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

英文原文

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

Time-Dependent Density-Functional Theory for the Stopping Power of an Interacting Electron

(2) where vq = −4πZ1 /q 2 is the Fourier transform of the bare Coulomb interaction and χq,k (ω ) is the double Fourier transform of the linear density-response function of a system of interacting electrons in the static ﬁeld of an impurity of charge Z1 at r = 0. In order to prove Eq. (2), we ﬁrst expand the right-hand side of Eq. (1) in powers of Z1 and we then exploit the symmetry properties of many-fold density-response functions to demonstrate that this expansion divided by the projectile velocity at its zero value exactly coincides (to all orders in Z1 ) with
Department of Physics and Institute for Condensed Matter Theory, Chonnam National University, Gwangju 500-757, Korea 2 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan 3 Materia Kondentsatuaren Fisika Saila, Zientzi Fakultatea, Euskal Herriko Unibertsitatea, 644 Posta Kutxatila, E-48080 Bilbo, Basque Country, Spain 4 Donostia International Physics Center and Centro Mixto CSIC-UPV/EHU, Donostia, Basque Country, Spain (Dated: February 2, 2008) Based on the time-dependent density-functional theory, we have derived a rigorous formula for the stopping power of an interacting electron gas for ions in the limit of low projectile velocities. If dynamical correlation between electrons is not taken into account, this formula recovers the corresponding stopping power of noninteracting electrons in an eﬀective Kohn-Sham potential. The correlation eﬀect, speciﬁcally the excitonic one in electron-hole pair excitations, however, is found to considerably enhance the stopping power for intermediately charged ions, bringing our theory into good agreement with experiment.

凝聚态物理实验第六章第二节

Other GMR systems
• Some other trilayer and multilayer structures (left column) that show GMR effect at room T (right common feature of all these GMR systems? • Ferromagnetic metal layers (Fe and Co) separated by nonmagnetic metal layers (Cu, Au, Cr)!
• MR materials have important technological applications.
Giant Magnetoresistance (GMR)
• Most MR devices are used to measure the external magnetic field by monitoring the change of resistance. • For practical MR applications, we need to find materials with large MR in relatively weak magnetic field at room temperatures.
• Compared to the paramagnetic state, the energy gain is: V = ¼ UD(EF)2E2 • For the FM state to be favored, the energy gain should be larger than energy cost: ¼ UD(EF)2E2 > ½ D(EF)E2
Magnetoresistance (MR)

电器功能故障与整车水管理

W技朮交淙]L5Tech ni c al Com muni c ati o n L______电器功能故障与整车水管理崔学平#,董金颖#,朱新武#,于奥洋#,崔一健2(1.北汽福田汽车股份有限公司，山东潍坊261061；2.山东建筑大学，山东济南250101)摘要：电器功能与整车水管理存在相关性，处理不到位会导致电器功能失效或其他问题，引起客户抱怨%本文阐述两例凝水产生导致的电器零件失效问题，并结合电气原理图进行分析，快速发现和解决问题，水管理问题应从开始设计初期加以考虑和预防％关键词：电器原理；水管理；风门电机；凝水；风道中图分类号：(463.6文献标志码：2文章编号：1003-0639(2021)04-0057-03The Correlation Between Electrical Device Malfunction and Vehicle Water ManagementCUI Xue-ping1%DONG Jin-ying1%ZHU Xin-wu1,YU Ao-yang1%CUI Yi-jian2(1.Beiqi Foton Motor Co.,Ltd.,Weifang261061； 2.ShanUong Jianzhu University,Jinan250101,China)Abstract：There are some correlation between electrical device function and vehicle water management,improper treatment will lead to device malfunction or other problems,which will cause customer complains.This paper expounds two failure problems of electrical parts caused by condensate water,analyze by the electrical schematic diagram,to find and solve the problems quickly.Water management should be considered and prevented at the beginning.Key words：electrical schematic diagram；water management；mode actuator；condensate water；duct崔学平(1975-),男，工程师，从事汽车电器空调开发与研究工作6!引言作为交通工具的车辆已经脱离简单的运输功能，人们对车辆细节方面越来越关注。

