-TaS2[67].The shorter mean-free path may be caused by the weaker exchange interaction or the more scattering between phonons and spinons.

They also showed more details of the field dependence of YbMgGaO4[66].First,being similar with Ref.[65],κshowed a significant increase with magnetic field,which means the magnetic scattering of phonons do exist in this system.Second,combining ultralowtemperature thermal conductivity measurement with magnetic torque and dc magnetization measurements,the authors pointed out that YbMgGaO4exhibited phase transitions at 1/3 andof the saturation moment,respectively,which indicated an up-up-down phase in the low-field region.

The thermal conductivity of YbMgGaO4responded to the magnetic field noteworthily,although Ref.[65]and Ref.[66]showed different results about the linear term.Meanwhile,the thermal conductivity of YbMgGaO4is less than non-magnetic cousin material LuMgGaO4.Together,these experimental results suggest that strong coupling between phonons and spin excitations does exist in YbMgGaO4and has significant impacts on heat transport significantly.Bearing this in mind,we can only separate the contribution of phonons and spinons in thermal conductivity when their coupling is neglected.So,even spinons do exist in YbMgGaO4,using equationκ=κphonon+κspinand using non-zero linear term inκas the evidence of the existence of spinons or not may need more careful considerations.

3.1T-TaS2

As a kind of van der Waals materials,1T-TaS2has two-dimensional Ta layers sandwiched by two S layers,and is considered as a fully gapped Z2 spin liquid or a Dirac spin liquid[68].As shown in Fig.24,undergoing a commensurate charge-density-wave transition at 180 K,each 13 Ta atoms make up a star-of-David cluster with effective spinSeff=1/2 at the center[69].With antiferromagnetic exchange interactions,the clusters construct a triangular structure which provide geometrical frustrations.In Ref.[70],it was pointed that no long-range magnetic order occurred even down to 70 mK byμSR measurements,suggesting 1T-TaS2to be a QSL candidate.

Fig.22.(a) and (b) Temperature dependence of thermal conductivity κ of YbMgGaO4 with different magnetic fields.(c)Temperature dependence of thermal conductivity of the nonmagnetic counterpart LuMgGaO4.(d) Magnetic response of κ of YbMgGaO4 at T=0.2 K[65].

Fig.23.(a) Temperature dependence of thermal conductivity κ of YbMgGaO4 along different directions: red squares for for in-plane κa and blue circles for out-of-plane κc.(b) Magnetic response of κ of YbMgGaO4 at different temperatures[66].

Yu et al.measured the thermal transport property of 1T-TaS2[72].They pointed out that the temperature dependence ofκ(T)/Tcould be well fitted by a power-law function,κ(T)/T～T1.69.This fitting without residual linear term means that only phonons can conduct heat and the power-law of phonon contribution is 1.69 rather than 2,indicating the spin-lattice coupling in 1T-TaS2.Meanwhile,it is clearly that the thermal conductivity did not change with magnetic field obviously even up to 9 T and the same fitting process gave similar result ofκ0/Tin different magnetic fields,as shown in Fig.25(c),which can be regarded as zero considering the experimental error.

Fig.24.Schematic crystal structure of 1T-TaS2.(a) The sandwich-like structure of S-Ta-S layers.(b) The triangular structure made of star-of-David clusters which host 13 Ta atoms for each star[71].

The authors listed some possibilities that could reconcile their data with QSL scenarios.For the scenario of gapless QSLs,there are two possibilities: i)The low-energy spin excitations may be exotic nodal bosonic excitations which contribute to an unknown powerlaw temperature dependence (～Tδ) toκ.ii) Nuclear quadrupole resonance measurements imply a highly inhomogeneous magnetic phase at all Ta sites[70].So the gapless excitations may still follow the Fermi distribution but are localized due to short correlation length ora strong scattering.For the scenario of a gapped QSL,being suggested in Ref.[68]as well,the authors argued that 1T-TaS2might be similar withκ-(BEDT-TTF)2Cu2(CN)3.Neither of them shows magnetic order nor spin freezing at low temperatures.Meanwhile,spin excitations contribute to the heat capacity but not to the linear term of the thermal conductivity in these two materials,indicating the scenario of gapped QSLs[51-53].In Ref.[53],it was pointed out that there was a gap which would close for magnetic field higher than about 4 T in the spin excitation spectrum ofκ-(BEDT-TTF)2Cu2(CN)3.But for 1T-TaS2,because of the large exchange interactionJ ≈0.13 eV estimated from the magnetic susceptibility[70],the gap is too large to be closed with a 9-T field.

However,a few months later,Ribak et al.suggested that the exchange couplingJthat contributed to the Curie-like term could be estimated asJ ≈0.1 meV[71].This value is much smaller than those of other QSL candidates and is three orders of magnitude smaller than the exchange interaction estimated fromχ,indicating that most of the spins do not contribute to the Curie-like part of the magnetic susceptibility.In Ref.[71],it is also shown that the linear termγexisting in the heat capacity,indicating that the low-energy spin excitations are gapless.So,to sum up,localized gapless spin excitations may be more likely than the gapped ones in 1T-TaS2if it is really a QSL material.

