Molecular dynamics study of a classical two-dimensional electron system Positional and orie

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二维电子气等离激元太赫兹波器件

二维电子气等离激元太赫兹波器件
，２，３秦㊀华１，黄永丹１，孙建东１，张志鹏１，余㊀耀１，李㊀想１，孙云飞４
１（中国科学院苏州纳米技术与纳米仿生研究所中国科学院纳米器件与应用重点实验室，江苏苏州２１５１２３；２中国科学院大学，北京１０００４９；３中国科学技术大学纳米技术与纳米仿生学院，江苏苏州２１５１２３；４苏州科技大学电子与信息工程学院，江苏苏州２１５００９）
Ａｂｓｔｒａｃｔ：Ｓｏｌｉｄｓｔａｔｅｔｅｒａｈｅｒｔｚｐｌａｓｍａｄｅｖｉｃｅｓａｒｅｂｅｃｏｍｉｎｇｏｎｅｏｆｔｈｅｉｍｐｏｒｔａｎｔｒｅｓｅａｒｃｈａｒｅａｓｉｎｗｈｉｃｈｂｏｔｈｓｏｌｉｄｓｔａｔｅｍｉｃｒｏｗａｖｅ／ｍｉｌｌｉｍｅｔｅｒｗａｖｅｅｌｅｃｔｒｏｎｉｃｓａｎｄｓｅｍｉｃｏｎｄｕｃｔｏｒｌａｓｅｒｔｅｃｈｎｏｌｏｇｉｅｓａｒｅｂｅｉｎｇｄｅｖｅｌｏｐｅｄａｎｄｍｅｒｇｅｄｔｏｗａｒｄｓｔｈｅｔｅｒａｈｅｒｔｚｆｒｅｑｕｅｎｃｙｒｅｇｉｍｅ．Ｉｎｔｈｉｓｒｅｖｉｅｗ，ｗｅｉｎｔｒｏｄｕｃｅｔｈｅｍａｎｉｐｕｌａｔｉｏｎ，ｅｘｃｉｔａｔｉｏｎ（２ＤＥＧ）ｐｌａｓｍｏｎｓｉｎＡｌＧａＮ／ＧａＮｈｅｔｅｒｏｓｔｒｕｃｔｕｒｅ，ａｎｄｒｅｐｏｒｔｔｈｅａｎｄｐｒｏｂｉｎｇｏｆｔｗｏｄｉｍｅｎｓｉｏｎａｌｅｌｅｃｔｒｏｎｇａｓ，ｍｏｄｕｌａｔｏｒｓａｎｄｅｍｉｔｔｅｒｓ．ｒｅｃｅｎｔｐｒｏｇｒｅｓｓｅｓｉｎｔｈｅｉｍｐｌｅｍｅｎｔａｔｉｏｎｏｆｐｌａｓｍｏｎｐｈｙｓｉｃｓｉｎｔｅｒａｈｅｒｔｚｄｅｔｅｃｔｏｒｓＴｈｅｃｏｕｐｌｉｎｇｂｅｔｗｅｅｎｔｈｅｐｌａｓｍｏｎｍｏｄｅｓａｎｄｔｈｅｔｅｒａｈｅｒｔｚｅｌｅｃｔｒｏｍａｇｎｅｔｉｃｗａｖｅｓｉｎｆｒｅｅｓｐａｃｅａｒｅｒｅａｌｉｚｅｄｂｙｕｓｉｎｇｇｒａｔｉｎｇｃｏｕｐｌｅｒ，ａｎｔｅｎｎａａｎｄｔｅｒａｈｅｒｔｚＦａｂｒｙＰ é ｒｏｔｃａｖｉｔｙｗｈｉｃｈｆｕｒｔｈｅｒｍｏｄｕｌａｔｅｓｔｈｅｔｅｒａｈｅｒｔｚｅｌｅｃｔｒｏｍａｇｎｅｔｉｃｍｏｄｅｓａｎｄｅｎｈａｎｃｅｓｔｈｅｃｏｕｐｌｉｎｇ．Ｔｈｅｄｉｓｐｅｒｓｉｏｎｒｅｌａｔｉｏｎｓｈｉｐｏｆｇａｔｅｃｏｎｔｒｏｌｌｅｄｐｌａｓｍｏｎｍｏｄｅｓａｒｅｖｅｒｉｆｉｅｄｉｎｇｒａｔｉｎｇｃｏｕｐｌｅｄ２ＤＥＧ．Ｓｔｒｏｎｇｃｏｕｐｌｉｎｇｂｅｔｗｅｅｎｔｈｅｐｌａｓｍｏｎｍｏｄｅｓａｎｄｔｈｅｔｅｒａｈｅｒｔｚｃａｖｉｔｙｍｏｄｅｓａｎｄｈｅｎｃｅｔｈｅｆｏｒｍａｔｉｏｎｏｆｐｌａｓｍｏｎｐｏｌａｒｉｔｏｎｍｏｄｅｓａｒｅｒｅａｌｉｚｅｄｉｎａｇｒｏｕｐｉｎｇｃｏｕｐｌｅｄ２ＤＥＧｅｍｂｅｄｄｅｄｉｎａ，ｔｅｒａｈｅｒｔｚｍｏｄｕｌａｔｉｏｎｗｉｔｈｈｉｇｈｍｏｄｕｌａｔｉｏｎＦａｂｒｙＰ é ｒｏｔｃａｖｉｔｙ．Ｂａｓｅｄｏｎｔｈｅｓａｍｅｇｒａｔｉｎｇｃｏｕｐｌｅｄ２ＤＥＧｄｅｐｔｈａｎｄｔｅｒａｈｅｒｔｚｐｌａｓｍｏｎｅｍｉｓｓｉｏｎａｒｅｏｂｓｅｒｖｅｄ．Ｉｎａｎｔｅｎｎａｃｄｅｆｆｅｃｔｃｈａｎｎｅｌ，ｂｏｔｈｒｅｓｏｎａｎｔａｎｄｎｏｎｒｅｓｏｎａｎｔｅｘｃｉｔａｔｉｏｎｏｆｌｏｃａｌｉｚｅｄｐｌａｓｍｏｎｍｏｄｅｓａｒｅｏｂｓｅｒｖｅｄｂｙｐｒｏｂｉｎｇｔｈｅｔｅｒａｈｅｒｔｚｐｈｏｔｏｃｕｒｒｅｎｔ／ｖｏｌｔａｇｅ．Ａｔｅｒａｈｅｒｔｚｓｅｌｆｍｉｘｉｎｇｍｏｄｅｌｉｓｄｅｖｅｌｏｐｅｄｆｏｒａｎｔｅｎｎａｃｏｕｐｌｅｄｆｉｅｌｄｅｆｆｅｃｔｔｅｒａｈｅｒｔｚｄｅｔｅｃｔｏｒａｎｄｐｒｏｖｉｄｅｓａｇｕｉｄｅｌｉｎｅｆｏｒｔｈｅｄｅｓｉｇｎａｎｄｏｐｔｉｍｉｚａｔｉｏｎｏｆｈｉｇｈｓｅｎｓｉｔｉｖｉｔｙｔｅｒａｈｅｒｔｚｄｅｔｅｃｔｏｒｓ．Ｏｕｒｓｔｕｄｉｅｓｉｎｄｉｃａｔｅ，ｈｉｇｈｓｐｅｅｄａｎｄｈｉｇｈｓｅｎｓｉｔｉｖｉｔｙｔｅｒａｈｅｒｔｚｄｅｔｅｃｔｏｒｓａｎｄｔｈｅｆｏｃａｌｐｌａｎｅａｒｒａｙｓｃａｎｂｅｔｈａｔｒｏｏｍｔｅｍｐｅｒａｔｕｒｅ，ｔｈｅｄｅｖｅｌｏｐｅｄｂｙｕｓｉｎｇｔｈｅｎｏｎｒｅｓｏｎａｎｔｐｌａｓｍｏｎｅｘｃｉｔａｔｉｏｎｉｎａｎｔｅｎｎａｃｏｕｐｌｅｄｆｉｅｌｄｅｆｆｅｃｔｃｈａｎｎｅｌ．Ｈｏｗｅｖｅｒｈｉｇｈｄａｍｐｉｎｇｒａｔｅｏｆｓｏｌｉｄｓｔａｔｅｐｌａｓｍａｗａｖｅｉｓｙｅｔｔｈｅｍａｉｎｈｕｒｄｌｅｔｏｏｖｅｒｃｏｍｅｆｏｒｐｌａｓｍｏｎｔｅｒａｈｅｒｔｚｅｍｉｔｔｅｒｓａｎｄｍｏｄｕｌａｔｏｒｓｂｏｔｈｏｆｗｈｉｃｈｒｅｌｙｏｎｔｈｅｒｅｓｏｎａｎｔｐｌａｓｍｏｎｅｘｃｉｔａｔｉｏｎ．Ｔｈｅｆｏｒｍａｔｉｏｎｏｆｈｉｇｈｑｕａｌｉｔｙｆａｃｔｏｒ，ｍａｎｉｐｕｌａｔｉｏｎｏｆｔｈｅｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓｏｆｐｌａｓｍｏｎｐｌａｓｍｏｎｃａｖｉｔｙｉｎｃｌｕｄｉｎｇｔｈｅｓｏｌｉｄｓｔａｔｅｐｌａｓｍａｐｈｙｓｉｃｓｃａｖｉｔｙ，ｕｔｉｌｉｚａｔｉｏｎｏｆｎｅｗｈｉｇｈｅｌｅｃｔｒｏｎｍｏｂｉｌｉｔｙｔｗｏｄｉｍｅｎｓｉｏｎａｌｅｌｅｃｔｒｏｎｉｃｍａｔｅｒｉａｌｓａｎｄｈｉｇｈｑｕａｌｉｔｙ，ｓｍａｌｌ，ｅｔｃ．ｗｏｕｌｄｂｅｔｈｅｆｏｃｕｓｏｆｆｕｔｕｒｅｒｅｓｅａｒｃｈ．ｍｏｄｅｖｏｌｕｍｅｔｅｒａｈｅｒｔｚｒｅｓｏｎａｎｔｃａｖｉｔｙＫｅｙｗｏｒｄｓ：ｔｗｏｄｉｍｅｎｓｉｏｎａｌｅｌｅｃｔｒｏｎｇａｓ；ｐｌａｓｍｏｎ；ｔｅｒａｈｅｒｔｚ；ｇａｌｌｉｕｍｎｉｔｒｉｄｅ

Singularity of the density of states in the two-dimensional Hubbard model from finite size

a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from ﬁnitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepﬁsh 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A ﬁnite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external ﬁeld”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some ﬁnite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at ﬁnite “exter-nal ﬁeld”was found in ﬁnite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at ﬁnite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a ﬁxed particle number.Hence the chem-ical potential can be thought of as an external ﬁeld for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at ﬁnite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at ﬁnite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed aﬀected,which in turn may lead to an extension of the Lifshitz-type singularities into the ﬁnite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tiﬁcation of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative diﬀerences from the 42lattice.In Section 4we analyse the ﬁnite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-ﬁlling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are deﬁned asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the conﬁgurations of Ising ﬁelds isdenoted by an integral.Since ˜z (µ)is not positive deﬁnite for Re(µ)=0we weight the ensemble of conﬁgurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice conﬁgurations reﬂect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given conﬁguration.In this regime the adiabatic approximation,which is the basis of the ﬁnite-temperature algorithm,is valid.The situa-tion diﬀers strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)conﬁgurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSﬁelds the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonﬁelds presented in Ref.[1].The auxilary HS ﬁelds do not have a kinetic energy term in the bosonic action which would suppress their rapidﬂuctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-ﬁlling,results in high uncontrolledﬂuctuations of the expansion coeﬃcients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coeﬃcients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coeﬃcients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of theﬁrst coeﬃcients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coeﬃcient)forβ=5−7.5for diﬀer-entµ0.The coeﬃcients are obtained using the bootstrap method on over10000conﬁgurations forβ=5increasing to over30000forβ=7.5.In spite of diﬀerent values of the average sign in these simulations,the coeﬃcients of the expansion(16)indicate good correspondence between coeﬃcients obtained with diﬀerent values of the update chemical potentialµ0:the normalized coeﬃcients taken from diﬀerentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeﬃ-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with diﬀerent values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identiﬁed by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect toﬁnd that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in theseﬁgures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for theﬁnite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Diﬀerentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which deﬁnes the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for diﬀerent values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the ﬁnite size eﬀects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at ﬁnite temperature.However,for suﬃciently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of theﬁnite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreﬂected in the diﬀerence between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourﬁnite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove signiﬁcant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearﬁts for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reﬂecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coeﬃcients and the Yang-Lee zeros,the smallnumber of zeros obtained with suﬃcient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaborateﬁttings of the scal-ing behaviour.Theﬁnite-size eﬀects may still be signiﬁ-cant,especially as the simulation temperature decreases,thus aﬀecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atﬁnite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepﬁsh,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creﬃeld,E.G.Klepﬁsh,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coeﬃcients for ex-pansion(14)at diﬀerent update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topﬁgure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the theﬁt of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。

