Semiclassical Dynamics of Electrons in Magnetic Bloch Bands a Hamiltonian Approach

The Behavior of Electrons in aSemiconductoris a fascinating topic that has been the subject of much research over the years. Semiconductors are materials that can partially conduct electricity, and their behavior is dependent on the presence or absence of electrons in their crystal structure.One of the key features of a semiconductor is its energy band structure. In a semiconductor, there are two types of energy bands - the valence band and the conduction band. The valence band is where the electrons that are bound to the atoms in the crystal structure exist, while the conduction band is where the electrons that can move freely exist.is affected by a number of different factors. One important factor is the concentration of impurities in the material. Impurities are atoms that have been deliberately added to the semiconductor to change its electrical properties. For example, adding boron to silicon creates a p-type semiconductor, while adding phosphorus creates an n-type semiconductor.In a p-type semiconductor, the concentration of impurities is such that there is a relative shortage of electrons, and the holes left behind by the missing electrons behave as if they were positive charges. This has the effect of attracting electrons to the area, creating a flow of current. In contrast, in an n-type semiconductor, the concentration of impurities is such that there is an excess of electrons, which can move freely through the material.Another important factor that affects the behavior of electrons in a semiconductor is temperature. At low temperatures, the behavior of electrons is dominated by their interaction with the crystal lattice of the material. As the temperature increases, however, electrons can gain enough energy to move into the conduction band, where they can move freely and conduct electricity.In addition to temperature and impurities, the behavior of electrons in a semiconductor is also affected by the presence of electric fields. When an electric field is applied to a semiconductor, it can cause the electrons to move in a particular direction, which can be used to create a current, or to control the behavior of the material.is a complex topic, and there is still much to learn about the interactions between electrons and the crystal lattice of the material. However, the ability to control this behavior has led to the development of many of the technologies that we take for granted today, from computer chips to solar cells.In conclusion, the behavior of electrons in a semiconductor is a fascinating topic that has been the subject of much research over the years. Understanding the interaction between electrons and the crystal lattice of the material is key to our ability to control the behavior of semiconductors and to develop new technologies that rely on their unique properties.。

a r X i v :c o n d -m a t /9811182v 1 [c o n d -m a t .m t r l -s c i ] 13 N o v 1998Quadratic electronic response of a two-dimensional electron gasA.Bergara 1,J.M.Pitarke 1,and P.M.Echenique 21MateriaKondentsatuaren Fisika Saila,Zientzi Fakultatea,Euskal Herriko Unibertsitatea,644Posta Kutxatila,48080Bilbo,Basque Country,Spain2MaterialenFisika Saila,Kimika Fakultatea,Euskal Herriko Unibertsitatea,1072Posta Kutxatila,20080Donostia,Basque Country,Spain(February 1,2008)AbstractThe electronic response of a two-dimensional (2D)electron system represents a key quantity in discussing one-electron properties of electrons in semicon-ductor heterojunctions,on the surface of liquid helium and in copper-oxide planes of high-temperature superconductors.We here report an evaluation of the wave-vector and frequency dependent dynamical quadratic density-response function of a 2D electron gas (2DEG),within a self-consistent fieldapproximation.We use this result to find the Z 31correction to the stopping power of a 2DEG for charged particles moving