3D Density of States The density of states
半导体物理与器件——Terms汉译英.docx
半导体物理与器件--- Terms(术语)U1 Terms:Semic on ductor physics and devices 半导体物理与器件,Space lattice 空间晶格'unit cell 晶胞'primitive cell 原胞,basic crystal structures 基本晶格结构(five), Miller indices 密勒指数' atomic bonding 原子价键U2 Terms:quantum mechanics 量了力学'energy quanta 能量子,wave-particle duality 波粒二象性,the uncertainty principle测不准原理/海森堡不确定原理Schrodinger's wave equation 薛定灣波动方程,eletrons in free Space 自由空间中的电子the infinite potential well 无限深势阱,the step potential function 阶跃势函数,the potential barrier 势垒.U3 Terms:Pauli exclusion principle 泡利不相容原理,quantum state 量了态. allowed energy band 允带'forbidden energy band 禁带. con ducti on band 导带'vale nee band 价带,hole 空穴'electron 电子.effective mass 有效质量.density of states function 状态密度函数,the Fermi-Dirac probability function 费米■狄拉克概率函数' the Boltzmann approximation 波尔兹曼近彳以‘ the Fermi energy 费米能级.U4 Terms:charge carriers 载流子'effective density of states function 有效族犬态密度函数,intrinsic 本征的,the intrinsic carrier concentration 本征载流子浓度'the intrinsic Fermi level 本征费米能级.charge n eutrality 电中性状态'compe nsated semic on ductor 补偿半导体‘ degenerate 简并的,non-degenerate 非简并的'position of E F费米能级的位置U5 Terms:drift current 漂移电流,diffusion current 扩散电流,mobility 迁移率,lattice scattering 晶格散身寸,ionized impurity scattering 电离杂质散射'velocity saturation饱和速度’conductivity 电导率‘resistivity 电阻率.graded impurity distribution 杂质梯度分布'the induced electric field 感生电场'the Einstein relation 爱因斯坦关系, the hall effect 霍尔效应U6 Terms:non equilibrium excess carriers 非平衡过剩载流子'carrier generation and recombination 载流子的产生与复合'excess minority carrier 过剩少子,lifetime 寿命'low-level injection 小注入,ambipolar transport 双极输运'quasi-Fermi energy 准费米能级.U7 Terms:the space charge region 空间电荷区'the built-in potential 内建电势,the built-in potential barrier 内建电势差,the space charge width 空间电荷区宽度,zero applied bias 零偏压,reverse applied bias 反偏'on esided jun ction 单边突变结.U8 Terms:the PN junction diode PN 结二极管,minority carrier distribution 少数载流子分布'the ideal-diode equation 理想二极管方程,the reverse saturation current density 反向饱和电流密度.a short diode 短二极管,generation-recombination current 产生・复合电流,the Zener effect 齐纳效应,the avalanche effect 雪崩效应,breakdown 击穿.U9 Terms:Schottky barrier diode (SBD)肖特基势垒二极管,Schottky barrier height 肖特基势垒高度.Ohomic contact 欧姆接触'heterojunction 异质纟吉'homojunction 单质纟吉,turn-on voltage 开启电压,narrow-bandgap 窄带隙'wide-bandgap 宽带隙,2-D electron gas 二维电子气U10 Terms:bipolar transistor 双极晶体管,base基极,emitter发射极,collector集电极. forward active regi on 正向有源区'in verse active region 反向有源区' cut-off 截止,saturation 饱和,current gain电流增益'common・base 共基,commorvemitter 共身寸.base width modulation 基区宽度调制效应,Early effect 厄利效应'Early voltage厄利电压Ull Terms:Gate 栅极,source 源极,drain 漏极'substrate 基底. work function difference 功函数差threshold voltage 阈值电压'flat-band voltage 半带电压enhancement mode 增强型'depletion mode 耗尽型strong inversion 强反型'weak inversion 弱反型,transconductance 跨导'l-V relationship 电流■电压关系。
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.。
与森林相关的词汇及对照英文翻译
森林forest2006年6月13日16:45出处:FanE『翻译中国』词汇由:tradoser提供森林forest原始林virgin forest次生林sec on dary forest天然林n atural forest人工林forest pla ntati on, man-made forest速生丰产林fast-growing and high-yield plantation 乔林high forest 中林composite forest矮林coppice forest针叶林coni ferous forest阔叶林broad leaved forest落叶阔叶林deciduous broadleaved forest常绿阔叶林evergree n broadleaved forest热带雨林tropical rain forest红树林man grove forest林种forest category用材林timber forest防护林protecti on forest经济林non-timber product forest培育木材以外的其它林产品为主的森林。
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石墨烯功函数
石墨烯功函数中的特定函数1. 功能介绍石墨烯功函数中的特定函数是用来描述石墨烯材料中的电子能级分布和电子行为的函数。
通过计算和分析这些特定函数,可以了解石墨烯材料的电子结构、导电性质以及与其他材料的界面相互作用等重要信息。
这些特定函数在石墨烯相关研究领域具有广泛的应用,包括电子输运、器件设计和材料工程等方面。
2. 常见的特定函数2.1 密度态密度(Density of States, DOS)密度态密度(Density of States, DOS)是描述材料中电子能级分布的函数。
对于石墨烯材料来说,密度态密度函数表示了在给定能量范围内每个能级上的电子数目。
通过计算密度态密度函数,可以了解石墨烯中的能带结构、能级分布以及电子行为等信息。
在石墨烯中,由于其特殊的晶格结构和电子能带结构,密度态密度函数呈现出特殊的形态。
石墨烯的能带结构包括两个能带,分别是价带和导带,它们在费米能级附近相交,形成两个锐利的能带峰。
在费米能级附近,石墨烯的密度态密度函数呈现出线性关系,即呈现出线性色散关系。
这种特殊的线性色散关系是石墨烯具有优异电子输运性能的重要原因之一。
2.2 偏态密度(Partial Density of States, PDOS)偏态密度(Partial Density of States, PDOS)是描述材料中特定原子或分子轨道上电子能级分布的函数。
在石墨烯材料中,偏态密度函数用于描述石墨烯中不同原子轨道上的电子能级分布。
通过计算偏态密度函数,可以了解不同原子轨道上的电子密度和电子行为等信息。
石墨烯是由碳原子构成的,因此石墨烯的偏态密度函数主要描述碳原子轨道上的电子能级分布。
在石墨烯中,碳原子的3个p轨道参与形成π键,而s轨道和另外两个p轨道不参与键合。
因此,石墨烯的偏态密度函数主要包括s轨道和p轨道的贡献。
通过计算和分析石墨烯的偏态密度函数,可以了解碳原子轨道上的电子密度和电子行为,进而揭示石墨烯的电子结构和导电性质等重要信息。
晶格振动模式密度定义
晶格振动模式密度定义晶格振动模式密度(Phonon Density of States,简称PDOS)是描述晶体中原子振动模式的一种物理量。
晶体中的原子在平衡位置附近以小振幅做简谐振动,这些简谐振动构成了晶体的振动模式。
PDOS给出了不同频率的振动模式在能量空间中的分布情况,反映了晶体各种振动模式的丰度和分布情况。
PDOS对于研究晶体的热力学性质、热传导、声学性质等都具有重要意义。
它是计算晶体热容、热导率、声子散射等性质的基础。
此外,PDOS还可以用于研究晶体的相变、物理化学性质以及材料的设计和优化。
PDOS的具体定义如下:设晶体中的原子总数为N,晶格振动模式的总数为M,则PDOS可以定义为每单位频率范围内单位原子数的平均数,即:PDOS(ω) = (1/ N) * ∑(m=1 to M) δ(ω - ω_m)其中,δ(ω-ω_m)为狄拉克函数,当ω等于ω_m时取值为1,否则取值为0。
PDOS可以分为各向同性的PDOS和各向异性的PDOS。
各向同性PDOS是指晶体中各个晶向上的振动模式在一些频率范围内的分布情况,它是晶体的各向同性介质的特征。
各向异性PDOS是指晶体中不同晶向上的振动模式在一些频率范围内的分布情况,它反映了晶体的各向异性效应,比如晶体的声子色散关系。
在实际计算中,PDOS通常通过量子力学计算或者分子动力学模拟得到。
对于固体材料,计算PDOS是一个复杂的过程,需要考虑晶胞、原子的排列方式、晶格常数等诸多因素。
目前,常用的计算方法包括密度泛函理论(DFT)、哈密顿动力学模拟(HMD)等。
根据计算得到的PDOS,可以进一步研究晶体的声子态密度(Phonon Density of States,简称PhDOS),PhDOS是PDOS的积分,表示在一些频率以下的所有振动模式的能量状态密度。
总结起来,晶格振动模式密度(PDOS)是指描述晶体中不同频率的振动模式在能量空间中的分布情况。
它是了解晶体物理、热学以及声学性质的重要指标,可以通过理论计算或者模拟得到。
P和As掺杂Mn_(4)Si_(7)第一性原理计算
第50卷第2期2021年2月人 工 晶 体 学 报JOURNALOFSYNTHETICCRYSTALSVol.50 No.2February,2021P和As掺杂Mn4Si7第一性原理计算钟 义,张晋敏,王 立,贺 腾,肖清泉,谢 泉(贵州大学大数据与信息工程学院,新型光电子材料与技术研究所,贵阳 550025)摘要:采用第一性原理计算方法,对本征Mn4Si7以及P和As掺杂的Mn4Si7的电子结构和光学性质进行计算解析。
计算结果表明本征Mn4Si7是带隙值为0.810eV的间接带隙半导体材料,P掺杂Mn4Si7的带隙值增大为0.839eV,As掺杂Mn4Si7的带隙值减小为0.752eV。
掺杂使得Mn4Si7的能带结构和态密度向低能方向移动,同时使得介电函数的实数部分在低能区明显增大,虚数部分几乎全部区域增加且8eV以后趋向于零。
此外掺杂还增加了高能区的消光系数、吸收系数、反射系数以及光电导率,明显改善了Mn4Si7的光学性质。
关键词:第一性原理;Mn4Si7;掺杂;能带结构;态密度;光学性质中图分类号:O469 文献标志码:A 文章编号:1000 985X(2021)02 0273 05First PrinciplesCalculationsofPandAsDopedMn4Si7ZHONGYi,ZHANGJinmin,WANGLi,HETeng,XIAOQingquan,XIEQuan(InstituteofNewOptoelectronicMaterialsandTechnology,CollegeofBigDataandInformationEngineering,GuizhouUniversity,Guiyang550025,China)Abstract:TheelectronicstructureandopticalpropertiesofintrinsicandP,AsdopedMn4Si7werecalculatedwiththefisrt principlescalculationmethod.TheresultshowsthattheintrinsicMn4Si7isanindirectsemiconductormaterialwithagapof0.810eV,thePdopedMn4Si7bandgapincreasesto0.839eV,andtheAsdopedMn4Si7bandgapdecreasesto0.752eV.Dopingcausesashifttothelowenergyregion,andcausesanincreaseoftherealpartofdielectricfunctionnotablyinthelowenergyregionandanincreaseoftheimaginarypartinalmostallregion,imaginarypartdecreasestozeroafter8eV.Besides,dopingobviouslyincreasestheextinctioncoefficient,absorptioncoefficientreflectioncoefficientandphotoconductivityinthehighenergyregion,andimprovestheopticalpropertiesoftheMn4Si7.Keywords:first principle;Mn4Si7;doping;bandstructure;densityofstate;opticalproperty 收稿日期:2020 11 26 基金项目:国家自然科学基金(61264004);贵州省科学技术基金(黔科合基础[2018]1028);贵州大学研究生重点课程(贵大研[2015]026);贵州省高层次创新型人才培养项目(黔科合人才[2015]4015);贵州省留学回国人员科技活动择优资助项目(黔人项目资助合同[2018]09) 作者简介:钟 义(1994—),男,湖南省人,硕士研究生。
半导体物理与器件第四版课后习题答案3