Boundary Conditions in an Electric Current Contact

a r X i v :c o n d -m a t /0201054v 1 [c o n d -m a t .m t r l -s c i ] 4 J a n 2002Boundary Conditions in an Electric Current ContactO.Yu.Titov ∗CICATA—IPN,Av.Jos´e Siurob 10,Col.Alameda,76040Quer´e taro,Qro.,M´e xicoJ.Giraldo †Grupo de F´ısica de la Materia Condensada,Universidad Nacional de Colombia,A.A.60739,Bogot´a ,ColombiaYu.G.Gurevich ‡Depto.de F´ısica,CINVESTAV—IPN,Apdo.Postal 14–740,07000M´e xico, D.F.,M´e xicoIn most electronic devices,electric current of both types (electrons and holes)ﬂows througha ually the boundary conditions have been formulated exclusively for open circuit.The boundary conditions proposed here bypass this limitation by the ﬁrst time,as far as we are aware.Besides,these new boundary conditions correctly describe current ﬂow in a circuit,i.e.,closed circuit conditions,which are the usual operation conditions for electronic devices and for the measurement of many transport properties.We also have generalized the case (as much as it is possible in a classical treatment),so self-consistent boundary conditions to describe current-ﬂow through a contact between two arbitrary conducting media are developed in the present work.These boundary conditions take into account a recently developed theory:inﬂuence of temperature space inhomogeneity due to the interfaces and quasi-particles temperature-mismatch on thermo-generation and recombination.They also take into account surface resistance,surface recombination rates and possible temperature discontinuities at the interface due to ﬁnite surface thermo-conductivity.The temperature diﬀerence between current-carriers and phonon subsystems is also included in this approach.PACS numbers:73.25.+i,73.30.+y,73.40.-cThis work addresses the problem of deﬁning appropriate boundary conditions (BCs)in a closed electronic cir-cuit of any type in the simplest general case.It is a continuation and generalization of previous studies published elsewhere.1,2,3,4,5They have been undertaken to clarify diﬀerent processes arising in connection with real working conditions of electronic devices in general,6,7and in the search for applications to design new optoelectronic and thermoelectric devices.8,9,10In spite of great advances in the ﬁeld of contemporary electronics,the eﬀects of any boundary between two arbitrary materials have been studied,in practice,only for open circuits.The presence of both charge carriers (like electrons and holes in semiconducting materials;Cooper pairs in superconductors;etc...)in circuits containing heterojunctions,makes a big diﬀerence when the circuit is closed.Additionally,an intrinsic nonlinear eﬀect due to the temperature diﬀerence between electrons and holes,from one side,and phonons,from the other (hot electrons),is usually forgotten (see,for example Ref.11,12,13).This eﬀect has been studied in Ref.5.Actually,one can mention very few works handling closed circuits and non-equilibrium carriers (see Ref.14and reference therein).In more recent works 1,2,4it has been shown that it is compulsory to take into account non-equilibrium carriers in bipolar and p -type semiconductors even in a linear approximation by electric ﬁeld;furthermore,recombination starts to play an important role even in the linear regime and cannot be neglected.3,4,5In practice,people use zero-current BCs to deﬁne the functionality of solid state electronic devices in general.6,7This is an ideal assumption that works well in many cases,particularly when there is no diﬀerent kind of charge and heat carriers and interfaces involved.For a proper description of real performance conditions in heterojunctions,it should not be ignored that all devices are working in a mode such that electric current ﬂows through the ends of a semiconducting structure and in a whole closed circuit,completed with another semiconductor,a normal metal or a superconducting material.Typically the semiconductors have both charge carriers,electrons and holes,and important recombination processes take place at the ends.All these facts demand a more stringent choice of BCs,and in particular of non-vanishing current BCs at the borders between diﬀerent elements of a circuit to setup the correct working conditions of solid state electronic devices in general.These BCs become more important nowdays when accurate measurements of current to the level of holes and electrons are feasable.10In what follows,general BCs at the contact between two arbitrary conducting media will be obtained.Next we exemplify the usual case of a metal-semiconductor boundary.Finally we comment our results.BCs(see,for example,Ref.15)can be obtained from the transport equations for non-equilibrium carriers.16In the static case they look like:div j n=eR n,div j p=−eR p.(1) Here R n,p are electron and hole recombination rates;j n=σn −d˜ϕn dx ,j p=σp −d˜ϕp dx ,(2) are electron and hole electric currents;σn,p—electron and hole conductivity;˜ϕn,p=ϕ∓µn,pτn ,R p=δpτn+δpNotice that the recombination rate R in Eq.(8)is x dependent.Each one of the contacting media changes its properties in a distance close to the Debye’s radius and becomes inhomogeneous in that region.This explains the x dependence of R in Eq.(8).Let us now integrate Eq.(1)with dx fromεtoδ,and with dεfrom−δto+δ.One obtains:j n(+0)=σs n{˜ϕn(−0)−˜ϕn(+0)−αs n[T n(+0)−T n(−0)]}+eR n+sj p(+0)=σs p ˜ϕp(−0)−˜ϕp(+0)−αs p[T p(+0)−T p(−0)] −eR p+s.(11) Hereσs n andσs p are electron and hole surface conductivity:(σs n,p)−1=limδ→0+δ−δ(σn,p)−1(ξ)dξ,(12)αs n,p are thermopower surface coeﬃcients:αs n,p=1We can simplify Eq.(15)due to the absence of surface recombination in a metal(it has only electrons as charge carriers):R n−s=R p−s=0.Therefore,from Eq.(17)it follows that:R s=R n+s=R p+s.(18) There are no positive charge carriers in metals,which means that electron current in the metal,j m n(−0),should be equal to the whole current j0.In this case,the BCs(15)reduce to:j0=σs n ϕm−ϕs(0)−µm e+∆εce +µs(0)e−αs n[T s n(0)−T m n(0)] +eR s,j s p(0)=−eR s.(20)The last condition in(20)is equal to the second condition in(7).Theﬁrst equation in(7)can be rewritten asj s n(0)−j0=eR s.(21) Notice that theﬁrst equation(20)follows from Eqs.(19)and(21).All this yields the new BCs for a metal-semiconductor junction which are now formed by Eq.(19),Eq.(21),and the second of equations(20).In conclusion,we have formulated general BCs,corresponding to currentﬂow through the boundary between two conducting media.These conditions consider possible jumps of the electron’s and hole’s Fermi quasilevels and of the electric potentials at the boundary;they also take into account the surface recombination rates.The general procedure has been applied to a usual contact,namely the metal-semiconductor boundary,but the method is valid for all types of contacts(n+-n,p+-p,p-n,Schottky barrier,etc...)between diﬀerent conducting materials.This work was partially supported by CONACyT,Mexico.JG has received support from DINAIN(Universidad Nacional de Colombia),Colombia.∗Electronic address:oleg.titov@†Electronic address:jgiraldo@.co‡Electronic address:gurevich@ﬁs.cinvestav.mx1Yu.G.Gurevich,G.N.Logvinov,O.I.Lyubimov,and O.Yu.Titov,Phys.Rev.B51,6999(1995).2Yu.G.Gurevich,J.Thermoelectricity2,5(1997).3I.N.Volovichev and Yu.G.Gurevich,Semiconductors35,306(2001).4Yu.G.Gurevich,G.N.Logvinov,G.Espejo,O.Yu.Titov,A.Meriuts,Semiconductors34,755(2000).5Yu.G.Gurevich and I.N.Volovichev,Phys.Rev.B60,7715(1999).6J.Singh,Semiconductor devices:basic principles(John Wiley and Sons,New York,2001).7S.M.Sze,Semiconductor devices:physics and technology(John Wiley and Sons,New York,1985).8R.Aguado and L.Kouwenhoven,Phys.Rev.Lett.84,1986(2000);C.W.Hoyt et al.,ibid,85,3600(2000).9Francis J.DiSalvo,Science285,703(1999);G.D.Mahan and L.M.Wood,Phys.Rev.Lett.80,4016(1998).10A.Fujiwara and Y.Takahashi,Nature40,560(2001).11Jan Tautc,Photo and Thermoelectric Eﬀects in Semiconductors(Pergamon Press,N.-Y.,Oxford,London,Paris,1962).12A.I.Anselm,Introduction to Semiconductor Theory(Mir-Moscow,Prentice-Hall,Englewood Cliﬀs,NJ,1981).13A.F.Ioﬀe,Physics of Semiconductors(Infosearch,London,1960).14I.I.Balmush,Z.M.Dashevsky,A.I.Kasiyan,Thermoelectric Phenomena in Multilayer Semiconductor Structures(Shtiintsa, Kishenev,1992)[in russian].ndau and E.M.Lifshits,Course of theoretical Physics,Vol.8:Electrodynamics of Continuous Media(Pergamon, New York,1984).16V.L.Bonch-Bruevich,S.G.Kalashnikov,Physics of Semiconductors(VEB Deutscher Verlag der Wissen-Schaften,Berlin, 1982)[in german].17Yu.G.Gurevich,O.L.Mashkevich,Phys.Rep.181,327(1989).18D.A.Neamen,Semiconductor Physics and Devices:Basic Principles(Richard D.Irwin,Boston,1992).19V.P.Silin,A.A.Rukhadze,Electromagnetic Properties of the Plasma and Related Media(Atomizdat Moscow,1961)[in russian].。