at a fixed distance from the plane of the 2D sheet,Z 1being the projectile charge.We reproduce,in the high-density limit,previous full nonlinear calculations of the stopping powerof a 2DEG for slow antiprotons,and we go further to calculate the Z 31correc-tion to the stopping power of a 2DEG for a wide range of projectile velocities.Our results indicate that linear response calculations are,for all projectile velocities,less reliable in two dimensions than in three dimensions.Typeset using REVT E XI.INTRODUCTIONSince the pioneering work of Bohm and Pines1,the conduction electrons in a metal have been described as a three-dimensional(3D)gas of electrons in a neutralizing uniform positive charge.The dynamical linear density-response function of a3D electron gas(3DEG)was evaluated by Lindhard2in the so-called random-phase approximation(RPA),in which each electron is assumed to move in the externalfield plus the inducedfield of all electrons.The wave-vector and frequency dependent dynamical quadratic density-response function of a 3DEG has also been evaluated3,by going beyond linear response theory.The knowledge of this quantity has been proved to be of great importance in discussing the properties of electrons in a variety of3D systems4,and,in particular,in explaining the experimentally observed difference between the electronic energy losses of protons and antiprotons moving through a solid5,6.The suggested existence of two-dimensional electron layers in metal-insulator-semiconductor structures and on the surface of liquid helium led several years ago to a great activity in the study of a two-dimensional electron gas7,where the electrons are con-fined to a plane and neutralized by an inert uniform rigid positive plane background.The 2D electron system has also been considered in discussing the physics of new-class materials such as copper-oxide planes of high-temperature superconductors8.It has been found that electrons confined to a2D layer show sometimes interesting properties not shared by a3D electron system.For instance,for a2D metal the plasma frequency goes to zero in the long-wavelength limit,in contrast to the3D situation9.Stern10evaluated the dynamical linear density-response function of a2DEG in the RPA,and calculated the plasmon dispersion and the asymptotic screened Coulomb potential.Also,much effort has gone into studying the ground state energy11and the excitation spectrum of a2DEG12.The stopping power for a fast particle moving parallel to a2DEG wasfirst evaluated in the RPA,within linear response theory,by Horing et al13,and the effect on this quantity offinite temperature14, localfield corrections15and recoil16has also been considered.Nonlinear calculations of thestopping power for slow protons and antiprotons have been performed only very recently17,18 on the basis of a scattering theory approach,the scattering cross sections being calculated for a statically screened potential.In this paper we present results for the dynamical quadratic electronic density-response of a2DEG to longitudinal externalfields of arbitrary wave-vector and frequency,which we evaluate on the same level of approximation as the RPA linear density-response function of Stern10.In order to illustrate the usefulness of the knowledge of this quantity,we consider, as an example,the stopping power of a2DEG,and we use the quadratic density-response function to evaluate the Z31nonlinear correction to the stopping power of a2DEG for particles of charge Z1e moving at afixed distance from the2D plasma.II.DYNAMICAL ELECTRONIC RESPONSEWe consider a uniform electron system of density n0at zero temperature.Linear and quadratic density-response functions of this system,χ(x,x′)and Y(x,x′,x′′),with x=(r,t), may be introduced in connection with the response to the presence of a time-varying external influence,after an expansion of the induced electron density in powers of the external poten-tial V(x).According to time-dependent perturbation theory,the induced electron density is given,up to second order in V(x),by(we use atomic units throughout,i.e.,e2=¯h=m e=1)n ind(x)= d x′χ(x,x′)V(x′)+ d x′ d x′′Y(x,x′,x′′)V(x′)V(x′′),(2.1) whereχ(x,x′)=−iΘ(t−t′)<Ψ0|[˜ρH(x),˜ρH(x′)]|Ψ0>(2.2) andY(x,x′,x′′)=Θ(t−t′)Θ(t′−t′′)<Ψ0|[[˜ρH(x),˜ρH(x′)],˜ρH(x′′)]|Ψ0>/2+Θ(t−t′′)Θ(t′′−t′)<Ψ0|[[˜ρH(x),˜ρH(x′′)],˜ρH(x′)]|Ψ0>/2.