Chapter 33.1If o a were to increase, the bandgap energy would decrease and the material would begin to behave less like a semiconductor and more like a metal. If o a were to decrease, the bandgap energy would increase and thematerial would begin to behave more like an insulator._______________________________________ 3.2Schrodinger's wave equation is:()()()t x x V xt x m ,,2222ψ⋅+∂ψ∂- ()tt x j ∂ψ∂=, Assume the solution is of the form:()()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-=ψt E kx j x u t x exp , Region I: ()0=x V . Substituting theassumed solution into the wave equation, we obtain:()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-⎩⎨⎧∂∂-t E kx j x jku x m exp 22 ()⎪⎭⎪⎬⎫⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-∂∂+t E kx j x x u exp ()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-⋅⎪⎭⎫ ⎝⎛-=t E kx j x u jE j exp which becomes()()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-⎩⎨⎧-t E kx j x u jk m exp 222 ()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-∂∂+t E kx j x x u jkexp 2 ()⎪⎭⎪⎬⎫⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-∂∂+t E kx j x x u exp 22 ()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-+=t E kx j x Eu exp This equation may be written as()()()()0222222=+∂∂+∂∂+-x u mE x x u x x u jk x u kSetting ()()x u x u 1= for region I, the equation becomes:()()()()021221212=--+x u k dx x du jk dxx u d α where222mE=α Q.E.D.In Region II, ()O V x V =. Assume the same form of the solution:()()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-=ψt E kx j x u t x exp , Substituting into Schrodinger's wave equation, we find:()()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-⎩⎨⎧-t E kx j x u jk m exp 222 ()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-∂∂+t E kx j x x u jkexp 2 ()⎪⎭⎪⎬⎫⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-∂∂+t E kx j x x u exp 22 ()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-+t E kx j x u V O exp ()⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-=t E kx j x Eu exp This equation can be written as:()()()2222x x u x x u jk x u k ∂∂+∂∂+- ()()02222=+-x u mEx u mV OSetting ()()x u x u 2= for region II, this equation becomes()()dx x du jk dxx u d 22222+ ()022222=⎪⎪⎭⎫ ⎝⎛+--x u mV k O α where again222mE=α Q.E.D._______________________________________3.3We have()()()()021221212=--+x u k dx x du jk dxx u d α Assume the solution is of the form: ()()[]x k j A x u -=αexp 1()[]x k j B +-+αexp The first derivative is()()()[]x k j A k j dxx du --=ααexp 1 ()()[]x k j B k j +-+-ααexp and the second derivative becomes()()[]()[]x k j A k j dxx u d --=ααexp 2212 ()[]()[]x k j B k j +-++ααexp 2Substituting these equations into the differential equation, we find()()[]x k j A k ---ααexp 2()()[]x k j B k +-+-ααexp 2(){()[]x k j A k j jk --+ααexp 2()()[]}x k j B k j +-+-ααexp ()()[]{x k j A k ---ααexp 22 ()[]}0exp =+-+x k j B α Combining terms, we obtain()()()[]222222αααα----+--k k k k k ()[]x k j A -⨯αexp()()()[]222222αααα--++++-+k k k k k ()[]0exp =+-⨯x k j B α We find that00= Q.E.D. For the differential equation in ()x u 2 and the proposed solution, the procedure is exactly the same as above._______________________________________ 3.4We have the solutions ()()[]x k j A x u -=αexp 1()[]x k j B +-+αexp for a x <<0 and()()[]x k j C x u -=βexp 2()[]x k j D +-+βexp for 0<<-x b .The first boundary condition is ()()0021u u =which yields0=--+D C B AThe second boundary condition is201===x x dx dudx du which yields()()()C k B k A k --+--βαα()0=++D k β The third boundary condition is ()()b u a u -=21 which yields()[]()[]a k j B a k j A +-+-ααexp exp ()()[]b k j C --=βexp()()[]b k j D -+-+βexp and can be written as()[]()[]a k j B a k j A +-+-ααexp exp ()[]b k j C ---βexp()[]0exp =+-b k j D β The fourth boundary condition isbx a x dx dudx du -===21 which yields()()[]a k j A k j --ααexp()()[]a k j B k j +-+-ααexp ()()()[]b k j C k j ---=ββexp()()()[]b k j D k j -+-+-ββexp and can be written as ()()[]a k j A k --ααexp()()[]a k j B k +-+-ααexp()()[]b k j C k ----ββexp()()[]0exp =+++b k j D k ββ_______________________________________ 3.5(b) (i) First point: πα=aSecond point: By trial and error, πα729.1=a (ii) First point: πα2=aSecond point: By trial and error, πα617.2=a_______________________________________3.6(b) (i) First point: πα=aSecond point: By trial and error, πα515.1=a (ii) First point: πα2=aSecond point: By trial and error, πα375.2=a_______________________________________ 3.7ka a aaP cos cos sin =+'αααLet y ka =, x a =α Theny x x xP cos cos sin =+'Consider dy dof this function.()[]{}y x x x P dy d sin cos sin 1-=+⋅'- We find()()()⎭⎬⎫⎩⎨⎧⋅+⋅-'--dy dx x x dy dx x x P cos sin 112y dydxx sin sin -=- Theny x x x x x P dy dx sin sin cos sin 12-=⎭⎬⎫⎩⎨⎧-⎥⎦⎤⎢⎣⎡+-'For πn ka y ==, ...,2,1,0=n 0sin =⇒y So that, in general,()()dk d ka d a d dy dxαα===0 And 22 mE=α Sodk dEm mE dk d ⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛=-22/122221 α This implies thatdk dE dk d ==0α for an k π= _______________________________________ 3.8(a) πα=a 1π=⋅a E m o 212()()()()2103123422221102.41011.9210054.12---⨯⨯⨯==ππa m E o19104114.3-⨯=J From Problem 3.5 πα729.12=aπ729.1222=⋅a E m o()()()()2103123422102.41011.9210054.1729.1---⨯⨯⨯=πE18100198.1-⨯=J 12E E E -=∆1918104114.3100198.1--⨯-⨯= 19107868.6-⨯=Jor 24.4106.1107868.61919=⨯⨯=∆--E eV(b) πα23=aπ2223=⋅a E m o()()()()2103123423102.41011.9210054.12---⨯⨯⨯=πE18103646.1-⨯=J From Problem 3.5, πα617.24=aπ617.2224=⋅a E m o()()()()2103123424102.41011.9210054.1617.2---⨯⨯⨯=πE18103364.2-⨯=J 34E E E -=∆1818103646.1103364.2--⨯-⨯= 1910718.9-⨯=Jor 07.6106.110718.91919=⨯⨯=∆--E eV_______________________________________3.9(a) At π=ka , πα=a 1π=⋅a E m o 212()()()()2103123421102.41011.9210054.1---⨯⨯⨯=πE19104114.3-⨯=JAt 0=ka , By trial and error, πα859.0=a o ()()()()210312342102.41011.9210054.1859.0---⨯⨯⨯=πoE19105172.2-⨯=J o E E E -=∆11919105172.2104114.3--⨯-⨯= 2010942.8-⨯=Jor 559.0106.110942.81920=⨯⨯=∆--E eV (b) At π2=ka , πα23=aπ2223=⋅a E m o()()()()2103123423102.41011.9210054.12---⨯⨯⨯=πE18103646.1-⨯=JAt π=ka . From Problem 3.5, πα729.12=aπ729.1222=⋅a E m o()()()()2103123422102.41011.9210054.1729.1---⨯⨯⨯=πE18100198.1-⨯=J23E E E -=∆1818100198.1103646.1--⨯-⨯= 19104474.3-⨯=Jor 15.2106.1104474.31919=⨯⨯=∆--E eV_______________________________________3.10(a) πα=a 1π=⋅a E m o 212()()()()2103123421102.41011.9210054.1---⨯⨯⨯=πE19104114.3-⨯=JFrom Problem 3.6, πα515.12=aπ515.1222=⋅a E m o()()()()2103123422102.41011.9210054.1515.1---⨯⨯⨯=πE1910830.7-⨯=J 12E E E -=∆1919104114.310830.7--⨯-⨯= 19104186.4-⨯=Jor 76.2106.1104186.41919=⨯⨯=∆--E eV (b) πα23=aπ2223=⋅a E m o()()()()2103123423102.41011.9210054.12---⨯⨯⨯=πE18103646.1-⨯=JFrom Problem 3.6, πα375.24=aπ375.2224=⋅a E m o()()()()2103123424102.41011.9210054.1375.2---⨯⨯⨯=πE18109242.1-⨯=J 34E E E -=∆1818103646.1109242.1--⨯-⨯= 1910597.5-⨯=Jor 50.3106.110597.51919=⨯⨯=∆--E eV_____________________________________3.11(a) At π=ka , πα=a 1π=⋅a E m o 212()()()()2103123421102.41011.9210054.1---⨯⨯⨯=πE19104114.3-⨯=JAt 0=ka , By trial and error, πα727.0=a oπ727.022=⋅a E m o o()()()()210312342102.41011.9210054.1727.0---⨯⨯⨯=πo E19108030.1-⨯=Jo E E E -=∆11919108030.1104114.3--⨯-⨯= 19106084.1-⨯=Jor 005.1106.1106084.11919=⨯⨯=∆--E eV (b) At π2=ka , πα23=aπ2223=⋅a E m o()()()()2103123423102.41011.9210054.12---⨯⨯⨯=πE18103646.1-⨯=JAt π=ka , From Problem 3.6,πα515.12=aπ515.1222=⋅a E m o()()()()2103423422102.41011.9210054.1515.1---⨯⨯⨯=πE1910830.7-⨯=J23E E E -=∆191810830.7103646.1--⨯-⨯= 1910816.5-⨯=Jor 635.3106.110816.51919=⨯⨯=∆--E eV_______________________________________3.12For 100=T K, ()()⇒+⨯-=-1006361001073.4170.124gE164.1=g E eV200=T K, 147.1=g E eV 300=T K, 125.1=g E eV 400=T K, 097.1=g E eV 500=T K, 066.1=g E eV 600=T K, 032.1=g E eV_______________________________________3.13The effective mass is given by1222*1-⎪⎪⎭⎫⎝⎛⋅=dk E d mWe have()()B curve dkE d A curve dk E d 2222> so that ()()B curve m A curve m **<_______________________________________ 3.14The effective mass for a hole is given by1222*1-⎪⎪⎭⎫ ⎝⎛⋅=dk