固体物理总结 Charles Kittel第七版

● A solid is said to be a crystal if the atoms are arranged in such a way that their positions areexactly periodic●Bravais lattice : all lattice points are equivalent,and hence by necessity all atoms in the crystal areof the same kind.●Non-Bravais lattice:some of the lattice points are nonequivalent●Choose the origin of coordinates at a certain lattice point, say A●Position R=n1a+n2b (n1,n2)is a pair of integers values depend on the lattice point.●Lattice vetors: the set of all vectors expressed by this equation.●Primitive Unit cell: the area of the parallelogram whose sides are the basis vectors a and b is aunit cell of the lattice.●TIPS:1) All unit cells have the same area2)each unit cell has one lattice point●Nonprimitive cell:1,the area of the non primitive cell is an integral multiple of the primitivecell.2,no connection should be drawn between nonprimitive cells and no-bravais lattices.●14 bravais lattice ,7crystal systems.●In all non simple lattices the unit cells are nonprimitive●Inversion center. Acell has an inversion center if there is a point at which the cell remainsinvariant when the mathematical transformation r-->-r is performed on it.All bravais lattices are inversion symmetric.●Reflection plane : A plane in a cell such that, when a mirror reflection in this plane is performed,the cell remains invariant.●Rotation axis: this is an axis such that, if the cell is rotated around it through some angle,the cellremains invariant. The axis is called n-fold if the angle of rotation is 2pi/n.● A nonbravais lattice is one in which, with each lattice site, there is associated a cluster of atomscalled the basis.●The symmetry of the basis ,called point-group symmetry,refers to all possible rotations(includinginversion and reflection)which leave the basis invariant. 32 different groups●SPACE-GROUP SYMMETRY: combine the rotation symmetries of the point groups with thetranslation symmetries. 72.●CRYSTAL DIRECTION: choose an origin on the line. Choose the latttice vector joining A to anypoint on the line. R=n1a+n2b+n3c the direction is now specified by the integral triplet[n1,n2,n3].the triplet [n1,n2,n3]is the smallest integer of the same relative ratios.●Collective all directions to the[n1,n2,n3] by <n1,n2,n3>●MILLER INDICES: 1,find intercepts with the axes along the basis vectors a,b,c, Let theseintercepts be x,y,and z. Fraction triplete (x/a,y/b,z/c),invert (a/x,b/y,c/z).●Curly bracket {hkl}represents to all the plane labeled by the same Miller indices(hkl)●● A Bravais lattice is a lattice in which every lattice point has exactly the same environment.●Five Bravais lattices in two dimensions 5、Hexagonal lattice1、Square lattice2、Rectangular lattice3、Oblique lattice4、Centered Rectangular lattice●Unit cell :1、primitive:Single lattice point per cell .Smallest area in 2D, or Smallest volume in3D ; 2、Conventional&Non-primitive: More than one lattice point per cell .Integral multiples of the area of primitive cell SC both 1 and 2●Seven crystal systems divided into fourteen Bravais lattices:Triclinic三斜1：Simple Angles and edge unequalMonoclinic单斜2:Simple,Base-Centered One2-foldrotation axis Orthorhombic正交Tetragonal正方Cubic立方Trigonal三角Hexgonal六角●3Common Unit Cell:SC BCC FCCSimple BC FCCV olume,conventional cellLattice points per cellV olume,primitive cellLattice points per unit volumeNumber of nearest neighborsNearest-neighbor distanceNumber of second neighborsSecond neighbor distancePacking fraction●To distinguish a lattice direction from a lattice point, the triple is enclosed in square brackets [ ...]is used.[n1n2n3]which is the smallest reduced integer of the same relative ratios.As to negative directions we should add a bar on n●Cubic has highest symmetric directions●Miller Indices are a symbolic vector representation for the orientation of an atomic plane in acrystal lattice.●Coordination Number (CN)(配位数) : The Bravais lattice points closest to a given point are thenearest neighbours：A simple cubic has coordination number 6; a body-centered cubic lattice, 8;and a face-centered cubic lattice,12.●Atomic Packing Factor (APF) is defined as the volume of atoms within the unit cell divided bythe volume of the unit cell.●BCC lattice + single atom basis=SC lattice + basis of 2 atoms at (0,0,0) and (1/2,1/2,1/2)Li NaK Rb Cs Fe Nb W●FCC lattice + single atom basis=SC lattice + basis of 4 atoms at (0,0,0), (1/2,1/2,0)(1/2,0,1/2),and (0,1/2,1/2) Cu Ni Pb●Sodium chloride structure consists of equal numbers of sodium and chlorine ions placed atalternate points of a simple cubic lattice. Each ion has six of the other kind of ions as its nearest neighbours. e.g.LiF,NaBr,KCl,LiI,●Cesium chloride consists of equal numbers of cesium and chlorine ions, placed at the points of abody-centered cubic lattice so that each ion has eight of the other kind as its nearest neighbors.●Diamond structure:The diamond lattice is consist of two interpenetrating face centered bravaislattices.There are eight atom in the structure of diamond.Each atom bonds covalently to 4 others equally spread about atom in 3d.●Zinc Sulfide structure● perofskite structure(钙钛矿结构) CHAPTER 2● The condition of constructive interference(相长干涉) ● Bragg Law● diffraction condition● (1)Show that the reciprocal lattice vector G=hb1+kb2+lb3 is orthogonal to he plane representedby Miller indices (h,k,l) (2) prove that the distance between two adjacent parallel planes of the direct lattice is d hkl =2π/|Ghkl|. (3)Show that the diffraction condition is equivalent to the Bragg law .● Reciprocal lattice vector correspond to a set of crystal planes.● Brillouim zone surface describes all k vectors that are constructively diffracted by the crystal. ● Properties of reciprocal lattice:(1) Reciprocal lattice is also a Bravais lattice in the samesymmetry class.(2)Every crystal structure has two lattices associated with it, the crystal lattices and reciprocal lattice. (3)Reciprocal lattice vector correspond to a set of crystal planes.(4) Reciprocal lattice of the reciprocal lattice is the original direct lattice. CHAPTER 3 ● Cohesive (binding) energy U:the energy that must be addedto the crystal to separate its components into neutral free atoms at rest ； Energy of free atoms - Crystal energy • Magnitude ~ 1-10eV except for the inert gas crystals (0.02-0.2eV) • U ≤Eion(Ionization energy = Binding energy of valence electrons) • U controls the melting temperature and bulk modulus ●V(r): the net potential energy of interaction as function of r. r : he distance between atoms, ions, or molecules.a,b: proportionality constant of attraction and repulsion, respectively. m, n: constant characteristics of each type of bond and type of structure. ● Types of bonds (a) Van der Waals (Molecular) Electrons localized among atoms (Inert gas crystals : He, Ne, Ar, Kr, Xe, Rn ) Fluctuating dipole –dipole interactionOn average spherically symmetric distribution of electron charge with the positive nucleus in the center.But thermal fluctuations (finite T) cause instantaneous electric dipole momentTransparent Insulators –completely filled outer electron shells \\FCC structures except for He3and He4\\Weakly bonding –van der Waals bondingMODEthe Lennard-Jones potential(b) Covalent Electrons shared by the neighboring atoms HCl, NH3Covalent bonding takes place between atoms with small differences in electronegativity which are close to each other in periodic tableThe covalent bonding is formed by sharing of outer shell electrons (i.e., s and p electrons) between atoms rather than by electron transfer.Gd π2=()m na b V r r r-=+(c) Metallic Electrons free to move through sample Metallic bonding is the typeof bonding found in metal elementsThis is the electrostatic force of attraction between positively charged ions and delocalized outer electrons(d) Ionic Electrons transferred to adjacent atoms （Alkali halides 金属卤化物Li, Na, K, Rb, Cs ，F, Cl, Br, I ）Electron configuration(电子组态) closed electronic shellsionic bonding is produced whenever an element w/. a relatively low ionization energy is combined with an element w/. a high electron affinity.Madelung constant α: geometric sum(depends on relative distance, number, and sign of neighboring atoms )CalculationNotice:Higher coordination number gives larger Madelung constant\\It depends on the structure of the crystal but not unit cell dimensions. CHAPTER 41.Born-Oppenheimer approximation adiabatic approximation(绝热近似)2.Monoatomic Chain● elastic waves.:A solid is composed of discrete atoms, however when the wavelength is very long,one may disregard the atomic nature and treat the solid as a continous medium. ●● The force on the sth atom;:The force to the right: The force to the left: F=● Our wavelike solutions on the other hand are uncoupled oscillations called normal modes; each khas a definite w given by above eqn. and oscillates independently of the other modes.So the number of modes is expected to be the same as the number of equations N. ;● Periodic boundary conditions (Born-Karman) : (i)N atom form a cirque, which make all atoms tobe equivalent.(ii)The motion of atom can be seen as linearity, due to numbers of atoms are large. ● In three dimensionsp atoms per primitive cell \3p vibration branchesAcoustic (3) : LA (longitudinal TA1(transverse) TA2(transverse) Optic (3p-3) : LO (longitudinal) TO(transverse)N primitive cell: 3pN vibration modes ；Acoustic (3N) : LA (longitudinal)TA1(transverse)TA2(transverse) Optic ((3p-3)N): LO (longitudinal) TO(transverse)● Phonon ：Specifying the vibrational states of the crystal by specifying number of phonon in each()()222()()...........2r a r a r a d V d V V r V a r a dr dr ==-⎛⎫⎛⎫=+-++ ⎪ ⎪⎝⎭⎝⎭∑∑∑∑⎪⎭⎫ ⎝⎛+==j k j j j kj k n k E E )(21)(ωstate ki 。

