Correlation between electrons and vortices in quantum dots

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纳米氧化镧负载碳上用于催化

1. INTRODUCTION During the past decades, oxygen reduction reaction (ORR) has always been considered as one of the most important electrochemical reactions because of its wide applications in advanced energy storage and conversion devices, such as fuel cells and metal-air batteries.1,2 However, because of the sluggish kinetics of ORR, there is great need for developing eﬃcient electrocatalysts to promote the ORR process. Carbonsupported platinum (Pt/C) is currently thought as the most active electrocatalyst for the ORR process in both alkaline and acidic environments, which has been developed and utilized in various fuel cells and metal-air batteries, but the high cost and limited availability of Pt hinder its further application.3−5 Therefore, it is still a key challenge to develop an ORR catalyst

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Liang Guo Stephen L.Hodson Timothy S.FisherXianfan Xu1e-mail:xxu@ School of Mechanical Engineering and Birck Nanotechnology Center,Purdue University,West Lafayette,IN47907Heat Transfer AcrossMetal-Dielectric Interfaces During Ultrafast-Laser Heating Heat transfer across metal-dielectric interfaces involves transport of electrons and pho-nons accomplished either by coupling between phonons in metal and dielectric or by cou-pling between electrons in metal and phonons in dielectric.In this work,we investigate heat transfer across metal-dielectric interfaces during ultrafast-laser heating of thin metalﬁlms coated on dielectric substrates.By employing ultrafast-laser heating that cre-ates strong thermal nonequilibrium between electrons and phonons in metal,it is possible to isolate the effect of the direct electron–phonon coupling across the interface and thus facilitate its study.Transient thermo-reﬂectance measurements using femtosecond laser pulses are performed on Au–Si samples while the simulation results based on a two-temperature model are compared with the measured data.A contact resistance between electrons in Au and phonons in Si represents the coupling strength of the direct electron–phonon interactions at the interface.Our results reveal that this contact resist-ance can be sufﬁciently small to indicate strong direct coupling between electrons in metal and phonons in dielectric.[DOI:10.1115/1.4005255]Keywords:interface thermal resistance,ultrafast laser,thermo-reﬂectance,two-temper-ature model,electron–phonon coupling1IntroductionInterface heat transfer is one of the major concerns in the design of microscale and nanoscale devices.In metal,electrons,and pho-nons are both energy carriers while in dielectric phonons are the main energy carrier.Therefore,for metal-dielectric composite structures,heat can transfer across the interface by coupling between phonons in metal and dielectric or by coupling between electrons in metal and phonons in dielectric through electron-interface scattering.Phonon–phonon coupling has been simulated mainly by the acoustic mismatch model and the diffuse mismatch model[1].As for electron–phonon coupling,there are different viewpoints.Some studies have assumed that electron–phonon coupling across a metal-dielectric interface is negligible and heat transfer occurs as electron–phonon coupling within metal and then phonon–phonon coupling across the interface[2].Electron–phonon coupling between metal(Cr,Ti,Al,Ni,and Pt)and SiO2 has exhibited negligible apparent thermal resistance using a parallel-strip technique[3].On the other hand,comparison between simulations and transient thermal reﬂectance(TTR) measurements for Au-dielectric interfaces reveals that energy could be lost to the substrate by electron-interface scattering dur-ing ultrafast-laser heating,and this effect depends on electron temperature and substrate thermal properties[4–6].In this study,we employ TTR techniques to investigate inter-face heat transfer for thin goldﬁlms of varying thicknesses on sili-con substrates.(Here,we consider silicon as a dielectric since heat is carried by phonons in silicon.)Similar work has been reported [5].In our model,we consider two temperatures in metal and also the temperature in the dielectric substrate.This allows us to inves-tigate the effect of both the coupling between electrons in metal and phonons in the dielectric substrate,and the coupling between phonons in metal and phonons in the dielectric substrate,and allows us to isolate the effect of the electron–phonon coupling across the interface that can be determined from the TTR mea-surement.Experimentally,we employ pulse stretching to mini-mize the effect of nonequilibrium among the electrons.As a result,the experimental data can be well-explained using the com-putational model.The thermal resistance between electrons in Au and phonons in Si,which quantiﬁes the direct electron–phonon coupling strength,is calculated from the measured data.The results reveal that in the thermal nonequilibrium state,this electron–phonon coupling at the interface is strong enough to dominate the overall interface heat transfer.2TTR MeasurementAu–Si samples of varying Au thicknesses were prepared by thermal evaporation at a pressure of the order of10À7Torr.The thicknesses of the goldﬁlms are39,46,60,77,and250nm,meas-ured using an atomic force microscope.The pump-and-probe technique is used in a collinear scheme to measure the thermo-reﬂectance signal.The laser pulses are generated by a Spectra Physics Ti:Sapphire ampliﬁed femtosecond system with a central wavelength of800nm and a repetition rate of5kHz.The wave-length of the pump beam is then converted to400nm with a sec-ond harmonic crystal.The pump pulse has a pulse width(full width at half maximum-FWHM)of390fs measured by the sum-frequency cross-correlation method and is focused onto the sam-ple with a spot radius of20.3l m.The probe beam has a central wavelength of800nm and a pulse width of205fs measured by autocorrelation and is focused with a spot radius of16.9l m.This pump pulse width is intentionally stretched from the original pulse width of50fs to minimize the inﬂuence of thermal nonequili-brium among electrons since the electron thermalization time in Au can be of the order of100fs[7].This thermalization time is pump wavelength and pumpﬂuence dependent,and can be of the order of10fs if higher laserﬂuence is used[8,9].Our experiments did show the importance of pulse stretching.Figure1shows the TTR measurement results for the sample of thickness77nm with different pumpﬂuences before and after stretching the pulse.The plots show the normalized relative reﬂectance change(ÀD R/R)1Corresponding author.Contributed by the Heat Transfer Division of ASME for publication in the J OURNAL OF H EAT T RANSFER.Manuscript received May18,2011;ﬁnal manuscript received September30,2011;published online February13,2012.Assoc.Editor: Robert D.Tzou.with the delay time between the pump and the probe pulses to show the contrast in cooling rates.With a shorter pulse (Fig.1(a )),a steep initial drop is seen in the signal,which is attributed to the behavior of nonequilibrium among electrons.Since the TTM to be used for simulation assumes a well-deﬁned tempera-ture for electrons,i.e.,the electrons in gold have reached thermal equilibrium (not necessarily a uniform temperature),the model cannot predict the fast initial drop in the signals in Fig.1(a ).As will be shown later,the signals obtained by stretching the pulse can be predicted well using the TTM.3Two-Temperature Model for Thermal Reflectance MeasurementsUltrafast-laser heating induces thermal nonequilibrium between electrons and phonons in metal,which can be described by the TTM [10–13].We note that the heterogeneous interface consid-ered here involves three primary temperature variables (two in the metal and one in the dielectric).The “two-temperature”model is applied to the metal side.For investigating electron–phonon and phonon–phonon coupling at the interface,two thermal resistances are deﬁned:R es (its reciprocal)indicates the coupling strength between electrons in metal and phonons in dielectric,while R ps indicates the coupling strength between phonons in metal and phonons in dielectric.(Large thermal resistance corresponds to weak coupling.)The resulting governing equations,initial,and interface conditions areC e @T e @t ¼k e @2T e@x2ÀG ðT e ÀT p ÞþS (1a )C p @T p @t ¼k p @2T p @x 2þG ðT e ÀT p Þ(1b )C s @T s @t ¼k s @2T s@x(1c )T e ðt ¼0Þ¼T p ðt ¼0Þ¼T s ðt ¼0Þ¼T 0(2)Àk e@T e @xx ¼L ¼T e ÀT s R es x ¼L(3a )Àk p @T px ¼L ¼T p ÀT s ps x ¼L(3b )Àk s@T sx ¼L ¼T e ÀT s es x ¼L þT p ÀT s ps x ¼L(3c )The subscripts e ,p ,and s denote electrons in metal,phonons in metal,and phonons in the dielectric substrate,respectively.C is the volumetric heat capacity,k is the thermal conductivity,G is the electron–phonon coupling factor governing the rate of energy transfer from electrons to phonons in metal,and L is the thickness of the metal layer.At the front surface of the metal layer insula-tion boundary condition is used due to the much larger heat ﬂux caused by laser heating relative to the heat loss to air.At the rear surface of the substrate,since the thickness of the substrate used is large enough (1l m)so that there is no temperature rise during the time period of consideration,the insulation boundary condition is also applied.Thermal properties of phonons in both metal and dielectric are taken as temperature-independent due to the weak temperature dependence.The thermal conductivity of phonons in metal is much smaller than that of the electrons and is taken in this work as 0.001times the bulk thermal conductivity of gold (311W/(mK)).The volumetric heat capacity of the metal phonon is taken as that of the bulk gold.C e is taken as proportional to T e [14]with the proportion coefﬁcient being 70J/(m 3K 2)[15],and k e is calculated by the model and the data used in Ref.[13]which is valid from the room temperature to the Fermi temperature (6.39Â104K in Au,[14]).G can be obtained using the model derived in Ref.