Two-Dimensional Gas of Massless Dirac Fermions in Graphene

Two-Dimensional Gas of Massless Dirac Fermions in Graphene K.S. Novoselov1, A.K. Geim1, S.V. Morozov2, D. Jiang1, M.I. Katsnelson3, I.V. Grigorieva1, S.V. Dubonos2, A.A. Firsov21Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester, M13 9PL, UK2Institute for Microelectronics Technology, 142432, Chernogolovka, Russia3Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, the NetherlandsElectronic properties of materials are commonly described by quasiparticles that behave as nonrelativistic electrons with a finite mass and obey the Schrödinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective “speed of light” c∗ ≈106m/s. Our studies of graphene – a single atomic layer of carbon – have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at halfinteger filling factors; b) graphene’s conductivity never falls below a minimum value corresponding to the conductance quantum e2/h, even when carrier concentrations tend to zero; c) the cyclotron mass mc of massless carriers with energy E in graphene is described by equation E =mcc∗2; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of π due to Berry’s phase.Graphene is a monolayer of carbon atoms packed into a dense honeycomb crystal structure that can be viewed as either an individual atomic plane extracted from graphite or unrolled single-wall carbon nanotubes or as a giant flat fullerene molecule. This material was not studied experimentally before and, until recently [1,2], presumed not to exist. To obtain graphene samples, we used the original procedures described in [1], which involve micromechanical cleavage of graphite followed by identification and selection of monolayers using a combination of optical, scanning-electron and atomic-force microscopies. The selected graphene films were further processed into multi-terminal devices such as the one shown in Fig. 1, following standard microfabrication procedures [2]. Despite being only one atom thick and unprotected from the environment, our graphene devices remain stable under ambient conditions and exhibit high mobility of charge carriers. Below we focus on the physics of “ideal” (single-layer) graphene which has a different electronic structure and exhibits properties qualitatively different from those characteristic of either ultra-thin graphite films (which are semimetals and whose material properties were studied recently [2-5]) or even of our other devices consisting of just two layers of graphene (see further). Figure 1 shows the electric field effect [2-4] in graphene. Its conductivity σ increases linearly with increasing gate voltage Vg for both polarities and the Hall effect changes its sign at Vg ≈0. This behaviour shows that substantial concentrations of electrons (holes) are induced by positive (negative) gate voltages. Away from the transition region Vg ≈0, Hall coefficient RH = 1/ne varies as 1/Vg where n is the concentration of electrons or holes and e the electron charge. The linear dependence 1/RH ∝Vg yields n =α·Vg with α ≈7.3·1010cm-2/V, in agreement with the theoretical estimate n/Vg ≈7.2·1010cm-2/V for the surface charge density induced by the field effect (see Fig. 1’s caption). The agreement indicates that all the induced carriers are mobile and there are no trapped charges in graphene. From the linear dependence σ(Vg) we found carrier mobilities µ =σ/ne, whichreached up to 5,000 cm2/Vs for both electrons and holes, were independent of temperature T between 10 and 100K and probably still limited by defects in parent graphite. To characterise graphene further, we studied Shubnikov-de Haas oscillations (SdHO). Figure 2 shows examples of these oscillations for different magnetic fields B, gate voltages and temperatures. Unlike ultra-thin graphite [2], graphene exhibits only one set of SdHO for both electrons and holes. By using standard fan diagrams [2,3], we have determined the fundamental SdHO frequency BF for various Vg. The resulting dependence of BF as a function of n is plotted in Fig. 3a. Both carriers exhibit the same linear dependence BF = β·n with β ≈1.04·10-15 T·m2 (±2%). Theoretically, for any 2D system β is defined only by its degeneracy f so that BF =φ0n/f, where φ0 =4.14·10-15 T·m2 is the flux quantum. Comparison with the experiment yields f =4, in agreement with the double-spin and double-valley degeneracy expected for graphene [6,7] (cf. caption of Fig. 2). Note however an anomalous feature of SdHO in graphene, which is their phase. In contrast to conventional metals, graphene’s longitudinal resistance ρxx(B) exhibits maxima rather than minima at integer values of the Landau filling factor ν (Fig. 2a). Fig. 3b emphasizes this fact by comparing the phase of SdHO in graphene with that in a thin graphite film [2]. The origin of the “odd” phase is explained below. Another unusual feature of 2D transport in graphene clearly reveals itself in the T-dependence of SdHO (Fig. 2b). Indeed, with increasing T the oscillations at high Vg (high n) decay more rapidly. One can see that the last oscillation (Vg ≈100V) becomes practically invisible already at 80K whereas the first one (Vg <10V) clearly survives at 140K and, in fact, remains notable even at room temperature. To quantify this behaviour we measured the T-dependence of SdHO’s amplitude at various gate voltages and magnetic fields. The results could be fitted accurately (Fig. 3c) by the standard expression T/sinh(2π2kBTmc/heB), which yielded mc varying between ≈ 0.02 and 0.07m0 (m0 is the free electron mass). Changes in mc are well described by a square-root dependence mc ∝n1/2 (Fig. 3d). To explain the observed behaviour of mc, we refer to the semiclassical expressions BF = (h/2πe)S(E) and mc =(h2/2π)∂S(E)/∂E where S(E) =πk2 is the area in k-space of the orbits at the Fermi energy E(k) [8]. Combining these expressions with the experimentally-found dependences mc ∝n1/2 and BF =(h/4e)n it is straightforward to show that S must be proportional to E2 which yields E ∝k. Hence, the data in Fig. 3 unambiguously prove the linear dispersion E =hkc∗ for both electrons and holes with a common origin at E =0 [6,7]. Furthermore, the above equations also imply mc =E/c∗2 =(h2n/4πc∗2)1/2 and the best fit to our data yields c∗ ≈1⋅106 m/s, in agreement with band structure calculations [6,7]. The employed semiclassical model is fully justified by a recent theory for graphene [9], which shows that SdHO’s amplitude can indeed be described by the above expression T/sinh(2π2kBTmc/heB) with mc =E/c∗2. Note that, even though the linear spectrum of fermions in graphene (Fig. 3e) implies zero rest mass, their cyclotron mass is not zero. The unusual response of massless fermions to magnetic field is highlighted further by their behaviour in the high-field limit where SdHO evolve into the quantum Hall effect (QHE). Figure 4 shows Hall conductivity σxy of graphene plotted as a function of electron and hole concentrations in a constant field B. Pronounced QHE plateaux are clearly seen but, surprisingly, they do not occur in the expected sequence σxy =(4e2/h)N where N is integer. On the contrary, the plateaux correspond to half-integer ν so that the first plateau occurs at 2e2/h and the sequence is (4e2/h)(N + ½). Note that the transition from the lowest hole (ν =–½) to lowest electron (ν =+½) Landau level (LL) in graphene requires the same number of carriers (∆n =4B/φ0 ≈1.2·1012cm-2) as the transition between other nearest levels (cf. distances between minima in ρxx). This results in a ladder of equidistant steps in σxy which are not interrupted when passing through zero. To emphasize this highly unusual behaviour, Fig. 4 also shows σxy for a graphite film consisting of only two graphene layers where the sequence of plateaux returns to normal and the first plateau is at 4e2/h, as in the conventional QHE. We attribute this qualitative transition between graphene and its two-layer counterpart to the fact that fermions in the latter exhibit a finite mass near n ≈0 (as found experimentally; to be published elsewhere) and can no longer be described as massless Dirac particles. 2The half-integer QHE in graphene has recently been suggested by two theory groups [10,11], stimulated by our work on thin graphite films [2] but unaware of the present experiment. The effect is single-particle and intimately related to subtle properties of massless Dirac fermions, in particular, to the existence of both electron- and hole-like Landau states at exactly zero energy [912]. The latter can be viewed as a direct consequence of the Atiyah-Singer index theorem that plays an important role in quantum field theory and the theory of superstrings [13,14]. For the case of 2D massless Dirac fermions, the theorem guarantees the existence of Landau states at E=0 by relating the difference in the number of such states with opposite chiralities to the total flux through the system (note that magnetic field can also be inhomogeneous). To explain the half-integer QHE qualitatively, we invoke the formal expression [9-12] for the energy of massless relativistic fermions in quantized fields, EN =[2ehc∗2B(N +½ ±½)]1/2. In QED, sign ± describes two spins whereas in the case of graphene it refers to “pseudospins”. The latter have nothing to do with the real spin but are “built in” the Dirac-like spectrum of graphene, and their origin can be traced to the presence of two carbon sublattices. The above formula shows that the lowest LL (N =0) appears at E =0 (in agreement with the index theorem) and accommodates fermions with only one (minus) projection of the pseudospin. All other levels N ≥1 are occupied by fermions with both (±) pseudospins. This implies that for N =0 the degeneracy is half of that for any other N. Alternatively, one can say that all LL have the same “compound” degeneracy but zeroenergy LL is shared equally by electrons and holes. As a result the first Hall plateau occurs at half the normal filling and, oddly, both ν = –½ and +½ correspond to the same LL (N =0). All other levels have normal degeneracy 4B/φ0 and, therefore, remain shifted by the same ½ from the standard sequence. This explains the QHE at ν =N + ½ and, at the same time, the “odd” phase of SdHO (minima in ρxx correspond to plateaux in ρxy and, hence, occur at half-integer ν; see Figs. 2&3), in agreement with theory [9-12]. Note however that from another perspective the phase shift can be viewed as the direct manifestation of Berry’s phase acquired by Dirac fermions moving in magnetic field [15,16]. Finally, we return to zero-field behaviour and discuss another feature related to graphene’s relativistic-like spectrum. The spectrum implies vanishing concentrations of both carriers near the Dirac point E =0 (Fig. 3e), which suggests that low-T resistivity of the zero-gap semiconductor should diverge at Vg ≈0. However, neither of our devices showed such behaviour. On the contrary, in the transition region between holes and electrons graphene’s conductivity never falls below a well-defined value, practically independent of T between 4 and 100K. Fig. 1c plots values of the maximum resistivity ρmax(B =0) found in 15 different devices, which within an experimental error of ≈15% all exhibit ρmax ≈6.5kΩ, independent of their mobility that varies by a factor of 10. Given the quadruple degeneracy f, it is obvious to associate ρmax with h/fe2 =6.45kΩ where h/e2 is the resistance quantum. We emphasize that it is the resistivity (or conductivity) rather than resistance (or conductance), which is quantized in graphene (i.e., resistance R measured experimentally was not quantized but scaled in the usual manner as R =ρL/w with changing length L and width w of our devices). Thus, the effect is completely different from the conductance quantization observed previously in quantum transport experiments. However surprising, the minimum conductivity is an intrinsic property of electronic systems described by the Dirac equation [17-20]. It is due to the fact that, in the presence of disorder, localization effects in such systems are strongly suppressed and emerge only at exponentially large length scales. Assuming the absence of localization, the observed minimum conductivity can be explained qualitatively by invoking Mott’s argument [21] that mean-free-path l of charge carriers in a metal can never be shorter that their wavelength λF. Then, σ =neµ can be re-written as σ = (e2/h)kFl and, hence, σ cannot be smaller than ≈e2/h per each type of carriers. This argument is known to have failed for 2D systems with a parabolic spectrum where disorder leads to localization and eventually to insulating behaviour [17,18]. For the case of 2D Dirac fermions, no localization is expected [17-20] and, accordingly, Mott’s argument can be used. Although there is a broad theoretical consensus [18-23,10,11] that a 2D gas of Dirac fermions should exhibit a minimum 3conductivity of about e2/h, this quantization was not expected to be accurate and most theories suggest a value of ≈e2/πh, in disagreement with the experiment. In conclusion, graphene exhibits electronic properties distinctive for a 2D gas of particles described by the Dirac rather than Schrödinger equation. This 2D system is not only interesting in itself but also allows one to access – in a condensed matter experiment – the subtle and rich physics of quantum electrodynamics [24-27] and provides a bench-top setting for studies of phenomena relevant to cosmology and astrophysics [27,28].1. Novoselov, K.S. et al. PNAS 102, 10451 (2005). 2. Novoselov, K.S. et al. Science 306, 666 (2004); cond-mat/0505319. 3. Zhang, Y., Small, J.P., Amori, M.E.S. & Kim, P. Phys. Rev. Lett. 94, 176803 (2005). 4. Berger, C. et al. J. Phys. Chem. B, 108, 19912 (2004). 5. Bunch, J.S., Yaish, Y., Brink, M., Bolotin, K. & McEuen, P.L. Nanoletters 5, 287 (2005). 6. Dresselhaus, M.S. & Dresselhaus, G. Adv. Phys. 51, 1 (2002). 7. Brandt, N.B., Chudinov, S.M. & Ponomarev, Y.G. Semimetals 1: Graphite and Its Compounds (North-Holland, Amsterdam, 1988). 8. Vonsovsky, S.V. and Katsnelson, M.I. Quantum Solid State Physics (Springer, New York, 1989). 9. Gusynin, V.P. & Sharapov, S.G. Phys. Rev. B 71, 125124 (2005). 10. Gusynin, V.P. & Sharapov, S.G. cond-mat/0506575. 11. Peres, N.M.R., Guinea, F. & Castro Neto, A.H. cond-mat/0506709. 12. Zheng, Y. & Ando, T. Phys. Rev. B 65, 245420 (2002). 13. Kaku, M. Introduction to Superstrings (Springer, New York, 1988). 14. Nakahara, M. Geometry, Topology and Physics (IOP Publishing, Bristol, 1990). 15. Mikitik, G. P. & Sharlai, Yu.V. Phys. Rev. Lett. 82, 2147 (1999). 16. Luk’yanchuk, I.A. & Kopelevich, Y. Phys. Rev. Lett. 93, 166402 (2004). 17. Abrahams, E., Anderson, P.W., Licciardello, D.C. & Ramakrishnan, T.V. Phys. Rev. Lett. 42, 673 (1979). 18. Fradkin, E. Phys. Rev. B 33, 3263 (1986). 19. Lee, P.A. Phys. Rev. Lett. 71, 1887 (1993). 20. Ziegler, K. Phys. Rev. Lett. 80, 3113 (1998). 21. Mott, N.F. & Davis, E.A. Electron Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1979). 22. Morita, Y. & Hatsugai, Y. Phys. Rev. Lett. 79, 3728 (1997). 23. Nersesyan, A.A., Tsvelik, A.M. & Wenger, F. Phys. Rev. Lett. 72, 2628 (1997). 24. Rose, M.E. Relativistic Electron Theory (John Wiley, New York, 1961). 25. Berestetskii, V.B., Lifshitz, E.M. & Pitaevskii, L.P. Relativistic Quantum Theory (Pergamon Press, Oxford, 1971). 26. Lai, D. Rev. Mod. Phys. 73, 629 (2001). 27. Fradkin, E. Field Theories of Condensed Matter Systems (Westview Press, Oxford, 1997). 28. Volovik, G.E. The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).Acknowledgements This research was supported by the EPSRC (UK). We are most grateful to L. Glazman, V. Falko, S. Sharapov and A. Castro Netto for helpful discussions. K.S.N. was supported by Leverhulme Trust. S.V.M., S.V.D. and A.A.F. acknowledge support from the Russian Academy of Science and INTAS.43µ (m2/Vs)0.8c4P0.4 22 σ (1/kΩ)10K0 0 1/RH(T/kΩ) 1 2ρmax (h/4e2)1-5010 Vg (V) 50 -10ab 0 -100-500 Vg (V)50100Figure 1. Electric field effect in graphene. a, Scanning electron microscope image of one of our experimental devices (width of the central wire is 0.2µm). False colours are chosen to match real colours as seen in an optical microscope for larger areas of the same materials. Changes in graphene’s conductivity σ (main panel) and Hall coefficient RH (b) as a function of gate voltage Vg. σ and RH were measured in magnetic fields B =0 and 2T, respectively. The induced carrier concentrations n are described by [2] n/Vg =ε0ε/te where ε0 and ε are permittivities of free space and SiO2, respectively, and t ≈300 nm is the thickness of SiO2 on top of the Si wafer used as a substrate. RH = 1/ne is inverted to emphasize the linear dependence n ∝Vg. 1/RH diverges at small n because the Hall effect changes its sign around Vg =0 indicating a transition between electrons and holes. Note that the transition region (RH ≈ 0) was often shifted from zero Vg due to chemical doping [2] but annealing of our devices in vacuum normally allowed us to eliminate the shift. The extrapolation of the linear slopes σ(Vg) for electrons and holes results in their intersection at a value of σ indistinguishable from zero. c, Maximum values of resistivity ρ =1/σ (circles) exhibited by devices with different mobilites µ (left y-axis). The histogram (orange background) shows the number P of devices exhibiting ρmax within 10% intervals around the average value of ≈h/4e2. Several of the devices shown were made from 2 or 3 layers of graphene indicating that the quantized minimum conductivity is a robust effect and does not require “ideal” graphene.ρxx (kΩ)0.60 aVg = -60V4B (T)810K12∆σxx (1/kΩ)0.4 1ν=4 140K 80K B =12T0 b 0 25 50 Vg (V) 7520K100Figure 2. Quantum oscillations in graphene. SdHO at constant gate voltage Vg as a function of magnetic field B (a) and at constant B as a function of Vg (b). Because µ does not change much with Vg, the constant-B measurements (at a constant ωcτ =µB) were found more informative. Panel b illustrates that SdHO in graphene are more sensitive to T at high carrier concentrations. The ∆σxx-curves were obtained by subtracting a smooth (nearly linear) increase in σ with increasing Vg and are shifted for clarity. SdHO periodicity ∆Vg in a constant B is determined by the density of states at each Landau level (α∆Vg = fB/φ0) which for the observed periodicity of ≈15.8V at B =12T yields a quadruple degeneracy. Arrows in a indicate integer ν (e.g., ν =4 corresponds to 10.9T) as found from SdHO frequency BF ≈43.5T. Note the absence of any significant contribution of universal conductance fluctuations (see also Fig. 1) and weak localization magnetoresistance, which are normally intrinsic for 2D materials with so high resistivity.75 BF (T) 500.2 0.11/B (1/T)b5 10 N 1/2025 a 0 0.061dmc /m00.04∆0.02 0c0 0 T (K) 150n =0e-6-3036Figure 3. Dirac fermions of graphene. a, Dependence of BF on carrier concentration n (positive n correspond to electrons; negative to holes). b, Examples of fan diagrams used in our analysis [2] to find BF. N is the number associated with different minima of oscillations. Lower and upper curves are for graphene (sample of Fig. 2a) and a 5-nm-thick film of graphite with a similar value of BF, respectively. Note that the curves extrapolate to different origins; namely, to N = ½ and 0. In graphene, curves for all n extrapolate to N = ½ (cf. [2]). This indicates a phase shift of π with respect to the conventional Landau quantization in metals. The shift is due to Berry’s phase [9,15]. c, Examples of the behaviour of SdHO amplitude ∆ (symbols) as a function of T for mc ≈0.069 and 0.023m0; solid curves are best fits. d, Cyclotron mass mc of electrons and holes as a function of their concentration. Symbols are experimental data, solid curves the best fit to theory. e, Electronic spectrum of graphene, as inferred experimentally and in agreement with theory. This is the spectrum of a zero-gap 2D semiconductor that describes massless Dirac fermions with c∗ 300 times less than the speed of light.n (1012 cm-2)σxy (4e2/h)4 3 2 -2 1 -1 -2 -3 2 44Kn7/ 5/ 3/ 1/2 2 2 210 ρxx (kΩ)-4σxy (4e2/h)0-1/2 -3/2 -5/2514T0-7/2 -4 -2 0 2 4 n (1012 cm-2)Figure 4. Quantum Hall effect for massless Dirac fermions. Hall conductivity σxy and longitudinal resistivity ρxx of graphene as a function of their concentration at B =14T. σxy =(4e2/h)ν is calculated from the measured dependences of ρxy(Vg) and ρxx(Vg) as σxy = ρxy/(ρxy + ρxx)2. The behaviour of 1/ρxy is similar but exhibits a discontinuity at Vg ≈0, which is avoided by plotting σxy. Inset: σxy in “two-layer graphene” where the quantization sequence is normal and occurs at integer ν. The latter shows that the half-integer QHE is exclusive to “ideal” graphene.。