(2.3)Here|Ψ0>denotes the normalized ground state,˜ρH is the densityfluctuation operator ˜ρH=ˆρH−n0,whereˆρH is the exact Heisenberg density operator in the unperturbed system,andΘ(x)is the Heaviside step function accounting for causality.In a self-consistentfield or random-phase approximation(RPA),it is assumed that the electron density induced by an external potential can be replaced by the electron density induced in a non-interacting electron gas by the sum of the external potential,V(x),and the potential created by the induced electron density itself,V ind(x).Hence,in this approx-imation:n ind(x)= d x′χ0(x,x′) V(x′)+V ind(x′)+ d x′ d x′′Y0(x,x′,x′′) V(x′)+V ind(x′) V(x′′)+V ind(x′′) .(2.4) Hereχ0(x,x′)and Y0(x,x′,x′′)are’free-particle’linear and quadratic density-response func-tions,andV ind(x)= d x′v(x,x′)n ind(x′),(2.5) where v(x,x′)represents the instantaneous Coulomb interaction.Introducing Eq.(2.5)into Eq.(2.4)and keeping in Eq.(2.4)only terms up to second order in V(x),onefinds the following integral equations for RPA linear and quadratic density-response functions:χ(x1,x2)=χ0(x1,x2)+ d x′1 d x′2χ0(x1,x′1)v(x′1,x′2)χ(x′2,x2)(2.6) andY(x1,x2,x3)= d x′2 d x′3Y0(x1,x′2,x′3)K(x′2,x2)K(x′3,x3)+ d x′1 d x′′1χ0(x1,x′1)v(x′1,x′′1)Y(x′′1,x2,x3),(2.7) where K(x,x′)is the so-called inverse dielectric function:K(x,x′)=δ(x−x′)+ d x′′v(x,x′′)χ(x′′,x′).(2.8) The integral equations(2.6)and(2.7)have,within many-body perturbation theory, the simple diagrammatic interpretation shown in Fig.1,where empty and full bubblesrepresent non-interacting and interacting linear density-response functions,χ0(x1,x2)and χ(x1,x2),respectively.Similarly,empty and full triangles represent non-interacting and interacting quadratic density-response functions,Y0(x1,x2,x3)and Y(x1,x2,x3),respec-tively,and dashed lines represent the Coulomb interaction,v(x,x′).Thus,RPA linear and quadratic density-response functions are represented diagrammatically by summing over the infinite set of diagrams containing one(see Fig.1a)and three(see Fig.1b)strings of empty bubbles,respectively.In the case of a homogeneous2DEG,the electrons are free to move in two spatial di-mensions,having their motion constrained in the third dimension.Thus,assuming time invariance,we define the Fourier transformsχq= d2r1 d t1e−i[q·(r1−r2)−ω(t1−t2)]χ(r1,t1;r2,t2)(2.9) andY q1,q2= d2r1 d t1 d2r2 d t2e−i[q1·(r1−r2)−ω1(t1−t2)]e−i[(q1+q2)·(r2−r3)−(ω1+ω2)(t2−t3)]×Y(r1,t1;r2,t2;r3,t3),(2.10) where r1,r2and r3represent two-dimensional position-vectors in the2D plane,and q is the trimomentum q=(q,q0).Hence,within the RPA wefind:χq=χ0q+χ0q v qχq(2.11) andY q,−q1=Y0q,−q1K q1K q−q1+χ0q v q Y q,−q1,(2.12)whereK q=1+v qχq(2.13) and v q=2π/|q|.For a non-interacting Fermi gas,the ground state is obtained byfilling all the plane wave states inside the Fermi sphere of radius q F=√distance(n−10=πr2s).As in the case of a3DEG19,wefind non-interacting linear and quadratic density-response functions to be20χ0q=2 d2k q0−(ωk+q−ωk)+iη+(1−n k)n k+q(2π)2[n k(1−n k+q)(1−n k+q1)(q0+ωk−ωk+q+iη)(q01+ωk−ωk+q1+iη)+n k+q(1−n k)(1−n k+q1)(−q0+ωk+q−ωk−iη)(−q0+q01+ωk+q−ωk+q1−iη)+n k+q1(1−n k)(1−n k+q)(−q01+ωk+q1−ωk−iη)(q0−q01+ωk+q1−ωk+q+iη)+(q1→q−q1)],(2.15)where n q=Θ(q F−|q|),ωk=k2/2,andηis a positive infinitesimal.Analytical evaluation of Eq.(2.12)results in the non-interacting linear density-response function of Stern10.As for the non-interacting quadratic density-response function,wefirst sum occupation numbers in Eq.(2.15)tofindY0q,−q1=− d2k q0+ωk−ωk+q+iη1−q0+ωk−ωk+q−iη1−q01+ωk−ωk+q1−iη1whereI q,q1=1A1−A cosχ−sgn A arctan sinχ A1−A cosχΘ(A2−q2F) +arctan A1sinχA21−q2F4|q||q1|sinχ.