[16].In this work,the value of G at the room tem-perature is taken as 4.6Â1016W/(m 3K)[17],and its dependence on electron and phonon temperatures follows [16].The laser heat-ing source term S is represented by the model used in [13]asS ¼0:94ð1ÀR ÞJ t p ðd þd b Þ1Àexp ÀL d þd bexp Àx d þd b À2:77t t p2"#(4)which assumes all the absorbed laser energy is deposited in the metal layer.J is the ﬂuence of the pump laser,R is the surface re-ﬂectance to the pump,t p is the pulse width (FWHM),d is the opti-cal penetration depth,and d b is the electron ballistic length (around 100nm in Au,[18]).R es and R ps are treated as free pa-rameters for ﬁtting the experimental data.The wavelength of the probe laser in the experiment is centered at 800nm.For this wavelength,the incident photon energy is below the interband transition threshold in Au,which is around 2.47eV [18],and the Drude model can be used to relate the tem-peratures of electrons and phonons to the dielectric function and then the index of refraction,which is expressed as [19]e ¼e 1Àx 2px ðx þi x s Þ(5)x is the frequency of the probe laser and x p is the plasma fre-quency (1.37Â1016rad/s in Au evaluated using the data in Ref.[14]).x s is the electron collisional frequency,the inverse of the electron relaxation time.The temperature dependence ofelectricalFig.1TTR measurement results for the Au–Si sample of Authickness 77nm with different ﬂuences.(a )Results before pulse stretching;(b )results after pulse stretching.resistivity indicates that x s is approximately proportional to pho-non temperature at high temperature [14]and from the Fermi liq-uid theory,its variation with electron temperature is quadratic (T e 2)[20].Therefore,x s is related to T e and T p approximately asx s ¼A ee T 2e þB ep T p(6)A ee is estimated from the low-temperature measurement [21]andB ep is usually estimated from the thermal or electrical resistivity near the room temperature [14].In this work,A ee is taken as the lit-erature value 1.2Â107s À1K À2[6]while e 1and B ep are evaluated by ﬁtting the room-temperature value of the complex dielectric con-stant at 800nm wavelength provided in Ref.[22],which are found to be 9.7and 3.6Â1011s À1K À1,respectively.The complex index of refraction n 0þin 00is the square root of the dielectric ing Eqs.(5)and (6),n 0and n 00are evaluated as 0.16and 4.90,respectively,which agree with the empirical values [23].The re-ﬂectance is then calculated from n 0and n 00by the method of transfer matrix [24],which considers multiple reﬂections in thin ﬁlms.4Results and DiscussionThe results of TTR measurements with a pump ﬂuence of 147J/m 2are plotted in Fig.2.The fast decrease of the reﬂectance indicates that energy transfer between electrons and phonons in metal,followed by a relatively slow decrease after several ps which indicates electrons and phonons have reached thermal equi-librium.The initial cooling rates are smaller for samples with thicknesses less than the electron ballistic length since the electron temperature is almost uniform across the thin ﬁlm,and coupling with phonons within the metal ﬁlm and the dielectric substrate is the only cooling mechanism.For a thicker sample of thickness 250nm,the initial decrease is much faster due to thermal diffu-sion in the gold ﬁlm caused by a gradient of the electron tempera-ture in the ﬁlm.We investigate the effect of R es and R ps using the thermo-reﬂectance signal.Two values of R ps ,1Â10À10m 2K/W and 1Â10À7m 2K/W,are used,each with a parameterized range of values for R es .Figure 3shows the calculated results for the sample with a 39nm-thick gold ﬁlm.Little difference can be seen between Figs.3(a )and 3(b )while different cooling rates are obtained with varying R es in either plot,indicating that the cooling rate is not sensitive to the coupling strength between phonons in metal and dielectric.Note that an interface resistance of 1Â10À10m 2K/W is lower than any reported value,indicating a very high coupling strength between the phonons in metal and dielectric.Conversely,the results vary greatly with the coupling strength between electrons in metal and phonons in dielectric at the interface.This is because the lattice (phonon)temperature rise in metal is much smaller than the elec-tron temperature that the interface coupling between phonons in metal and dielectric does not inﬂuence the surface temperature,which directly determines the measured reﬂectance.On the other hand,the temperature rise of electrons is much higher,and conse-quently,the cooling rate is sensitive to R es .The relatively high sensitivity of R es to that of R ps demonstrates that the former can be isolated for the study of the coupling between electrons in metal and phonons in dielectric.We now use the measured TTR data to estimate R es ,the thermal resistance between electrons in metal and phonons in dielectric.R es is adjusted by the least square method to ﬁt the simulation results with the measured data,and the results are shown in Fig.4.We note that it is impossible to ﬁt the measured results using insu-lation interface condition (i.e.,no coupling or extremely large thermal resistance between electrons in metal and phonons in the dielectric substrate),which will signiﬁcantly underestimate the cooling rate.For thin samples,we ﬁnd that the value of R es is of the order of 10À10to 10À9m 2K/W.This value is below the ther-mal resistances of representative solid–solid interfaces measured in thermal equilibrium [25].This indicates that the direct coupling between electrons in metal and phonons in dielectric is strong.It is also noted that the resistance values increases with the thickness of the gold ﬁlm,indicating a decrease in the coupling strength between electrons in metal and the dielectric substrate.This could be due to the lower electron temperature obtained in thicker ﬁlms,and a decrease of the coupling strength with a decrease in the electron temperature [5].For the sample of thickness 250nm,R es has little effect on the simulation result since the interface is too far from the absorbing surface to inﬂuence the surface tempera-ture,and therefore it is not presented here.The agreement between the ﬁtted results and the measured data is generally good.The small discrepancy between the measured and the ﬁtted results can result from inaccuracy in computingtheFig.2TTR measurement results on Au–Si samples of varying AuthicknessesFig.3Simulation results with varying R es for the Au–Si sample of Au thickness 39nm.(a )R ps 51310210m 2K/W;(b )R ps 5131027m 2K/W.absorption or the temperature.Figure 1(b)shows the normalized TTR measurement results on the sample of thickness 77nm with three laser ﬂuences.It is seen that small variations in the shape of the TTR signals can be caused by different laser ﬂuences and thus the maximum temperature reached in the ﬁlm.Absorption in metal,multiple reﬂections between the metal surface and the Au–Si inter-face,and possible deviations of the properties of thin ﬁlms from those of bulk can all contribute to uncertainties in the temperature simulation;therefore affecting the calculated reﬂectance.With the values of R es shown in Fig.4,the calculation shows that the highest electron temperature,which is at the surface of 39nm–thick gold ﬁlm,is about 6700K.The highest temperature of electrons is roughly inversely proportional to the thickness of the ﬁlms for the four thinner ﬁlms.The highest temperature of elec-trons is much less than the Fermi temperature and thus ensures the validity of the linear dependence of C e on T e [14].The highest temperature for the lattice in metal is about 780K,also in the 39nm-thick gold ﬁlm.This large temperature difference between electrons and lattice indicates that the interface heat transfer is dominated by the coupling between electrons in metal and the phonons in the dielectric substrate.As shown in Fig.4,the meas-ured R es is very low,of the order of 10À10to 10À9m 2K/W.Even if R ps ,which is not determined in this study,is also that low (note that 10À10to 10À9m 2K/W is lower than any reported values),because of the large difference in temperatures between electrons and the phonons in metal,the interface heat transfer rate (Eqs.(3a )–(3c ))due to the coupling between electrons in metal and the substrate is much larger than that due to the coupling between phonons in metal and the substrate.5ConclusionsIn conclusion,TTR measurements using femtosecond laser pulses are performed on Au–Si samples and the results are analyzed using the TTM model.It is shown that due to the strong nonequilibrium between electrons and phonons during ultrafast-laser heating,it is possible to isolate the effect of the direct electron–phonon coupling across the interface,allowing investiga-tion of its ing stretched femtosecond pulses is shown to be able to minimize the nonequilibrium effect among electrons,and is thus more suitable for this study.The TTR measurement data can be well-represented using the TTM parison between the TTR data and the TTM results indicates that the direct coupling due to electron-interface scattering dominates the interface heat transfer during ultrafast-laser heating of thin ﬁlms.AcknowledgmentThis paper is based upon work supported by the Defense Advanced Research Projects Agency and SPAWAR Systems Cen-ter,Paciﬁc under Contract No.N66001-09-C-2013.The authors also thank C.Liebig,Y.Wang,and W.Wu for helpful discussions.NomenclatureA ee ¼coefﬁcient in Eq.(6),s À1K À2B ep ¼coefﬁcient in Eq.(6),s À1K À1C ¼volumetric heat capacity,J/(m 3K)G ¼electron–phonon coupling factor,W/(m 3K)i ¼unit of the imaginary number J ¼ﬂuence of the pump,J/m 2k ¼thermal conductivity,W/(mK)L ¼metal ﬁlm thickness,mn 0¼real part of the complex index of refractionn 00¼imaginary part of the complex index of refraction R ¼interface thermal resistance,m 2K/W;reﬂectance S ¼laser source term,W/m3Fig.4Comparison between the measurement and the simulation results for Au–Si samples of different Au thicknesses.The open circle represents the meas-ured data and the solid line represents the simulation results.(a )39nm ﬁtted by R es 55310210m 2K/W;(b )46nm ﬁtted by R es 56310210m 2K/W;(c )60nm ﬁtted by R es 51.231029m 2K/W;and (d )77nm ﬁtted by R es 51.831029m 2K/W.T¼temperature,Kt¼time,st p¼pulse width of the pump(FWHM),sx¼spatial coordinate,me¼complex dielectric constante1¼constant in the Drude modeld¼radiation penetration depth,md b¼electron ballistic depth,mx¼angular frequency of the probe,rad/sx p¼plasma frequency,rad/sx s¼electron collisional frequency,rad/sSubscripts0¼initial statee¼electron in metales¼electron in metal and phonon in dielectricp¼phonon in metalps¼phonon in metal and phonon in dielectrics¼phonon in dielectricReferences[1]Cahill,D.G.,Ford,W.K.,Goodson,K.E.,Mahan,G.D.,Majumdar,A.,Maris,H.J.,Merlin,R.,and 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Dielectric susceptibility of the Coulomb-glass