E d m p We have that()()B curve dkEd A curve dk E d 2222> so that ()()B curve m A curve m p p **<_______________________________________ 3.15Points A,B: ⇒<0dk dEvelocity in -x directionPoints C,D: ⇒>0dk dEvelocity in +x directionPoints A,D: ⇒<022dk Ednegative effective massPoints B,C: ⇒>022dkEd positive effective mass _______________________________________3.16For A: 2k C E i =At 101008.0+⨯=k m 1-, 05.0=E eV Or ()()2119108106.105.0--⨯=⨯=E J So ()2101211008.0108⨯=⨯-C3811025.1-⨯=⇒CNow ()()38234121025.1210054.12--*⨯⨯==C m 311044.4-⨯=kgor o m m ⋅⨯⨯=--*31311011.9104437.4o m m 488.0=* For B: 2k C E i =At 101008.0+⨯=k m 1-, 5.0=E eV Or ()()2019108106.15.0--⨯=⨯=E JSo ()2101201008.0108⨯=⨯-C 3711025.1-⨯=⇒CNow ()()37234121025.1210054.12--*⨯⨯==C m 321044.4-⨯=kg or o m m ⋅⨯⨯=--*31321011.9104437.4o m m 0488.0=*_______________________________________ 3.17For A: 22k C E E -=-υ()()()2102191008.0106.1025.0⨯-=⨯--C 3921025.6-⨯=⇒C()()39234221025.6210054.12--*⨯⨯-=-=C m31108873.8-⨯-=kgor o m m ⋅⨯⨯-=--*31311011.9108873.8o m m 976.0--=* For B: 22k C E E -=-υ()()()2102191008.0106.13.0⨯-=⨯--C 382105.7-⨯=⇒C()()3823422105.7210054.12--*⨯⨯-=-=C m3210406.7-⨯-=kgor o m m ⋅⨯⨯-=--*31321011.910406.7o m m 0813.0-=*_______________________________________ 3.18(a) (i) νh E =or ()()341910625.6106.142.1--⨯⨯==h E ν1410429.3⨯=Hz(ii) 141010429.3103⨯⨯===νλc E hc 51075.8-⨯=cm 875=nm(b) (i) ()()341910625.6106.112.1--⨯⨯==h E ν1410705.2⨯=Hz(ii) 141010705.2103⨯⨯==νλc410109.1-⨯=cm 1109=nm_______________________________________ 3.19(c) Curve A: Effective mass is a constantCurve B: Effective mass is positive around 0=k , and is negativearound 2π±=k . _______________________________________ 3.20()[]O O k k E E E --=αcos 1 Then()()()[]O k k E dkdE ---=ααsin 1()[]O k k E -+=ααsin 1 and()[]O k k E dk E d -=ααcos 2122Then221222*11 αE dk Ed m o k k =⋅== or212*αE m =_______________________________________ 3.21(a) ()[]3/123/24lt dn m m m =*()()[]3/123/264.1082.04oom m =o dn m m 56.0=*(b)o o l t cnm m m m m 64.11082.02123+=+=*oo m m 6098.039.24+=o cn m m 12.0=*_______________________________________ 3.22(a) ()()[]3/22/32/3lh hh dp m m m +=*()()[]3/22/32/3082.045.0o om m +=[]o m ⋅+=3/202348.030187.0o dp m m 473.0=*(b) ()()()()2/12/12/32/3lh hh lh hh cpm m m m m ++=*()()()()om ⋅++=2/12/12/32/3082.045.0082.045.0 o cp m m 34.0=*_______________________________________ 3.23For the 3-dimensional infinite potential well, ()0=x V when a x <<0, a y <<0, and a z <<0. In this region, the wave equation is:()()()222222,,,,,,z z y x y z y x x z y x ∂∂+∂∂+∂∂ψψψ()0,,22=+z y x mEψ Use separation of variables technique, so let ()()()()z Z y Y x X z y x =,,ψSubstituting into the wave equation, we have222222zZXY y Y XZ x X YZ ∂∂+∂∂+∂∂ 022=⋅+XYZ mEDividing by XYZ , we obtain021*********=+∂∂⋅+∂∂⋅+∂∂⋅ mEz Z Z y Y Y x X XLet01222222=+∂∂⇒-=∂∂⋅X k x X k x X X xx The solution is of the form: ()x k B x k A x X x x cos sin +=Since ()0,,=z y x ψ at 0=x , then ()00=X so that 0=B .Also, ()0,,=z y x ψ at a x =, so that ()0=a X . Then πx x n a k = where ...,3,2,1=x n Similarly, we have2221y k y Y Y -=∂∂⋅ and 2221z k zZ Z -=∂∂⋅From the boundary conditions, we find πy y n a k = and πz z n a k = where...,3,2,1=y n and ...,3,2,1=z n From the wave equation, we can write022222=+---mE k k k z y xThe energy can be written as()222222⎪⎭⎫ ⎝⎛++==a n n n m E E z y x n n n z y x π _______________________________________ 3.24The total number of quantum states in the 3-dimensional potential well is given (in k-space) by()332a dk k dk k g T ⋅=ππ where222 mEk =We can then writemEk 2=Taking the differential, we obtaindE Em dE E m dk ⋅⋅=⋅⋅⋅⋅=2112121 Substituting these expressions into the density of states function, we have()dE E mmE a dE E g T ⋅⋅⋅⎪⎭⎫ ⎝⎛=212233 ππ Noting thatπ2h=this density of states function can be simplified and written as()()dE E m h a dE E g T ⋅⋅=2/33324π Dividing by 3a will yield the density of states so that()()E h m E g ⋅=32/324π _______________________________________ 3.25For a one-dimensional infinite potential well,222222k a n E m n ==*π Distance between quantum states()()aa n a n k k n n πππ=⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛+=-+11Now()⎪⎭⎫ ⎝⎛⋅=a dkdk k g T π2NowE m k n *⋅=21dE Em dk n⋅⋅⋅=*2211 Then()dE Em a dE E g n T ⋅⋅⋅=*2212 π Divide by the "volume" a , so ()Em E g n *⋅=21πSo()()()()()EE g 31341011.9067.0210054.11--⨯⋅⨯=π ()EE g 1810055.1⨯=m 3-J 1-_______________________________________ 3.26(a) Silicon, o n m m 08.1=*()()c nc E E h m E g -=*32/324π()dE E E h m g kTE E c nc c c⋅-=⎰+*232/324π()()kT E E c nc cE E h m 22/332/33224+*-⋅⋅=π()()2/332/323224kT hm n⋅⋅=*π ()()[]()()2/33342/33123210625.61011.908.124kT ⋅⋅⨯⨯=--π ()()2/355210953.7kT ⨯=(i) At 300=T K, 0259.0=kT eV()()19106.10259.0-⨯= 2110144.4-⨯=J Then ()()[]2/3215510144.4210953.7-⨯⨯=c g25100.6⨯=m 3- or 19100.6⨯=c g cm 3-(ii) At 400=T K, ()⎪⎭⎫⎝⎛=3004000259.0kT034533.0=eV()()19106.1034533.0-⨯= 21105253.5-⨯=J Then()()[]2/32155105253.5210953.7-⨯⨯=c g2510239.9⨯=m 3- or 191024.9⨯=c g cm 3-(b) GaAs, o nm m 067.0=*()()[]()()2/33342/33123210625.61011.9067.024kT g c ⋅⋅⨯⨯=--π ()()2/3542102288.1kT ⨯=(i) At 300=T K, 2110144.4-⨯=kT J ()()[]2/3215410144.42102288.1-⨯⨯=c g2310272.9⨯=m 3- or 171027.9⨯=c g cm 3-(ii) At 400=T K, 21105253.5-⨯=kT J ()()[]2/32154105253.52102288.1-⨯⨯=c g2410427.1⨯=m 3-181043.1⨯=c g cm 3-_______________________________________ 3.27(a) Silicon, o p m m 56.0=* ()()E E h mE g p-=*υυπ32/324()dE E E h mg E kTE p⋅-=⎰-*υυυυπ332/324()()υυυπE kTE pE E hm 32/332/33224-*-⎪⎭⎫ ⎝⎛-=()()[]2/332/333224kT hmp-⎪⎭⎫ ⎝⎛-=*π ()()[]()()2/33342/33133210625.61011.956.024kT ⎪⎭⎫ ⎝⎛⨯⨯=--π ()()2/355310969.2kT ⨯=(i)At 300=T K, 2110144.4-⨯=kT J ()()[]2/3215510144.4310969.2-⨯⨯=υg2510116.4⨯=m3-or 191012.4⨯=υg cm 3- (ii)At 400=T K, 21105253.5-⨯=kT J()()[]2/32155105253.5310969.2-⨯⨯=υg2510337.6⨯=m3-or 191034.6⨯=υg cm 3- (b) GaAs, o p m m 48.0=*()()[]()()2/33342/33133210625.61011.948.024kT g ⎪⎭⎫ ⎝⎛⨯⨯=--πυ ()()2/3553103564.2kT ⨯=(i)At 300=T K, 2110144.4-⨯=kT J()()[]2/3215510144.43103564.2-⨯⨯=υg2510266.3⨯=m 3- or 191027.3⨯=υg cm 3-(ii)At 400=T K, 21105253.5-⨯=kT J()()[]2/32155105253.53103564.2-⨯⨯=υg2510029.5⨯=m 3-or 191003.5⨯=υg cm 3-_______________________________________ 3.28(a) ()()c nc E E h m E g -=*32/324π()()[]()c E E -⨯⨯=--3342/33110625.61011.908.124πc E E -⨯=56101929.1 For c E E =; 0=c g1.0+=c E E eV; 4610509.1⨯=c g m 3-J 1-2.0+=c E E eV; 4610134.2⨯=m 3-J 1-3.0+=c E E eV; 4610614.2⨯=m 3-J 1- 4.0+=c E E eV; 4610018.3⨯=m 3-J 1- (b) ()E E h m g p-=*υυπ32/324()()[]()E E -⨯⨯=--υπ3342/33110625.61011.956.024E E -⨯=υ55104541.4 For υE E =; 0=υg1.0-=υE E eV; 4510634.5⨯=υg m 3-J 1-2.0-=υE E eV; 4510968.7⨯=m 3-J 1-3.0-=υE E eV; 4510758.9⨯=m 3-J 1-4.0-=υE E eV; 4610127.1⨯=m 3-J 1-_______________________________________ 3.29(a) ()()68.256.008.12/32/32/3=⎪⎭⎫ ⎝⎛==**pnc m m g g υ(b) ()()0521.048.0067.02/32/32/3=⎪⎭⎫ ⎝⎛==**pncmm g g υ_______________________________________3.30 Plot_______________________________________ 3.31(a) ()()()!710!7!10!!!