从爱因斯坦到霍金的宇宙尔雅答案

1.5 、爱因斯坦生平与科学贡献（一）1、爱因斯坦的第一任妻子是（A、米列娃）。

2、爱因斯坦在哪个国家上的大学（B、瑞士）3、爱因斯坦是哪个民族的（C、犹太人）4、数学家希尔伯特和闵可夫斯基是小学同学。

（√）5、爱因斯坦大学毕业的时候没有文凭（×）1.6、爱因斯坦生平与科学贡献（二）1、爱因斯坦在（D、1914）年到达柏林，开始在柏林大学任教。

2、在质能方程?中，c 表示什么？（C、光速）3、爱因斯坦的博士论文的主题是（C、分子大小的测量）4、爱因斯坦逝世的时候拥有哪两个国家的国籍？（B、美国和瑞士）5、爱因斯坦1902-1909 年在下面哪个单位工作？（D、专利局）6、爱因斯坦任职的第一个大学是格丁根大学。

（×）7、爱因斯坦的儿子拿到过诺贝尔奖（×）1.7、迈克尔逊实验、洛伦兹变换与相对论的建立1、“真空中的光速对任何观察者来说都是相同的”是什么原理？（A、光速不变原理）2、“洛伦兹变换”最初用来调和19 世纪建立起来的经典电动力学同（）的矛盾。

窗体顶端（C、牛顿力学）3、谁通过实验证明了光速在不同惯性系和不同方向上都是相同的？（C、迈克尔逊）4、爱因斯坦提出相对论主要参考了哪个实验（B、斐索实验）5、从“伽利略变换”能推出“洛伦兹收缩”。

（×）6、恒星以v 的速度运动，恒星发出光的速度是。

（×）7、洛伦兹提出“洛伦兹收缩”是为了解决迈克尔逊实验和光行差现象的矛盾。

（√）1.8、相对论的几个结论1、银河系的直径约多少光年？（C、10 万）2、“相对论”的结论不包括下面哪一项？（D、安培定律）3、除太阳以外离我们最近的恒星是（C、比邻星）。

4、“双生子佯谬”是谁提出来的？（A、郎之万）5、火车以0.9 倍光速在运动，在火车上的人以0.9 倍光速同方向运动，我们就会看到火车上的人速度超过光速。

（×）6、“相对论”的质量公式最先是由爱因斯坦给出的。

第2章夸克与轻子 (2)

第二章夸克与轻子Quarks and leptons2.1 粒子园The particle zoo学习目标Learning objectives：我们怎样发现新粒子？能否预言新粒子？什么是奇异粒子？大纲参考：3.1.1￣太空入侵者宇宙射线是由包括太阳在内的恒星发射而在宇宙空间传播的高能粒子。

如果宇宙射线粒子进入地球大气层，就会产生寿命短暂的新粒子和反粒子以及光子。

所以，就有“太空入侵者”这种戏称。

发现宇宙射线之初，大多数物理学家都认为这种射线不是来自太空，而是来自地球本身的放射性物质。

当时物理学家兼业余气球旅行者维克托·赫斯（Victor Hess）就发现，在5000m高空处宇宙射线的离子效应要比地面显著得多，从而证明这种理论无法成立。

经过进一步研究，表明大多数宇宙射线都是高速运动的质子或较小原子核。

这类粒子与大气中气体原子发生碰撞，产生粒子和反粒子簇射，数量之大在地面都能探测到。

通过云室和其他探测仪，人类发现了寿命短暂的新粒子与其反粒子。

μ介子（muon）或“重电子”（符号μ）。

这是一种带负电的粒子，静止质量是电子的200多倍。

π介子（pion）。

这可以是一种带正电的粒子（π＋）、带负电的粒子（π－）或中性不带电粒子（π0），静止质量大于μ介子但小于质子。

K介子（kaon）。

这可以是一种带正电的粒子（K＋）、带负电的粒子（K－）或中性不带电粒子（K0），静止质量大于π介子但小于质子。

科学探索How Science Works不同寻常的预言An unusual prediction在发现上述三种粒子之前，日本物理学家汤川秀树（Hideki Yukawa）就预言，核子间的强核力存在交换粒子。

他认为交换粒子的作用范围不超过10－15m，并推断其质量在电子与质子之间。

由于这种离子的质量介于电子与质子之间，所以汤川就将这种粒子称为“介子”（mesons）。

一年后，卡尔·安德森拍摄的云室照片显示一条异常轨迹可能就是这类粒子所产生。

Class-1_531605726

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e-
பைடு நூலகம்
electrons from ideal gas with charge e
Successes and problems of the Drude model
Successes: • explanation of electrical conductivity and thermal conductivity • Quantitative explanation of the Wiedemann-Franz law (by luck!) • The Hall effect and magnetoresistance • ac conductivity of metals Problems: • heat capacity puzzle: Classical particle: u = 3/2 nkBT, cv = 3/2 nkB = const., experiment cv ~ T • The susceptibility puzzle:
Sommerfeld model = Drude model + quantum mechanics = Free Electron Gas + Schrodinger equation + Fermi statistics
• Sommerfeld kept Drude’s concept of Free Electron Gas, and incorporate quantum mechanics into it. • Free Electron Gas model is also called Drude-Sommerfeld Model