a rX iv:c ond-ma t/992145v1[c ond-m at.dis-nn]1Fe b 1999Dielectric susceptibility of the Coulomb-glassA.D´ıaz-S´a nchez,1,3M.Ortu˜n o,1M.Pollak,2A.P´e rez-Garrido,1and A.M¨o bius 31Departamentode F´ısica,Universidad de Murcia,E-30071Murcia,Spain,2Departmentof Physics,University of California at Riverside,Riverside,California 92521,3Institut f¨u r Festk¨o rper-und Werkstoﬀorschung,D-01171Dresden,Germany (February 1,2008)Abstract We derive a microscopic expression for the dielectric susceptibility χof a Coulomb glass,which corresponds to the deﬁnition used in classical elec-trodynamics,the derivative of the polarization with respect to the electric ﬁeld.The ﬂuctuation–dissipation theorem tells us that χis a function of the thermal ﬂuctuations of the dipole moment of the system.We calculate χnu-merically for three–dimensional Coulomb glasses as a function of temperature and frequency.PACS number(s):72.80.Ng;71.27.+a;71.55.JvTypeset using REVT E XI.INTRODUCTIONThe subject of this paper is the calculation of the dielectric susceptibility in Coulomb glasses,a term used for Anderson insulators with Coulomb interactions between the local-ized electrons.We consider situations deep in the insulating phase,when quantum energies t arising from tunneling are much smaller than the other important energies in the prob-lem,i.e.,Coulomb interactions and random energyﬂuctuations.The model also applies to systems in the quantum Hall regime far away from the peaks,for which the conductivity is exponentially small as compared to e2/h and the conduction mechanism is by variable range hopping between localized states.The model can be easily extended to granular metals in the insulating regime.Previous calculations of the dielectric susceptibility of the Coulomb glass have used an ex-pression directly obtained from the analogy between Coulomb and spin glasses[1].However, the non-local character of the processes involved in Coulomb glasses makes this expression of the dielectric susceptibility inappropriate.Furthermore,it does not corresponds to the standard deﬁnition of the dielectric susceptibility.Theﬁrst aim of this work is to present a microscopic expression of the dielectric susceptibilityχvalid for the Coulomb glass,and to apply theﬂuctuation–dissipation theorem to this expression.We also want to calculateχat very low temperatures.To this end we have to take into account that interactions,and specially those of long–range character,drastically change the properties of systems with localized states[2,3].Most properties of these systems are aﬀected by electron correlations, and such eﬀects cannot be described by one–particle densities of states or excitations.To deal with complex excitations,methods were developed[4–12]to obtain the low lying states and energies of Coulomb glasses.The paper is organized as follows:Section II introduces the Coulomb glass models used for the numerical calculations.Section III presents the derivation of a microscopic expression for the dielectric susceptibility of Coulomb glasses.Section IV describes how the low-energy many-particle states are obtained numerically and gives a method to calculate the depen-dence of the dielectric susceptibility with the frequency.In section V,we present the results obtained for the dielectric susceptibility of Coulomb glasses at very low temperatures and the dependence on temperature,and frequency.Finally,in section VI we extract some conclusions.II.MODELSOur next results apply to any model of a Coulomb glass but to be deﬁnite we performed numerical simulations for the two most common models:the standard model with a uniform random potential distribution and the classical impurity band model(CIB).Efros and Shklovskii proposed a practical model to represent Coulomb glass problems with localized electronic states,which has been widely used and extended[2,3].This model is represented by the standard tight-binding Hamiltonian:H= iφi n i+ i<j e2r ij(1) where n i∈{0,1}denotes the occupation number of site i.We use rationalized units,unlike in most works on the Coulomb glass,because in this problem they constitute the most convenient choice.We will consider sites at random positions,with a densityρequals to 1,and simulate the disorder by a ramdon potential in each siteφi,uniformly distributed between−W/2and W/2.r ij is the distance between sites i and j according to periodic boundary conditions in the sense of[13]in the perpendicular directions to the applied electric ﬁeld.Charge neutrality is achieved by a background compensation charge−K at each lattice site.The classical impurity band model(CIB)is a realistic representation of a lightly doped semiconductor in which the random potential arises from the minority impurities[14].Here we consider an n-type,partially compensated semiconductor with donor concentration N D, and acceptor concentration N A=KN D.The Hamiltonian is given by:H=e22 i=j(1−n i)(1−n j)r iν(2)where the donor occupation number n i equals1for occupied donors,and0for ionized donors. The indexνruns over the acceptors and r iν=|r i−rν|,with r i being the donor coordinates and rνthe acceptor coordinates.We chose the donor densityρequal to1and imposed again periodic boundary conditions.For numerical reasons we constrained the nearest neighbor distance to be larger than0.5.We used K=0.5because the interaction eﬀects are largest there.III.SUSCEPTIBILITY OF THE COULOMB GLASSWe now proceed to obtain a proper microscopic expression for the dielectric susceptibility of Coulomb glasses.It is applicable to an arbitrary three–dimensional model of Coulomb glass as long as the interaction between charges goes as1/r.A certain analogy between the spin glass and the Coulomb glass lead to an incorrect expression for the dielectric susceptibility of the Coulomb glass.In some sense the local spin s i in the spin glass is analogous to the site occupation n i in the Coulomb glass.If only n i=0and n i=1are allowed due to strong on-site interaction,then the analogy is with spins s i=1/2.The magneticﬁeld in the spin glass is then analogous to the chemical potential in the Coulomb glass.However,this analogy should not be pushed too far and does not apply to polarizabilities.The magnetic polarizability is the change of total spin induced by the magneticﬁeld,but the electric polarizability is the change in the electric polarization induced by an electricﬁeld,not the change in the total occupation number induced by a change of a global potential as the direct analogy would have it.The basic diﬀerence between the two susceptibilities can be understood more clearly by realizing that the magnetic polarization s i T comes fromﬁeld-inducedﬂips of spatiallyﬁxed spins,where ... T refers to thermal average.An analogous electric polarization can come fromﬂips of local dipoles,but in many systems it involves aﬁeld induced displacement of charges (there is no magnetic equivalent to this because there are no magnetic charges).Such a polarizability is thus not represented by n i T responding to a potential(as in[1]and[15])but by i x i n i T responding to a ﬁeld (as in [16]).The dielectric susceptibility χthen is:χ=1∂E (3)where E is the total electric ﬁeld.Our ﬁrst aim is to obtain a microscopic expression for Coulomb glasses of the classical deﬁnition of the dielectric susceptibility.Let us assume that we apply an electric displace-ment D .This will induce a polarization P equal toP =e∂φj ∆φj (5)where ∆φj is the change in potential at site j .The partial derivative appearing in this expression is proportional to the local susceptibility χij :∂ n i TT χij (6)where T is the temperature,and the Boltzmann constant k B is taken to be 1throughout the paper.The change in potential corresponding to a uniform electric displacement is ∆φi =Dx i /ǫ0,so the ratio between P and D is:∂PT N ij x i χij x j ≡χ0.(7)To calculate the dielectric susceptibility numerically it is convenient to apply the ﬂuctuation-dissipation theorem to χ0in order to rewrite it in terms of thermal ﬂuctua-tions of the dipole moment.Taking into account the expression for the thermal average ofthe site occupation, n i T,it is easy to obtain that its derivative with respect to the potential in j,i.e.,the local susceptibilityχij,is equal to theﬂuctuation in the electron occupancy of the two sites involved:χij= n i n j T− n i T n j T(8) Using this equation in expression(7)forχ0,we arrive ate2χ0=( d2 T− d 2T)(9)T Nwhere d=e i x i n i is the dipolar moment of the sample.The dielectric susceptibility is a function of the thermalﬂuctuation of the dipolar moment.Our computer simulations can per force involve only systems of mesoscopic size.For the macroscopic susceptibility we can imagine building a macroscopic system of many mesoscopic cubes of linear size L arranged toﬁll the space.Each of these samples corresponds to a particular realization of the random positions and energies of the sites involved.The total electricﬁeld that a given microscopic sample feels is the sum of the appliedﬁeld,D,and the inducedﬁeld.If the appliedﬁeld is uniform,and the polarizabilities of all the samples were the same,the polarization would also be uniform and the inducedﬁeld would come only from the boundary of the sample.We then have(in our units):ǫ0E=D−P(10) and get from Eqs.(9)and(10):χ0χ=random media has been given in[17,18].Here we shall avoid the inherent complications of such a computation and assume that Eq.(11)is approximately valid even at higher frequencies if we use forχ0a value averaged over many computer realizations.The relation σ=iωǫmeans that aﬁnite dc conductivity implies an inﬁnite dc dielectric susceptibility.A proper calculation of the DC conductivity can be done by percolation in conﬁguration space, but it is a diﬃcult problem requiring huge numerical eﬀorts so that for three-dimensional systems we could only consider very small samples.Our approximate calculation of the divergence of the susceptibility allows us to estimate the variation of the DC conductivity with temperature as we will see.IV.NUMERICAL PROCEDUREA.Low-energy conﬁgurationsWe calculate the dielectric susceptibility at very low temperatures making use of the ground state and the very low-energy conﬁgurations of the systems.With the procedure that we brieﬂy discuss bellow we obtain theﬁrst5.000many-particle conﬁgurations and calculate their dipoleﬂuctuations,Eq.