(2.19)Here A=q0/|q|−|q|/2,A1=q01/|q1|−|q1|/2,and G=2[f q,q1−f−q,−q+q1+(q1→q−q1)](2.21)andf q,q1=Θ(q2F−A2)1q2F−A2q2F−A2.(2.22)In particular,at low frequencies an expansion of H q,q1in powers of the frequency q0gives, after retaining only thefirst-order terms,H L q,q1=Θ(4q2F−|q|2)4(|q|cosχ−|q1|)4q2F−|q|2q0.(2.23)In the static limit(q0→0)bothfirst and second order contributions to the absorptionprobability,Imχq and H q,q1,are proportional to the frequency q0,as in the case of a3DEG.III.ELECTRONIC STOPPING POWERWe consider an ion of charge Z1moving with constant velocity v at afixed distance h from a2DEG of density n0.The Coulomb potential of this moving particle has the formV(r,z;t)=Z1|r−v t+(z−h)ˆk|−1,(3.1) where r represents,as in Eqs.(2.9)and(2.10),a two-dimensional position-vector in the 2D plane,and z denotes the coordinate normal to the2DEG which we consider to be located at z=0.Hence,in order to obtain the induced potential,we introduce this time-varying external potential into Eq.(2.1),and Eq.(2.1)into Eq.(2.5).We note that density-response functions of a2DEG have their z arguments localized to the2D plane by positionalδfunctions,we Fourier transform,andfind,up to second order in Z1:V ind(r,z;t)=Z1 d2q(2π)2 d2q1v d2r d zδ(r−v t)δ(z−h)∇V ind(r,z;t)·v.(3.3)Substituting Eq.(3.2)into Eq.(3.3),we have23S=−Z21(2π)2ωe−2|q|h v q Im[χq,ω]v q−Z31(2π)2ω d2q1Y q,−q1=K q Y0q,−q1K q1K q−q1,(3.6)whereK q= 1−χ0q v q −1.(3.7) The linear contribution to the stopping power of Eq.(3.4),which is proportional to Z21,was evaluated in the RPA by Horing et al13at T=0,and similar calculations were presented by Bret and Deutsch14atfinite temperature.On the other hand,the quadratic contribution to the stopping power of Eq.(3.4)is,in the RPA and for a geometry with the ion-beam in-plane within the2D electron layer(h=0),equivalent to the result derived in Ref.6,within many-body perturbation theory,as the energy loss per unit path length of the projectile,the integration space being now in two dimensions instead of three dimensions. This contribution to the stopping power,which is proportional to Z31,discriminates between the energy loss of a proton and that of an antiproton,and appears as a consequence of losses to one-and two-step electronic excitations generated by both linearly and quadratically screened ion potentials,as discussed in Ref.6.In the case of slow intruders(v→0),only the low-frequency form of the response enters in the evaluation of Eq.(3.4).At zero frequencies both linear and quadratic density-response functions are real,Im χ0q and H q,q1being at low frequencies proportional to the frequency q0.Thus,retaining only thefirst-order terms in the frequencies,both Z21and Z31contributions to the stopping power are found to be proportional to the velocity of the projectile,as in a3DEG.For the RPA quadratic contribution to the stopping power wefind, after insertion of Eq.(2.23)into Eq.(3.4),the following result:S L (3)=8vZ31 ∞0d|q|4q2F−|q|2∞0d|q1| π0dχe−(|q|+|q1|+|q−q1|)h f L1+f L2f L2=−Θ(2q F−|q|)|q|(|q|cosχ−|q1|)represent the full nonlinear contribution to the stopping power for antiprotons reported in Refs.18and27,respectively,multiplied by a factor of−1,showing an excellent agreement, in the high-density limit,with our Z31nonlinear contribution.For r s≥2,higher order corrections become important,and for r s>3,Z21contributions(dashed line)are smaller than the Z31correction,indicating that the external potential cannot be treated,for these electron densities,as a small perturbation.In the case of a projectile moving at a given h distance above the2D plasma the external perturbation is obviously diminished and,in particular,for h=1/q F the quadratic stopping power is smaller than the linear one for all electron densities(see the inset of Fig.2).The full nonlinear contribution to the stopping power for protons reported in Refs.18and27for h=0is