英语文献

Three-dimensional model and simulation of vacuum arcs under axial magnetic fieldsLijun Wang,a)Shenli Jia,Xin Zhou,Haijing Wang,and Zongqian ShiState Key Laboratory of Electrical Insulation and Power Equipment,Xi’an Jiaotong University,Xi’an710049,China(Received15October2011;accepted13December2011;published online26January2012)In this paper,a three-dimensional(3d)magneto-hydro-dynamic(MHD)model of axial magneticﬁeld vacuum arcs(AMFVAs)is established.Based on this model,AMFVAs are simulated andanalyzed.Three-dimensional spatial distributions of many important plasma parameters and electriccharacteristics in AMFVAs can be obtained,such as ion number density,ion temperature,electrontemperature,plasma pressure,current densities along different directions(x,y,and z),ion velocitiesalong different directions,electricﬁelds strength along different directions,and so on.Simulationresults show that there exist signiﬁcant spiral-shaped rotational phenomena in the AMFVAs,thiskind of rotational phenomenon also can be veriﬁed by the many related experiments(AMFVAsphotographs,especially for stronger AMF strength).For current simulation results of AMFVAs,themaximal rotational velocity at anode side is about1100m/s.Radial electricﬁeld is increased fromarc center to arc edge;axial electricﬁeld is decreased from cathode side to anode side.Radialelectricﬁeld at arc edge can be larger than axial electricﬁeld.Azimuthal electricﬁeld in mostregions is much smaller than radial and axial electricﬁeld,but it can reach about1.19kV/m.Radialmagneticﬁeld is the smallest one compared with other components,it reaches to maximum value atthe position near to anode,it can inﬂuence arc characteristics.V C2012American Institute of Physics.[doi:10.1063/1.3677881]I.INTRODUCTIONVacuum arcs(VAs)exist in the wide applications, including vacuum interrupters,1,2vacuum remelting metal-lurgy,3vacuum thrusters,4,5vacuum coating,6pulsed power technology,7and so on.For all of above applications,it is very important to understand the basic mechanisms of vac-uum arcs.For VAs in vacuum interrupters(VIs),there exist two kinds of control technology:axial magneticﬁelds (AMFs)and transverse magneticﬁelds(TMFs).At present, the AMFs technology is more popular for higher current inter-ruption,because it can keep VAs in the diffuse status under the conditions of very high current.VAs under AMFs have been studied by many researchers based on experiments.8–16 In order to further understand VAs mechanisms,VAs have been modeled and simulated.17–23At present,most simula-tions are based on2d axial symmetric model.17–21In the pa-per,22the deﬂection effect of VAs under combined action of AMFs and external magneticﬁeld from bus bar has been modeled and simulated.However,the model is still simpliﬁed as two-dimensional conﬁguration.In fact,AMFVAs in com-mercial VIs are not completely axial symmetrical conﬁgura-tion,the interelectrode plasma under the control of AMFs not only exists the signiﬁcant rotation phenomena but also anode melting pool can appear rotational phenomena.24–26More-over,the motion of cathode spots and electrode erosion of high-current VAs are also not completely symmetrical.26In order to further understand AMFVAs,three-dimensional model is needed.Hartmann et al.23have simulated AMFVAs based on3d conﬁguration,but the3d model is not detailed described.In this paper,based on two-ﬂuid theory and Maxell equation,3d magnetic hydrodynamic(MHD)model of AMFVAs will be established.3d spatial characteristics of AMFVAs also will be simulated and analyzed.The rotation phenomena of interelectrode plasmas also will be researched and analyzed as an emphasis.The paper is arranged as follow-ing.In Sec.II,a3d MHD model of AMFVAs is established. Physical and mathematical models are detailed described in this section.In Sec.III,the spatial distributions of typical sim-ulation results will be given and analyzed.In Sec.IV,the related results will be discussed.The conclusions will be given in the Sec.V.II.3D MODELS OF AMFVASA.Physical and mathematical modelAMFVAs are modeled as Fig.1.Interelectrode plasma is considered as aﬂuid under AMFs.The cathode spots and anode are considered as boundary conditions of interelec-trode plasma.Physical model of AMFVAs and model assumption have been described in the earlier2d model.20,21 In this3d MHD model,Cartesian coordinates are adopted.Based on two-ﬂuid theory and Maxwell equations,the mathematical model of AMFVAs can be obtained: Mass Conservation equation:@n i u x@xþ@n i u y@yþ@n i u z@z¼0:(1)Here,n i is ion number density,u x is ion velocity along x direction,u y is ion velocity along y direction,u z is ion veloc-ity along z direction.a)Electronic mail:lijunwang@.1070-664X/2012/19(1)/013507/10/$30.00V C2012American Institute of Physics19,013507-1PHYSICS OF PLASMAS19,013507(2012)Momentum conservation equation along x direction:m i n i u x @u x @x þu y @u x @y þu z@u x@z¼j y B z Àj z B y ÀÁÀ@n i kT i @x þ@n e kT e@xþr Áðs i Þx ;(2)m i is the copper ion mass,j y is current density along y direc-tion,j z is current density along z direction,B z is AMF,B y ismagnetic ﬁeld along y direction,k is Boltzmann constant,T i is ion temperature,T e is electron temperature,s i is viscosity stress tensor,and n e is electron number density.Momentum conservation equation along y direction:m i n i u x@u y @x þu y @u y @y þu z @u y@z¼j z B x Àj x B z ðÞÀ@n i kT i @y þ@n e kT e@yþr Áðs i Þy :(3)B x is magnetic ﬁeld along x direction;j x is current densityalong x direction;B x is magnetic ﬁeld along x direction.Momentum conservation equation along z direction:m i n i u x@u z @x þu y @u z @y þu z @u z@z¼j x B y Àj y B x ÀÁÀ@n i kT i @z þ@n e kT e@zþr Áðs i Þz ;(4)l i is ion viscosity coefﬁcient.20Ion energy conservation equation:r Áðu *ðm i n i E i þp i ÞÞ¼r Áðk i r T i Þþr Áðs i Áu *ÞþZen i u *ÁE *þ3km e n e ei m i ðT e ÀT i Þþu *ÁÀn ee rj *þg T n e k r T e (5)Here,E i ¼C vi T i þu *22,C vi is ion speciﬁc heat at constant vol-ume;u *is ion velocity vector;p i is ion pressure;k i is ion thermal conductivity;20and ei is electron-ion collision frequency.Electron energy conservation equation:r Áðv *ðn e m e E e þp e ÞÞ¼r Á32j *ÁkT ee þk e r T e À3km e n e ei m i ðT e ÀT i Þþj 2x þj 2y þj 2zr þg Tk ej x @T e @x þj y @T e @y þj z @T e@z :(6)Here,E e ¼C ve T e þv *22,C ve is electron speciﬁc heat at con-stant volume;v !is electron velocity vector;p e is electronpressure;k e is electron thermal conductivity;20r is electrical conductivity;g T is a coefﬁcient.Relationship between current density and electrons,ions velocity:j *¼en e ðu *Àv *Þ;(7)j *is current density vector;e is electron charge.Magnetic transportation equation along x direction:Àr Á1l 0r r B x¼@@y v x B y Àv y B x ÀÁÀ@@z v z B x Àv x B z ðÞÀ1en 2e @n e @y Á@p e @z À@n e @z Á@p e @yÀ1l 0r 2@B x @z À@B z @x Á@r@z À@B y @x À@B x @y Á@r @y !þ1l 0r 2@r @x @B x @x þ@r @y @B x @y þ@r @z @B x@z;(8)v x ,v y ,and v z are electron velocities along x ,y ,and z direc-tions,respectively.l 0is vacuum magnetic permeability;p eis electron pressure.Magnetic transportation equation along y direction:Àr Á1l 0r r B y¼@@z v y B z Àv z B y ÀÁÀ@@x v x B y Àv y B x ÀÁÀ1en 2e @n e @z Á@p e @xÀ@n e @x Á@p e@z À1l 0r @B y @x À@B x @y Á@r @x À@B z @y À@B y @z Á@r@z !