Murayama et al..eported their studies on 1T-TaS2with the different disorder levels (pure,Se-substituded and electron-irradiated) in Ref.[67].The thermal transport measurements showed that the linear term did exist inκbut it would be suppressed by impurities.The authors estimated the mean-free path of spinons usingκspin=1/3Cspin〈vs〉lspinand gotlspin≈5 nm,in which the inter-spin distance was about 1 nm.It is strange that all kinds of crystal did not respond to high magnetic field obviously,only with a slight enhancement.Meanwhile,the linear term did not disappear in 12-T magnetic field,indicating that even in such high field,itinerant gapless spin excitations still existed in 1T-TaS2.These results are different from YbMgGaO4where the linear term ofκis present in zero field but absent after turning into the fully-polarized state in high magnetic field.

Fig.25.(a) and (b) Temperature dependence of thermal conductivity κ of 1T-TaS2 with different magnetic fields.(c)Linear term of the thermal conductivity κ0/T obtained by optimal fitting of κ as κ(T)=κ0×T +κ1×T1.69[72].

Fig.26.(a) and (b) Thermal conductivity of two different pure 1T-TaS2 samples.(c) and (d) Thermal conductivity of two different Se-doped 1T-TaS2-xSex samples.(e) Thermal conductivity of electron-irradiated 1T-TaS2.Insets show the thermal conductivity κ plotted as κ/T vs.T2 in the ultralow-temperature region[67].

In the heat capacity measurements (Fig.27),all the samples show thatγ,the linear term in the heat capacity,is always present and increases with magnetic field at first until about 2.5 T then decreases in high field,indicating that not only gapless itinerant excitations,but also gapless localized excitations exist in 1T-TaS2,as explained in the following.Comparing different crystals with different disorder levels,the authors argued that there were two kinds of gapless magnetic excitations: the itinerant excitations were sensitive to doping and would disappear with strong disorder,but the localized ones were robust against disorder.It is worth mentioning thatγ,which is contributed by itinerant and localized excitations at the same time,do not change with magnetic field monotonously,so examining how the thermal conductivity responds to magnetic field may need more data than just at 0 T and 12 T.

4.Na2BaCo(PO4)2

Over time,research in QSLs was puzzled by the structure disorder,such as site mixing and lattice distortions that made the system turned into a randomsinglet state or spin-glass state.For example,the disorder caused by random site mixing of Mg2+/Zn2+and Ga3+makes the system deviate from QSL state in YbMgGaO4and YbZnGaO4[47],or even some nonmagnetic doping in 1T-TaS2will suppress the magnetic excitations[67].Therefore,a QSL candidate material with perfect structure and without intrinsic disorder has been long awaited.

Fig.27.Linear term in the heat capacity, γ,for different 1T-TaS2 samples (red and blue for pure ones,yellow for Se-doped samples,pink for electron-irradiated 1T-TaS2)[67].

In 2019,Zhong et al.reported a new QSL candidate material with geometrical frustrations,Na2BaCo(PO4)2,and listed a series of experimental characteristics revealing that it is an ideal system to realize the QSL state[73].The structure is shown in Fig.28.The magnetic ions,Co2+with spinSeff=1/2[74],construct the triangular lattice in theabplane.Because the nearest spins in the same layer interact antiferromagnetically through a Co-O-O-Co super-superexchange path but the interaction between the nearest spins in the neighboring layers is supersuper-superexchange through Co-O-O-O-Co chain,the intralayer coupling is much stronger than the interlayer coupling.In this sense,the system can be treated as a two-dimensional magnet.

Thermal.ransport.easurement.n Na2BaCo(PO4)2was reported one year later,by Li et al.[75]..he ultralow-temperature thermal conductivity was measured down to 70 mK (Fig.29).As expected,in the low-temperature range (up to 500 mK),κcan be well fitted byκ/T=a+bT2.From the finite linear term,one can also estimate the mean-free path of the itinerant spin excitations,which is about 7 times of the inter-spin distance.Under a 14-T field,no matter alonga-axis orc-axis,the data follow a simple power law asκ～T3and are obviously larger than the zero-field data.This behavior is similar to YbMgGaO4,meaning that the system turns into a fully-polarized state in high field and there is a strong spin-phonon coupling at zero field.It is worth noting here that the zero-field data plotted inκ/T vs.T2shows a weakly abnormally upturning feature below 100 mK,whose origin is still unknown at the moment.