结构化学第二章

cos q cos f 抖 sin f + r r 抖 r sin q f q
（2.1.11）
抖 = sin q sin f 抖 y
cos q sin f 抖 cos f + + r r 抖 r sin q f q
（2.1.12）
抖 sin q = cos q 抖 z r r q
（2.1.13）
这样就可以根据直角坐标(x,y,z)和球坐标(r,θ,φ)之间的变换关系推出球坐标形式的物理量算符.例如角动量沿z轴分量的算 Ù 符( M z )可由(2.1.11)、（2.1.12）式推得如下
tan f = y / x
按偏微分关系
（2.1.6）
抖 = 抖 x
骣r 鼢抖骣q 抖骣f 珑鼢 + + 珑鼢珑x 抖桫 x 抖桫 x 桫 r q
f
（2.1.7）
将（2.1.4）式对x求偏导，并按（2.1.3）式关系带如，得
骣r ÷ ¶ 2r ç ÷= 2 x = 2r sin q cos f ç ÷ ç¶ x 桫
ì ï ï Y1,± 1 = í ï ï î
3 py = sin q sin f 4p 3 px = sin q cos f 4p
量子数的允许值
2.2.1主量子数n（The principal quantum number, n）在解R方程中,为了使解得的函数Rnl收敛,必须使 4 2
me En = 2 8e0 h2
= - 2.178 10- 18 J = - 13.595eV
若以电子质量me代替折合质量μ，那么
E1 = - 2.180? 10
- 18
J
13.606eV