also represented in Fig.2by circles and crosses,respectively,showing large differences with our Z31calculations for all electron densities.These differences appear as a consequence of perturbation theory failing to describe electronic states bound to the proton,which in2D systems can be supported by arbitrarily weak attractive potentials28.As the velocity of the projectile(v)and/or the distance from the2D plasma(h)increase, the ion potential becomes a relatively smaller perturbation and the Z31contribution to the stopping power for antiprotons may,therefore,be expected to approximately describe the full nonlinear contribution to the stopping power for arbitrary values of v and h,as long as r s<2,and also for lower densities(r s≥2)as the velocity and the h distance increase. Substitution of the full RPA linear and quadratic2DEG density-response functions of Eqs.(3.5)and(3.6)into Eq.(3.4)results in the quadratic stopping power plotted by a solid line in Fig.3,as a function of the impinging projectile velocity,for r s=1and h=0,and also for r s=1and h=1/q F(see the inset of Fig.3).The dotted line represents the low-velocity limit,as obtained from Eq.(3.8),and dashed lines represent linear RPA contributions to the stopping power of Eq.(3.4).The quadratic contribution to the stopping power of a2DEG presents properties not shared by the3DEG.First,the range of validity of the linear velocity dependence of this contribution to the stopping power(see Fig.3)persists only up to velocities much smallerthan the Fermi velocity,in contrast to the3D situation in which the linear velocity de-pendence persists up to velocities approaching the Fermi velocity6.Second,at velocities around the plasmon threshold velocity,for which the projectile has enough energy to excite a plasmon29,the ratio between Z31and Z21contributions to the stopping power increases, again in contrast to the3D situation.Furthermore,we have found that the increase of this ratio at the plasmon threshold velocity becomes dramatic as the electron density de-creases.Of course,the ratio between quadratic and linear contributions to the stopping power decreases very rapidly with h(see the inset of Fig.3),but the relative,and for large r s dramatic,increase of this ratio at the plasmon threshold velocity persists for all values of h that are not much larger than1/q F.For a2DEG the group velocity of the plasmon wave nearly coincides with the plasmon threshold velocity of the projectile(see Fig.4);also, the inclusion of short-range correlations,which are ignored in the RPA,is known to have a substantial effect on the plasmon dispersion30.We interpret the anomalous enhancement of the ratio between quadratic and linear contributions to the stopping power at the plasmon threshold velocity and small electron densities(r s>1)as a result of neglecting,within the RPA,short-range correlations between the electrons of a2D system,since these correlations are non-negligible for all values of h as long as r s is not small.V.CONCLUSIONSWe have presented an analytical evaluation of the wave-vector and frequency dependent non-interacting quadratic density-response function of a2DEG.We have used this result to find,within a self-consistentfield approximation,the Z31correction to the stopping power of a2DEG for charged recoiless particles moving at afixed distance from the2D plasma sheet. We have reproduced,in the high-density limit,previous full nonlinear calculations of the stopping power of a2DEG for slow antiprotons,and we have gone further to calculate the Z31correction to the stopping power for a wide range of projectile velocities.We have found that the Z31contribution to the stopping power of a2DEG presents properties not sharedby the3DEG.On the one hand,the range of validity of the linear velocity dependence of the Z31contribution to the stopping power persists only up to velocities much smaller than the Fermi velocity,and,on the other hand,an anomalous enhancement of the ratio between quadratic and linear contributions to the stopping power at the plasmon threshold velocity is found,within the RPA,at small electron densities(r s>1).Also,our results indicate that linear response calculations are,for all projectile velocities,less reliable in2D than in 3D.ACKNOWLEDGMENTSWe acknowledge partial support by the University of the Basque Country,the Basque Unibertsitate eta Ikerketa Saila,the Spanish Ministerio de Educaci´o n y Cultura,and Iber-drola SA.REFERENCES1D.Bohm and D.Pines,Phys.Rev.92,609(1953).2J.Lindhard,K.Dan.Vidensk.Selsk.Mat.