Trying to find out more details of the magnetic excitations at ultralow temperatures,Li et al.measured heat capacitycpdown to 50 mK,as shown in Fig.30(a)[75].At zero field,there is a small but sharp peak atTN=148 mK,which represents the establishment of antiferromagneitc (AFM) order in common.The ac magnetic susceptibility (χac) measurements down to 50 mK shows that there is no long-range magnetic ordering nor spin-glass state because of no frequency dependence ofχac.The magnetic ordering at low temperatures is still suppressed by strong quantum fluctuation even after the AFM phase transition.This phase transition may be related to the abnormal feature at ultralow temperatures inκ(T)/T.With more details about the magnetic field dependence ofκ,cpandχ,an Up-Up-Down phase in the ultralow-temperature range is observed,revealing the effect of geometrical frustration on quantum spin state transitions.

III.CONCLUSIONS

In this paper we review the important role that thermal transport measurements,a powerful microscopic probe,play in the study of some strongly correlated electronic systems.For cuprates Nd-LSCO,thermal transport measurements indicate that whether the pseudogap occurs or not,the system keeps its metal nature and holes well-defined fermionic excitations.Seebeck coefficient measurements prove that the upturn of resistivity and the increase of Hall coefficient in pseudogap phase are caused by the loss of carrier density rather than the localization of excitations.Furthermore,the particle-hole asymmetry in strange metal phase was observed through Seebeck coefficient measurements and is described by“skewed”marginal Fermi liqui model successfully.For heavy fermion system CeIrIn5,which shows superconductivity at low temperatures,intraplane and out-of-plane thermal conductivity measurements show its superconducting gap structure.Being different from CeCoIn5with adx2-y2symmetry,CeIrIn5has a hybrid gap structure where there is a nodal line in the equator and two nodal points at the poles.Furthermore,measurements in doped samples support the presence of the nodal line but cannot provide information about the nodal points.Applying magnetic field in thea-bplane reveals point nodes at the poles and suggests the presence of subphases under fields.For QSL candidates,thermal conductivity measurements are considered to be one of the most powerful tools to detect spinons.But for most of the candidates,there are strong discrepancies about the results and conclusions,not only on the existence of spinons or not,but also on the magnetic response or the magnitude ofκ.For YbMgGaO4,the strong evidence for the spinphonon coupling may restrict detecting the existence of spinons by the linear term in the thermal conductivity.

Fig.28.Schematic crystal structure of Na2BaCo(PO4)2[67].

Fig.29.Thermal conductivity of Na2BaCo(PO4)2 with different magnetic fields.(a) is plotted as κ/T vs.T and (b)-(d)are plotted as κ/T vs.T2[75].The inset in (b) is plotted as κ/T vs.T2 in the ultralow temperature region.

Fig.30.(a)Temperature dependence of the heat capacity of Na2BaCo(PO4)2 in different magnetic fields.(b)Magnetization curves of Na2BaCo(PO4)2 at different temperatures[75].

The coupling between spin and lattice may be a dilemma for thermal transport research in QSLs.In the strong coupling condition,it may be problematic to treat their contributions to the thermal conductivityκindependently asκ=κspin+κlattice.However,as discussed in Ref.[76],when spin and lattice are decoupled by the magnetic field,heat hardly transports from spin to lattice.This means that if spin-lattice coupling is weak and the above equation is reliable,it is hard to observe heat transport relying on the spin part because the sensors are only coupled with the lattice rather than the spins.Therefore the decoupling of spin and lattice and the observability of spin heat conductivity are contradictory.For the coupling condition like YbMgGaO4,it is still debatable whether the linear term in the thermal conductivity represents the existence of spinons.For decoupling conditions,evidence about spin current caused by temperature gradient rather than heat current carried by spins may be a new experimental hallmark for spinons[77].

Thermal transport measurements is a powerful probe for strongly correlated systems,but it is harder to perform than electrical transport measurements.Thermal relaxation time is longer than electrical’s,which calls for more stable environments,including temperature and magnetic field.Meanwhile,being different from electrical measurements that deal with electrical signals directly,measuring thermal signals needs to turn them to electrical ones by standard resistances or thermal couples.This disadvantage calls for more elements and brings more errors in.Considering the small temperature gradient (no more than 3% to 5% of environment temperature,which means tens of millilKelvin at ultralow temperatures),measuring thermal signals accurately is a great challenge.As mentioned in Ref.[25],the authors measuring Seebeck coefficient by ac technique.Lock-in amplifier can improve precision by filtering and amplifying thermal signals,providing a possible way to make thermal transport measurements more accurate.We believe that with more sustained efforts,this field will develop more rapidly and provide more and more reliable and important information on the low-energy excitations of the strongly correlated electronic systems.

ACKNOWLEDGMENTS

The work was supported by National Key Projects for Research and Development of China with Grant No.2021YFA1400400,the National Natural Science Foundation of China with Grants No.12225407 and 12074174,China Postdoctoral Science Foundation with Grants No.2022M711569 and 2022T150315,Jiangsu Province Excellent Postdoctoral Program with Grant No.20220ZB5,and Fundamental Research Funds for the Central Universities.