(9).Weﬁnd the low-energy many-particle conﬁgurations by means of a three-steps algorithm [10].This comprises local search[4,5],thermal cycling[11],and construction of“neighbour-ing”states by local rearrangements of the charges[4,5].The eﬃciency of this algorithm is illustrated in Ref.[10].In theﬁrst step we create an initial set S of metastable states. We start from states chosen at random and relax these states by a local search algorithm which ensures stability with respect to excitations from one up to four sites.In the second step this set S is improved by means of the thermal cycling method,which combines the Metropolis and local search algorithms.The third step completes the set S by systemati-cally investigating the surroundings of the states previously found.At the end we only keep conﬁgurations with aﬁx number of electrons,so we work with canonical ensembles.B.Frequency dependenceAtﬁnite frequencies only transitions with characteristic timeτIJ shorter than the inverse of the frequency contribute to the susceptibility.Thus,for a given frequency,we consider two conﬁgurations as connected if theirτIJ is shorter than the inverse of the frequency,and we group the conﬁgurations in clusters according to these connections.The characteristic transition time between conﬁgurations I and J is[2],τIJ=ω−10exp 2 r ij/a exp(E IJ/T)/Z(12) In this equation,the quantityω0is a constant of the order of the phonon frequency,ω0∼1013s−1.The sum is the minimized sum over all hopping distances between sites which change their occupation in the transition I→J.a denotes the localization radius,E IJ= max(E I,E J)where E I is the energy of the state I,and Z is the partition function.We calculate the susceptibility of each cluster through Eq.(9),assuming thermal equilib-rium in the cluster.The glassy nature of our systems is responsible for the existence of the clusters,which indicate the non-ergodicity of the systems for times shorter than the critical time connecting all the conﬁgurations in a single cluster.Each realization of the systems will be in a given cluster and will not see the other clusters.The probability to be in a cluster depends on the history of the system and is very diﬃcult to estimate.In order to obtain averages of the susceptibility,we will assume that the weight of each cluster is proportional to its partial partition function,which constitutes the simplest possible assumption.The results areﬁnally averaged over many diﬀerent disorder realizations.V.RESULTSIf we take into account all types of transitions,including the slowest ones,Coulomb glass behaves like a conductor and is able to screen fully as its susceptibility diverges.But small samples may not have excitations which carry electrons across the entire sample and produce nearly equipotential surfaces at the two opposite edges.So we must consider samples abovea certain critical size at which the steady state susceptibilityχdiverges.We found that this critical size is about200sites for both models(Eqs.(2)and(1)).Above this size the results are practically independent of size.Fig.1shows the average value ofχ0as a function of frequency for several temperatures. The plots are for the standard model of size N=256.The localization radius is a=0.2, which is maintained throughout the paper.At low frequenciesχ0increases with T while at high frequencies it decreases with T.The reason is that at smallωhopping extends over many conﬁgurations,and the main eﬀect of T is to enhance the transition rates.At large ωhopping is between two optimal conﬁgurations for that frequency(or even two sites)and the main eﬀect of T is to equalize the occupation probabilities.This bears analogy with uncorrelated hopping conductivity which increases strongly with T asω→0and behaves as1/T at high frequency.The result for the CIB model are very similar to those for the standard model and so we do not show them explicitly.The results for the CIB model roughly correspond to those for the standard model with an eﬀective disorder energy of approximately3.As already mentioned,an accurate calculation of the frequency dependent macroscopic susceptibility requires the distribution f(χ0),not only the average value ofχ0.Fig.2shows the integrated distribution P(χ0)= χ00f(χ′0)dχ′0for N=256at T=0.01for diﬀerent values of the frequency,ω→0(dotted curve),ω=103s−1(dashed curve),ω=107s−1 (long dashed curve).The solid curve is a plot of the function1−exp{−(xλ)α}with the parametersλ=1.5andα=0.6.In the range examined this form of integrated distribution ﬁts(with varyingλandα)fairly well our data for all T andω.The broad character of the distribution indicates large mesoscopicﬂuctuations and shows a need to examine in the future the accuracy of our approximation by taking proper account of the distribution ofχ0.We deﬁne a critical time for saturationτc as the inverse of the frequency for which the value of the susceptibility is95%of the asymptotic value for extremely long times.We studied the T dependence of this critical timeτc.For all temperatures considered the values ofτc are extremely large,which is a sign of the glassy nature of our systems.Sinceτc islong and close to the saturation ofχ0,we expect our results to be rather accurate for this study.Fig.3plots the logarithm ofτc vs.T−1/2for four sizes of the standard model and for two sizes of the CIB model.The data areﬁtted quite well by straight lines indicating that a similar mechanism which gives rise to the T−1/2conductivity[4]is also eﬀective in the dielectric susceptibility.This should of course be expected because of the close connection between the two properties.The square root of the slope of each straight line yields a characteristic temperature T0which has been often associated with variable range hopping theory in a Coulomb gap.In that theory T0is given by:e2T0=βVI.CONCLUSIONSWe derived a microscopic expression for the dielectric susceptibilityχapplicable to hop-ping systems including systems where interactions are important.The expression is par-ticularly suitable for low frequencies.It corresponds to the expression used in classical electrodynamics.Si and Varma[20]have recently study the same problem in the metallic limit of the metal-insulator transition for two dimensional systems,and obtain than the static compressibility vanishes at the transition.Some previous works[1,15]used expression based on an analogy between spin and Coulomb glasses.We argues that these analogies cannot be extended to the susceptibility.The fundamental reason is that unlike in spin glasses the susceptibility in the hopping systems arises from non-local processes.Theﬂuctuation–dissipation theorem tells us that the dielectric susceptibility is a function of the thermalﬂuctuations of the dipole moment of the system,instead of theﬂuctuations of the charge density,result that one obtains when using analogy between Coulomb and Spin glasses.We calculateχnumerically for three–dimensional Coulomb glass systems as a function of temperature and frequency.We found thatχdiverges as the frequency tends to zero.One has to consider sizes larger than a critical one of approximately N=200for the CIB model and for the standard model with W=3.The logarithm of the critical time for saturation varies proportionally to T−1/2,the same dependence as in variable range hopping.The characteristic temperature for this dependence is approximately equal to0.9,a factor of three smaller than the theoretical predictions for the equivalent constant appearing in variable range hopping.ACKNOWLEDGMENTSWe acknowledgeﬁnancial support from the DGES project number PB96-1118,SMWK, and DFG(SFB393).A great part of this work was performed during A.D.-S.’s visit at the IFW Dresden;A.D.-S.thanks the IFW for its hospitality.REFERENCES[1]J.H.Davies,P.A.Lee,and T.M.Rice,Phys.Rev.B29,4260(1984);W.Xue andP.A.Lee,Phys.Rev.B38,9093(1988)[2]M.Pollak and M.Ortu˜n o,in Electron-Electron Interactions in Disordered Systems,edited by A.L.Efros and M.Pollak(North-Holland,Amsterdam,1985),p.287.[3]B.I.Shklovskii and A.L.Efros,Electronic Properties of Doped Semiconductors(Springer,Berlin,1984).[4]A.P´e rez-Garrido,M.Ortu˜n o,E.Cuevas,J.Ruiz,and M.Pollak,Phys.Rev.B55,R8630(1997).[5]A.M¨o bius and M.Pollak,Phys.Rev.B53,16197(1996).[6]M.Mochena and M.Pollak,Phys.Rev.Lett.67,109(1991).[7]M.Schreiber and K.Tenelsen,Europhys.Lett.21,697(1993).[8]J.Talamantes and D.Espericueta,Model.Simul.Mater.Sci.1,761(1993).[9]A.D´ıaz-S´a nchez,A.M¨o bius,M.Ortu˜n o,A.P´e rez-Garrido,and M.Schreiber,Phys.Stat.Sol.b205,17(1998).[10]A.D´ıaz-S´a nchez,A.M¨o bius,M.Ortu˜n o,A.Neklioudov,and M.Schreiber,to be pub-lished.[11]A.M¨o bius, A.Neklioudov, A.D´ıaz-S´a nchez,K.H.Hoﬀmann, A.Fachat,andM.Schreiber,Phys.Rev.Lett.79,4297(1997).[12]A.M¨o bius and P.Thomas,Phys.Rev.B55,7460(1997).[13]N.Metropolis,A.W.Rosenbluth,M.N.Rosenbluth,A.H.Teller,and E.Teller,J.Chem.Phys.21,1087(1953).[14]A.L.Efros and B.I.Shklovskii in Electron-Electron Interactions in Disordered Systems,edited by A.L.Efros and M.Pollak(North-Holland,Amsterdam,1985),p.409.[15]E.R.Grannan and C.C.Yu,Phys.Rev.Lett.71,3335(1993).[16]M.Pollak and T.H.Geballe,Phys.Rev.122,1742(1961)[17]M.Pollak,Proc.R.Soc.Lond.A325,383(1971).[18]H.A.Pohl and M.Pollak,J.Chem.Phys.66,4031(1977).[19]V.L.Nguyen,Sov.Phys.Semicond.18,207(1984).[20]Qimiao Si,C.M.Varma,cond-mat/9805264FIGURESFIG.1.Averaged values ofχ0obtained for the standard model,plotted against frequency, for several values of the temperature as follows:0.006(•),0.008( ),0.01( ),0.012( ),0.014 (◭),0.016( ),0.018(◮),and0.02(◦).The disorder energy is W=2and the localization radius is a=0.2.FIG.2.Accumulated distribution probability ofχ0for N=256andω→0(dotted curve),ω=103s−1(dashed curve),ω=107s−1(long dashed curve).The solid curve corresponds to the ﬁt explained in the text.χav is the average value at each frequency.FIG.3.Logarithm of critical timeτc for reaching the static susceptibility as a function of T−1/2 for four sizes of the standard model,N=64(•),128( ),216( ),512( ),and for two sizes of the CIB model,N=216(◭)and512( ).FIG.4.Dielectric susceptibilityχas a function of frequency for several temperatures,T=0.006 (•),0.008( ),0.01( ),0.012( ),0.014(◭),0.016( ),0.018(◮),and0.02(◦).The other parameters and the model considered are the same as in Fig.1.。