-=-=i i i i i N g N g W()()()()()()()()()()()()1201238910!3!7!78910===(b) (i) ()()()()()()()()12!10!101112!1012!10!12=-=i W 66=(ii) ()()()()()()()()()()()()1234!8!89101112!812!8!12=-=i W 495=_______________________________________ 3.32()⎪⎪⎭⎫ ⎝⎛-+=kT E E E f F exp 11(a) kT E E F =-, ()()⇒+=1exp 11E f()269.0=E f (b) kT E E F 5=-, ()()⇒+=5exp 11E f()31069.6-⨯=E f(c) kT E E F 10=-, ()()⇒+=10exp 11E f ()51054.4-⨯=E f_______________________________________ 3.33()⎪⎪⎭⎫ ⎝⎛-+-=-kT E E E f F exp 1111or()⎪⎪⎭⎫ ⎝⎛-+=-kT E E E f F exp 111(a) kT E E F =-, ()269.01=-E f (b) kT E E F 5=-, ()31069.61-⨯=-E f(c) kT E E F 10=-, ()51054.41-⨯=-E f_______________________________________3.34(a) ()⎥⎦⎤⎢⎣⎡--≅kT E E f F F exp c E E =; 61032.90259.030.0exp -⨯=⎥⎦⎤⎢⎣⎡-=F f 2kT E c +; ()⎥⎦⎤⎢⎣⎡+-=0259.020259.030.0exp F f 61066.5-⨯=kT E c +; ()⎥⎦⎤⎢⎣⎡+-=0259.00259.030.0exp F f 61043.3-⨯=23kT E c +; ()()⎥⎦⎤⎢⎣⎡+-=0259.020259.0330.0exp F f 61008.2-⨯=kT E c 2+; ()()⎥⎦⎤⎢⎣⎡+-=0259.00259.0230.0exp F f 61026.1-⨯=(b) ⎥⎦⎤⎢⎣⎡-+-=-kT E E f F F exp 1111()⎥⎦⎤⎢⎣⎡--≅kT E E F exp υE E =; ⎥⎦⎤⎢⎣⎡-=-0259.025.0exp 1F f 51043.6-⨯= 2kT E -υ; ()⎥⎦⎤⎢⎣⎡+-=-0259.020259.025.0exp 1F f 51090.3-⨯=kT E -υ; ()⎥⎦⎤⎢⎣⎡+-=-0259.00259.025.0exp 1F f 51036.2-⨯=23kTE -υ; ()()⎥⎦⎤⎢⎣⎡+-=-0259.020259.0325.0exp 1F f 51043.1-⨯= kT E 2-υ;()()⎥⎦⎤⎢⎣⎡+-=-0259.00259.0225.0exp 1F f 61070.8-⨯=_______________________________________3.35()()⎥⎦⎤⎢⎣⎡-+-=⎥⎦⎤⎢⎣⎡--=kT E kT E kT E E f F c F F exp exp and()⎥⎦⎤⎢⎣⎡--=-kT E E f F F exp 1 ()()⎥⎦⎤⎢⎣⎡---=kT kT E E F υexp So ()⎥⎦⎤⎢⎣⎡-+-kT E kT E F c exp ()⎥⎦⎤⎢⎣⎡+--=kT kT E E F υexp Then kT E E E kT E F F c +-=-+υOr midgap c F E E E E =+=2υ_______________________________________ 3.3622222ma n E n π =For 6=n , Filled state()()()()()2103122234610121011.92610054.1---⨯⨯⨯=πE18105044.1-⨯=Jor 40.9106.1105044.119186=⨯⨯=--E eV For 7=n , Empty state()()()()()2103122234710121011.92710054.1---⨯⨯⨯=πE1810048.2-⨯=Jor 8.12106.110048.219187=⨯⨯=--E eV Therefore 8.1240.9<<F E eV_______________________________________ 3.37(a) For a 3-D infinite potential well()222222⎪⎭⎫ ⎝⎛++=a n n n mE z y x π For 5 electrons, the 5th electron occupies the quantum state 1,2,2===z y x n n n ; so()2222252⎪⎭⎫ ⎝⎛++=a n n n m E z y x π()()()()()21031222223410121011.9212210054.1---⨯⨯++⨯=π1910761.3-⨯=Jor 35.2106.110761.319195=⨯⨯=--E eV For the next quantum state, which is empty, the quantum state is 2,2,1===z y x n n n . This quantum state is at the same energy, so 35.2=F E eV(b) For 13 electrons, the 13th electronoccupies the quantum state 3,2,3===z y x n n n ; so ()()()()()2103122222341310121011.9232310054.1---⨯⨯++⨯=πE 1910194.9-⨯=Jor 746.5106.110194.9191913=⨯⨯=--E eVThe 14th electron would occupy the quantum state 3,3,2===z y x n n n . This state is at the same energy, so 746.5=F E eV_______________________________________ 3.38The probability of a state at E E E F ∆+=1 being occupied is()⎪⎭⎫ ⎝⎛∆+=⎪⎪⎭⎫ ⎝⎛-+=kT E kT E E E f F exp 11exp 11111 The probability of a state at E E E F ∆-=2being empty is()⎪⎪⎭⎫ ⎝⎛-+-=-kT E E E f F 222exp 1111⎪⎭⎫ ⎝⎛∆-+⎪⎭⎫ ⎝⎛∆-=⎪⎭⎫ ⎝⎛∆-+-=kT E kT E kT E exp 1exp exp 111or()⎪⎭⎫ ⎝⎛∆+=-kT E E f exp 11122so ()()22111E f E f -= Q.E.D. _______________________________________3.39(a) At energy 1E , we want01.0exp 11exp 11exp 1111=⎪⎪⎭⎫ ⎝⎛-+⎪⎪⎭⎫ ⎝⎛-+-⎪⎪⎭⎫ ⎝⎛-kT E E kT E E kT E E F F FThis expression can be written as01.01exp exp 111=-⎪⎪⎭⎫ ⎝⎛-⎪⎪⎭⎫ ⎝⎛-+kT E E kT E E F F or()⎪⎪⎭⎫⎝⎛-=kT E E F 1exp 01.01Then()100ln 1kT E E F += orkT E E F 6.41+= (b)At kT E E F 6.4+=, ()()6.4exp 11exp 1111+=⎪⎪⎭⎫ ⎝⎛-+=kT E E E f F which yields()01.000990.01≅=E f_______________________________________ 3.40 (a)()()⎥⎦⎤⎢⎣⎡--=⎥⎦⎤⎢⎣⎡--=0259.050.580.5exp exp kT E E f F F 61032.9-⨯=(b) ()060433.03007000259.0=⎪⎭⎫⎝⎛=kT eV31098.6060433.030.0exp -⨯=⎥⎦⎤⎢⎣⎡-=F f (c) ()⎥⎦⎤⎢⎣⎡--≅-kT E E f F F exp 1 ⎥⎦⎤⎢⎣⎡-=kT 25.0exp 02.0or 5002.0125.0exp ==⎥⎦⎤⎢⎣⎡+kT ()50ln 25.0=kTor()()⎪⎭⎫⎝⎛===3000259.0063906.050ln 25.0T kT which yields 740=T K_______________________________________ 3.41 (a)()00304.00259.00.715.7exp 11=⎪⎭⎫ ⎝⎛-+=E for 0.304%(b) At 1000=T K, 08633.0=kT eV Then()1496.008633.00.715.7exp 11=⎪⎭⎫ ⎝⎛-+=E for 14.96%(c) ()997.00259.00.785.6exp 11=⎪⎭⎫ ⎝⎛-+=E for 99.7% (d)At F E E =, ()21=E f for all temperatures_______________________________________ 3.42(a) For 1E E =()()⎥⎦⎤⎢⎣⎡--≅⎪⎪⎭⎫ ⎝⎛-+=kT E E kTE E E fF F11exp exp 11Then()611032.90259.030.0exp -⨯=⎪⎭⎫ ⎝⎛-=E fFor 2E E =, 82.030.012.12=-=-E E F eV Then()⎪⎭⎫ ⎝⎛-+-=-0259.082.0exp 1111E for()⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛---≅-0259.082.0exp 111E f141078.10259.082.0exp -⨯=⎪⎭⎫ ⎝⎛-=(b) For 4.02=-E E F eV,72.01=-F E E eVAt 1E E =,()()⎪⎭⎫⎝⎛-=⎥⎦⎤⎢⎣⎡--=0259.072.0exp exp 1kT E E E f F or()131045.8-⨯=E f At 2E E =,()()⎥⎦⎤⎢⎣⎡--=-kT E E E f F 2exp 1 ⎪⎭⎫ ⎝⎛-=0259.04.0expor()71096.11-⨯=-E f_______________________________________ 3.43(a) At 1E E =()()⎪⎭⎫⎝⎛-=⎥⎦⎤⎢⎣⎡--=0259.030.0exp exp 1kT E E E f F or()61032.9-⨯=E fAt 2E E =, 12.13.042.12=-=-E E F eV So()()⎥⎦⎤⎢⎣⎡--=-kT E E E f F 2exp 1 ⎪⎭⎫ ⎝⎛-=0259.012.1expor()191066.11-⨯=-E f (b) For 4.02=-E E F ,02.11=-F E E eV At 1E E =,()()⎪⎭⎫⎝⎛-=⎥⎦⎤⎢⎣⎡--=0259.002.1exp exp 1kT E E E f F or()181088.7-⨯=E f At 2E E =,()()⎥⎦⎤⎢⎣⎡--=-kT E E E f F 2exp 1 ⎪⎭⎫ ⎝⎛-=0259.04.0expor ()71096.11-⨯=-E f_______________________________________ 3.44()1exp 1-⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-+=kTE E E f Fso()()2exp 11-⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-+-=kT E E dE E df F⎪⎪⎭⎫ ⎝⎛-⎪⎭⎫⎝⎛⨯kT E E kT F exp 1or()2exp 1exp 1⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-+⎪⎪⎭⎫ ⎝⎛-⎪⎭⎫⎝⎛-=kT E E kT E E kT dE E df F F (a) At 0=T K, For()00exp =⇒=∞-⇒<dE dfE E F()0exp =⇒+∞=∞+⇒>dEdfE E FAt -∞=⇒=dEdfE E F(b) At 300=T K, 0259.0=kT eVFor F E E <<, 0=dE dfFor F E E >>, 0=dEdfAt F E E =,()()65.91110259.012-=+⎪⎭⎫ ⎝⎛-=dE df (eV)1-(c) At 500=T K, 04317.0=kT eVFor F E E <<, 0=dE dfFor F E E >>, 0=dEdfAt F E E =,()()79.511104317.012-=+⎪⎭⎫ ⎝⎛-=dE df (eV)1- _______________________________________ 3.45(a) At midgap E E =,()⎪⎪⎭⎫⎝⎛+=⎪⎪⎭⎫ ⎝⎛-+=kT E kTE E E f g F2exp 11exp 11Si: 12.1=g E eV, ()()⎥⎦⎤⎢⎣⎡+=0259.0212.1exp 11E for()101007.4-⨯=E fGe: 66.0=g E eV ()()⎥⎦⎤⎢⎣⎡+=0259.0266.0exp 11E for()61093.2-⨯=E f GaAs: 42.1=g E eV ()()⎥⎦⎤⎢⎣⎡+=0259.0242.1exp 11E for()121024.1-⨯=E f(b) Using the results of Problem 3.38, the answers to part (b) are exactly the same as those given in part (a)._______________________________________3.46(a) ()⎥⎦⎤⎢⎣⎡--=kT E E f F F exp ⎥⎦⎤⎢⎣⎡-=-kT 60.0exp 108or ()810ln 60.0+=kT()032572.010ln 60.08==kT eV ()⎪⎭⎫⎝⎛=3000259.0032572.0Tso 377=T K(b) ⎥⎦⎤⎢⎣⎡-=-kT 60.0exp 106()610ln 60.0+=kT()043429.010ln 60.06==kT ()⎪⎭⎫⎝⎛=3000259.0043429.0Tor 503=T K_______________________________________ 3.47(a) At 200=T K,()017267.03002000259.0=⎪⎭⎫⎝⎛=kT eV⎪⎪⎭⎫ ⎝⎛-+==kT E E f F F exp 1105.019105.01exp =-=⎪⎪⎭⎫ ⎝⎛-kT E E F()()()19ln 017267.019ln ==-kT E E F 05084.0=eV By symmetry, for 95.0=F f , 05084.0-=-F E E eVThen ()1017.005084.02==∆E eV (b) 400=T K, 034533.0=kT eV For 05.0=F f , from part (a),()()()19ln 034533.019ln ==-kT E E F 10168.0=eVThen ()2034.010168.02==∆E eV _______________________________________。
一份关于二维电子气(2DEG)的讲义
where r is the vector in plane of 2DEG. Throughout our considerations we will assume that all the distances are much larger than interatomic distance and thus we will use the effective
Density of States
The density of states g( ) is defined as number of states per the energy interval , + d . It is clear that
g( ) = δ( − α)
α
where α is the set of quantum numbers characterizing the states. In the present case it includes the subband quantum number n, spin quantum number σ, valley quantum number
v (for n-type materials), and in-plane quasimomentum k. If the spectrum is degenerate
with respect to spin and valleys one can define the spin degeneracy νs and valley degeneracy
n-AlGaAs i-GaAs
EC
EF EC
EF
EV
EV
EF
卡梅伦液压数据手册(第 20 版)说明书
iv
⌂
CONTENTS OF SECTION 1
☰ Hydraulics
⌂ Cameron Hydraulic Data ☰
Introduction. . . . . . . . . . . . . ................................................................ 1-3 Liquids. . . . . . . . . . . . . . . . . . . ...................................... .......................... 1-3
4
Viscosity etc.