Mxene

MXene:A New Family of Promising Hydrogen Storage Medium Qianku Hu,*,†,‡Dandan Sun,†Qinghua Wu,†Haiyan Wang,†Libo Wang,†Baozhong Liu,†Aiguo Zhou,*,†and Julong He§†School of Material Science and Engineering,Henan Polytechnic University,454000Jiaozuo,P.R.China‡Department of Geosciences,Stony Brook University,11794Stony Brook,New York,United States§State Key Laboratory of Metastable Materials Science and Technology,Yanshan University,066004Qinhuangdao,P.R.China *Web-Enhanced Feature*Supporting Informationthe hydrogen molecules(3.4wt%hydrogen storage capacity)reversibly under ambient conditions.Meanwhile,the hydrogenC)were also evaluated,and the results were similar to those ofmembers was expected to be a good candidate for reversibleNowadays hydrogen storage and transport remain a great challenge for its vehicle applications.1,2Storing hydrogen in solid materials is more practical,safe,and economic than that in gaseous or liquid phases.1−4According to the interaction nature between hydrogen and host materials,solid-state storage materials can be classiﬁed into two categories:chemisorption of dissociated hydrogen atoms and physisorption of intact hydrogen molecules.Either approach has its own disadvantages. For chemisorption,strong bonding(40−80kJ/mol)5,6between hydrogen atoms and host materials(mainly metal hydrides or complex chemical hydrides)makes it diﬃcult to release hydrogen at moderate temperatures.For physisorption,an ideal sorbent material should possess two fundamental properties:high speciﬁc surface area and suitable binding energy(20−30kJ/mol,corresponding to∼0.2−0.3eV)5,7with hydrogen molecules.The sorption-based storage materials studied now mainly comprise carbon-base materials8−12 (including nanotubes,fullerenes,graphene,and nanoporous carbon),metal−organic frameworks(MOFs),13,14and covalent organic frameworks(COFs).15Almost all these materials meet theﬁrst requirement.For example,both sides of graphene can be utilized to store hydrogen,10,11and the highest Brunauer−Emmett−Teller surface area of MOF tested to date reach about 7000m2/g.16However,the weak binding strength of hydrogen (4−10kJ/mol,physisorption mostly by van der Waals forces)5 is the largest obstacle for the practical application of these materials.17At present these sorption-based hydrogen storage materials can only operate around liquid nitrogen temperature. Metal decorations were employed to increase the bindingenergy of hydrogen on sorption-based materials.A great deal oftheoretical calculation has been conducted to investigate the eﬀects of metal decorations(including alkali,alkaline earth,and transition metals).18−28The transition metal results are more exciting.Via Kubas-type interaction,29the binding energies of hydrogen with the transition metals lies between strong chemisorption and weak physisorption and comes into a desirable range.However,because of their large cohesive energies,transition-metal adatoms have a tendency of aggregating into clusters,20which would signiﬁcantly reduce the hydrogen storage capacity.30Some calculations show that boron-doping or vacancy defects on carbon adsorbents may prevent the clustering behavior of metal adatoms.27,31,32 However,it is very diﬃcult to fabricate such metal-well-dispersed carbon adsorbents with boron-doping or vacancy defects.Despite these problems,the Kubas-type hydrogen storage mode is still a promising direction.A material,with lightweight,high speciﬁc surface area,no metal-clustering behavior and exhibiting Kubas interactions to store hydrogen,is very hopeful to meet the gravimetric storage capacity targetReceived:September25,2013Revised:November20,2013Published:November21,2013(5.5wt %by 2015)33set by the U.S.DOE.However,the discovery and synthesis of such material is of great challenge.Recently,a new family of graphene-like 2D materials,named as MXene,was prepared by exfoliating the counterpart MAX phases in hydro ﬂuoric acid.34,35MAX phases are a large family (>60members)of layered ternary transition-metal carbides or nitrides with a chemical formula M n +1AX n (n =1,2,or 3),where M is a transition metal,A is an A-group element (mostly IIIA or IVA group),and X is C and/or N.36,37To date,the as-synthesized MXene phases include Ti 2C,Ti 3C 2,Ta 4C 3,(Ti 0.5Nb 0.5)2C,(V 0.5Cr 0.5)3C 2,and Ti 3CN.34,35For a new material,it is very important scienti ﬁcally and technically to explore its basic properties and potential applications.Good electrical conductivities of bare MXenes were predicted theoretically and can be tuned by termination/functionalization with di ﬀerent groups.34,35,38−40The electrochemical intercala-tion behaviors of Li ions in MXene structures were also investigated experimentally and theoretically,which prove MXene phases are very promising as anode materials of Li-ion battery.41−44In this paper,we investigated the possibility of employing MXene phases as hydrogen storage media by ﬁrst-principles calculations.We chose Ti 2C as a representative of MXenes on the basis of the following reasons:(i)Titanium is a commonly used decoration element and has been proved to be e ﬀective for hydrogen storage in carbon-based materials.(ii)The 2D Ti 2C phase has already been synthesized (though with ﬂuorine (F)and/or hydroxyl (OH)termination).35(iii)Except for Sc 2C,Ti 2C possesses the highest surface area per weight among all possible MXene phases,and thus it is expected to have high gravimetric hydrogen storage capacities.Figure 1gives the crystal structures of bulk Ti 2AlC and 2D Ti 2C.In the layeredTi 2AlC structure,Ti −Al bond is relatively weaker than Ti −C bond.Consequently,the layers of Al atoms can be selectively etched by hydro ﬂuoric acid,which results in the formation of the 2D Ti 2C sheets.35The 2D Ti 2C structure is composed of sharing-edge Ti 6C octahedrons,in which,C atoms occupy the octahedral interstitial sites between near-close-packed Ti atoms.One Ti 2C sheet can be simply considered as one graphene sheet coated with a Ti atoms sheet on each side.These Ti atoms indeed are constituent elements of Ti 2C,and thus the problem of aggregation of Ti atoms can be avoided.With the cleavage of Ti −Al bond,all Ti atoms of Ti 2C lie in an unsaturated coordination state,which is an indispensable condition for metal atoms to exhibit the Kubas interactions.5,45On the basis of aforementioned reasons,we think 2D Ti 2C (even most MXene phases)is very likely to be a reversible and high-gravimetric-capacity hydrogen storage material operated under ambient conditions.The purpose of the calculations in this paper is to con ﬁrm this speculation.■THEORETICAL METHODS Our ﬁrst-principles total energy pseudopotential calculationswere performed using the density functional theory (DFT)asimplemented in CASTEP code.46The exchange and correlation energy is described by the local density approximation (LDA)functional.Vanderbilt ultrasoft pseudopotentials were em-ployed within a plane wave basis set with the cuto ﬀenergy of 480eV.The numerical integration of the Brillouin zone was performed using 6×6×2(unit cell)and 2×2×1(supercell)Monkhorst −Pack (MP)k -point sampling.Both structural parameters and atomic positions with no constraints were fully relaxed using the BFGS minimization method until the convergence tolerance (energy <5.0×10−6eV,force <0.01eV/Å,stress <0.02GPa and displacement <5.0×10−4Å)was reached.The binding energy (E b )of H 2with the Ti2C host material was calculated according to the followingequation:=+−+E E nE E n()/n b host H host H 22where E host is the total energy of the host structure (bare Ti 2Cstructure or already adsorbed with some H 2or H),E H 2is thetotal energy of an free H 2,E host+n H 2is the total energy of thehost structure adsorbed with new hydrogen molecules,and n is the number of new adsorbed hydrogen molecules.The binding energy of H atom to the surface of Ti 2C was also calculated like this.■RESULTS AND DISCUSSIONTi 2C Model.The 2D Ti 2C structure was constructed by removing the Al element from the parent Ti 2AlC structure.Thereafter,a vacuum space with a thickness of 20Åwas inserted between the neighboring slabs to avoid arti ﬁcial interactions between them.The optimized lattice constant a for the Ti 2C model is 3.002Å,which is in good agreement with the other result of 3.007Å.38A Ti 2C 3×3periodic supercell (containing nine carbon and eighteen titanium atoms,shown in Figure 2a)was used as thehost material.To investigate the hydrogen adsorption,it is important to ﬁrst ﬁnd the favorable adsorption sites.On Ti2C surfaces,there exist three types of high-symmetry sites:top site over a Ti atom at the top surface;hollow_1site above the center of three Ti atoms at the top surface,under which there exists one Ti atom at the bottom surface;and hollow_2site above the center of three Ti atoms at the top surface,under which there exists one C atom at the bottom surface.Hydrogen at Ti2C 3×3Supercell.We ﬁrst studied theadsorption of a single hydrogen molecule on the Ti 2C 3×3supercell.A H 2molecule was put parallel or perpendicularlyabove the three adsorption sites respectively,and thus six models were constructed,which are shown in Figure 2a.After geometry optimization,the H 2initially locating parallel above the three sites and perpendicularly above the hollow_2site is thermally unstable and dissociates into two H atoms.Nearly all dissociated H atoms occupy the hollow_1site.The calculations (Supporting Information)for one hydrogen atom locating above the top,hollow_1and hollow_2sites,respectively,show that the hollow_1site is the most stable one.Thus it canbeFigure 1.(a)Crystal structure of bulk Ti 2AlC.The black solid line labels out the unit cell.(b)Side views and (c)top views of 2D Ti 2C structure.Two Ti 2C octahedrons were labeled out by green color.understood that the dissociated H atoms prefer to occupy the hollow_1site.We turn our eyes onto the stable A and B adsorption sites in Figure 2a where the H 2molecules are still intact and perpendicular to the surface.However,if initially the H 2molecule is put not perpendicular to the A or B sites but is tilted by an angle (>10°),the H 2molecules also dissociate just like the case of parallel to the surface.Therefore,on the basis of the above calculated results,we can presume that the ﬁrst batch of H 2molecules arriving onto the clean surface of 2D Ti 2C on the hydrogenation process dissociate into H atoms and then the dissociated H atoms occupy the hollow_1site.A model,labeled as Ti 2C(3×3)_18H shown in Figure 2b,was constructed to represent this situation.In this model,the H atoms are adsorbed on the hollow_1sites on both sides of 2D Ti 2C.The calculated binding energy is 5.027eV per H atom with an H −Ti distance of 1.977Å.The large binding energy means the interaction of dissociated hydrogen atoms with the Ti 2C surface is of strong chemisorption.Hydrogen at Ti 2C(3×3)_18H Model.Next,we studied the adsorption of H 2molecules on the Ti 2C(3×3)_18H structure.Similarly,six models were constructed with one H 2molecule parallel or perpendicularly above the top,hollow_1and hollow_2sites,respectively,as shown in Figure 3.After geometry optimization,the H 2molecule parallel above the hollow_2site (labeled as F)relaxes to parallel above the top site (labeled as D).The H 2molecules for the other ﬁve con ﬁgurations (A,B,C,D,and E;see labels in Figure 3)still keep the initial sites and orientation.For the A,B,C,and E con ﬁgurations,the calculated binding energies are in the range 30−70meV,and the H −H bond length increases slightly by 0.01−0.04Åfrom the value 0.769Åof an isolated H 2.The weak binding energy and the nearly unchanged H −H bond length illuminate the H 2are physisorbed on the A,B,C,and E sites.An interesting and exciting result is about the con ﬁguration that the H 2is adsorbed parallel on the top site (D in Figure 3).For this con ﬁguration,the binding energy is calculated to be 0.237eV,which falls into the desired range 0.2−0.3eV.From the binding energy and elongated H −H bond length,we speculated this interaction nature should be of Kubas type,which would be con ﬁrmed later by the results of density of states (DOS).At room temperatures,the hydrogen molecules by physisorption are very little.Therefore,we can presume that under ambient conditions the second batch of H 2molecules arriving on the surface of 2D Ti 2C are adsorbed only parallel on the top site.A model,labeled as Ti 2C(3×3)_18H_18H 2,was constructed as shown in Figure 4a to represent this situation.Inthis model,onto every top site on both sides was laid one H2molecule whose axis is parallel to the surface.The initial orientation of the axes of all the H 2molecules was set to be the same.After optimization,one-third of the H 2molecules rotated 60°clockwise around the c axis,one-third of the H 2molecules rotated 60°counterclockwise around the c axis,and the remaining one-third kept in the original orientation.Theoptimized structure of Ti 2C(3×3)_18H_18H 2is shown inFigure 4b.The optimized structure is more symmetrical thanthe initial structure.The average binding energy for the H2molecules by Kubas-type interaction is 0.272eV per H 2with an H −H bond length of 0.823Å.Hydrogen at Ti 2C(3×3)_18H_18H 2Model.Now weconsider the hydrogen adsorption by physical forces at liquidnitrogen temperatures.At low temperatures,the hydrogen molecules by Kubas-type interaction and the hydrogen atoms by chemisorption are bound tightly to the Ti 2C surfaces.Therefore,the Ti 2C(3×3)_18H_18H 2model was used as the host material to investigate the physical adsorption of hydrogen molecules.For this model,H 2molecules have occupied the top sites.Only the hollow_1and hollow_2sites could be usedtoFigure 2.(a)Adsorption of a single hydrogen molecule at di ﬀerent sites of Ti 2C 3×3supercell.The blue arrows indicate the moving direction ofthe dissociated H atoms.The black solid line labels out the 3×3supercell.(b)Ti 2C(3×3)_18Hmodel.Figure 3.Adsorption of a single hydrogen molecule at di ﬀerent sites of Ti 2C(3×3)_18H model.The calculated binding energies for H 2atdiﬀerent sites were given.The blue arrow indicates the movingdirection of the unstable H 2molecule.The black solid line labels outthe3×3supercell.The legend of di ﬀerent atoms is the same to that in Figure 2adsorb H 2molecules.Four con ﬁgurations,as shown in Figure 5,were constructed in which one H 2molecule is put parallel or perpendicularly above the hollow_1and hollow_2sites respectively.All four adsorption sites are found to be stable.The weak binding energies for the four sites mean the nature of physical interaction.For the hollow_1or hollow_2site,thebinding energies of the parallel and the perpendicularcon ﬁgurations are very close.Thus,at liquid nitrogentemperatures,both the parallel and the perpendicularcon ﬁgurations are possible.Whereas the perpendicularcon ﬁguration has a little higher binding energy than thecorresponding parallel con ﬁguration,the two perpendicularcon ﬁgurations were chosen to represent the physisorption ofH 2molecules above the hollow_1and hollow_2sitesrespectively.A model was constructed to represent this situation,in whichevery hollow_1or hollow_2site on both sides of Ti2C(3×3)_18H_18H 2is occupiedby one H 2molecule with its axisperpendicular to the site.After optimization,the H2moleculesabovethe hollow_1site are stable.However,the H 2moleculesabove the hollow_2site depart from the surfaces along the c -axis direction.These should arise from the short distance and the resulting repulsion between the H 2molecules above thehollow_1site and the neighboring hollow_2site.A Ti2C(3×3)_18H_36H2model with theH 2molecules only occupying thehollow_1site of Ti 2C(3×3)_18H_18H 2was constructed.Partsa andb of Figure 6give the top and side views of thismodel.Figure 4.(a)Initial structure of Ti 2C(3×3)_18H_18H 2model.(b)Optimized structure of Ti 2C(3×3)_18H_18H 2model.The black solid line labels out the 3×3supercell.The legend of di ﬀerent atoms is the same to that in Figure2.Figure 5.Adsorption of a single hydrogen molecule at di ﬀerent sites of Ti 2C(3×3)_18H_18H 2model.The calculated binding energies for H 2at di ﬀerent sites were given.The black solid line labels out the 3×3supercell.Figure 6.(a)Top views and (b)side views of the Ti 2C(3×3)_18H_36H 2model that possesses the maximum hydrogen storage capacity.The black solid line labels out the 3×3supercell.The average binding energy is calculated to be 0.109eV per H 2for these 18hydrogen molecules above the hollow_1site.Maximum Hydrogen Storage Capacity.Now all thepossible sites on the Ti 2C surface have been considered to bind hydrogen.In the Ti 2C(3×3)_18H_36H 2structure,one adsorption site only binds one H 2molecule.We attempted to attach more H 2molecules to di ﬀerent adsorption sites.But unfortunately it failed.The new added H 2molecule directly ﬂies away or pushes the neighboring H 2molecule o ﬀand then occupies its position.Therefore,the Ti 2C(3×3)_18H_36H 2model possesses the maximum hydrogen storage capacity.In this supercell model,18H atoms (1.7wt %)are bound by strong chemical forces,36H atoms (3.4wt %)are bound in molecule form by weak physical forces,and the remaining 36H atoms (3.4wt %)are bound in molecule form by Kubas-type interaction.Under ambient conditions,desorption of chem-isorbed hydrogen cannot take place,and physisorbed hydro-gens are not easy to bind.Only the hydrogen bound by Kubas-type interaction could be adsorbed and released reversibly under ambient conditions.The reversible capacity of 3.4wt %is still considerable and signi ﬁcant for practical applications.Density of States and Mulliken Populations.The Kubas-type interaction between H 2and transition metals has the following features:19,22,45,47(i)Adsorbed H 2molecules keep intact and the bond length is elongated approximately 10%from the bond length 0.75Åof a free H 2molecule.(ii)The binding energy is between the physisorption and the chemisorption and usually lies in the range 0.2−0.8eV.(iii)The bond axis of the adsorbed H 2molecule is not perpendicular to the transition metal.In this paper,the hydrogen molecules adsorbed upon the top sites possess all these features.Hence,we conclude that the binding nature of these hydrogen molecules with Ti 2C should be of the Kubas-type interaction.This Kubas-type interaction is associated with the electron donation of H 2σorbitals into the empty d orbitals of a transition metal,and simultaneously the electron back-donation from the ﬁlled metal d orbitals into the H 2σ*antibonding orbitals.23,45Thus the Kubas-type interaction involves the orbital hybridization between the transition metal d orbitals and the H 2σorbitals (including bonding and antibonding).Figure 7gives the partial density of states (PDOS)for the s orbitals of one H 2molecule above the top site and the 3d orbitals of the underlying Ti atom in the model of Ti 2C(3×3)_18H_36H 2.It can be seen that the Ti 3d orbitals are hybridized with the H 2σorbitals in the range −10to −6eV.And the peaks from −2to 0eV correspond to the hybridization of the Ti 3d with the H 2σ*orbitals.These hybridizations are very similar to the results obtained by otherresearches for the adsorption of H 2on Ti atom.19,47The approximately 10%elongation of H −H bond length arises from the decrease of bonding-orbital electrons and theincrease of antibonding-orbital electrons.If excessive electronsare donated from Ti 3d orbitals to H2σ*antibonding orbitals,the H 2molecule will be unstable and then dissociate.It is the reason that in this study the ﬁrst H 2arriving at Ti atomdissociates into H atoms.This phenomenon is also observed in other literatures.19,20,48,49The Mulliken charge population calculations show every dissociated H atom gains 0.33e totally from the three nearest Ti atoms.This means that every Ti atom donates 0.33e totally to the three nearest H atoms.Meanwhile,due to the smaller electronegativity of Ti than C,every Ti atom donates about 0.35e to the three nearest C atoms.Thus when the second H 2arrives,the Ti atom has no enough charges to destabilize the dihydrogen state.As a result,the second arriving H 2does not dissociate and locates above the top site in a molecular form.And even,no more charges in the Ti atom can be transferred to bind extra H 2molecules.It is the reason thatone Ti atom can bind 4−6H2molecules in other literatures,19,22,47,48but one Ti atom in the 2D Ti 2C can onlybind one H 2molecule by Kubas-type interaction.Ab Initio Molecular Dynamic Simulations.From thebinding energies results,we speculated that the hydrogen molecules bound by Kubas-type interaction could adsorb and desorb reversibly under ambient conditions.To verify this and ascertain the exact desorption temperature,desorption behaviors of hydrogen on Ti 2C were investigated by ab initio molecular dynamic (MD)simulations using the Nosealgorithm.The simulation temperature was set to be 300and400K.The total simulation time was set to be 1.5ps with a time step of 1.0fs.Figure 8gives the snapshots of the Ti 2C(3×3)_18H_36H 2model after 1.5ps molecular dynamic simulations at 300and 400K.(Movies 1,2,3,and 4in mpg format for top and side views of desorption processes of hydrogen are available as Web Enhanced Objects.)At both 300and 400K,all the H 2molecules by chemisorption still stay at the initial sites and all the H 2molecules by physisorption depart from the surfaces.These phenomena can be easily understood from the binding energy results.For the eighteen H 2molecules by Kubas-type interaction,the results at 300K are di ﬀerent with those at 400K.At 400K,nearly half of the H 2molecules by Kubas-type interaction depart from the surface.It should be reminded that the simulation system does not reach the balancebecause the 1.5ps simulation time is not long enough (however,already quite costly in computation time).Thus the temperature 400K provides enough energy for the release of the H 2molecules bound by the Kubas-type interaction.At 300K,only three H 2molecules ﬂy away.And an important phenomenon was observed on the simulation process of 300K.When a released H 2molecule ﬂies over the region of a vacanttop site,the vacant top site can catch and adsorb this H 2molecule again.Thus at 300K mostly the top sites should be saturated with H 2molecules.From the above discussions,the adsorption and desorption of hydrogen by Kubas-typeinteraction can be accomplished in the narrow temperature range 300−400K.Therefore,the MD results give clear evidence that 2D Ti2C is a reversible hydrogen storage material under ambientconditions.Figure 7.Partial density of states (PDOS)for thes orbitalsof one H 2molecule above the top site and the 3d orbitals of the underlying Ti atom in Ti 2C(3×3)_18H_36H 2model.The black dash line represents the Fermi level.Perspectives of MXene Phases as Hydrogen Storage Materials.An important experimental result should be noted that because MXene phases were made in aqueous hydro ﬂuoric acid,the as-fabricated 2D Ti 2C are chemically terminated with ﬂuorine (F)and/or hydroxyl (OH)groups.34,35The binding interactions are so strong.Thus,although e ﬀort has been made,42bare MXene phases with no surface termination have not been prepared.From our point of view,two routes are possible to obtain bare MXene structures:(i)removing the F and OH groups from the synthesized MXenes by chemical or physical methods;(ii)ﬁnding a new way to exfoliate parent MAX phases.This is a big challenge and task for materials scientists and chemists.When the writing of this paper came to a close,we realized that the F-or OH-terminated MXene phases may be also a good hydrogen storage material due to the electrostatic interactions between the F (or OH)anions and the adsorbed H 2.The corresponding calculations are now in progress.When the A elements are removed from the corresponding MAX family,which includes more than 60members,theoretically there exist more than 20MXene phases.These MXene phases have structures and compositions similar to Ti 2C and thus are also expected to be good hydrogen storage materials.To test it,the hydrogen storage properties of 2D Sc 2C and V 2C were calculated in a simple way.We replaced Ti atoms in Ti 2C(3×3)_18H,Ti 2C(3×3)_18H_18H 2,and Ti 2C-(3×3)_18H _36H 2models by Sc and V atoms,respectively.The optimized geometry structures (Supporting Information)for hydrogen adsorbed on Sc 2C and V 2C surfaces are similar to those on the Ti 2C surface.The hydrogens are also bound by three modes:physisorption,chemisorption,or Kubas-type interaction.The binding energies for the H 2on Sc 2C and V 2C by Kubas-type interaction were calculated to be 0.164and 0.242eV per H 2respectively,which are also suitable.Therefore,the studies in this paper opened the door of a house that contains a series of reversible and high-gravimetric-capacity hydrogen storage materials operated under ambient conditions.And the hydrogen adsorption and desorption behaviors could be adjusted by using di ﬀerent MXene phases as hydrogen storage materials.■CONCLUSIONSIn summary,using ﬁrst-principles total energy pseudopotential calculations,we systematically investigated the possibility of 2D Ti 2C structure (a representative MXene)as hydrogen storage materials.The calculations show that hydrogen can be adsorbed on di ﬀerent sites on both sides of Ti 2C layered structure.Considering all adsorbed hydrogen molecules and atoms,the maximum hydrogen storage capacity was calculated to be 8.6wt %,which meets the gravimetric storage capacity target (5.5wt %by 2015)set by the U.S.DOE.These hydrogen are bound by three modes:chemisorption of the H atom (1.7wt %),physisorption of the H 2molecule (3.4wt %),and Kubas-type binding of the H 2molecule (3.4wt %)with calculated binding energies of 5.027,0.109,and 0.272eV,respectively.The binding energy of 0.272eV for the H 2molecule by Kubas-type interaction just falls into the desired range for a reversible hydrogen storage material under ambient conditions.Ab-initioMD simulations con ﬁrmed that the hydrogen molecules bound by Kubas-type interaction can be adsorbed and released reversibly in the temperature range 300−400K.The di ﬀerent binding energy values for the three modes imply that 2D Ti 2C can store hydrogenat low,room,and high temperatures.The hydrogen storage properties of Sc 2C and V 2C MXene phases were also evaluated in a simple way.The results are similar to that for Ti 2C and are also fascinating.Therefore,MXene phases including more than 20members should be a new family of hydrogen storage materials.Experiments areexpected to con ﬁrm the results of this work.And furtherexperimental and computational investigations should be conducted on other MXene phases.Discovery of MXene phases with better hydrogen storage performances is anticipated in the near future.■ASSOCIATED CONTENT *Supporting Information Adsorption geometries and total energies and structural parameters of all the optimized con ﬁgurations involved in this paper.This material is available free of charge via the Internet at .*Web-Enhanced Features Four animations in mpg format are available in the HTML version of the paper.■AUTHOR INFORMATION Corresponding Authors*Q.Hu:e-mail,hqk@.*A.Zhou:e-mail,zhouag@.NotesThe authors declare no competing ﬁnancial interest.■ACKNOWLEDGMENTS This work was supported by National Natural Science Foundation of China (Grant Nos.51202058,50802024,Figure 8.Top and side snapshots of Ti 2C(3×3)_18H_36H2modelafter 1.5ps moleculardynamic simulations at 300and400K.Severalhydrogen molecules departing too far from the surfaces were not given.The legend of di ﬀerent atoms is the same to that in Figure 2.。