外文翻译(英文)用氧化物粉末制NTC热敏电阻的微观结构与导电性能的关系(外文)

Correlation between Microstructure andConductance in NTC Thermistors Produced from Oxide PowdersG.D.C.Csete de Gyo rgyfalva,*A.N.Nolte and I.M.ReaneyDepartment of Engineering Materials,Sir Robert Had®eld Building,University of She eld,She eld,S13JD,UKAbstractA detailed study of spinel-structured Ni 1Àx Mn 2+x O 4formed by a mixed oxide route has shown that when x %0a high proportion of NiO is residual in the sin-tered ceramic.Wickham (Wickham,D.G.,Solid phase equilibria in the system NiO±Mn 2O 3±O 2.J.Inorg.Chem.,1964,26,1369±1377)demonstrated that the spinel phase decomposes in air above 900 C.Sintering in this system is usually per-formed around 1200 C.Decomposition of the spi-nel phase is therefore inevitable.The e ect of decomposition on the microstructure and electrical properties of Ni 1Àx Mn 2+x O 4based ceramics is discussed.#1999Elsevier Science Limited.All rights reservedKeywords :microstructure,electrical conductivity,spinels,thermistors.1IntroductionNegative temperature coe cient (NTC)thermistors are found in an ever increasing number of electrical and electronic products.Ni 1Àx Mn 2+x O 4,where x denotes the deviation from the stoichiometric 1:1NiO:Mn 2O 3ratio,o ers a range of properties that are suitable for most temperature sensing applications.When x =0,(nickel manganite,NiMn 2O 4),the solid solution has an inverse cubic spinel structure,based on a 2Â2Â2array of face centred cubic (fcc)oxygen subunits.When x =1,Mn 3O 4is present which is a tetragonally distorted spinel.The properties routinely used to characterise NTC thermistors are resistance,R 1and R 2,at 25 C (T 1)and 85 (T 2)and a B value (with units of tem-perature in Kelvin)which is a measure of the sensi-tivity of the device over a given temperature range:BT 1T 2T 2ÀT 1ln R 1R 2IThe exact mode of conduction in nickel manga-nite is poorly understood,but several modelsinvoke the small polaron theory.1,2Small polaron conduction is sometimes referred to as a `hopping'mechanism,as it involves the transfer of polarisa-tion from one cation to another.In the nickel manganite system,it has been postulated that the mixed valence,Mn 4+,Mn 3+cations present on the octahedral sites give rise to these small polaron pathways.1The octahedral cations in the spinel structure lie in chains along some <110>direc-tions.These vectors represent the smallest inter-cationic distances within the unit cell.Another important parameter when considering applications for Ni 1Àx Mn 2+x O 4ceramics is their thermal stability or aging characteristics (changes in conductance over long periods,i.e.lifetime of the component).Reports indicate that better ther-mal stability is found in tetragonal ceramics rather than cubic materials though the conductivity of the latter is 10to 100times higher.2,3This could be explained by a reduction in the concentration of Mn 4+compared to Mn 2+and Mn 3+or possibly by the presence of planar defects such as ferroelas-tic domain walls.4Ni 1Àx Mn 2+x O 4ceramics have been prepared by the carbonate and oxalate methods,in addi-tion to the more conventional mixed oxide route.5Irrespective of the preparation route,sintering (typically around 1200 C)is always carried out above the decomposition temperature in air for the system ($900 C)as discussed by Wickham.5Con-sequently,ceramics ®red using conventional pro-cessing will contain multiple phases,e.g.NiO fromJournal of the European Ceramic Society 19(1999)857±860#1999Elsevier Science LimitedPrinted in Great Britain.All rights reservedP I I :S 0955-2219(98)00331-80955-2219/99/$-see front matter857*To whom correspondence should be addressed.Fax:+44-(0)114-222-5943;e-mail:mtp96gdc@she the decomposed spinel and Mn-rich regions,3,6,7in accordance with the equation:xiwn III2y43x xiy 3Àx3xi II 3À3x a 3Àxwn II 2x a 3Àx wn III2O4 x6O2PIt is the intention of this paper to demonstrate how the degree of decomposition from single phase in¯uences conductivity and,in particular,aging. X-ray di raction and transmission electron micro-scopy will be used to monitor the degree of decomposition and accelerated aging tests(470 C) will be performed.2Experimental ProcedureThe NiO and Mn2O3powders in a1:1Mn2O3:NiO molar ratio were weighed out using an electronic balance( 0.01g)and transferred to a poly-propylene vessel with a charge of ZrO2milling media(the weight of ZrO2varied with the weight of the batch being processed).The batch was mil-led for6h to reduce particle size distribution to a mean of6"m and a maximum of12"m then drawn through a suction®lter.The resulting slurry was dried in a70 C oven overnight.The dried powder was calcined in a mullite crucible at900 C for16h and subjected to a further6h milling under the above conditions.One one cm diameter pellets were pressed from the powders and sintered at 1250 C,achieving densities better than95%. Microstructural and structural characterisation were carried out using transmission electron microscopy(TEM)and X-ray di raction(XRD), respectively.XRD was performed on solid cera-mics and loose powders using a Phillips PW1050 di ractometer with a Cu K source.A0.02 step size was used at a scan rate of0.5 minÀ1.TEM samples were prepared by grinding the ceramic to a thickness of20"m and ion beam milling to per-foration.Images were obtained using JEOL200CX and3010TEMs:the latter was equipped with a LINK energy dispersive X-ray detector. Accelerated aging tests were carried out using a non-induction wound furnace held at470 C.Tem-perature¯ux was monitored in the furnace using a thermocouple mounted immediately adjacent to the test piece.Platinum wires leading to a high precision HP4284A LCR meter were used to make contact to the electroded surface of the cera-mic.Changes in the resistance of the leads and contacts as a function of temperature were taken into account by performing a closed circuit run.Typically,temperature varied within a 0.2 C range over10h.3Results and DiscussionWickham,5in his study of the Ni1Àx Mn2+x O4solid solution,demonstrated that above900 C decom-position occurs resulting in the formation of NiO and a Mn-rich spinel phase.The higher the tem-perature above the onset of the decomposition reaction,the more rapid the rate.In order to study the decomposition reaction in more detail and its potential e ect on electrical properties,single phase ceramics(within the sensitivity of conventional XRD)were fabricated,as demonstrated in Fig.1. Figure2shows a series of XRD traces from single phase samples heat treated at1000,1100and 1200 C for1h.The evolution of peaks corres-ponding to NiO can be observed in accordance with the predictions of Wickham.6The relative intensities of the NiO peaks(marked)increase with increasing temperature.Figure3is a bright®eld(BF)TEM image showing a typical region of spinel grains in single phase material.The grain boundaries and interiors are free from second phase.Inset in Fig.3is a <110>zone axis di raction pattern(ZADP)from one of the spinel grains in the image.Figure4isaFig.1.XRD trace of single phase ceramic.Note absence ofNiOpeaks.Fig.2.XRD spectra of samples held at1000,1100,1200 c for1h.NiO peaks are marked.858G.D.C.Csete de GyoÈrgyfalva et al.BFTEM image obtained from a sample decom-posed for9h at1250 C.Inset is a<110>ZADP pattern from the imaged region.The fundamental re¯ections can be indexed according to a<110> zone axis from rock salt structured NiO.The weak re¯ections at half integer positions arise from regions of spinel phase,observed as dark contrast. Rock salt(NiO)and spinel structured compounds invariably exhibit a cube//cube orientation rela-tionship.Oxides with the rock salt structure are based around single fcc oxygen subunits whereas spinel structured compounds have a2Â2Â2fcc oxygen sublattice.In order to study the aging characteristics of the ceramics as a function of decomposition,con-ductance measurements were performed over10h at470 C 0.2 C.Figure5shows the change in conductance normalised to the initial value,against time at470 C for(A)single phase spinel and(B) `partially'decomposed spinel(heat treated for9h at1250 C).The single phase sample showed a negligible drift in resistivity over the test period, whereas the`partially'decomposed sample exhib-ited a steady decline in conductance.Di erences in the absolute starting values can be attributed to small variations in the dimensions of the samples. Figures6and7are XRD traces showing the samples before and after the accelerated aging experiments.Figure6,which corresponds to Fig.5(A)(decomposed),shows a reduction intheFig.4.BFTEM image of spinel regions in a NiO matrix.Insetis a<110>ZADP from the NiO.Faint re¯ections are presentat half integer positions arising from the dark regions ofspinel.Fig.5.Graph showing normalised conductance versus time at470 C for(A)single phase and(B)decomposed(9h at1250 C)material.Fig.6.XRD spectra of single phase sample(A)before and(B)after acceleratedaging.Fig.7.XRD spectra of decomposed sample(A)before and(B)after acceleratedaging.Fig.3.BFTEM image of spinel grains in single phase mate-rial.Inset is a<110>zone axis di raction pattern(ZADP)from a spinel grain.NTC thermistors produced from oxide powders859intensity of the NiO peaks(A)before and(B)after the experiment.However,Fig.7,which corre-sponds to Fig.5(B)(single phase),shows traces that are identical(A)before and(B)after.It is thought that the accelerated aging at470 C leads to NiO being re-absorbed into the ceramic during the lifetime of the experiment.It is proposed that the decomposition reaction occurs homogeneously throughout the ceramic,and the NiO is intimately mixed with the spinel phase,as evidenced by Fig.4. The reverse process may therefore occur relatively quickly because of the short di usion distances involved(of the order of nm according to Fig.4). However,it should be noted that aging at room temperature may be related to di erent phenomena than suggested by these accelerated tests.4Conclusions.The reaction between NiO and Mn2O3pro-ceeds forwards slowly at temperatures less than900 C,but will reverse as temperature increases above this value..The rate of decomposition increases with increasing temperature resulting in amicrostructure of intimately mixed NiO and Mn-rich spinel..Initial investigations indicate that a single phase ceramic gives rise to substantial improvements in thermal stability under accelerated aging.References1.Brabers,V.A.M.and Terhell,J.,Electrical conductivityand cation valencies in nickel manganite.Phys.Stat.Sol.(a),1982,69,325±332.2.Dorris,S.E.and Mason,T.O.,Electrical properties andcation valences in Mn3O4.J.Am.Ceram.Soc.,1988, 71(5),379±385.3.Rousset,A.,Larange,A.,Brieu,M.,Couderc,J.andLegros,R.,In¯uence de la microstructure sur la stabilite electrique des thermistance.C.T.N Journ.de Phys.III, 1992,4,833±845.4.Macklen,E.D.,Electric conductivity and cation distribu-tion in nickel manganite.J.Phys.Chem.Solids,1986, 47(11),1073±1079.5.Wickham,D.G.,Solid phase equilibria in the systemNiO±Mn2O3±O2.J.Inorg.Chem.,1964,26,1369±1377.6.Feltz,A.,Topfer,J.and Schirrmeister,F.,Conductivitydata and preparation routes for NiMn2O4thermistor ceramics.J.Eur.Ceram.Soc.,1992,9,187±191.7.Jung,J.,Topfer,J.,Murbe,J.and Feltz,A.,Micro-structure and phase development in NiMn2O4spinel ceramics during isothermal sintering.J.Europ.Ceram.Soc.,1990,6,351±359.860G.D.C.Csete de GyoÈrgyfalva et al.。