Steam data....................................................................................................................................................................................... 6
1 Liquid Flow.............................................................................. 1-4
Viscosity. . . . . . . . . . . . . . . . . ...................................... .......................... 1-5 Pumping. . . . . . . . . . . . . . . . . ...................................... .......................... 1-6 Volume-System Head Calculations-Suction Head. ........................... 1-6, 1-7 Suction Lift-Total Discharge Head-Velocity Head............................. 1-7, 1-8 Total Sys. Head-Pump Head-Pressure-Spec. Gravity. ...................... 1-9, 1-10 Net Positive Suction Head. .......................................................... 1-11 NPSH-Suction Head-Life; Examples:....................... ............... 1-11 to 1-16 NPSH-Hydrocarbon Corrections.................................................... 1-16 NPSH-Reciprocating Pumps. ....................................................... 1-17 Acceleration Head-Reciprocating Pumps. ........................................ 1-18 Entrance Losses-Specific Speed. .................................................. 1-19 Specific Speed-Impeller. .................................... ........................ 1-19 Specific Speed-Suction...................................... ................. 1-20, 1-21 Submergence.. . . . . . . . . ....................................... ................. 1-21, 1-22 Intake Design-Vertical Wet Pit Pumps....................................... 1-22, 1-27 Work Performed in Pumping. ............................... ........................ 1-27 Temperature Rise. . . . . . . ...................................... ........................ 1-28 Characteristic Curves. . ...................................... ........................ 1-29 Affinity Laws-Stepping Curves. ..................................................... 1-30 System Curves.. . . . . . . . ....................................... ........................ 1-31 Parallel and Series Operation. .............................. ................. 1-32, 1-33 Water Hammer. . . . . . . . . . ...................................... ........................ 1-34 Reciprocating Pumps-Performance. ............................................... 1-35 Recip. Pumps-Pulsation Analysis & System Piping...................... 1-36 to 1-45 Pump Drivers-Speed Torque Curves. ....................................... 1-45, 1-46 Engine Drivers-Impeller Profiles. ................................................... 1-47 Hydraulic Institute Charts.................................... ............... 1-48 to 1-52 Bibliography.. . . . . . . . . . . . ...................................... ........................ 1-53
一份关于二维电子气(2DEG)的讲义
Density of States
The density of states g( ) is defined as number of states per the energy interval , + d . It is clear
where α is the set of quantum numbers characterizing the states. In the present case it includes the subband quantum number n, spin quantum number σ, valley quantum number
ns
=
ox
edox
(Vg
− Vt)
where Vt is the threshold voltage for the barrier’s creation Another important systems with 2DEG involve modulation-doped GaAs-AlGaAs het-
v (for n-type materials), and in-plane quasimomentum k. If the spectrum is degenerate
with respect to spin and valleys one can define the spin degeneracy νs and valley degeneracy
erostructures. The bandgap in AlGaAs is wider than in GaAs. By variation of doping it is possible to move the Fermi level inside the forbidden gap. When the materials are
半导体一些术语的中英文对照
半导体一些术语的中英文对照离子注入机ion implanterLSS理论Lindhand Scharff and Schiott theory 又称“林汉德-斯卡夫-斯高特理论”。
沟道效应channeling effect射程分布range distribution深度分布depth distribution投影射程projected range阻止距离stopping distance阻止本领stopping power标准阻止截面standard stopping cross section 退火annealing激活能activation energy等温退火isothermal annealing激光退火laser annealing应力感生缺陷stress-induced defect择优取向preferred orientation制版工艺mask-making technology图形畸变pattern distortion初缩first minification精缩final minification母版master mask铬版chromium plate干版dry plate乳胶版emulsion plate透明版see-through plate高分辨率版high resolution plate, HRP超微粒干版plate for ultra-microminiaturization 掩模mask掩模对准mask alignment对准精度alignment precision光刻胶photoresist又称“光致抗蚀剂”。
负性光刻胶negative photoresist正性光刻胶positive photoresist无机光刻胶inorganic resist多层光刻胶multilevel resist电子束光刻胶electron beam resistX射线光刻胶X-ray resist刷洗scrubbing甩胶spinning涂胶photoresist coating后烘postbaking光刻photolithographyX射线光刻X-ray lithography电子束光刻electron beam lithography离子束光刻ion beam lithography深紫外光刻deep-UV lithography光刻机mask aligner投影光刻机projection mask aligner曝光exposure接触式曝光法contact exposure method接近式曝光法proximity exposure method光学投影曝光法optical projection exposure method 电子束曝光系统electron beam exposure system分步重复系统step-and-repeat system显影development线宽linewidth去胶stripping of photoresist氧化去胶removing of photoresist by oxidation等离子[体]去胶removing of photoresist by plasma 刻蚀etching干法刻蚀dry etching反应离子刻蚀reactive ion etching, RIE各向同性刻蚀isotropic etching各向异性刻蚀anisotropic etching反应溅射刻蚀reactive sputter etching离子铣ion beam milling又称“离子磨削”。
能带结构的第一性原理计算实验报告(硅、铜)
硅晶体能带结构的第一性原理计算班级:材料科学与工程3班学号:3015208064姓名:黄慧明一、实验目的通过实际操作初步的了解和掌握Materials Studio,基本掌握CASTEP 模块的操作步骤。
通过学习Materials Studio 软件,能够独立的进行简单的固体结构模型的构造和相关电子结构的计算和分析。
加深对课堂知识的直观认识,包括能带结构和相关的基本概念等。
二、实验原理第一性原理的理论计算的主要理论基础是量子力学的基本方程和相对论效应,在第一性原理发展过程中,相继提出变分原理、泡利不相容原理、密度泛函理论等。
其基本思路就是它的基本思想,是将多原子构成的实际体系理解为由电子和原子构成的多粒子系统,运用量子力学等基本物理原理最大限度的对问题进行“非经验”处理。
在第一性原理的计算过程中运用了三个近似:非相对论近似(忽略了电子运动的相对论效应);Born-Oppenheimer 近似,核固定近似;单电子近似。
密度泛函理论的主要目标就是用电子密度取代波函数做为研究的基本量。
用电子密度更方便处理。
在密度泛函理论(DFT)中,单电子运动的薛定谔方程按原子单位可表示为)()()()](2[22r k r r V mk k ψεψ=+∇-这里,电荷密度用单电子波函数表示∑=rk r r n 2)()(ψ单电子有效势为)][(′|′r -r |′)r ρ()(ρ93KS ][r V r d r v V xc ++=⎰三、实验内容运用Materials Studio 软件,采用其中的第一性原理计算软件(CASTEP),计算分析不同类型物质(石墨烯、Si、Cu、ZnO)的能带结构、电子态密度和电荷密度。
四、实验步骤1、模型构建建立一个新的project,并在其中建立一个3D工作区域,在菜单栏选择File |Import,显示出Import Document对话框,在对话框中选择Example|Documents |3D model|Si.xsd(硅晶胞模型)并打开,在3D窗口中右击鼠标,选择Display Style,在对话框中选择Ball and stick,并且调节球棍模型尺寸即可得到未修正的硅晶胞原始模型(图1)。
flow3d单词翻译
Vvect刚体初速度的矢量Delete source bitmap files--删除原位图文件Frame rate--帧速率A VI capture--动画捕捉A VI filename--动画文件名FLOW-3D (R) --FLOW-3D 简体中文版Interface version --接口版本Solver version--求解器版本Number of Processors--处理器数量Total Physical Memory (RAM) --物理内存总数(RAM) f3dtknux_license_file--授权许可文件Host Name--主机名F3D_VERSION --软件版本Operating System--操作系统Type--类型Porous--孔隙Porosity --孔隙率Lost foam--消失模Standard--标准Thermal conductivity--导热率Material name--材料名称Custom--自定义Surface area multiplier--面积倍增Unit system--系统单位Solid properties --固体属性Initial conditions--初始化条件Surface properties--表面属性Solids database--固体数据库Surface roughness--表面粗糙度Temperature--温度Temperature variables--温度变化Saturation temperature --饱和温度Units=CGS --单位=公制Solutal expansion coefficient --溶质膨胀系数Ratio of solute diffusion coefficient ---比溶质扩散系数Surface tension --表面张力Gas constant--气体常量Thermal conductivity --导热率Surface tension coeff--表面张力系数Critical solid fraction--关键凝固比率Solidus temperature--固相线温度Phase change--相变Material name --材料名称Thermal properties --热性质Custom --自定义Constant thinning rate--不断变薄率Units=SI -单位=国际单位制Partition coefficient--分隔系数Dielectric constant --介电常数Specific heat --比热Eutectic temperature --低共熔温度Coherent solid fraction --凝固Thermal expansion --热膨胀Unit System --系统单位Units=custom --单位=自定义Units=slugs --单位=斯勒格Reference temperature--起始温度Latent heat of vaporization--汽化潜热Reference solute concentration--参考溶质浓度Pure solvent melting temperature --熔点温度Liquidus temperature--液相温度Viscosity --黏度Solidification--凝固Vapor specific