þ1l 0r 2@r @x @B y @x þ@r @y @B y @y þ@r @z @B y@z :(9)Components form of current density:j x ¼À1l 0@B y @z À@B z @y j y ¼À1l 0@B z @x À@B x@z j z ¼À1l 0@B x @yÀ@B y @x :8>>>>>>>><>>>>>>>>:(10)Because B z is considered as uniform distribution in this pa-per,@B z@y and @B z @y is equal to 0.FIG.1.(Color online)3d MHD model of AMFVAs.013507-2Wang et al .Phys.Plasmas 19,013507(2012)In this paper,magnetic ﬁelds B x ,B y along x and y direc-tions can be calculated by magnetic ﬁeld transportation equa-tions (Eqs.(8)and (9)).Then,current densities along x ,y ,and z directions can be obtained from magnetic ﬁeld distribu-tions (see Eq.(10)).After that,electric ﬁelds along different directions (x ,y ,and z )can be obtained by generalized ohm’s laws.In order to get radial and azimuthal electric ﬁelds,Car-tesian coordinate system need be changed as cylindrical coordinate system.In this paper,the electric ﬁled can be obtained by gener-alized ohm’s law:r u ¼Àj *r þu *ÂB *À1n e e j *ÂB *þ1n e e r P e þg T k r T e e E *¼Àr u :8><>:(11)Because relationship between current density and electrons,ions velocity,which is shown in Eq.(7).Generalized Ohm’s law can be derived as the following equationr u ¼Àj *r þv *ÂB *þ1n e er P e þg T k r T ee :(12)Therefore,electric ﬁelds along x ,y and z directions can bedescribed byE x ¼j x Àðv y ÁB z Àv z ÁB y ÞÀ1e @p e Àg T k @T e E y ¼j y r Àðv z ÁB x Àv x ÁB z ÞÀ1n e e @p e @y Àg T k e @T e@y E z ¼j z r Àðv x ÁB y Àv y ÁB x ÞÀ1n e e @p e @z Àg T k e @T e @z:8>>>>>>>>><>>>>>>>>>:(13)FIG.2.(Color online)Ion number density,ion pressure,and plasma pressure in 3d space (a)and xoz plane (b)(d ¼41mm;h ¼10mm;I ¼25kA;B z ¼6mT/kA).013507-3Vacuum arcs under axial magnetic fields Phys.Plasmas 19,013507(2012)B.Boundary conditions 1.Cathode sideFor cathode side,electron and ion temperature are both considered as 3eV.From Kutzner and Miller’s experimental results,27mean charge number of ions is 1.85.In the simula-tion,uniform current density of cathode side are assumed,current density is described by j 0¼I 0/p R 02,R 0is radius of cathode side,it is related to area occupied by cathode spots.According to computational ﬂuid dynamic (CFD)theory,for inlet (cathode side)condition of subsonic ﬂow,the stagna-tion pressure,stagnation temperature,and the direction of plasma ﬂow should be speciﬁed,here,ﬂow direction of nor-mal to cathode surface is assumed.Boundary conditions of magnetic transportation equa-tion along x direction at cathode side:B x ¼l 0j 0y2:(14)Boundary conditions of magnetic transportation equation along y direction at cathode side:B y ¼Àl 0j 0x2:(15)2.Radial boundaryBoundary conditions of magnetic transportation equa-tion along x direction at radial boundary:B x ¼l 0I 0y2p R :(16)Boundary conditions of magnetic transportation equation along y direction at radial boundary:B y ¼Àl 0I 0x2p R 2:(17)In Eqs.(16)and (17),R means radial boundary of arccolumn.FIG.3.(Color online)Ion velocity components (axial velocity,radial velocity,and azimuthal velocity)in 3d space (a)and xoz plane (b)(d ¼41mm;h ¼10mm;I ¼25kA;B z ¼6mT/kA).013507-4Wang et al .Phys.Plasmas 19,013507(2012)Boundary conditions of electron energy equation at ra-dial boundary:@T e@n¼0:(18)3.Anode sideAnode is considered as equipotential surface,according to this condition and the generalized ohms law,boundary condition of anode side for magnetic ﬁeld transport equation B x is obtained:@B x @z ¼Àl 0r @u sh @y þv z B x Àv x B z þ1n e e @P e @y þg T k e Á@T e@y:(19)Boundary condition of anode side for magnetic ﬁeld trans-port equation B x is obtained:@B y @z ¼l 0r @u sh @x þv y B z Àv z B y þ1n e e @P e @x þg T k e Á@T e@x:(20)In Eqs.(19)and (20),/sh means anode sheath potential.For electron energy equation,only the adiabatic condi-tion is considered:@T e@z¼0:(21)The ion pressure of the anode side is assumed according to the critical ﬂow condition (ion Mach number is equal to 1).21C.Simulation methodsIn the simulation,computational ﬂuid dynamic (CFD)software Fluent is used.The control volume method and coupled solver are adopted.Due to the larger number of mesh grids,parallel calculation technology is used.The cal-culation task is decomposed into 8parts in the simulation.III.SIMULATION RESULTS AND ANALYSISIn this paper,the electrode diameter d is considered as 41mm;gap distance h is 10mm;arc current is 25kA;AMF is considered as uniform distribution in thewholeFIG.4.(Color online)Ion velocity vector at anode side plane (d ¼41mm;h ¼10mm;I ¼25kA;B z ¼6mT/kA).FIG.5.(Color online)Ion temperature and electron temperature in 3d space (a)and xoz plane (b)(d ¼41mm;h ¼10mm;I ¼25kA;B z ¼6mT/kA).013507-5Vacuum arcs under axial magnetic fields Phys.Plasmas 19,013507(2012)interelectrode,whose intensity is in the reasonable range for VIs (3mT/kA;6mT/kA,and 10mT/kA).A.Ion number density and plasma pressureThe ion number density,ion pressure,and plasma pres-sure in 3d space and xoz plane are shown in Fig.2.Accord-ing to the current simulation results of AMFVAs,because only the external AMF (radial magnetic ﬁeld produced by bus bar or coil is not considered)is considered,many solved variables (ion number density,pressure,ion and electron temperature,and so on)have the similar distributions at xoz and yoz plane.So,in this section,the distributions of most variables are given only at xoz plane.From Fig.2,it can be seen that the ion number density is in the order of 1022m À3,ion number density of near cathode side is larger than that of near anode side.Ion number density at arc center also is larger than that at arc edge.According to the color change of ion density,it can be seen that the gradient of ion number density along x direction is larger than that along z direction.The ion number density located at the center of cathode is about 3.58Â1022m À3.Ion number density at anode center is about 1.6Â1022m À3.Ion number density at arc edge is about 0.9Â1022m À3.This kind of distribution of ion density also can be veriﬁed by many photographs of Vas.9,25,26In the most interelectrode arc region,ion pressure and plasma pressure are in the order of 104Pa.Plasma pressure includes ion and electron pressure,from Fig.2,it also can be seen that the electron pressure is larger than ion pressure.The gra-dient of plasma pressure along x direction also is larger than gradient along z direction.According to arc photographs,italso can be found that light intensity of vacuum arc at center position is much stronger than that at edge region.B.Ion velocity contourIon velocity components in 3d space and xoz plane are shown in Fig.3.From Fig.3,it can be seen that the axial (z direction)velocity is larger than radial and azimuthal ve-locity.In the most arc column region,the axial velocity is between 1000m/s and 3000m/s,the radial velocity is very small,and at anode side,the radial velocity is also smaller than 400m/s.In the most arc column region,the azimuthal velocity is much larger than radial velocity.At anode side,the maximal rotation velocity can reach to about 1100m/s.Under the mid-gap plane,the rotation velocity has the inverse direction compared with anode side plane.At the cathode side,the only axial velocity is considered,that is to say,the radial and rotational velocities are considered as zero.According to the simulation results of different compo-nents of ions velocity,it can be seen that the ions have the spiral-shaped motion orbit,which is because the direction of azimuthal velocity appears transition at mid-gap plane region.But,the rotation velocity at anode side is much larger than other region.From distribution of xoz plane,at anode side,the rotation velocity along x direction is ﬁrst increased and then decreased.C.Ion velocity vector at anode surfaceIon velocity vector at anode side plane is shown in Fig.4.In anode side plane,the axial component of ion veloc-ity is not included.From Fig.4,it also can be seen that intheFIG.6.