The Role of the Exchange-Correlation Potential in ab initio Electron Transport Calculations

a r X i v :c o n d -m a t /0609637v 2 [c o n d -m a t .m e s -h a l l ] 6 M a r 2007The Role of the Exchange-Correlation Potential in ab initioElectron Transport CalculationsSan-Huang Ke,1Harold U.Baranger,2and Weitao Yang,11Department of Chemistry,Duke University,Durham,NC 27708-03542Department of Physics,Duke University,Durham,NC 27708-0305(Dated:March 5,2007)The eﬀect of the exchange-correlation potential in ab initio electron transport calculations is investigated by constructing optimized eﬀective potentials (OEP)using diﬀerent energy functionals or the electron density from second-order perturbation theory.We calculate electron transmission through two atomic chain systems,one with charge transfer and one without.Dramatic eﬀects are caused by two factors:changes in the energy gap and the self-interaction error.The error in conductance caused by the former is about one order of magnitude while that caused by the latter ranges from several times to two orders of magnitude,depending on the coupling strength and charge transfer.The implications for accurate quantum transport calculations are discussed.PACS numbers:73.40.Cg,72.10.-d,85.65.+hThe calculation of electron transport through sin-gle molecules directly from quantum mechanics is cur-rently being intensively investigated for both funda-mental physics and applications in molecular electron-ics [1].In such a calculation,properties of the par-ticular molecule must be incorporated into an accu-rate transport model.A frequently used theoretical ap-proach is the single-particle Green function (GF)method [2]combined with a density functional theory (DFT)[3]electronic structure calculation.In this approach [4,5,6,7,8],the atomic structure of the entire lead-molecule-lead system is taken into account explicitly [7,8].Despite its advantages and high eﬃciency for large systems,several aspects of this approach remain prob-lematic [9,10,11,12,13,14].Here we address one aspect:we show that an improved description of electron-electron exchange and correlation within Kohn-Sham DFT dra-matically changes the predicted conductance.In the standard GF+DFT approach,all electron-electron interaction eﬀects are incorporated through the self-consistent DFT calculation,while the transmis-sion calculation is simply single-particle.Consequently,as emphasized by others [11,13],exchange-correlation corrections to the expression for the current are ne-glected.But even before considering those corrections,self-interaction error (SIE)is a potentially serious prob-lem within the GF+DFT method itself [3,12,14]:it leads to an overly extended charge distribution and,therefore,inaccurate molecule-lead charge transfer,espe-cially for weakly coupled systems.In ab initio transport calculations,SIE will directly aﬀect the position of the chemical potential in the molecular HOMO-LUMO gap (“gap”for short),as well as the broadening of the HOMO and LUMO orbitals,possibly producing large errors in the conductance [12].It is thus critical to improve the xc potential so as to eliminate the eﬀects of SIE.Another well-known problem with DFT is that the predicted gap is too small,often leading to a signiﬁcant overestimation of the conductance.The solution to this problem relies on a quasiparticle calculation or the construction of a SIE-free functional with a nonlocal xc potential.In this paper,we focus on eliminating the SIE and revealing the magnitude of the errors caused by the two problems.One way to eliminate SIE is to use Hartree-Fock (HF)theory:the exact treatment of exchange eliminates SIE.However,because HF involves a single determinant and lacks dynamical screening,the LUMO orbital is not phys-ically meaningful and the gap is too large for extended systems and large molecules.Hybrid functionals,like B3LYP [15,16],are a possible compromise:these mix the HF exchange potential with the local eﬀective potential obtained from the local density approximation (LDA)or generalized gradient approximation.Although B3LYP is a signiﬁcant improvement over LDA for almost all molec-ular systems,SIE still remains [17].The optimized eﬀective potential (OEP)approach is a direction for improving DFT calculations [18],in which the (local)eﬀective potential is expressed as an implicit density functional in terms of the Kohn-Sham orbitals.OEP enables one to construct a local xc potential from any energy functional,such as the HF exact exchange (EXX)or B3LYP functionals,or from an electron den-sity obtained from a more accurate theory [19],such as second-order many-body perturbation theory (MP2).Most OEP calculations to date use the HF energy func-tional (EXX-OEP).This simplest exchange-only OEP approach improves systematically the electronic struc-ture of various semiconductors [20]:the band gaps are signiﬁcantly improved over those of both LDA and HF,although the underlying reason is still open [21,22].In this paper,we implement the OEP approach in DFT-based ab initio transport calculations and inves-tigate,for the ﬁrst time,the eﬀect of diﬀerent xc potentials—LDA,HF,EXX-OEP,B3LYP,B3LYP-OEP,and MP2-OEP.Our purpose in using OEP is to con-struct a local xc potential which is SIE free (EXX-OEPFIG.3:(color online)Electric ﬁeld induced electron transfer (change in Q )between two H 8clusters separated by 8˚A (as shown in the inset).For such a large separation,the electron transfer should be an integer.For HF,this is the case;how-ever,LDA and B3LYP calculations show a substantial SIE.the peaks in T (E )become sharper and the conductance decreases.Note that the spread in conductance values becomes larger:now the maximum diﬀerence (between LDA and EXX-OEP)is about a factor of 10.To show the eﬀect from the gap,we ﬁrst examine the real transport gap of the H 16molecule by calcu-lating the ionization potential (I )and electron aﬃnity (A )using the delta self-consistent ﬁeld method (∆SCF)and the outer valence Green function method (OVGF)[24].The result for I −A is ∆SCF(HF,LDA,B3LYP)=4.7eV,6.0eV,5.9eV,and OVGF=6.6eV.Note that all the results are substantially smaller than the 8eV HF gap,indicating that it is too large.In particular,the large diﬀerence between ∆SCF(HF)and HF shows that HF does not give a good description for this long-chain molecule because of the lack of screening/correlation,de-spite the fact that it works well for very small molecules (the screening is very weak there,see the database at /cccbdb).On the other hand,I −A is signiﬁcantly larger than all the DFT gaps (2∼3eV)in the second class,indicating that they are too small (EXX-OEP does not work well here).A rough estima-tion of the eﬀect of the gap is the diﬀerence in conduc-tance between EXX-OEP and HF,both of which lack correlation and are SIE free.In Fig.2,this diﬀerence is about one order of magnitude for both strong and weak coupling;because the HF gap is too large,this rough estimation is probably an overestimate.So far we have discussed the SIE and gap issues for the system without molecule-lead charge transfer,where SIE causes overly broadened HOMO and LUMO states.For systems with charge transfer,SIE is a more signiﬁcant problem because it may lead to too much charge trans-fer,particularly in weakly coupled systems.To directly demonstrate this,we calculate,by using Mulliken popu-lation analysis,the charge transfer between two weakly coupled H atomic chains induced by a strong electric ﬁeld (see Fig.3).Each chain