-Fys.Medd.28(8),1(1954).3R.Cenni and P.Saracco,Nucl.Phys.A487,279(1988).4C.F.Richardson and N.W.Ashcroft,Phys.Rev.B50,8170(1994);J.M.Rommel and G.Kalman,Phys.Rev.B54,3518(1996).5J.M.Pitarke,R.H.Ritchie,P.M.Echenique,and E.Zaremba,Europhys.Lett.24,613 (1993);J.M.Pitarke,R.H.Ritchie,and P.M.Echenique,Nucl.Instrum.Methods B24, 613(1993).6J.M.Pitarke,R.H.Ritchie,and P.M.Echenique,Phys.Rev.B52,13882(1995).7T.Ando,A.Fowler,and F.Stern,Rev.Mod.Phys.54,437(1982).8J.R.Engelbrecht and M.Randeira,Phys.Rev.Lett.65,1032(1990);P.W.Anderson, Phys.Rev.Lett.64,1839(1990).9R.H.Ritchie,Phys.Rev.106,874(1957);R.H.Ritchie,Surf.Sci.34,1(1973).10F.Stern,Phys.Rev.Lett.18,546(1967).11A.K.Rajagopal and J.C.Kimball,Phys.Rev.B15,2819(1977).12A.V.Chaplik,Zh.Eksp.Teor.Fiz.60,1845(1971);G.F.Giuliani and J.J.Quinn,Phys. Rev.B26,4421(1982);L.Zheng and S.D.Sarma,Phys.Rev.B53,9964(1996).13N.J.M.Horing,H.C.Tso,and G.Gumbs,Phys.Rev.B36,1588(1987).14A.Bret and C.Deutsch,Phys.Rev.E48,2994(1993).15Y.-N.Wang and T.-C.Ma,Phys.Lett.A200,319(1995).16A.Bergara,I.Nagy,and P.M.Echenique,Phys.Rev.B55,12864(1997).17A.Krakovsky and J.K.Percus,Phys.Rev.B52,R2305(1995).18Y.-N.Wang and T.-C.Ma,Phys.Rev.A55,2087(1997).19C.D.Hu and E.Zaremba,Phys.Rev.B37,9268(1988);A.Bergara,I.Campillo,J.M. Pitarke,and P.M.Echenique,Phys.Rev.B56,15654(1997).20Differences between the retarded density-response functions of Eqs.(2.14)and(2.15),on the one hand,and time-ordered response functions,as defined within many-body theory (see,e.g.,G.D.Mahan,Many-Particle Physics(Plenum,New York,1981)),on the other hand,lie inω+iη,which should be replaced in Eqs.(2.14)and(2.15)byω+iηsgn(ω)[ωbeing either q0,q01or(q0−q01)]in order to obtain their time-ordered counterparts.21I′q,q1,I′−q,−q+q1and I′−q1,q−q1in Eq.(2.17)should be multiplied by sgn(q0)sgn(q01),sgn(q0)sgn(q0−q01)and sgn(q01)sgn(q0−q01),respectively,to obtain the real part of the corresponding time-ordered quadratic response function.Similarly,the right-hand side of Eq.(2.21)should be multiplied by sgn(q0)in order to obtain the imaginary part of the time-ordered quadratic response function.22P.M.Echenique,F.Flores,and R.H.Ritchie,Solid State Phys.43,229(1990).23The real parts ofχq and Y q,−q1give no contribution to the integrals of Eq.(3.4),sinceRe[χq]=Re[χ−q]and Re[Y q,q1]=Re[Y−q,−q1].24I.Nagy,Phys.Rev.B51,77(1995).25P.Hohenberg and W.Kohn,Phys.Rev.136,B864(1964);W.Kohn and L.J.Sham,ibid. 140,A1133(1965).26P.M.Echenique,R.M.Nieminen,and R.H.Ritchie,Solid State Commun.37,779(1981); P.M.Echenique,R.M.Nieminen,J.C.Ashley,and R.H.Ritchie,Phys.Rev.A33,897 (1986).27E.Zaremba,P.M.Echenique,and I.Nagy,to be published.28E.N.Economou,Green’s Functions in Quantum Physics(Springer-Verlag,Berlin,1990), pg.64.29At long wavelengths the plasma frequency in the2DEG varies like q1/2,in contrast to the 3D case(see,e.g.,A.L.Fetter,Ann.Phys.(N.Y.)81,367(1973)).At short wavelengths, the group velocity of the2D plasmon is nearly constant and approaches the plasmon threshold velocity,v th.For r s=1,v th=2.02v0(v0is Bohr’s velocity).30A.Gold and L.Calmels,Phys.Rev.B48,11622(1993).FIGURESFIG.1.Diagrammatic interpretation of the RPA integral equations(2.6)and(2.7)for(a)linear and(b)quadratic density-response functions,respectively.Electron-hole empty and full bubbles represent non-interacting and interacting linear density-response functions,respectively.Empty and full triangles represent non-interacting and interacting quadratic density-response functions, respectively.The interacting RPA quadratic density-response function is obtained by summing over the infinite set of diagrams that combine three strings of empty bubbles(two-electron loops) through an empty three-electron loop.Dashed lines represent the electron-electron bare Coulomb interaction.FIG.2.Low-velocity limit of the Z31stopping power,as obtained from Eq.