(Color online)Radial and azimuthal magnetic ﬁeld in 3d space (a)and xoz plane (b)(d ¼41mm;h ¼10mm;I ¼25kA;B z ¼6mT/kA).013507-6Wang et al .Phys.Plasmas 19,013507(2012)most arc center region,the ion motion exists signiﬁcant rota-tion;the radial velocity is very small.From center to edge,the rotation velocity is ﬁrst increased and then decreased.The rotation direction is along anticlockwise.However,at the arc edge region,the radial velocity is higher,and rotation phenomenon of ions is not more signiﬁcant than that in other region.This kind of rotation is related to the action of elec-tromagnetic force in the arc column.Maximum rotational velocity is about 1100m/s.D.Ion temperature and electron temperatureIon and electron temperatures in 3d space and xoz plane are shown in Fig.5.From Fig.5,it can be seen that from cathode side to anode side,electron temperature is increased,which is mainly due to the joule heating effect.The increase of ion temperature is mainly due to the energy exchange between ions and electrons by frequent collisions.This con-clusion also can be obtained from 2d model.20,21E.Radial and azimuthal magnetic fieldRadial and azimuthal magnetic ﬁelds in 3d space and xoz plane are shown in Fig.6.From Fig.6,it can be seen that the azimuthal magnetic ﬁeld is increased from arc center to arc edge.From the distribution of azimuthal magnetic ﬁeld,it also can be deduced that anode side appears current pared azimuthal and radial magnetic ﬁeld,it can be seen that the radial magnetic ﬁeld is much smaller than azimuthal magnetic ﬁeld.The radial magnetic ﬁeld at arc axis is zero and is ﬁrst increased then decreased from arc center to arc edge.The radial magnetic ﬁeld exist positive and negative value in the column,which shows that the direction of radial magnetic ﬁeld is changed in the interelec-trode.At anode side,the radial magnetic ﬁeld is higher,which is related to current constriction at anode side.The maximum radial magnetic ﬁeld is about 10.9mT.Here,ra-dial magnetic ﬁeld is generated by the arc column,not includes that generated from electrodesystem.FIG.7.(Color online)Current density along different directions (axial,radial,and azimuthal)in 3d space (a)and xoz plane (b)(d ¼41mm;h ¼10mm;I ¼25kA;B z ¼6mT/kA).013507-7Vacuum arcs under axial magnetic fields Phys.Plasmas 19,013507(2012)F.Current density distributions along different directionsCurrent density along different directions in 3d space and xoz plane is shown in Fig.7.From Fig.7,it can be seen that the axial current density appear signiﬁcant nonuniform phe-nomenon at anode pared with radial and azimuthal current density,axial current density is dominant in the arc column region.The maximum values of axial,radial,and azi-muthal current density are 2.44Â107A/m 2,1.16Â107A/m 2,and 0.93Â107A/m 2,respectively.Different from the distribu-tion of axial current density,radial and azimuthal current den-sities are ﬁrst increased then decreased from arc center to arc edge.From Fig.7,it also can be seen that the polarity of radial and azimuthal current density is changed in the column.G.Current density vector at anode sideCurrent density vector at anode side plane is shown in Fig.8.At this plane,only radial and azimuthal current den-sity is included,axial current density is not included.From Fig.8,it can be seen that the azimuthal current density is directed along anticlockwise,which is the same as that of ion azimuthal velocity.However,the radial current density is directed from center to edge,which is contrary to the direction of ion radial velocity.But,near cathode side,the direction of radial current is opposite to that near anode side (Fig.7).Compared with ion velocity vector (Fig.4),the val-ues of radial current density are close to that of azimuthal current density,the rotational phenomenon is not signiﬁcant than that of ion velocity distribution.Rotational motion of ions is more signiﬁcant than radial motion.H.Electric field distributions along different directionsElectric ﬁeld distributions along different directions in3d and xoz plane are shown in Fig.9.From Fig.9,it can beseen that axial electric ﬁeld is between 1kV/m and 5kV/m in the arc column region.From axial electric ﬁeld distribu-tion,it can be seen that the axial electric ﬁeld is decreased from cathode side to anode pared Fig.9with Fig.7,it can be seen that the distribution of axial electric ﬁeld is different from that of axial current density.Their dif-ference is not contradictory,which is because electric con-ductivity also is increased from cathode side to anode side.Moreover,the increase of electric conductivity is faster than the increase of axial current density.From arc center to arc edge,radial electric ﬁeld is increased from 0to 4kV/m.The distribution of azimuthal electric ﬁeld is similar to that of az-imuthal current density.The maximal azimuthal electric ﬁeld is about 1.18kV/m.From Fig.9,it also can be seen that the azimuthal electrical ﬁeld is so strong that it cannot be neglected,it can inﬂuence VAs characteristics.In order to analyze radial electric ﬁeld E r distribution in Fig.9.E r is given by the Eq.(22):E r ¼j r r Àðv h ÁB z Àv z ÁB h ÞÀ1n e e @p e @r Àg T k e @T e @r ¼j r r À½ðu h À1n e e j h ÞÁB z Àðu z À1n e e j z ÞÁB h Þ À1n e e @p e @r Àg T k e @T e@r:(22)From Eq.(22),it can be seen that the radial electric ﬁeld is related to the distributions of radial current density j r ,azi-muthal electron velocity v h ,axial electron velocity v z ,axial magnetic ﬁeld B z ,azimuthal magnetic ﬁeld B h ,radial gradi-ent of electron pressure p e ,and radial gradient of electron temperature T e .According to our evaluation,the inﬂuence of electron pressure and electron temperature on E r distribution is very small.So,E r is mainly inﬂuenced be the ﬁrst term and the second term of right side in Eq.(22).In vacuum arc,due to lower electron density,the Hall Effect is very signiﬁcant,current constriction appears from cathode to anode,not only non-uniform axial current density j z but also j r and j h (in Fig.7).Because j r and j h is smaller than j z in the most arc column region.The decisive inﬂuence should be from v z times B h .If other inﬂuence factors are not considered,the distribution of E r is approximately similar to the distribution of v z times B h .If we considered all inﬂuence factors,the distribution in Fig.9can be obtained.Londer and Ulyanov also 28have analyzed electric ﬁeld and potential along radial position based on simpler assumption,they also found this kind of distribution.At anode surface,radial electric ﬁeld should be equal to zero,but in this paper,the researched region is only arc col-umn,there exists anode sheath between anode surface and adjacent arc column plasma,so,the radial electric ﬁeld in the edge of arc column is likely not to be zero.I.Influence of AMF strength on ion velocity at anode planeAMF is the important inﬂuence factor for VAs character-istics.Fig.10shows that the inﬂuence of difference AMF strengths on ions rotational velocity at anode side plane.With the increase of AMF strengths,the ions rotationalvelocityFIG.8.