contains 8H atoms separated byFIG.with (a)2.8energy the the Li cluster to the H-chain are listed.The SIE-infected functionals place the chemical potential near a molecular resonance,while the SIE-free functionals place it near the middle of the gap.1˚A ,and the separation between the two chains is 8˚A.Because of the very large separation,the physical elec-tron transfer must be an integer.In HF,the electron transfer is indeed always an integer,showing that it is SIE free.For LDA,the result is almost linear in the elec-tric ﬁeld (full SIE),while B3LYP signiﬁcantly improves upon LDA but is still not accurate (partial SIE).The transmission through system B,in which there may be substantial charge transfer,is shown in Fig.4for two values of the molecule-lead separation.Molecule-lead charge transfer determines the position of the chemical potential (ﬁxed in the lead)in the molecular gap,and therefore the resulting conductance.The charge transfer and conductance are listed in the ﬁgure.Note the strik-ingly diﬀerent behavior of the two groups of function-als:for functionals with SIE (LDA,B3LYP,and B3LYP-OEP),the chemical potential enters the HOMO reso-nance because the charge transfer is large,while for func-tionals without SIE (HF,EXX-OEP,and MP2-OEP),the chemical potential is at the middle of the gap because the charge transfer is near zero.Consequently,the conduc-tance given by these two groups of functionals are very4diﬀerent,up to three orders of magnitude.For the smaller separation,2.8˚A in Fig.4(a),the func-tionals with SIE give a charge transfer of about0.5e,and the resulting conductance is around0.8G0.When the separation is increased to4.0˚A[panel(b)],the peaks be-come sharper,and the conductance in all cases decreases by more than an order of magnitude.In contrast,the charge transfer resulting from the SIE functionals de-creases only slightly,showing clearly that it is an arti-fact of SIE.Despite the quantitative diﬀerences between the stronger and weaker coupling,the broad features in the two cases are the same:the biggest step in conduc-tance(a factor of∼30)is between the SIE functionals and MP2-OEP followed by two smaller decreases,ﬁrst from MP2-OEP to EXX-OEP and then further to HF, each by about a factor of10.Here the eﬀect from the gap is also about one order of magnitude,from comparing EXX-OEP and HF as for system A.While it is clear,in terms of SIE,that the EXX-OEP and MP2-OEP calculations improve signiﬁcantly the standard GF+DFT calculation,it is not obvious which one of the SIE-free functionals–HF,EXX-OEP, or MP2-OEP–gives a conductance closest to the truth. MP2-OEP provides a near-exact local eﬀective potential for Kohn-Sham DFT,but itsﬁnite xc potential disconti-nuity is not included in the gap.As a result,its gap is too small.EXX-OEP gives a slightly larger gap which,how-ever,is still too small,and correlation is absent.HF,on the other hand,yields too large a gap,and correlation is also absent.Therefore,in terms of transport,EXX-OEP and MP2-OEP probably overestimates the conductance while HF probably underestimates it.The error seems to be about a factor of10.Finally,we relate the present calculation to the more rigorous time-dependent DFT formalism(TDDFT)[25, 26].In principle,unlike DFT,TDDFT can treat the elec-tronic structure of excited states[27].A Landauer-like form for the steady-state current can be derived from TDDFT[26]:in the linear-response regime(zero bias), the current is a Kohn-Sham term plus a correction from dynamical xc eﬀects.The eﬀective potential in our cal-culation can be regarded as the long time limit of that in the Kohn-Sham term,which is the major part of the current.The missing dynamical xc eﬀect is an open issue studied in[11,13].Our results on the eﬀects of the xc potential are helpful for improving TDDFT calculations within the adiabatic approximation.In summary,by implementing the OEP approach in an ab initio transport calculation,we have systematically investigated the eﬀect of diﬀerent local and nonlocal xc potentials.Dramatic eﬀects,up to orders of magnitude, originate from two factors–the SIE and the energy gap. The former will dominate for systems with charge trans-fer and can be eliminated by using a SIE-free OEP po-tential,while the latter is diﬃcult to treat within DFT and leads to a typical overestimation of the conductance by about a factor of10.Possible solutions are either to perform transport calculations using quasiparticle states, like those in GW approximation,or to develop SIE-free energy functionals without the discontinuity problem. We thank Aron Cohen,Kieron Burke,and Qin Wu for valuable conversations.This work was supported in part by the NSF(DMR-0506953).[1]J.R.Heath and M.A.Ratner,Physics Today56,43(2003).[2]S.Datta,Electronic Transport in Mesoscopic Systems(Cambridge University Press,Cambridge,1995).[3]R.Parr and W.Yang,Density-Functional Theory ofAtoms and Molecules(Oxford University Press,New York,1989).[4]J.Taylor,H.Guo,and J.Wang,Phys.Rev.B63,245407(2001).[5]P.S.Damle,A.W.Ghosh,and S.Datta,Phys.Rev.B64,201403(R)(2001).[6]Y.Xue,S.Datta,and M.Ratner,Chem.Phys.281,151(2002).[7]M.Brandbyge,J.-L.Mozos,P.Ordej´o n,J.Taylor,andK.Stokbro,Phys.Rev.B65,165401(2002).[8]S.-H.Ke,H.Baranger,and W.Yang,Phys.Rev.B70,085410(2004).[9]F.Evers,F.Weigend,and M.Koentopp,Phys.Rev.B69,235411(2004).[10]K.Burke,R.Car,and R.Gebauer,Phys.Rev.Lett.94,146803(2005).[11]N.Sai,M.Zwolak,G.Vignale,and M.DiVentra,Phys.Rev.Lett.94,186810(2005).[12]C.Toher,A.Filippetti,S.Sanvito,and K.Burke,Phys.Rev.Lett.95,146402(2005).[13]M.Koentopp,K.Burke,and F.Evers,Phys.Rev.B73,121403(R)(2006).[14]B.Muralidharan,A.W.Ghosh,and S.Datta,Phys.Rev.B73,155410(2006).[15]A.D.Becke,J.Chem.Phys.98,5648(1993).[16]C.Lee,W.Yang,and R.Parr,Phys.Rev.B37,785(1988).[17]Y.Zhang and W.Yang,J.Chem.Phys.109,2604(1998).[18]W.Yang and Q.Wu,Phys.Rev.Lett.89,143002(2002),and references therein.[19]Q.Wu and W.Yang,J.Chem.Phys.118,2498(2003),and references therein.[20]M.St¨a dele,M.Moukara,J.A.Majewski,P.Vogl,andA.G¨o rling,Phys.Rev.B59,10031(1999).[21]S.Sharma,J.K.Dewhurst,and C.Ambrosch-Draxl,Phys.Rev.Lett.95,136402(2005).[22]M.Gr uning,A.Marini,and A.Rubio,J.Chem.Phys.124,154108(2006).[23]NWChem,A Computational Chemistry Package for Par-allel Computers,Paciﬁc Northwest National Lab,Rich-land,Washington,USA(2003).[24]Ortiz,J.Chem.Phys.89,6348(1988).[25]G.Stefanucci and C.-O.Almbladh,Europhys.Lett.67,14(2004).[26]G.Stefanucci,S.Kurth,and E.K.U.Gross,cond-mat/0607333(2006).[27]E.Runge and E.K.U.Gross,Phys.Rev.Lett.52,997(1984).。

凝聚态物理Chap2-transport

helpful in establishing a phenomenological physical picture
Very often transport measurements are the first to be carried out on new materials.
Phenomena discovered by transport
Common motive force F: Electrical: E
Common current J: Charge: Je
Thermal: -TJQ Spin: JM
L is the transport coefficient
The transport coefficients measure the ability of electrons and phonons to “transport” a “steady current” under the influence of an external “motive force”.
• Fractional Quantum Hall Effect beautiful electron dance induced by strong electron correlations • Most precise method to measure h • potential application in electronics
Features of transport measurements
• transport measurements are relatively easy • can be applied on all kinds of materials • can be carried out in extreme physical conditions • have close connections to applications • transport phenomena are very rich • transport results are very informative and instructive, very