(3.8)(solid line)for h=0and Z1=1,divided by the velocity,as a function of r s;the corresponding Z21stopping power is represented by a dashed line.Full nonlinear contributions to the stopping power for antiprotons [Z1=−1](stars and squares)and protons[Z1=1](circles ans crosses)have been obtained by subtracting RPA linear calculations from the results of Ref.18(stars and circles)and Ref.27[with XC contributions to the DFT scattering potential excluded](squares and crosses),and dividing by Z1;thus,the negative values at r s>1simply mean that the full nonlinear stopping power lies,in the case of protons,below the linear results.The inset exhibits Z21and Z31contributions to the stopping power(dashed and solid lines,respectively)for h=1/q F and Z1=1.FIG.3.Full RPA Z31stopping power,as obtained from Eq.(3.4)(solid line)for Z1=1,h=0 and r s=1,as a function of the velocity of the projectile.The corresponding Z21stopping power is represented by a dashed line,and the dotted line represents the low-velocity limit of the Z31term. The inset shows the same results for h=1/q F.FIG.4.Two-dimensional RPA plasmon dispersion relation(solid line)and maximum energy transfer(ωmax=qv)at the plasmon threshold velocity(dotted line),as functions of the wave number.The electron density parameter has been taken to be r s=1,thus the plasmon threshold velocity being v th=2.02v0(v0is Bohr’velocity).==+. . . . . .+Figure 1a+=++==+. . . . . .+......Figure 1b0,511,522,5300,20,40,60,811,21,4Fig. 4ωq v t h =2.02。

Unit 2 Classification of MaterialsSolid materials have been conveniently grouped into three basic classifications: metals, ceramics, and polymers. This scheme is based primarily on chemical makeup and atomic structure, and most materials fall into one distinct grouping or another, although there are some intermediates. In addition, there are three other groups of important engineering materials —composites, semiconductors, and biomaterials.译文：译文：固体材料被便利的分为三个基本的类型：金属，陶瓷和聚合物。

固体材料被便利的分为三个基本的类型：金属，陶瓷和聚合物。

固体材料被便利的分为三个基本的类型：金属，陶瓷和聚合物。

这个分类是首先基于这个分类是首先基于化学组成和原子结构来分的，化学组成和原子结构来分的，大多数材料落在明显的一个类别里面，大多数材料落在明显的一个类别里面，大多数材料落在明显的一个类别里面，尽管有许多中间品。

尽管有许多中间品。

除此之外，此之外， 有三类其他重要的工程材料－复合材料，半导体材料和生物材料。

有三类其他重要的工程材料－复合材料，半导体材料和生物材料。

Composites consist of combinations of two or more different materials, whereas semiconductors are utilized because of their unusual electrical characteristics; biomaterials are implanted into the human body. A brief explanation of the material types and representative characteristics is offered next.译文：复合材料由两种或者两种以上不同的材料组成，然而半导体由于它们非同寻常的电学性质而得到使用；生物材料被移植进入人类的身体中。

半导体器件机理英文Semiconductor Device Mechanisms.Semiconductors are materials that have electrical conductivity between that of a conductor and an insulator. This unique property makes them essential for a wide range of electronic devices, including transistors, diodes, and solar cells.The electrical properties of semiconductors are determined by their electronic band structure. In a semiconductor, the valence band is the highest energy band that is occupied by electrons, while the conduction band is the lowest energy band that is unoccupied. The band gap is the energy difference between the valence band and the conduction band.At room temperature, most semiconductors have a relatively large band gap, which means that there are very few electrons in the conduction band. This makessemiconductors poor conductors of electricity. However, the electrical conductivity of a semiconductor can be increased by doping it with