(Color online)Current density vector at anode side plane (d ¼41mm;h ¼10mm;I ¼25kA;B z ¼6mT/kA).013507-8Wang et al .Phys.Plasmas 19,013507(2012)also is increased.When the AMF strengths are increased from 3mT/kA to 6mT/kA,10mT/kA,the maximum rotational ve-locity is increased from about 800m/s to 1100m/s and 1200m/s.That is to say,with the increase of AMF strength,the rotational phenomena will become more signiﬁcant.From Fig.10,it also can be seen than the maximum values of ions rotational velocity move toward outside along arc edge.From Fig.10,it also can be seen that the signiﬁcant rotation phe-nomena looks like a ring,the inner and outer diameters of rings are increased with the increase of AMF strength.IV.DISCUSSIONSEven if the axial velocity is dominant and larger than ra-dial and azimuthal velocities,ion ﬂow orbit shows spiral-shaped motion,which is because the direction of ion rotation velocity is changed from clockwise to anticlockwise near mid-gap plane.According to momentum conservation equa-tion,this kind of phenomena is mainly related to the electro-magnetic force.For azimuthal momentum equation,theelectromagnetic force j r xB z and j z xB r will inﬂuence the rota-tion direction of ions.Because j z and B z keep invariable direc-tion,the change of j r and B r directions leads to spiral-shaped orbit of ions motion.These kinds of ion rotational phenomena also were found by Hartmann et al .’s simulation results.23About AMFVAs,from arc photographs,it can be seen that the inclination of many plasma jets at arc edge,this kind of inclination is likely to be related to the rotation of ions.Moreover,with the increase of AMF strengths,this kind of in-clination will become more signiﬁcant.For commercial cup-shaped AMF electrode,the inclination angle of plasma jets before peak-current moment is smaller than that after peak-current moment,which is because the AMF strength is lagged from arc current due to an eddy effect.This kind of inclination can be related to the ratio between rotational velocity and axial velocity.In order to verify simulation results,the further experiments are needed.At present,many arc photographs are taken along the direction which is perpendicular to arc col-umn,only inclination phenomena can be found.If the arc pho-tographs can be taken along the direction which is inparallelFIG.9.(Color online)Electric ﬁeld distributions along different directions (axial,radial,and azimuthal)in 3d space (a)and xoz plane (b)(d ¼41mm;h ¼10mm;I ¼25kA;B z ¼6mT/kA).013507-9Vacuum arcs under axial magnetic fields Phys.Plasmas 19,013507(2012)to arc column,the signiﬁcant rotation can be found like Fig.4and Fig.10.As the results acted by arc column,anode elec-trode is melted and evaporated,from anode erosion traces,the signiﬁcant rotation phenomena also can be found.24–26The rotation of anode melting pool is likely to be not only related to the electromagnetic force but also related to the arc plasma rotation in the interelectrode;this kind of inﬂuence need to be further analyzed.V.CONCLUSIONSIn this paper,3d MHD model of AMFVAs has been established.Based on this model,AMFVAs have been simu-lated and analyzed.Because of the bigger number of mesh grids,parallel processing technology is adopted.Simulation results show that arc plasma exist signiﬁcant rotational phe-nomena,which also can be veriﬁed by the related experi-ments;the inclination of plasma jets in arc photographs is related to the rotation of ions.According to current simulation conditions,the maximal rotation velocity is about 1100m/s for AMF 6mT/kA;axial velocity is dominant,radial velocity is much smaller than rotational velocity in arc column region;radial electric ﬁeld is increased from arc center to arc edge,ra-dial electric ﬁeld value at arc edge is very large,which is close to or larger than axial electric ﬁeld value.For magnetic ﬁelds along different directions,radial component are the smallest one,but it does inﬂuence arc plasma characteristics.The model is promising and also can be used to simulate the deﬂection and unsymmetrical phenomena of VAs.The further experiments are needed to verify the simulation results.ACKNOWLEDGMENTSThis work is supported by National Natural Science Foundation of China (50907045),State Key Laboratory of Electrical Insulation and Power Equipment Fund (EIPE10306),and Doctoral Fund of Ministry of Education of China (200806981052and 20090201110015).1fferty.Vacuum Arcs:Theory and Application (John Wiley and Sons,New York,1976).2P.G.Slade,The Vacuum Interrupter:Theory,Design,and Application (Taylor &Francis,New York,2008).3P.Chapelle,J.P.Bellot,H.Duval,A.Jardy,and D.Ablitzer,J.Phys.D:Appl.Phys.35,137(2002).4A.Anders,J.Schein,and N.Qi,Rev.Sci.Instrum.71,827(2000).5M.Keidar,J.Schein,K.Wilson,A.Gerhan,M.Au,B.Tang,L.Idzkowski,M.Krishnan,and I.I.Beilis,Plasma Sources Sci.Technol.14,661(2005).6R.L.Boxman,I.I.Beilis,E.Gidalevich,and V.N.Zhitomirskyu,IEEE Trans.Plasma Sci.33,1618(2005).7D.C.Hagerman and A.H.Williams,Rev.Sci.Instr.30,182(1959).8A.M.Chaly,A.A.Logatchev,K.K.Zabello,and S.M.Shkol’nik,IEEE Trans.Plasma Sci.31(5),884,(2003).9S.Jia,Z.Shi,L.Wang,Z.Wang,and M.Rong,IEEE Trans.Plasma Sci.32(5),2113(2004).10D.Gentsch and W.Shang,IEEE Trans.Plasma Sci.33(5),1605(2005).11W.Shang,E.Schade,H.Fink,I.Kleberg,E.Schade,and D.L.Shmelev,IEEE Trans.Plasma Sci.31(5),923(2003).12E.D.Taylor,P.G.Slade,and M.B.Schulman,IEEE Trans.Plasma Sci.31(5),909(2003).13W.Hartmann,wall,R.Renz,N.Wenzel,and W.Wietzorek,XXIII ISDEIV (Romania,2008)pp.259-263.14K.Steinke and M.Lindmayer,IEEE Trans.Plasma Sci.33(5),1600(2005).15S.Yanabu,T.Tsutsumi,K.Yokokura,and E.Kaneko,IEEE Trans Plasma Sci.17(5),717(1989).16H.Schellekens,XXth ISDEIV (Tours,France,2002),pp.38–43.17R.L.Boxman,J.App.Phys.48(6),2338(1977).18I.I.Beilis,M.Keidar,R.L.Boxman,and S.Goldsmith,J.Appl.Phys.83(2),709(1998).19E.Schade and D.L.Shmelev,IEEE Trans.Plasma Sci.31(5),890(2003).20L.Wang,S.Jia,Z.Shi,and M.Rong,J.Phys.D:Appl.Phys.38,1034(2005).21L.Wang,S.Jia,Z.Shi,and M.Rong,J.Appl.Phys.100,113304(2006).22L.Wang,S.Jia,K.Liu,L.Wang,and Z.Shi,Phys.Plasmas 16(10),103502(2009).23W.Hartmann,A.Hauser,wall,R.Renz,and N.Wenzel,23rd ISDEIV (2008),pp.398-401.24L.Wang,S.Jia,Y.Liu,B.Chen,D.Yang,and Z.Shi,J.Appl.Phys.107,113306(2010).25D.Yang,L.Wang,S.Jia,L.Wang,Z.Shi,and Y.Li,IEEE Trans.Plasma.Sci.38(2),206(2010).26L.Wang,L.Wang,S.Jia,Z.Shi,D.Yang,and Y.Liu,IEEE Trans.Plasma.Sci.38(7),1682(2010).27J.Kutzner and ler,J.Phys.D:Appl.Phys.25,686(1992).28Y.I.Londer and K.N.Ulyanov,IEEE Trans.Plasma.Sci.35(4),897(2007).FIG.10.(Color online)Inﬂuence of AMF strength on ions rotational velocity at anode side plane (d ¼41mm;h ¼10mm;I ¼25kA)(a)B z ¼3mT/kA;(b)B z ¼6mT/kA;(c)B z ¼10mT/kA.013507-10Wang et al .Phys.Plasmas 19,013507(2012)。