电针镇痛的累积效应与脑内孤啡肽前体和前阿黑皮素基因表达的关系_王俊英

方案，可简化和规范临床治疗。

参考文献［1］杨冬梓．疑难妇产科学［M］．北京：科学技术文献出版社，2006：368－371．［2］张家礼，陈国权．金匮要略理论与实践［M］．北京：人民卫生出版社，2009：253－255．［3］经燕，王清．许润三治疗子宫内膜异位症、子宫腺肌病经验总结［J］．中日友好医院学报，2004，18（2）：104．［4］杨硕，尤昭玲．尤昭玲教授中医治疗子宫内膜异位症的诊疗心得［J］．中华现代临床医学杂志，2008，6（12）：1093－1094．收稿日期：2010－07－24基金项目：国家自然科学基金资助项目（30472241，30973796，90709031），国家重点基础研究发展计划（973计划）资助项目（2007CB512505，2006CB504506，2005CB523306）作者简介：王俊英（1983－），女，博士研究生，研究方向：针刺作用神经生物学机制。

通讯作者：刘俊岭（1946－），男，研究员，研究方向：经穴脏腑相关及针刺作用神经生物学机制［5］尤庆华．浅述从瘀论治慢性盆腔炎［J］．陕西中医，2007，28（3）：360－361．［6］乐杰．妇产科学［M］．北京：人民卫生出版社，2008：246．［7］张玉珍．中医妇科学［M］．北京：中国中医药出版社，2002：314．［8］魏绍斌，曹亚芳．从：“湿热瘀结”论治子宫内膜异位症探讨［J］．中国中医基础医学杂志，2006，12（10）：757－759．［9］魏绍斌，曾倩．盆炎康栓治疗慢性盆腔炎临床观察［J］．四川中医，2006，24（4）：83－84．［10］魏绍斌，杨心弦，郭蓉晓，等．内异康复栓直肠给药对子宫内膜异位症模型大鼠血液流变的影响［J］．中医药学刊，2006，24（2）：242－243．［11］王妍，要永卿，季晓黎，等．从调理冲任论温盒灸在治疗盆腔炎反复发作中的作用［J］．吉林中医药，2009，29（6）：499－500．［12］魏绍斌．盆腔炎反复发作的中医治疗探讨［J］．中国实用妇科与产科杂志，2008，24（4）：253－255．［13］郎景和．子宫内膜异位症研究的任务与展望（之二）［J］．中华妇产科杂志，2006，41（10）：649－651．［14］王清．对子宫内膜异位症中医治疗的见解［J］．中日友好医院学报，2002，16（4）：250－251．电针镇痛的累积效应与脑内孤啡肽前体和前阿黑皮素基因表达的关系王俊英，孟凡颖，陈淑萍，高永辉，刘俊岭（中国中医科学院针灸研究所，北京100700）摘要：目的：探讨累加电针治疗神经源性痛大鼠的作用机制。

电荷密度图像清晰度函数：解决晶体相位问题的新思路

Image definition evaluation functions for X-ray crystallography: A new perspective on the phase problem Hui LI*, Meng HE* and Ze ZHANG Beijing University of Technology, Beijing 100124, People’s Republic of China CAS Key Laboratory of Nanosystem and Hierarchical Fabrication, National Center for Nanoscience and Technology, Beijing 100190, People’s Republic of China Zhejiang University, Hangzhou 310014, People’s Republic of China Correspondence e-mail: huilicn@, mhe@ In memoriam of Ms.ion It is interesting to compare X-ray crystallography with the optical imaging process (Miao et al., 2015). In an optical imaging system, an object is illuminated by visible light, and the scattered light is then combined by the lenses to form an image which represent the object. In the case of X-ray crystallography, a crystal is illuminated by X-ray beam and the diffraction data is then measured directly since no appropriate “lens” is available for X-ray. One has to reconstruct the electron density image of the crystal from the diffraction data, which has been the core theme of X-ray crystallography since its advent. Theoretically, the electron density image of a crystal is just the inverse Fourier synthesis of its structure factors. Unfortunately, phases of structure factors can rarely recorded in diffraction experiments, and usually, only amplitudes of structure factor can be deduced from the diffraction data. Therefore, the electron density distribution of crystals cannot be reconstructed by a straightforward Fourier synthesis. Even worse, the phases “lost” in the diffraction experiments are more important than amplitudes of structure factors in defining the electron density distribution (Lattman and Derosier, 2008). This is so-called “phase problem” of X-ray crystallography. Fourier synthesis can be made after assigning an arbitrary phase set to the observed moduli, but the resultant Fourier map generally will not represent the real electron density distribution in the crystal. If all possible phase sets for the observed moduli can be tested in a limited time, the correct phase set and subsequently the electron density image which represents the crystal must be included in these tests. Then the “phase problem” is converted to a question how to pick out the correct electron density image among all the possible Fourier synthesis maps. Such a question is reminiscent of the auto-focusing process which is extensively performed in optical imaging devices. In the auto-focusing process, the image definition evaluation functions are calculated in real time and focusing is then tuned accordingly. The image definition evaluation functions describe the properties of photographs mathematically (Chen et al., 2013) and the properly focused image is then picked out based on the evaluation functions. Inspired by this, we try to establish evaluation functions for electron density image reconstructed from diffraction data. Hopefully, the evaluation function can pick out the electron density image which represents the crystal. For this purpose, the evaluation function should describe the general characteristic of electron density distribution in real crystals, and varies sensitively with phase sets assigned to the moduli to distinguish the faithfully reconstructed electron density distribution from the artifacts. The commonly recognized characteristics of electron density in crystals are atomicity and positivity. Actually, the atomicity and positivity have been extensively utilized in many (if not all) methods for structure solutions by X-ray diffraction. For example, the Patterson method (Patterson, 1934) and the recently popular charge flipping algorithm (Oszlányi and A. Sütő, 2004; 2008) are based on atomicity. Atomicity and positivity are also the prerequisite of the direct method (Woolfson, 1987; Woolfson and Fan, 1995), though apparently it works in reciprocal space. The

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j<k
an even number of vortices or magnetic ﬁeld ﬂuxes attached to them. When a particle moves around a closed loop it encircles the usual Aharonov-Bohm ﬂux due to the external magnetic ﬁeld which is partly cancelled by the vortices attached to the electrons. Therefore, the quasiparticles can be regarded as moving in an eﬀective magnetic ﬁeld which is much weaker than the applied magnetic ﬁeld. When constructing the CF wave function a Jastrow factor (zk − zl )2p was introduced for each pair of electron coordinates, quite similarly to Laughlin’s wave function. Subsequently, the lowest-Landau-level (LLL) projection procedure, was introduced with the consequence that the vortices are no longer rigidly bound to the electrons. Thus, the relative distribution of zeros and electrons becomes less restrictive, and their correlations in this composite fermion liquid were investigated numerically in recent papers.6,7,8 In the present paper, we investigate the electron-vortex correlations in a ﬁnite system by starting from exact many-body wave functions obtained by means of a direct numerical diagonalization. Our model system is a quantum dot containing a few (up to four) electrons. The numerical results are compared to those obtained from the analytically available rotating-electron-molecule9,10 (REM) wave functions. This recently formulated theory appeared as a competitor10 (or at least an alternative) to the CF approach. It is derived from a more solid theoretical background, and introduces no a priori requirements on the positions of the zeros of the wave function. The paper is organized as follows. The model, numerical procedure and the REM wave functions are described in Sec. II. The simple case of a two-electron quantum dot is described in Sec. III. Three- and four-electron dots are the subject of Secs. IV and V, respectively, and our conclusions are given in Sec. VI.
Correlation between electrons and vortices in quantum dots
M. B. Tavernier,1, ∗ E. Anisimovas,1, 2, † and F. M. Peeters1, ‡
Departement Natuurkunde, Universiteit Antwerpen (Campus Drie Eiken), Universiteitsplein 1, B-2610 Antwerpen, Belgium 2 Semiconductor Physics Institute, Goˇ stauto 11, LT-01108 Vilnius, Lithuania Exact many-body wave functions for quantum dots containing up to four interacting electrons are computed and we investigated the distribution of the wave function nodes, also called vortices. For this purpose, we evaluate the reduced wave function by ﬁxing the positions of all but one electron and determine the locations of its zeros. We ﬁnd that the zeros are strongly correlated with respect to each other and with respect to the position of the electrons and formulate rules describing their distribution. No multiple zeros are found, i. e. vortices with vorticity larger than one. Our exact calculations are compared to results extracted from the recently proposed rotating electron molecule (REM) wave functions.
(zj − zk )2p+1 exp −
1 4
l
|zl |2 ,
(1)
where units are used such that the magnetic length is set equal to unity. Here, z = x − iy is a complex number expressing the two-dimensional coordinates of the electrons. If one ﬁxes the coordinates of all electrons except one, the resulting wave function will have zeros of order 2p + 1 located at the positions of all ﬁxed electrons. The wave function (1) embodies the strong correlation between the electrons as the wave function (and the probability to ﬁnd an electron) in the vicinity of one of the ﬁxed electrons vanishes more quickly than prescribed by the Pauli exclusion principle alone. We also note that in Laughlin’s wave function all the zeros are rigidly bound to the electrons and there are no free zeros. In this respect, the wave function (1) is rather special since a different distribution of zeros (e. g., around or between the ﬁxed electrons) would also be able to serve the purpose of stronger correlation and reduced interaction energy. In the subsequently formulated composite fermion (CF) theory3,4,5 the strong correlations were dealt with in a diﬀerent way, by introducing weakly interacting quasiparticles. Also here, the zeros of the many-body wave function played a central role. A zero in the wave function can also be interpreted as a vortex, i.e., when going around a zero its phase changes by 2πn, and the winding number n equals the order of the zero. The new quasiparticles of the CF theory were interpreted as electrons with