heat --蒸气比热Density--密度Temperature sensitivity--温度敏感性Saturation pressure --饱和压力Temperature shift --温度变化Compressibility --可压缩性Contact angle --接触角度Latent heat of fusion (fluid 1) --熔解潜热(流体1)New fluid database --新流体数据库Accommodation coefficient --调节系数Strain dependent thinning rate --应变黏度系数Constant thickening rate --不断增厚率added to materials database --添加到材料库cannot be added. --不能被添加Record already exists in materials database--在材料库已经存在该记录. New saved in materials database--新保存到材料库中.Could not find material DB--没有发现材料数据Add--添加Close--关闭Add Mesh Points --添加网点Direction --方向New Point --新的点Mesh Block --网格块2-D advanced options --2-D 高级选项Option--选项Add --添加Type--类型Component--组Cancel--取消Browse --浏览Source --来源File name--文件名Advanced --高级Numerics--数值运算Advanced options--高级选项sigma --表面张力系数Air entrainment --卷气Activate air entrainment model --激活卷气模型Surface tension coefficient --表面张力系数Dialog--对话框Remove mesh constrains--清除网格限制Size of all cells --全部单元尺寸Total Cells--单元总数Baffle options --隔板选项Baffle index --主隔板Baffle color--隔板颜色Hide selected baffles --隐藏选中的隔板Use contour color--使用轮廓颜色Selection method--择伐作业Boundary type --边界类型Specified pressure --规定压力Grid overlay --网格重叠Specified velocity --指定速度Electric potential--电位Stagnation pressure --滞止压力V olume flow rate --体积流量Z flow direction vector--Z 流向Y flow direction vector --Y 流向X flow direction vector--X 流向Electric charge--电荷Mesh Block--网格块Add to component --添加为元件Specific heat --比热Simulate--仿真Stop preprocessor--停止预处理Block distribution--块分配Porous--孔隙Component --组Scalars--标量Add to component --添加为元件Cell size --单元尺寸Render space dimensions --渲染面积Cell size is empty--单元尺寸为空Create mesh block (Cylindrical) --创建网格块(柱状) Total number of cells --单元数量Cylinder subcomponent --子气缸Add to component--添加为元件Radius --半径Setting the default workspace location is required. You can change the location at any time from the Preferences menu.--需要设置本地默认工作区位置.你可以随时通过菜单来改变位置。
DOS态密度
态密度(Density of States,简称DOS)在DOS结果图里可以查看是导体还是绝缘体还是半导体,请问怎么看。
理论是什么?或者哪位老师可以告诉我这方面的知识可以通过学习什么获得。
不胜感激。
查看是导体还是绝缘体还是半导体,最好还是用能带图DOS的话看费米能级两侧的能量差谢希德。
复旦版的《固体能带论》一书中有,请参阅!另外到网上或者学校的数据库找找“第一性原理”方面的论文,里面通常会有一些计算分析。
下面有一篇可以下载的:ZnO的第一性原理计算hoffman的《固体与表面》对态密度的理解还是很有好处的。
下面这个是在版里找的,多看看吧:如何分析第一原理的计算结果用第一原理计算软件开展的工作,分析结果主要是从以下三个方面进行定性/定量的讨论:1、电荷密度图(charge density);2、能带结构(Energy Band Structure);3、态密度(Density of States,简称DOS)。
电荷密度图是以图的形式出现在文章中,非常直观,因此对于一般的入门级研究人员来讲不会有任何的疑问。
唯一需要注意的就是这种分析的种种衍生形式,比如差分电荷密图(d ef-ormation charge density)和二次差分图(difference charge density)等等,加自旋极化的工作还可能有自旋极化电荷密度图(spin-polarized charge density)。
所谓“差分”是指原子组成体系(团簇)之后电荷的重新分布,“二次”是指同一个体系化学成分或者几何构型改变之后电荷的重新分布,因此通过这种差分图可以很直观地看出体系中个原子的成键情况。
通过电荷聚集(accumulation)/损失(depletion)的具体空间分布,看成键的极性强弱;通过某格点附近的电荷分布形状判断成键的轨道(这个主要是对d轨道的分析,对于s 或者p轨道的形状分析我还没有见过)。
分析总电荷密度图的方法类似,不过相对而言,这种图所携带的信息量较小。
d-band-center的计算方法
d-带中心(d-band center )通常用于描述金属表面的电子结构,特别是过渡金属表面的 d-轨道电子的位置。
计算 d-带中心的方法因具体情况而异,但以下是一种常见的计算方法:
假设我们有一个金属表面,其晶格上有一个过渡金属(比如铜、铁等)。
我们关注过渡金属表面的 d-轨道电子。
1. 计算 d-轨道的态密度(Density of States ,DOS ): 使用第一性原理方法,
如密度泛函理论(Density Functional Theory ,DFT ),计算过渡金属表面的电子结构,得到 d-轨道的态密度。
2. 计算 d-带中心: 通过积分 d-轨道的态密度来计算 d-带中心。
d-带中心的计
算公式为:
d -band center =∑E d -band ×DOS (E )∑DOS d -band (E )
其中,E 是能量,DOS (E ) 是 d-轨道在能量 E 处的态密度。
这个积分通常是对 d-带上的所有能量点进行的。
通过计算每个能量点的能
量和对应的 d-轨道态密度,然后将它们相乘并累加,最后除以总的 d-态密度,得到 d-带中心。
计算 d-带中心的目的是描述 d-轨道电子在金属表面的位置,通常用电子在 d-轨道上的平均能量来表示。
请注意,具体计算方法可能因研究的体系和使用的计算工具而异。
在实际研究中,可以参考相关文献或使用专业的计算工具来进行 d-带中心的计算。
dos态密度
dos态密度DOS态密度(DensityofStates)是物理和材料科学的重要概念,它指的是每一个能量状态的可能性,即被称为“Density of States”的概念。
总的来说,DOS态密度是材料和物质的重要特性之一,它可以帮助我们了解和计算材料的电子特性,如电子密度、电子态密度、气体密度、电子结构、物质的理想性等等。
DOS态密度可以以能量作为自变量,并以多段可调锥或多段平板调制函数形式显式地表示,以表示能量态状态的概率密度变化趋势。
假设我们有一种物质,它的多个电子态的能量分布为E1、E2等,那么可以把这些能量看作函数f(E)的一个横轴,把不同电子态的可能性看作纵轴,通过形成所谓的DOS态密度图,可以直观地分析物质的电子态状态密度分布情况。
DOS态密度有着广泛的应用,它可以用于帮助我们了解材料的电子态状态分布,从而帮助我们分析材料的特性,提高材料的性能。
例如,它可以用于分析金属和半导体的电子态密度,比如金属的电子态密度曲线可以帮助我们了解金属的导电性能,而半导体的电子态密度曲线可以帮助我们了解半导体的介电性能。
此外,DOS态密度还可以用于研究绝热流动体和非绝热流体的热力学行为,用于研究各种材料的复杂光学性能,用于研究各种物质的拉曼谱等。
因此,DOS态密度是一个重要的概念,它可以帮助我们了解材料的电子态密度分布情况,从而提高材料的性能。
然而,DOS态密度的估算也是一个挑战,因为它是一个非线性的函数,而且因为它的复杂性,很难解决。
但是,有了计算机的帮助,我们可以更加准确和容易地计算DOS态密度,从而分析材料的性能。
总之,DOS态密度及其相关概念是物理和材料科学研究中一个重要的概念,它可以帮助我们了解材料的特性和性能,从而提高材料的性能。
态密度计算
态密度计算态密度(Density of States,DOS)是材料科学中常用的一个概念,用来描述材料中不同能级上的电子数目。
它是研究材料电子结构和相关物理性质的重要参数。
在固体材料中,电子的能级是连续的,而不是离散的。
态密度可以用来描述在给定能量范围内的电子能级的分布情况。
简单来说,态密度表示的是单位能量范围内存在的电子能级的数量。
态密度可以分为两类:自由电子态密度和带态密度。
自由电子态密度是指在不考虑晶格影响的情况下,单个电子在能量空间内的分布情况。
带态密度则是考虑了晶格效应,描述的是固体材料中电子能级的分布情况。
对于自由电子态密度,可以通过简单的数学推导得到。
在三维情况下,自由电子的态密度可以表示为:D(E) = V/(2π²)(2m/ħ²)^(3/2)√(E)其中,D(E)表示态密度,V表示体积,m表示电子质量,ħ表示约化普朗克常数,E表示能量。
在带态密度中,由于晶格的影响,电子的能级会发生分裂,形成能带结构。
带态密度的计算则需要考虑晶格的周期性。
对于简单的晶体,可以通过布里渊区的积分来计算带态密度。
带态密度的计算可以使用第一性原理方法,如密度泛函理论(DFT)等。
在DFT中,通过求解电子的薛定谔方程,可以得到材料的能带结构和带态密度。
态密度的计算在材料科学中有着广泛的应用。
例如,在设计新型材料时,通过计算不同能级上的态密度,可以预测材料的电子行为和物理性质。
在能源领域,态密度的计算可以帮助我们了解材料的导电性、光学性质等,从而指导材料的设计和优化。
总结起来,态密度是描述材料中电子能级分布情况的重要参数。
通过计算态密度,可以帮助我们了解材料的电子行为和物理性质,对材料的设计和优化具有重要意义。
无论是自由电子态密度还是带态密度,计算方法都有其特定的推导和应用。
态密度的研究将在材料科学领域中持续发展,为我们提供更多的理论基础和实验指导。
dft态密度计算
dft态密度计算
DFT(密度泛函理论)是一种用于计算材料电子结构和能带结构的理论方法。
在DFT计算中,态密度(Density of States,DOS)是一个重要的输出结果,它描述了材料中各个能级上的电子态密度分布情况。
DFT态密度的计算通常包括以下步骤:
1. 构建系统的电子密度分布:在DFT计算中,首先需要构建一个系统的电子密度分布,这可以通过迭代计算得到系统的总能量和电子波函数来实现。
2. 计算态密度:态密度可以通过波函数的能量和电子密度分布来计算。
具体来说,态密度可以表示为每个能级上的电子态数与该能级宽度之比。
3. 计算态密度与能量之间的关系:通过将态密度与能量之间的关系进行积分,可以得到整个能量范围内的态密度分布情况。
需要注意的是,DFT计算需要使用合适的交换-相关泛函(XC functional)来描述电子之间的相互作用。
不同的XC functional会导致不同的态密度结果,因此需要根据具体材料和问题选择合适的XC functional。
此外,DFT计算通常需要使用大规模的计算机资源,因此在实际应用中需要考虑计算效率和精度之间的平衡。
态密度与光生载流子
态密度与光生载流子
态密度(Density of states)和光生载流子(Photogenerated carriers)是半导体物理和光电器件研究中的重要概念。
1、态密度:
态密度是指给定能量的量子态在单位体积和单位能量间隔内的数量。
在半导体中,由于存在能带结构,态密度通常与能量的关系非常复杂。
一般来说,态密度越高,表示在给定能量范围内可以存在的量子态就越多。
2、光生载流子:
光生载流子是指半导体在吸收光子后,由价带电子跃迁到导带而产生的电子和空穴。
这些载流子可以在半导体内部自由移动,从而产生电流。
光生载流子的数量和性质对光电器件的性能有着重要影响。
在光电器件中,如太阳能电池、LED等,光生载流子的产生、分离和输运对器件的效率和工作机制有着决定性的影响。
同时,态密度也在很大程度上影响了这些器件的能带结构和光电性能。
因此,对态密度和光生载流子的理解和控制对于优化光电器件的性能至关重要。
DOS态密度
态密度(Density of States,简称DOS)之五兆芳芳创作在DOS结果图里可以查抄是导体仍是绝缘体仍是半导体,请问怎么看.理论是什么?或哪位老师可以告知我这方面的知识可以通过学习什么取得.不堪感谢.查抄是导体仍是绝缘体仍是半导体,最好仍是用能带图DOS的话看费米能级两侧的能量差谢希德.复旦版的《固体能带论》一书中有,请参阅!另外到网上或学校的数据库找找“第一性原理”方面的论文,里面通常会有一些计较阐发.下面有一篇可以下载的:ZnO的第一性原理计较hoffman的《固体与概略》对态密度的理解仍是很有利益的.下面这个是在版里找的,多看看吧:如何阐发第一原理的计较结果用第一原理计较软件开展的任务,阐发结果主要是从以下三个方面进行定性/定量的讨论: 1、电荷密度图(charge density); 2、能带结构(Energy Band Structure); 3、态密度(Density of States,简称DOS). 电荷密度图是以图的形式出现在文章中,很是直不雅,因此对于一般的入门级研究人员来讲不会有任何的疑问.唯一需要注意的就是这种阐发的种种衍生形式,比方差分电荷密图(def-ormation charge density)和二次差分图(difference charge density)等等,加自旋极化的任务还可能有自旋极化电荷密度图(spin-polarized charge density).所谓“差分”是指原子组成体系(团簇)之后电荷的重新散布,“二次”是指同一个别系化学成分或几何构型改动之后电荷的重新散布,因此通过这种差分图可以很直不雅地看出体系中个原子的成键情况.通过电荷聚集(accumulation)/损失(depletion)的具体空间散布,看成键的极性强弱;通过某格点邻近的电荷散布形状判断成键的轨道(这个主要是对d轨道的阐发,对于s或p轨道的形状阐发我还没有见过).阐发总电荷密度图的办法类似,不过相对而言,这种图所携带的信息量较小. 能带结构阐发现在在各个领域的第一原理计较任务中用得很是普遍了.但是因为能带这个概念自己的抽象性,对于能带的阐发是让初学者最感头痛的地方.关于能带理论自己,我在这篇文章中不想涉及,这里只考虑已得到的能带,如何能从里面看出有用的信息.首先当然可以看出这个别系是金属、半导体仍是绝缘体.判断的尺度是看费米能级和导带(也即在高对称点邻近近似成开口向上的抛物线形状的能带)是否相交,若相交,则为金属,不然为半导体或绝缘体.对于本征半导体,还可以看出是直接能隙仍是直接能隙:如果导带的最低点和价带的最高点在同一个k点处,则为直接能隙,不然为直接能隙.在具体任务中,情况要庞杂得多,并且各类领域中感兴趣的方面彼此相差很大,阐发不成能像上述阐发一样直不雅和普适.不过仍然可以总结出一些经验性的纪律来.主要有以下几点: 1)因为目前的计较大多采取超单胞(supercell)的形式,在一个单胞里有几十个原子以及上百个电子,所以得到的能带图往往在远低于费米能级处很是平坦,也很是密集.原则上讲,这个区域的能带其实不具备多大的解说/阅读价值.因此,不要被这种现象吓住,一般的任务中,我们主要关怀的仍是费米能级邻近的能带形状. 2)能带的宽窄在能带的阐发中占据很重要的位置.能带越宽,也即在能带图中的起伏越大,说明处于这个带中的电子有效质量越小、非局域(non-local)的程度越大、组成这条能带的原子轨道扩展性越强.如果形状近似于抛物线形状,一般而言会被冠以类sp带(sp-like band)之名.反之,一条比较窄的能带标明对应于这条能带的本征态主要是由局域于某个格点的原子轨道组成,这条带上的电子局域性很是强,有效质量相对较大. 3)如果体系为掺杂的非本征半导体,注意与本征半导体的能带结构图进行对比,一般而言在能隙处会出现一条新的、比较窄的能带.这就是通常所谓的杂质态(doping state),或依照掺杂半导体的类型称为受主态或施主态. 4)关于自旋极化的能带,一般是画出两幅图:majority spin和minority spin.经典的说,辨别代表自旋向上和自旋向下的轨道所组成的能带结构.注意它们在费米能级处的差别.