2 分子动力学第二章

2.2 力场函数的内容
一般而言，分子力场函数由成键项Ubonded、交叉项Ucross和非键项Unobonded三部分构成，所以分子总势能为：
U total = U bonded + U cross + U nobonded
成键项（Bonded Term）
• • • • 键伸缩能键角弯曲能二面角扭曲能离平面振动能 Us Ua Ut Uo
1
r12
2
3
r23
二面角扭曲能(torsion
angle)
• 单键旋转引起分子骨架扭曲所产生的能量变化.
U t = kn [1 ± cos(nϕ − ϕ 0 )] 1 2 U t = k (ϕ − ϕ 0 ) 2
1
4
2
3
离平面振动能(out of plane bending)
• 共平面的四个原子中有一个原子偏离平面进行微小振动所引起的分子能量变化.
• 利用LJ势描述原子间vdw作用的时候,通常有两种方式： • 1，指定反应参数 • 2，分别指定参数举例说明:水和甲醇混合液中的vdw类型
ε i j = ε iε j
1 σ ij = (σ i + σ j ) 2
Ai j = Ai Aj Bij = Bi B j
力场函数中参数的获得
• 有两种来源： 1，实验观测的数据 2，量子力学计算的数据 • 方法： 1，提出函数形式，并估计力常数的值 2，根据一些分子（或晶体）的结构+性质数据进行拟合 3，重复进行多次，并逐渐增加拟合的结构和性质，误差小于一定标准时即可拟合通常采用的软件：GULP。
• 分析力学所注重的不是力和加速度，而是具有更广泛意义的能量，同时又扩大了坐标的概念，因而使这种方法和结论便于运用到物理学的其它领域。 • 但是由于分析力学中数学推理较多，在历史上曾经发生过一些不良倾向，容易使人忘记力学的物理实质。 1. 拉格朗日力学 2. 哈密顿力学