凝聚态物理Chap1-el-ph

Review of conventional metal physics: Electrons and Phonons
• basic physical properties of normal metals
• non-interacting Fermi gas model
• Landau Fermi liquid theory
• ignore all interactions (except instantaneous collisions) • only consider kinetic energy • electrons were treated as independent classical particles • Maxwell-Boltzmann distribution
• Quantitative explanation of the Wiedemann-Franz law (by luck!)
• The Hall effect and magnetoresistance • ac conductivity and optical properties of metals Problems: • heat capacity puzzle: Classical particle: u = 3/2 nkBT, cv = 3/2 nkB = const., experiment cv ~ T • The susceptibility puzzle: does not change with temperature, non-Curie like ~ 1/T Drude model was a huge success given the little info he knew at the time

The yield of air fluorescence induced by electrons

∗ Corresponding author. e-mail-address: arqueros@gae.ucm.es
Preprint submitted to Elsevier Science
5 February 2008
of the ﬂuorescence traces of air showers viewing downward from the top of the atmosphere.
arXiv:astro-ph/0604498v2 19 Jun 2006
The yield of air ﬂuorescence induced by electrons
F. Arqueros ∗ , F. Blanco, A. Castellanos, M. Ortiz, J. Rosado
Departamento de F´ısica Ato´mica, Molecular y Nuclear, Facultad de Ciencias F´ısicas, Universidad Complutense, E-28040 Madrid, Spain.
In this paper, the air ﬂuorescence yield is calculated using well known theoretical properties and experimental data on the various cross sections involved in the ﬂuorescence emission. Particular attention has been paid to the role of low energy secondary electrons ejected in molecular ionizations. As a result of our analysis recipes are given which allows the calculation of the absolute value of the total ﬂuorescence yield (pressure and temperature dependent) as a function of electron energy in a wide interval ranging from the threshold to the Gev region. Comparison with available experimental data will be shown.

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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

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