如果费米能级与majority spin的能带图相交而处于minority spin的能隙中,则此体系具有明显的自旋极化现象,而该体系也可称之为半金属(half metal).因为majority spin与费米能级相交的能带主要由杂质原子轨道组成,所以也可以此为出发点讨论杂质的磁性特征. 5)做界面问题时,衬底资料的能带图显得很是重要,各高对称点之间有可能出现不合的情况.具体地说,在某两点之间,费米能级与能带相交;而在另外的k的区间上,费米能级正利益在导带和价带之间.这样,衬底资料就呈现出各项异性:对于前者,呈现金属性,而对于后者,呈现绝缘性.因此,有的任务是通过某种资料的能带图而选择不合的面作为生长面.具体的阐发应该结合试验结果给出.(如果我没记错的话,物理所薛其坤研究员曾阐发过$\beta$-Fe的(100)和(111)面对应的能带.有兴趣的读者可进一步查阅资料.)原则上讲,态密度可以作为能带结构的一个可视化结果.良多阐发和能带的阐发结果可以一一对应,良多术语也和能带阐发相通.但是因为它更直不雅,因此在结果讨论中用得比能带阐发更普遍一些.扼要总结阐发要点如下: 1)在整个能量区间之内散布较为平均、没有局域尖峰的DOS,对应的是类sp带,标明电子的非局域化性质很强.相反,对于一般的过渡金属而言,d轨道的DOS 一般是一个很大的尖峰,说明d电子相对比较局域,相应的能带也比较窄. 2)从DOS图也可阐发能隙特性:若费米能级处于DOS值为零的区间中,说明该体系是半导体或绝缘体;若有分波DOS跨过费米能级,则该体系是金属.此外,可以画出分波(PDOS)和局域(LDOS)两种态密度,加倍细致的研究在各点处的分波成键情况. 3)从DOS图中还可引入“赝能隙”(pseudogap)的概念.也即在费米能级两侧辨别有两个尖峰.而两个尖峰之间的DOS其实不为零.赝能隙直接反应了该体系成键的共价性的强弱:越宽,说明共价性越强.如果阐发的是局域态密度(LDOS),那么赝能隙反应的则是相邻两个原子成键的强弱:赝能隙越宽,说明两个原子成键越强.上述阐发的理论根本可从紧束缚理论出发得到解释:实际上,可以认为赝能隙的宽度直接和Hamiltonian矩阵的非对角元相关,彼此间成单调递增的函数关系. 4)对于自旋极化的体系,与能带阐发类似,也应该将majority spin和minority spin辨别画出,若费米能级与majority的DOS相交而处于minority的DOS的能隙之中,可以说明该体系的自旋极化. 5)考虑LDOS,如果相邻原子的LDOS在同一个能量上同时出现了尖峰,则我们将其称之为杂化峰(hybridized peak),这个概念直不雅地向我们展示了相邻原子之间的作用强弱.请教楼主:1、我一直不明白DOS图中的非键的概念.这里的非键,到底是什么意思?DOS图中能不克不及看出来?如何看?2、金属中除了金属键,电子都是以什么状态存在的?是非键吗?所谓的非键是不是就是我们过来所说的自由电子?仍是说,金属键的电子就是自由电子呢?金属中金属键占大部分啊,仍是说非键占大部分?3、离子键在DOS中能不克不及看出来?如何看?4、我曾看到文献上说,费米能邻近的非键是金属性的标记.这句话如何理解?1.其实DOS是固体物理的概念,而非键(以及成键和反键等)是结构化学的概念,但是现在经经常使用在同一个别系说明不合的问题.先说一下非键,然后在把它跟BAND和DOS结合起来.从结构化学的角度来说,份子轨道是由原子轨道线性组合而成.如果体系有n个原子轨道进行组合,就会产生n个份子轨道(即轨道数目守恒,其实从量子力学的角度,就是正交变换不会改动希尔伯特空间的维数).这些份子轨道的能量,可以高于,近似等于,或是低于原子轨道的能量,它们辨别对应于成键,非键,或是反键态.复杂的说,非键轨道跟组成它的原子轨道能量差未几,如果有电子排在该轨道上,则对体系成键能量上没有太大帮忙.由于固体中的每个能带都是有许多原子轨道组合而成,复杂的说,对于某一只能带,它的上半部对应化学上所谓的反键态,下半部分对应于成键态,而中部的区域对应于非键态.但是,由于能带是很是密集的,从而是连续(准连续的),对于某个具体的能级,往往很难说出具体是什么键态,如果这个能级不是对应于能带低,或是能带顶的话.并且,一般费米面邻近的能带不但仅由一种原子轨道扩展而成,而是不合种轨道杂化而成,要定量说明是比较难的.2.关于金属,粗糙的说,金属中的电子是以电子气的情况出现,散布于整个金属所在的空间.正价离子实通过对“公共”电子气的吸引而聚集在一起.从化学上讲,金属键可以看做是一种共价键,只是没有饱和性和标的目的性.但是这种理解太粗糙.从固体物理的角度,金属中电子散布跟半导体,绝缘体(也就是电介质)类似,对基态都是依照能量最低排在能带上.只不过,金属的费米能级穿过电子所在的能带(也就是电子没有占满该能带),从而使得费米面邻近的电子介入导电.所以,非键其实不是我们说的自由电子,两者没有必定的联系.金属中的电子也不是完全的自由电子,其波函数仍是受离子周期调制的布洛赫波,而非平面波.3.离子键等不克不及在DOS中看,我发过专门的帖子.单纯的从DOS最多可以定性的看出杂化,但是不克不及看出杂化轨道中的电子究竟偏向哪个原子,因此不克不及看出离子键或是共价键的情况.最近我师弟问我一个很垃圾杂志上用DOS阐发离子键或是共价键的文章,这个文章我看了一下,它的阐发是错的.4.按照我上面的说法,由于固体的“非键态”在DOS或是BAND的中部,当费米能级邻近是非键态时,换句话说,就是标明费米能级穿越了能带的中部,说明电子没有占满,因此是金属晶体,是金属性的标识.这么理解有道理.第一原理计较结果讨论(系列二)讨论一:电荷密度图(charge density),变型电荷密图(def-ormation charge density)和差分电荷密度图(difference charge density)等等,加自旋极化的任务还可能有自旋极化电荷密度图(spin-polarized charge density).所谓“变型”是指原子组成体系(团簇)之后电荷的重新散布,“差分”是指同一个别系化学成分或几何构型改动之后电荷的重新散布,因此通过这种差分图可以很直不雅地看出体系中个原子的成键情况.通过电荷聚集(accumulation)/损失(depletion)的具体空间散布,看成键的极性强弱;通过某格点邻近的电荷散布形状判断成键的轨道(这个主要是对d 轨道的阐发,对于s或p轨道的形状阐发我还没有见过).问题:我对这三种电荷图理解的不透彻,通过这三种电荷密度图能判断出是共价键和离子键吗?如果能,怎样判断出来?最好能给出三种电荷图加以说明.讨论二:对于成键阐发用的晶体轨道重叠计划图,如何阐发?谁会MULLIKEN电荷计划图,请列位虫友辅佐讨论这个问题,最好给个图,阐发一下.多谢.讨论三:TDOS,SDOS,SPDOS,LDOS,PDOS是从不合的正面去描写体系的电子结构,反响的意义也不合,大家谁知道TDOS,SDOS,SPDOS,LDOS,PDOS的区别?最好贴个图,一起阐发一下,配合学习.希望得到版主和列位虫友的支持,会的在温习一下,不会的就当学点新知识,大家配合学习.【讨论】关于用态密度来看体系成键的性质这里,先摆出我的不雅点,就是态密度跟体系成键性质(也就是局域键是共价,离子,金属,或是混杂键等等之类),并没有直接关系,但它可以从整体上(而非局域键)判断体系是金属,半导体或是绝缘体(这是能带论的底子不雅点之一).态密度的定义为(单位体积)单位能量上的状态数.复杂的说,就是在某个能量邻近,体系状态的散布的稠密程度.举个复杂的例子,比方罕有的氢份子和CO2份子的态密度,很显然,在不考虑能级各类展宽因素影响的情况下,都是DELTA函数.那么,从这一堆DELTA 函数,能看出来前者是共价键尔后者是共价键和离子键的一种混杂键吗?显然不克不及.(要注意的是,CO2中C-O 有离子成分的,不是纯的共价键,虽然离子成分较少.)对于固体,我们能从TDOS的带隙来判断体系的整体性质,比方是金属仍是半导体等等.但是无法给出更细的信息.那么LDOS和PDOS呢?它们能给出什么信息.以LDOS为例,我们可以给出体系某个原子的LDOS.我认为LDOS只能给出该原子原子轨道成键后的杂化情况,但是说明不了与周围原子的价键性质,也就是不克不及直接说明是离子键仍是共价键.其实这点很容易理解,比方,考虑A原子的S轨道和B原子的S轨道有杂化(为复杂起见不考虑其他轨道的杂化情况),这说明S和S电子之间有转移,但是是由A的S到B的S,仍是反过来?就不克不及仅仅的用LDOS 来判断了,必须考虑A和B具体的电负性.不过,由LDOS 的宽窄情况,可以看出来轨道的杂化程度,如果比较窄,则杂化不强烈(仍保持原子轨道DELTA函数的形式),如果比较宽,证明杂化比较强.其实能带中由孤立原子能级变成能带,就是杂化的进程,内层电子杂化不明显,能带较窄,而外层电子则相反.好了,对于这个小问题,就说这么多.说的也不一定对,如果哪个虫友有其他看法,跟我的不一样,还请赐教.根本上同意你的不雅点,只是一些概念上需要澄清:1.DOS 的概念是相对于固体而言的,小份子没有DOS这个概念,只有energy level. DOS用于说明轨道相互作用情况比较适合.2.关于“杂化”,化学上有特殊寄义,是同一原子的不合轨道再组合,老外叫它hybridization,而你这里实际应该是指mixing3.要甄别是“共价键”仍是“离子键”,用电荷密度来讲,可能更好一些.4.化学键没有绝对的共价与离子,只是看哪种作用力更明显一些罢了.呵呵,对不起,你有的不雅点我不是很同意,复杂说说我的看法,可以讨论一下. 1.份子也是有DOS的,不过DOS 是一些DELTA函数罢了.LS是做量子化学的,不知道用过没有用过(或听没听说过)ADF软件,计较后可以直接把DOS画出来.2.当然,原子轨道杂化(比方C的SP3杂化),是一个根本的定义.不过,现在就我看到的文献,对于不合原子的轨道叠加,也可以说是“杂化”(说成你所谓的mixing也行).例如,有文献经常说某个份子的HOMO是由A原子的3P轨道和B原子的2S轨道杂化而成,等等.这么说,似乎也不会引起误解.3.要鉴辨别子键跟共价键用电荷密度来说是更适合,这点我同意,这也是众所周知的事实.不过,我这里只是说明单纯从DOS上看不出来共价键或是离子的性质.并没有讨论用什么物理量(比方电荷差分密度,电荷变形密度等等)来看键的性质更适合.4.化学键没有绝对的共价键或是离子键,我同意.其实,不但仅是份子,固体中的情况也一样.比方罕有的离子晶体NACL,离子成分也不是100%的.你说的的第三、第四个问题我并没有在这个帖子中展宽讨论.可能让你有所误解,对不起.我觉的LS似乎微不雅图像不清楚.当然,这只是我的感到,未必准确.也可能是我的微不雅图像不清楚.“但是结合你的初始结构来看,ldos根本上仍是比较准确的”,这句话有问题,首先,即便是结合LDOS看结果,也应该是结合优化过的构型,而不克不及是初始构型.更重要的是,即便疏忽这个小的错误,这句话也不合错误.以最复杂的氢份子为例,其成键和反键态波函数只是差一个正负号,都是两个原子的S轨道杂化而成.因此从LDOS上,两个态所显示的都是两个S的重叠,无非是所处能量位置不合.因此,我即便把氢份子的结构(其实就是两点一线段)和LDOS给你,仅从这两个信息你也不克不及通过LDOS给出成键态和反键态的形状,自然不克不及给出键的标的目的.更不必说是多原子份子了.另外,“临近原子的相对位置,是可以看出是什么样的结合键的”,我不知道你这里的键是从份子轨道理论过来的,仍是从价键(VB)理论过来的.现在一般都是阐发份子轨道理论的结果.对于份子轨道,例如HOMO,HOMO-1,HOMO-2,......都是一个整体,每个轨道都代表一个电子在是空间的散布,单独看他们在某几个原子间的散布,并判断是什么键,似乎不合错误共价键在两个元素之间的局部区域电子密度会有较强的散布",这句话定性上是对的,如果差分电荷密度或是变形电荷密度集中在某两个原子间,证明共价键比较强.然而“你可以看到Ti和C之间区域电子密度较小,这是离子键的明显特征”,这句话的意思容易明白,但是,我觉得直接用原子的MULLIKEN电荷大小来说明离子性的强弱,可能更好一点.也许会有人说,MULLILKEN电荷其实禁绝确.的确,从其自己的算法来看,是这样.但是,用来定性阐发仍是有意义的.特别是相对大小的比较.因为“Ti和C之间区域电子密度较小”,怎么个小法才干体现离子键?这很不容易说清楚.最后,至于LDOS和PDOS的区别,前者偏重于具体原子的态密度,后者是整个晶胞的S,P,D的电子散布.其实,往往在使用LDOS中,我们也是把该原子的S,P,D电子分隔的,确切的说是LPDOS更适合,呵呵.当然,DOS的种类有良多中,比方TDOS,SDOS,SPDOS,LDOS,PDOS等等,这些DOS反应了体系电子结构的不合正面,表示了不完全相同的意义.以上都是最根本的概念,只要你熟悉固体物理,我想理解起来应该没有什么问题.量化从量子力学而来,从处理小份子开始,成立了一些描述问题的概念,电子散布的描述,引入能级概念,这时的能量散布只是一个个孤立的点;到了扩展体系或较大的体系时,一些能级距离逐突变得很小,以至难以区分,出了个“带”,此时才有了DOS.虽然两者类似,但仍是略有区此外,至于什么时候叫energy level,什么时候称DOS,见仁见智吧,团体认为仍是有区此外.至于“杂化”这个概念,无论从物理仍是化学的角度讲,原谅我不克不及接受你的不雅点,杂化(hybridization)不但仅是Pauling天才地玩出来的概念,它是有深刻的物理布景的,实际上它是量子力学一个假定(态叠加原理)的具体应用,这和“mixing"是两码事儿;其次,化学上是严格区别”杂化“与”mixing"(混杂,化学上多称为轨道组合),“杂化”在化学上是指同一原子不合轨道间的重新组合,而不克不及用来描述不合原子间轨道的混杂(mixing),玩物理的老外们多用state mixing,进口之后,时而才被被玩儿成“杂化”.很忸捏,我不必阿姆斯特丹的东西,虽然知道它不算太贵,但年计费很烦,虽然知道它的band可以计较固体,但仍是喜欢维也纳,萝卜青菜,各有所爱吧.谢谢你的讨论!呵呵,原子轨道原本多用STO暗示,只是便利于积分,于是采取GTO展开,形成各色基组,所以,你这里的说法又存在问题了采取基组原本就是拟合STO或原子轨道的,然后在此根本上,机关份子轨道,其实不是直接拿GTO线性组分解MO.我想再次强调的是,LCAO只是机关份子轨道的办法之一,没有人也没有那个原理限制你必须用LCAO,只是它用起来比较简练、便利、通用且易于程序化罢了,如果愿意,大可以采取其它机关方法.恕我直言和保持.1.DOS跟能级是有区此外.无论对于份子或是固体,都有能级(只不过固体的能级比较密集,是连续或是准连续,因而称之为带,或说,能带是固体能级的一个形象的说法),其实两者都是体系(份子或是固体)薛定谔方程的解,从量子力学的角度来看,并没有区别.我们其实也可以把固体中的某个能量叫做能级,比方罕有的费米能级,并没有任何妨害.而态密度是单位能量(单位体积)状态数的散布,对于体系的几何维度或是对称性等没有要求.在周期晶体,非周期固体,份子等等都可以应用.2.的确,现在hybridization或是mixing用的的确比较烂.不过,量子力学的态叠加原理不但仅可以用在同一个原子不合轨道的叠加.态叠加原理有着很是深刻的物理寄义.换句话说,LS所谓的mixing也是态叠加原理的体现.不过,我觉得你的看法仿佛不是这样的.3.我说的ADF指的就是ADF,而不是其中的BAND.ADF可以给出份子的DOS,而BAND可以给出固体的DOS.当然,BAND一般没人用.的确,我不认为辨别属于两个原子的轨道间的mixing是态叠加原理!说得口语些,描述体系的不合状态波函数间的组合仍是描述体系状态的波函数,这是态叠加原理.描述不合原子的轨道之间的混杂即成键是针对不合原子而言,从原子轨道这个条理讲,两个原子是不合体系,因此其轨道的mixing就不克不及算作态叠加原理,就像你不克不及把杂化轨道称为份子轨道一样,杂化轨道仍然是原子轨道,而此时的份子轨道不再是原子轨道.嘻嘻,有点牛角尖了,见谅!既然LS否定不合原子的轨道mixing是态叠加原理的体现.我想请教一下LS,那么在份子轨道理论中,是哪个量子力学基来源根底理(或其推论)可以包管体系的份子轨道可以展开为不合原子轨道的线性组合?我不是弄化学的,对结构化学或是量子化学不熟,不过一些问题可以讨论一下.将原子轨道线性组合为份子轨道只是机关份子轨道的一种手段,不需要什么量子力学原理来支撑,完全可以不必原子轨道来机关份子轨道,只要你能造出一个描述份子体系电子运动状态的函数就可以,不是有些软件还采取数值轨道的吗.....用GTO或是STO只是技巧上的问题,从物理上说,STO适合实际的电子波函数,但是从计较上说,GTO更便利数值运算.其实我上面说的很清楚了“份子轨道的展开不一定用原子轨道”,所以并没有一定用LCAO办法,希望你不要误解,呵呵.把份子轨道展开,究竟用什么基组,看计较的便利而定.可以是有一定物理意义的原子轨道,当然,也可以是没有直接物理意义的GTO基组(通过拟合STO等等手段来暗示)等等.甚至,只要你愿意,用平面波形式的基组也可以计较(比方用VASP计较超胞中的孤立份子).但是无论你用什么基组通过线性叠加来暗示份子轨道,其实面前都体现了态叠加原理.不然,从量子力学的角度,没有任何依据可以让你把份子轨道展开成某些函数的叠加形式.换句话说,只要你把波函数展开,无论用什么基组,都是态叠加原理的体相,选取基组中的有限项来实际的拟合份子轨道.呵呵.。