Appl. Environ. Microbiol.-2013-Hu-7763-9

Published Ahead of Print 4 October 2013. 10.1128/AEM.02467-13.2013, 79(24):7763. DOI:Appl. Environ. Microbiol. Ziye Hu, Theo van Alen, Mike S. M. Jetten and Boran KartalPlanctomycetesof Anaerobic Ammonium-Oxidizing Lysozyme and Penicillin Inhibit the Growth /content/79/24/7763Updated information and services can be found at: These include:SUPPLEMENTAL MATERIALSupplemental material REFERENCES/content/79/24/7763#ref-list-1at: This article cites 67 articles, 29 of which can be accessed free CONTENT ALERTSmore»articles cite this article), Receive: RSS Feeds, eTOCs, free email alerts (when new /site/misc/reprints.xhtml Information about commercial reprint orders: /site/subscriptions/To subscribe to to another ASM Journal go to: on June 17, 2014 by guest/Downloaded fromLysozyme and Penicillin Inhibit the Growth of Anaerobic Ammonium-Oxidizing PlanctomycetesZiye Hu,a Theo van Alen,a Mike S.M.Jetten,a,b Boran Kartal aDepartment of Microbiology,IWWR,Radboud University Nijmegen,Nijmegen,The Netherlands a;Department of Biotechnology,Delft University of Technology,Delft,The Netherlands bAnaerobic ammonium-oxidizing(anammox)planctomycetes oxidize ammonium in the absence of molecular oxygen with ni-trite as the electron acceptor.Although planctomycetes are generally assumed to lack peptidoglycan in their cell walls,recent genome data imply that the anammox bacteria have the genes necessary to synthesize peptidoglycan-like cell wall structures.In this study,we investigated the effects of two antibacterial agents that target the integrity and synthesis of peptidoglycan(ly-sozyme and penicillin G)on the anammox bacterium Kuenenia stuttgartiensis.The effects of these compounds were determined in both short-term batch incubations and long-term(continuous-cultivation)growth experiments in membrane bioreactors. Lysozyme at1g/liter(20mM EDTA)lysed anammox cells in less than60min,whereas penicillin G did not have any observable short-term effects on anammox activity.Penicillin G(0.5,1,and5g/liter)reversibly inhibited the growth of anammox bacteria in continuous-culture experiments.Furthermore,transcriptome analyses of the penicillin G-treated reactor and the control re-actor revealed that penicillin G treatment resulted in a10-fold decrease in the ribosome levels of the cells.One of the cell division proteins(Kustd1438)was downregulated25-fold.Our results suggested that anammox bacteria contain peptidoglycan-like com-ponents in their cell wall that can be targeted by lysozyme and penicillin G-sensitive proteins were involved in their synthesis. Finally,we showed that a continuous membrane reactor system with free-living planktonic cells was a very powerful tool to study the physiology of slow-growing microorganisms under physiological conditions.A naerobic ammonium-oxidizing(anammox)bacteria oxidizeammonium with nitrite as the terminal electron acceptor and with nitric oxide and hydrazine as intermediates(1).These micro-organisms contribute signiﬁcantly to the release ofﬁxed nitrogen back to the atmosphere(2–4)and are applied in wastewater treat-ment as an environmentally friendly and cost-effective method of nitrogen removal(5).All known anammox bacteria belong to the phylum Plancto-mycetes(6).Members of this phylum are unique in many aspects; for example,they have a complex cell compartmentalization(7), an unusual fatty acid composition of the phospholipids(8),and the lack of peptidoglycan on their cell wall(9,10).Peptidoglycan is a major cell wall component present in almost all bacteria(11)but was not detected in planctomycetes with biochemical assays(10, 12).Furthermore,classical peptidoglycan was not observed in anammox bacteria in ultrastructural studies(13,14).Planctomy-cetes,together with the chlamydiae and cell-wall-less mycoplas-mas,are the only known peptidoglycan-lacking microorganisms within the domain Bacteria(7).A peptidoglycan monomer is composed of a pentapeptide com-ponent and a glycan strand consisting of two connected amino sugar residues,N-acetylmuramic acid and N-acetylglucosamine(15).In bacterial cell walls,peptidoglycan monomers join together by concat-enated glycan strands,catalyzed by transglycosylases,and cross-linked short-stem peptides,catalyzed by transpeptidases,to form three-dimensional mesh-like layers that provide bacteria structural integrity and enable them to resist osmotic lysis(16).Penicillin-bind-ing proteins(PBPs)are involved in theﬁnal stage of peptidoglycan biosynthesis,the formation of peptidoglycan cross-links,cell separa-tion,and peptidoglycan maturation or recycling of monomers (17,18).PBPs that catalyze the cross-linkage of peptidoglycan can be divided into two classes on the basis of their activity:bifunctional PBPs and monofunctional PBPs.Bifunctional PBPs normallyhave both transglycosylase and transpeptidase activities,andmonofunctional PBPs have transpeptidase activity only(16).Thedisorder of PBPs or the peptidoglycan monomer itself thereforeleads to cell lysis and death for both Gram-positive and Gram-negative bacteria(11,18,19).There are several compounds,suchas lysozyme and␤-lactam antibiotics,that attack the integrity ofpeptidoglycan and as such are used as antibacterial agents.N-Acetylmuramide glycan hydrolase(lysozyme)hydrolyzes the gly-cosidic bonds of the peptidoglycan monomer(20,21),whereas ␤-lactam antibiotics,such as penicillin,inhibit the cross-linking of peptidoglycan by binding to PBPs(11).Some bacteria are re-sistant to penicillin or other types of␤-lactam antibiotics becausethey harbor␤-lactamases or have developed special PBPs with avery low afﬁnity for these antibiotics(22,23).Theoretically,anammox bacteria and all other planctomyceteswhich have peptidoglycan-lacking cell walls should not be sensi-tive to␤-lactam antibiotics,including penicillin G(24).Indeed,one of theﬁrst papers on the anammox bacteria reported thatpenicillin(penicillin V)had no inhibitory effect on anammoxactivity(25).Further,Güven et ed0.5g/liter penicillin G(aconcentration5times higher than the normal working concentra-tion of penicillin G)to inhibit possible heterotrophic denitriﬁerReceived23July2013Accepted30September2013Published ahead of print4October2013Address correspondence to Boran Kartal,kartal@science.ru.nl.Supplemental material for this article may be found at /10.1128/AEM.02467-13.Copyright©2013,American Society for Microbiology.All Rights Reserved.doi:10.1128/AEM.02467-13December2013Volume79Number24Applied and Environmental Microbiology p.7763–7763 on June 17, 2014 by guest / Downloaded fromactivity in an anammox culture and reported that the compound had no effect on anammox bacteria(26).Nevertheless,both of these studies were performed on enrichment cultures with less than80%anammox bacteria growing as bioﬁlm aggregates,which could have contained penicillin-degrading microorganisms or provided protection against antibiotics as a physical barrier.Fur-ther,they were conducted either as batch experiments(25)or by adding penicillin with long intervals without considering the fate of the added penicillin(26).The long-term effect of lysozyme on anammox bacteria has not been tested yet.Surprisingly,a recent metagenomic analysis indicated that the anammox organism Kuenenia stuttgartiensis encodes19out of21 genes that are necessary for peptidoglycan biosynthesis.Two of these(kustd1895and kuste2376)encode proteins that are homol-ogous to enzymes that are suggested to be monofunctional(trans-peptidase)PBPs(27).The absent two genes are homologous to PBP1a and PBP1b in Escherichia coli,which have both transgly-cosylase and transpeptidase activities and which have been sug-gested to be essential for cross-linking of sugar monomers(16).It should be noted that the reports on the function of both classes of proteins are derived from studies with laboratory strains that are not genetically related to anammox bacteria(e.g.,E.coli),and thefunction of proteins cannot be inferred directly through sequence comparison.Evidence from comparative genomic analysis of K.stutt-gartiensis and other planctomycetes as well as Gram-negative bac-teria also suggested that anammox bacteria could be genetically able to possess a Gram-negative bacterium-like cell wall structure (28).Nevertheless,this hypothesis cannot be supported by ultra-structural studies since the peptidoglycan layer,which was usually clearly visible in thin sections of Gram-negative bacteria,could not be observed in thin sections of anammox bacteria(29).Con-sequently,whether anammox bacteria have a cell wall containing a peptidoglycan-like component or not is still unknown.In the present work,we studied the effects of penicillin G and lysozyme on K.stuttgartiensis.To this end,we used a highly en-riched free-living planktonic cell culture(Ͼ95%enriched)in batch tests and continuous membrane reactors that recently be-came available(1,30).Streptomycin,a type of aminoglycoside antibiotic(protein synthesis inhibitor)targeting the30S subunit of the bacterial ribosome(31),was also used as a positive control. MATERIALS AND METHODSGenome analyses.All translated gene sequences of anammox species K. stuttgartiensis(27),Brocadia fulgida(32),and Scalindua profunda(33) were directly downloaded from the genome database at NCBI,JGI,or IMG/M.Downloaded sequences were submitted to the KEGG Automatic Annotation Server(KAAS)(34)for pathway mapping.Proteins of K.stutt-gartiensis that mapped to the peptidoglycan biosynthesis pathway were then retrieved and their sequences were used as queries in two indepen-dent BLAST searches using B.fulgida and S.profunda protein sequences as reference data sets,respectively.The protein sequences of B.fulgida and S. profunda which had the best hits with K.stuttgartiensis were collected and used as queries in a new BLAST search using the sequences in the NCBI protein database(nr)as the reference data set.Batch incubations and activity tests.The short-term inhibitory ef-fects of lysozyme(lysozyme from chicken egg white;Sigma-Aldrich),pen-icillin G(penicillin G potassium salt;Sigma-Aldrich),and streptomycin (streptomycin sulfate salt;Sigma-Aldrich)were tested in batch incuba-tions with previously described Kuenenia stuttgartiensis free-living plank-tonic cells(1,30).For determining the effect of lysozyme,10ml cells was incubated with0.25to1g/liter of lysozyme and with1to20mM EDTA (the concentrations of lysozyme and EDTA in each incubation are listed in Table S1in the supplemental material)for60min at37°C(35,36).If they were not completely lysed after incubation,the lysozyme-treated cells were pelleted by centrifugation for5min at1,200ϫg.Then,the pellet was washed with10ml of synthetic medium(37)without substrates3times or until it was free of EDTA and lysozyme.The cells were resuspended in10 ml synthetic medium containing2mM ammonium and nitrite,followed by anammox activity tests,as previously described(38),with modiﬁca-tions.In short,10ml cells was transferred to a30-ml serum bottle after the pH was adjusted to7.3.The bottle was sealed with a butyl rubber stopper and an aluminum crimp cap and then repeatedly vacuumed andﬂushed with Ar-CO2(95%/5%)to achieve anaerobic conditions before incuba-tion in a shaking incubator(250rpm)at30°C.Liquid samples(0.5ml) were taken every30min for ammonium and nitrite measurements until all nitrite was consumed.For penicillin G and streptomycin,the agents were added to10ml of cells and activity tests were performed immediately.Four different con-centrations of penicillin G(0.5,1,1.5,and2g/liter)and streptomycin(50, 100,150,and200mg/liter)were tested.Continuous culturing.The batch experiments were followed by ex-periments inﬁve successively operated2-liter(working volume)contin-uous membrane reactors that were carefully monitored for growth,activ-ity,and cell viability.Each reactor was inoculated with1liter of free-living K.stuttgartiensis cells and supplied with500ml/day of synthetic medium (37)containing45mM ammonium and nitrite as the inﬂuent.The reac-tors wereﬂushed continuously with Ar-CO2(95%/5%;10ml/min)to maintain anaerobic conditions.The temperature and pH were main-tained at30°C andϳ7.3,respectively,with a water bath and bicarbonate solution,respectively.Before the inhibitors were introduced to the reac-tor,the optical density at600nm(OD600)of the biomass was maintained at0.6to0.7,which represented the steady state of the culture,with a constant washout of cells(120ml/day)(Fig.1).On day13,the inhibitor was added to the inﬂuent and directly to all reactors except the control reactor to achieve the same concentration as the inﬂuent.The control reactor was operated for60days without adding any inhibitor(negative control).One of the reactors was inhibited by streptomycin(streptomycin sulfate salt;Sigma-Aldrich)at the working concentration of100mg/liter (positive control).This was followed by the operation of three reactors containing different concentrations(0.5,1,and5g/liter)of penicillin G. After the reactors treated with penicillin G became inactive,the cell wash-FIG1Effects of penicillin G and streptomycin on Kuenenia stuttgartiensis single-cell enrichment culture.Filled circles,no inhibitor;open circles,0.5 g/liter penicillin G;ﬁlled triangles,1g/liter penicillin G;open triangles,5g/liter penicillin G;ﬁlled squares,100mg/liter streptomycin.Hu et al. Applied and Environmental Microbiology on June 17, 2014 by guest / Downloaded fromout was stopped,fresh synthetic medium was supplied as the inﬂuent until all accumulated nitrite was consumed,and synthetic medium containing 45mM ammonium and nitrite but no penicillin G was supplied again to resuscitate the reactors.Analytical methods.Liquid samples from reactors and activity tests were pelleted by centrifugation for5min at16,000ϫg.The supernatants were transferred to new tubes and stored atϪ20°C until further analyses. The concentrations of ammonium and nitrite were measured colori-metrically as previously described(39).Protein concentrations were mea-sured using the biuret method,as described previously(40).FISH and phase-contrast microscopy.One milliliter of liquid sample was taken from the reactor deactivated by penicillin G and the negative-con-trol reactor.Sampleﬁxation andﬂuorescence in situ hybridization(FISH) were performed as described previously(41).Probe AMX820,speciﬁc for Kuenenia-and Brocadia-like anammox bacteria(42),was used to detect K. stuttgartiensis,and a mixture of probes EUB1to EUB4,speciﬁc for most bac-teria(43–45),was used to visualize most bacteria.DAPI(4=,6-diamidino-2-phenylindole)was used to stain the whole community DNA.For phase-contrast microscopy,15␮l of liquid sample was taken from both reactors at the same time point as that at which samples were ob-tained for FISH analyses and directly observed by use of an Axioplan2 imaging system(Carl Zeiss,Germany)withoutﬁxation.RNA isolation and transcriptome sequencing and analyses.Tran-scriptome sequencing was performed on samples from the control reactor and the reactor treated with0.5g/liter penicillin G.In short,equal amounts of cells were harvested on day35from both reactors,and total RNA was extracted with a RiboPure-Bacteria kit(Ambion)according to the manufacturer’s instructions.RNA quality in terms of the amount and size distribution was examined by an Agilent2100bioanalyzer(Agilent) before library construction.Sequencing library construction and tran-scriptome sequencing were performed using an Ion total transcriptome sequencing(RNA-Seq)kit and an Ion PGM200sequencing kit(Ion Tor-rent),respectively.Mapping of the transcriptome reads was performed with CLC Genomics Workbench software(CLC Bio,Denmark)using the RNA-Seq analysis tool with a minimum length of95%,a minimum identity of95%, and the genome of K.stuttgartiensis as a reference.All reads that mapped to rRNA and tRNA genes were excluded from the results.The number of reads per kilobase of the exon model per million mapped reads(RPKM) values of all protein-coding sequences(CDS)of penicillin G-treated sam-ples and control samples were retrieved and compared to each other to identify the expression level changes.Nucleotide sequence accession number.The transcriptome se-quences have been deposited in the Sequence Read Archive(SRA) under accession number PRJNA219373( /bioproject/219373).RESULTS AND DISCUSSIONAmong the19K.stuttgartiensis genes that were predicted to be involved in peptidoglycan biosynthesis,13(Table1)were sug-gested to be indispensably required,as determined by comparing the genomic data to data on the genes required for the peptidogly-can biosynthesis pathway using KAAS pathway mapping.The ge-nomes of the other two anammox organisms,freshwater species B.fulgida and marine species S.profunda,also encode all of the genes indicated by KAAS mapping to be required for peptidogly-can synthesis.For S.profunda,all genes essential for peptidoglycan biosynthesis had the highest sequence identity with K.stutt-gartiensis or anammox species KSU-1genes,suggested by a BLAST search using the sequences in the NCBI protein database (nr)as a reference data set.For B.fulgida,however,3genes(the UDP-N-acetylmuramoylalanine–D-glutamate ligase,pentapep-tide-transferase,and D-alanyl–D-alanyl ligase genes)did not have best BLAST hits with anammox bacteria(32).The analyses of the other available anammox genomes suggested that other anammox species,as well as K.stuttgartiensis,also had the genetic capacity to synthesize a peptidoglycan-like polymer.When K.stuttgartiensis cells were incubated with penicillin G and streptomycin in short-term activity tests,these compounds had no effect on the activity(Fig.2),which was in line with the previous observations(25).This is probably due to the fact that these compounds are growth inhibitors and the long doubling time of the anammox bacteria makes it impossible to determine their inhibitory effect in2to3h of short-term batch incubations. Furthermore,when1mM EDTA was used with1g/liter lysozyme, anammox cells were not lysed and there was no effect on anam-mox activity.However,when20mM EDTA was used with1g/liter lysozyme,complete lysis occurred in60min.Lysozyme is a very speciﬁc glycoside hydrolase that breaks down the1,4-␤-linkages between N-acetylmuramic acid and N-acetyl-D-glucosamine res-TABLE1Genes detected in the genome of Kuenenia stuttgartiensis predicted to be involved in peptidoglycan biosynthesis and their transcription levels under two different growth conditionsOpen readingframe Strand a Gene Description Expression value (RPKM b)Penicillin Gtreated Controlkustd1895R pbpA Penicillin-binding protein2 1.38 2.15 kuste2372R queA S-Adenosylmethionine-tRNA ribosyltransferase-isomerase 1.69 1.88 kuste2376F ftsI Division-speciﬁc transpeptidase,penicillin-binding protein3 6.078.85 kuste2378F murE UDP-N-acetylmuramyl tripeptide synthase7.268.46 kuste2379F murF UDP-N-acetylmuramoylalanyl-D-glutamyl-2,6-diaminopimelate–D-alanyl–D-alanine ligase 6.9712.34 kuste2380F mraY Phospho-N-acetylmuramoyl-pentapeptide transferase 2.48 3.85 kuste2383F murG Undecaprenyldiphospho-muramoylpentapeptide beta-N-acetylglucosaminyltransferase10.4412.28 kuste2385F murC UDP-N-acetylmuramate–L-alanine ligase 5.53 6.46 kuste2386F ddlA D-Alanine:D-alanine ligase 3.7210.31 kuste3293F mviN Putative virulence factor,ﬂippase 1.15 1.21 kuste3313R murA UDP-N-acetylglucosamine1-carboxyvinyltransferase(enolpyruvyl transferase) 5.99.54 kuste3480F murD UDP-N-acetylmuramoylalanine D-glutamate ligase 2.8 4.34 kuste3636R dacB D-Alanyl–D-alanine carboxypeptidase(penicillin-binding protein4) 1.25 1.62a R,reverse;F,forward.b RPKM,number of reads per kilobase of exon model per million mapped reads.Lysozyme and Penicillin Inhibit Anammox Bacteria7765 on June 17, 2014 by guest / Downloaded fromidues in a peptidoglycan monomer and between N-acetyl-D-glu-cosamine residues in chitodextrins(46,47).The complete lysis observed here indicated that the cell wall of the anammox bacteria contained molecules that lysozyme could target,most likely a pep-tidoglycan-like molecule.Due to this almost immediate lysis ef-fect,continuous cultures were not operated with lysozyme.Unlike Gram-positive bacteria,in which cross-linked peptidogly-can is in the outermost layer of the cell wall,in Gram-negative bacte-ria,there is an outer membrane containing lipopolysaccharides and protein outside the peptidoglycan layer(48).Therefore,for the lysis of Gram-negative bacteria,EDTA,which chelates divalent cations in the outer membranes and exposes the peptidoglycan layer to ly-sozyme,is necessary(36).Divalent cations are essential for normal interactions of cell wall components and in some cases also for the association of the surface protein or glycoprotein layer(S layer)with the outer membrane(49,50).An S layer is a2-dimensional layer composed of identical proteins or glycoproteins that cover an entire bacterial(Gram positive or Gram negative)or archaeal cell by attach-ing to the outermost cell wall membrane(51,52).The attachment of the S layer to the outermost membrane requires bivalent cations and could be disturbed by the presence of EDTA(51).The optimal EDTA concentration range for Gram-negative bacterial cell lysis is0.5to2 mM(53).In our experiments,however,lysis occurred only in the presence of a higher concentration of EDTA,suggesting that in the anammox bacterial cell wall the peptidoglycan-like structure might be covered with an additional layer(s)that could be similar to that in Gram-negative bacteria.Furthermore,a recent study showed that K. stuttgartiensis has an S layer as the outermost layer of the cell(67).On the basis of these observations,it could be conceivable that a higher concentration of EDTA is necessary to break down both the unique S layer and the outermost membrane of K.stuttgartiensis.To further assess the inhibitory effects of penicillin G and streptomycin,ﬁve continuous membrane reactors were operated as described above.The control reactor(i.e.,the experiment with-out inhibitors)was operated for60days without any loss of activ-ity or growth.Streptomycin,which was used as the positive con-trol,resulted in the washout of anammox bacteria(Fig.1)and nitrite accumulation.In the last3days of operation,the concen-tration of nitrite in the efﬂuent increased from0to over5mM. The bioreactor was completely inactivated within17days after the introduction of streptomycin into the reactor,which corresponds to the washout rate of the reactor(16.6days).Interestingly,at all tested concentrations,penicillin G(0.5,1, and5g/liter)also inhibited the growth of anammox bacteria,as determined by the OD600,and deactivated the reactor completely. This was observed as a rapid nitrite accumulation after approxi-mately3weeks of treatment,similar to the effect of streptomycin (Fig.1).These results show that the anammox bacteria are sensi-tive to␤-lactam antibiotics and,together with the results of the batch incubations with lysozyme,suggest that the cell walls of the anammox bacteria contain peptidoglycan-like polymers and PBPs are involved in their synthesis.Furthermore,when the biomass from the penicillin G-inhib-ited bioreactor was inspected with phase-contrast and epiﬂuores-cence microscopy(afterﬂuorescence in situ hybridization and DNA staining with DAPI),bloated anammox cells(approxi-mately twice the size of normal cells)were observed(Fig.3),indi-cating that K.stuttgartiensis is unable to grow or divide properly and undergoes plasmolysis.One of the genes in the K.stuttgartien-sis genome encodes cell division protein FtsI(Kuste2376)(27,54). This protein is also known as PBP3and is involved in septal peptidoglycan synthesis during cell division(55,56).It is conceiv-able that in the case of K.stuttgartiensis,penicillin also bound PBP 3and inhibited cell division of the anammox bacteria.When penicillin G addition was stopped and the reactor was supplied with mineral medium(37)without nitrite,the accumu-lated nitrite was completely removed within3to4days.Once nitrite was below the detection limit(Ͻ50␮M),ammonium and nitrite(45mM each)were supplied once more.This resulted in an increase of the optical density(OD600),indicating the anammox bacteria were growing again(Fig.1).Apparently,protein synthesis had not stopped completely in the penicillin-inhibited cells,and these were most likely able to synthesize new PBPs.During this period(between days45and60),the doubling times of the resus-citated reactors treated with0.5,1,and5g/liter penicillin G were calculated to be14,17,and18days,respectively.These doubling times were longer than the doubling time of the control reactor under steady-state growth,which was calculated to beϳ11days (57).We further investigated the effect of penicillin on the anam-mox bacteria by sequencing the transcriptome of K.stuttgartiensis from the control reactor and from the reactor that was treated with0.5g/liter penicillin G.When RNA quality was examined,it was revealed that rRNA quantity dropped signiﬁcantly(over10-fold;data not shown)after3weeks of penicillin G treatment, suggesting that the protein synthesis machinery of the cell was turned down.In total,in the penicillin G-treated sample,1,759 genes were neither up-nor downregulated,258were signiﬁcantly downregulated(over2-fold),and37of these were downregulated over5-fold(see Table S2in the supplemental material).The transcriptome analysis revealed that,in line with the observa-tion that cells stopped dividing,the anammox-speciﬁc cell division protein Kustd1438,which was reported to be a replacement of the protein for the cell division gene encoding the tubulin analogue FtsZ (54),was downregulated25-fold.All genes involved in peptidoglycan biosynthesis encoded by K.stuttgartiensis were expressed in both samples.However,none of them was signiﬁcantly up-or downregu-lated.Furthermore,mRNA transcribed from the gene(kusta0010)FIG2Short-term effects of penicillin G and streptomycin on anammox ac-tivity.White bars,biomass treated with penicillin G;gray bars,biomass treated with streptomycin.Hu et al. Applied and Environmental Microbiology on June 17, 2014 by guest / Downloaded fromencoding a membrane-bound lytic transglycosylase-like protein was detected in both samples.The function of this protein was suggested to be to remodel the peptidoglycan layer during cell enlargement and division by catalyzing the cleavage of the␤-1,4-glycosidic bond in peptidoglycan that is necessary for the insertion of new monomers (58,59).Besides,lytic transglycosylases could also bind to many types of PBPs(bifunctional and monofunctional)to form a complex that has been indicated to be involved in peptidoglycan biosynthesis (59–61).The ultimate result of the penicillin inhibition in the continu-ous cultures was the accumulation of nitrite in the efﬂuent of the reactor.Interestingly,one of the genes encoding a putative nitrite transporter(Kuste3055)was downregulated15-fold.It was previ-ously reported that this gene is by far the most expressed among the genes for nitrite transporters in K.stuttgartiensis(1,62).Ap-parently,the response of the cell to elevated nitrite concentrations was to shut down nitrite import into the cell.In K.stuttgartiensis, nitrite is reduced to nitric oxide(NO)by nitrite reductase NirS (Kuste4136)(62).Surprisingly,in the penicillin G-treated sample, Kuste4136was downregulated63-fold.Apparently,the cells re-sponded to high nitrite concentrations not by upregulating the nitrite-converting enzyme but by shutting down their catabolic machinery.Interestingly,this observed response was similar to the response of a nitrite reductase(NirK)-deﬁcient mutant of the aer-obic ammonium oxidizer Nitrosomonas europaea,which,in re-sponse to nitrite toxicity,downregulated its nitrite detoxiﬁcation genes(63).In contrast to the downregulation of258genes,only47genes were signiﬁcantly upregulated(over2-fold),and9of them were upregu-lated over5-fold(see Table S3in the supplemental material).The most upregulated functional gene was kustd1340(62),which encodes the second copy of Kustc0694.The product of kustc0694was previ-ously puriﬁed and identiﬁed to be hydrazine dehydrogenase(HDH), which is responsible for the four-electron oxidation of hydrazine to N2(1).The transcription levels of Kustc0694under both growth con-ditions were high(and it was among the transcripts with the highest levels of transcription detected,with an RPKM value of3,707in the penicillin G-treated sample and4,109in the control sample)but without signiﬁcant up-or downregulation.Until now,the physiolog-ical conditions where Kustd1340would be expressed have been un-known.Our results indicated that the kustd1340gene was at least transcribed to mRNA under extreme cellular stress.This indicated that the second copy of HDH could serve as a backup system under stress conditions.Future research will be aimed at better deﬁning these stress conditions and the puriﬁcation of the kustd1340gene product.FIG3Phase-contrast(A,B)and FISH(C,D)micrographs of Kuenenia stuttgartiensis single cells before(A,C)and after(B,D)3weeks of1-g/liter penicillin G treatment.The AMX820probe targeting Brocadia-and Kuenenia-like anammox bacteria was labeled with Cy3(red).Staining with DAPI(4=,6-diamidino-2-phenylindole dihydrochloride;light blueﬂuorescence)targeting double-stranded DNA is also shown.Barsϭ5␮m.Lysozyme and Penicillin Inhibit Anammox Bacteria 7767 on June 17, 2014 by guest / Downloaded from。