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3D Density of StatesThe density of states refers to the number of quantum states per unit energy. In other words, the density of states, denoted by ()g E , indicates how densely packed quantum states in a particular system. So, what is the importance of the density of states? Consider the expression ()g E dE . Integrating the density of the quantum states over a range of energy will produce a number of states.()()EEN E g E dE ∆=⎰Thus ()g E dE represents the number of states between and E dE . The number ofquantum states is important in the determination of optical properties of a material such as a semiconductor (i.e. carbon nanotubes as well as quantum dots).From the Schrodinger equation, we know that the energy of a particle is quantized and is given by222k E m =The variable k is related to the physical quantity of momentum. A particle’s energy is22221222m v p E mv m m===Relating the previous two equations yields22222k p p E k m m ==→=The momentum is a vector which has components in the x, y, and z directions. Therefore, k must also have direction components , , and x y z k k k . Since energy is not a vector, the more accurate expression for energy is222k E m =In a 3D system, then, the total energy is given by()22222x y z E k k k m=++Recall the result of the analysis of the 1D potential well. An electron can only exist in the well, and the wave function is given by()cos()sin()where:x A kx B kx n k aψπ=+=where a is the width of the barrier. For the cosine term in the wave function, n must be an odd integer, and for the sine term, n must be an even integer. Therefore, the wave function is only valid for all integers greater than zero. In three dimensions, each directional component of k would be, , and y x z x y z n n n k k k a a aπππ===The wave function is valid at regular intervals of aπ. Below is a plot of the validsolutions in k-space.Each green or red dot represents two quantum states (one for an electron with “spin up” and one for “spin down”). The red dots are those between the intervals of and +k k dk . To find the density of these states, we need to first examine the unit cell and its volume. The unit cell is the smallest shape which can be repeatedly be used to construct a lattice as in a diamond crystal, for example. Examining the image above and knowing that thequantum states are separated by an interval of aπ, the unit cell would be a cube of whichthe volume would be30V a π⎛⎫= ⎪⎝⎭The density of states problem is a problem of finding the number of states in the interval of and +E E dE . In k-space, the interval is simply and +k k dk . Because we areoperating three dimensions, k represents the radius of a sphere in k-space and dk is the thickness of the sphere. Hence, a shell is created which encloses a certain amount of quantum states in an infinitesimal interval. We use the sphere because k is directly related to E , and all the points on the sphere have equal energy. Since we want to find the density of states in an infinitesimal interval of energy, the shape used for the boundaries of the interval must represent equal energies. The volume of the shell isessentially the surface area of the inner sphere (of radius k ) multiplied by the thickness of the sphere (dk ).24V k dk π=Alternatively, we can find the volume by subtracting the volume inside the inner sphere from the volume inside the outer sphere (of radius k dk +). The result leaves a shell.()()()()3332233232244334333because and have no real meaning,the volume is 4343V k dk k k k dk kdk dk k dk dk k dk k dk πππππ=+-=+++-==Even though the k-space image displays valid wave equations solutions for both positive and negative integers of , , and x y z k k k , the wave function should only be valid for all positive values of , , and x y z k k k . Therefore, the entire shell’s volume is unnecessary; we really only need to consider an eighth of the entire volume (i.e. the upper right handquadrant which satisfies x > 0, y > 0, and z > 0 collectively). The revised shell volume is then,2214812V k dkk dk ππ⎛⎫= ⎪⎝⎭=The number of quantum states in an interval of dk is found by dividing the volume of the shell by the volume of a single state (i.e. the volume of the unit cell).()2332212()2k dk g k dk a a k dk πππ=⎛⎫ ⎪⎝⎭=Since we know the relation between k and E , we can find what we are searching: the density of states as a function of energy. Once again, the relationship between k and E is1222122212222k m mE dk dE m mE dE --=⎛⎫⎛⎫= ⎪⎪⎝⎭⎝⎭⎛⎫= ⎪⎝⎭Substituting the results into the density of states equation will give the density of states in terms of energy.3221322222()22()a g k dk k dka m mE mE g E dE dE ππ-=⎛⎫⎛⎫=⎪⎪⎝⎭⎝⎭Multiplying out the expression gives the final result for the density of states.()()()()12121122232413223233322222()2222()2m mE Eag E dE dEma EdE a m g E dE EdEπππ-----==⎛⎫= ⎪⎝⎭2D Density of StatesIn 2D, an electron is confined along one dimension but able to travel freely in the other two directions. In the image below, an electron would be confined in the z-direction but would travel freely in the XY plane.In the 3D density of states analysis, a spherical volume of width dk had to be used. However, in 2D, the problem of calculating becomes easier because we only need to operate in two dimensions. Instead of using the volume of a shell, the area of a ring with width of dk is used. Analogous to the sphere in three dimensions, the circle is used because all points on the circle are an equal distance from the origin; therefore, the circleindicates equal values of energy. The radius becomes 22x y k k k =+. In the 2D case, theunit cell is simply a square with side length ofaπ.Following the same procedure which was used in the 3D situation, the area of the unit cell is20A a π⎛⎫= ⎪⎝⎭Next, we need to find the area of the ring and then divide by the area of the unit cell. The area of a circle is 2r πwhere r is the radius. The area of the ring, then, is22()2A k dk k kdkπππ=+-=However, only positive values of k should be considered. The figure below highlights the correct area.Therefore, we only need one-fourth of the entire area of the ring. The revised area is1242A kdkkdk ππ==Dividing the ring area by the unit cell area, the density of states can be found.022()22(2)A g k dk A kdka a kdkπππ⎛⎫= ⎪⎝⎭=⎛⎫ ⎪⎝⎭=Since we know the relation between k and E , we can find what we are searching: the density of states as a function of energy. Once again, the relationship between k and E is1222122212222k m mE dk dE m mE dE --=⎛⎫⎛⎫= ⎪⎪⎝⎭⎝⎭⎛⎫= ⎪⎝⎭Substituting the results into the density of states equation will give the density of states in terms of energy.112222222222()()a mE m mE g E dE dEa m g E dE dEππ-⎡⎤⎛⎫⎛⎫⎢⎥= ⎪ ⎪⎝⎭⎝⎭⎢⎥⎣⎦=Notice that the 2D density of states, interestingly, does not depend on energy.1D Density of StatesThe density of states for a 1D quantum mechanical system exhibits a unique solution which has application in things such as nanowires and carbon nanotubes. In both the x and y directions, the electron is confined, but it moves freely in the z direction.Going from the 2D case to the 1D case, the unit cell becomes a line segment of lengthaπ. Just as a ring was used in the 2D system to find the number of quantum states in a differential interval of energy, so a line segment is used in the 1D situation. The line segment stretches from to k k dk + or to E E dE +, and the length of the line issimply dE . Dividing the infinitesimal line length (dE ) by the line length of the unit cell and then multiplying by two (i.e. accounting for the two electron spins), the density of states equation is0()2(2)2L g k dk L dk a adkππ⎛⎫= ⎪⎝⎭=⎛⎫ ⎪⎝⎭=Substituting dE for dk using the well defined relation yields1222112212212()22()(2)2()ag k dk dka m mE g E dE dE a m mEdE πππ-----=⎡⎤⎛⎫⎢⎥= ⎪⎝⎭⎢⎥⎣⎦⎛⎫= ⎪⎝⎭。