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arXiv:cond-mat/9906213v2 7 Dec 1999Moleculardynamicsstudyofaclassicaltwo-dimensionalelectronsystem:Positionalandorientationalorders

SatoruMuto1,HideoAokiDepartmentofPhysics,UniversityofTokyo,Hongo,Tokyo113-0033,Japan

1IntroductionMorethan60yearsago,Wignerpointedoutthatanelectronsystemwillcrys-tallizeduetotheCoulombrepulsionforlowenoughdensities(Wignercrystal-lization)[1].Althoughquantumeﬀectsplayanessentialroleinadegenerateelectronsystem,theconceptofWignercrystallizationcanbegeneralizedtoaclassicalcasewheretheFermienergyismuchsmallerthanthethermalen-ergy.Aclassicaltwo-dimensional(2D)electronsystemiswhollyspeciﬁedbythedimensionlesscouplingconstantΓ,theratiooftheCoulombenergytothekineticenergy.HereΓ≡(e2/4πǫa)/kBT,whereeisthechargeofanelec-tron,ǫthedielectricconstantofthesubstrate,athemeandistancebetweenelectronsandTthetemperature.ForΓ≪1thesystemwillbehaveasagaswhileforΓ≫1asasolid.Experimentally,GrimesandAdams[2]succeededinobservingatransitionfromaliquidtoatriangularlatticeinaclassical2Delectronsystemonaliquid-heliumsurfacearoundΓc=137±15,whichisingoodagreementwithnumericalsimulations[4–8].

Onthetheoreticalside,twoconspicuouspointshavebeenknownfor2Dsys-tems:(i)Mermin’stheoremdictatesthatnotruelong-rangecrystallineorderispossibleatﬁniteTinthethermodynamiclimit[9].Tobeprecise,the1/rCoulombinteractionistoolongrangedtoapplyMermin’sargumentsdirectly.Althoughtherehavebeensometheoreticalattempts[10,11]toextendthethe-oremtotheCoulombcase,norigorousproofhasbeenattained.(ii)AtheoryduetoKosterlitz,Thouless,Halperin,Nelson,andYoung(KTHNY)predictsthata“hexatic”phase,characterizedbyashort-rangepositionalorderandaquasi-long-rangeorientationalorder,existsbetweenaliquidphaseandasolidphase[12].BecausetheKTHNYtheoryisbasedonvariousassumptionsandapproximations,itsvalidityshouldbetestedbynumericalmethodssuchasamoleculardynamics(MD)simulation.Severalauthorshaveappliednumer-icalmethodstoclassical2Delectronsystems,buttheyarrivedatdiﬀerentconclusionsontheKTHNYprediction[5–8].

Inordertoaddressbothoftheaboveproblems,themostdirectwayistocalculatethepositionalandtheorientationalcorrelationfunctions,whichisexactlythemotivationofthepresentstudy[18].

2NumericalMethodAdetaileddescriptionofthesimulationisgivenelsewhere[17],soweonlyre-capitulateit.Weconsiderarectangularareawitharigiduniformneutralizingpositivebackgroundinperiodicboundaryconditions.TheaspectratiooftherectangleistakentobeLy/Lx=2/√positionalcorrelationfunctionisdeﬁnedbyC(r)≡󰀊ρ∗G(r)ρG(0)󰀋=󰀄󰀂i,jδ(r−|ri−rj|)1󰀂i,jδ(r−|ri−rj|)󰀁,

whereGisareciprocalvectorofthetriangularlatticewiththesummationtakenoverthesixG’sthatgivetheﬁrstpeaksofthestructurefactor[inset(a)ofFig.1].Theorientationalcorrelationfunctionisdeﬁnedby

C6(r)≡󰀊ψ∗6(r)ψ6(0)󰀋=󰀄󰀂

i,jδ(r−|ri−rj|)ψ∗6(ri)ψ6(rj)

󰀃n.n.

αe6iθα(r)withθα(r)beingtheangleofthevectorcon-

nectinganelectronatrandtheα-thnearestneighborwithrespecttoaﬁxedaxis.ThesummationistakenoverncnearestneighborsthataredeterminedbytheVoronoiconstruction[16].

3ResultsandDiscussionsLetusﬁrstlookatthepositionalandtheorientationalcorrelationfunctionsinFig.1forΓ=200andΓ=160,forwhichthesystemiswellinthesolidphase.Thepositionalcorrelationisseentodecayslowly,indicativeofanalgebraic(power-law)decayatlargedistances.Theround-oﬀinthecorrelationfunctionaroundhalfofthesystemsizeshouldbeaneﬀectoftheperiodicboundaryconditions.Thealgebraicdecayofthepositionalcorrelationindicatesthatthe2Delectronsolidhasonlyaquasi-long-rangepositionalorder.ThuswehaveobtainedanumericalindicationthatMermin’stheorem[9]remainsapplicabletothe1/rCoulombinteraction,whichisconsistentwiththeanalyticalbutapproximateresultsobtainedin[10,11].

Ontheotherhand,theorientationalcorrelationrapidlyapproachesaconstant,indicatingalong-rangeorientationalorder.Therefore,whilethe2Delectronsolidhasnotruelong-rangecrystallineorder,wecansaythatithasatopo-logicalorder.Fromasnapshotoftheconﬁguration[seeinset(b)inFig.1],wecanseethatalong-rangeorientationalorderispreservedsincedefects(5-or7-folddisclinations,etc)tendtoappearasdislocation(5-7combina-tionofdisclinations)pairs,i.e.,5-7-5-7disclinationquartetsthatonlydisturbtheorientationalcorrelationlocally.HerethecoordinationnumberisagaindeterminedfromtheVoronoiconstruction.