impurities.Donor impurities are atoms that have one more valence electron than the semiconductor atoms they replace. When a donor impurity is added to a semiconductor, the extra electron is donated to the conduction band, increasing the number of charge carriers and the electrical conductivityof the semiconductor.Acceptor impurities are atoms that have one lessvalence electron than the semiconductor atoms they replace. When an acceptor impurity is added to a semiconductor, the missing electron creates a hole in the valence band. Holes are positively charged, and they can move through the semiconductor by accepting electrons from neighboring atoms. This also increases the electrical conductivity of the semiconductor.The type of impurity that is added to a semiconductor determines whether it becomes an n-type semiconductor (witha majority of electrons as charge carriers) or a p-type semiconductor (with a majority of holes as charge carriers).The combination of n-type and p-type semiconductors is used to create a wide range of electronic devices,including transistors, diodes, and solar cells.Transistors.Transistors are three-terminal devices that can be used to amplify or switch electronic signals. The threeterminals are the emitter, the base, and the collector.In a bipolar junction transistor (BJT), the emitter is an n-type semiconductor, the base is a p-type semiconductor, and the collector is another n-type semiconductor. When a small current is applied to the base, it causes a large current to flow between the emitter and the collector. This makes BJTs ideal for use as amplifiers.In a field-effect transistor (FET), the gate is a metal electrode that is insulated from the channel. When avoltage is applied to the gate, it creates an electricfield that attracts or repels electrons in the channel. This changes the conductivity of the channel, which in turn controls the flow of current between the source and the drain. FETs are ideal for use as switches.Diodes.Diodes are two-terminal devices that allow current to flow in only one direction. The two terminals are the anode and the cathode.In a p-n diode, the anode is a p-type semiconductor and the cathode is an n-type semiconductor. When a voltage is applied to the diode, it causes electrons to flow from the n-type semiconductor to the p-type semiconductor, but not vice versa. This makes diodes ideal for use as rectifiers, which convert alternating current (AC) to direct current (DC).Solar Cells.Solar cells are devices that convert light energy into electrical energy. They are made of a semiconductor material, such as silicon, that has a p-n junction.When light strikes the solar cell, it creates electron-hole pairs in the semiconductor. The electrons areattracted to the n-type semiconductor, while the holes are attracted to the p-type semiconductor. This creates a voltage difference between the two semiconductors, which causes current to flow.Solar cells are used to power a wide range of devices, including calculators, watches, and satellites. They are also used to generate electricity for homes and businesses.Conclusion.Semiconductors are essential for a wide range of electronic devices. Their unique electrical properties make them ideal for use in transistors, diodes, and solar cells. As semiconductor technology continues to develop, we canexpect to see even more innovative and efficient electronic devices in the future.。

——电材专业英语课文翻译Semiconductor Materials• 1.1 Energy Bands and Carrier Concentration• 1.1.1 Semiconductor Materials•Solid-state materials can be grouped into three classes—insulators(绝缘体）, semiconductors, and conductors. Figure 1-1 shows the electrical conductivities δ(and the corresponding resistivities ρ≡1/δ)associated with(相关）some important materials in each of three classes. Insulators such as fused（熔融）quartz and glass have very low conductivities, in the order of 1E-18 to 1E-8 S/cm;固态材料可分为三种：绝缘体、半导体和导体。

图1－1 给出了在三种材料中一些重要材料相关的电阻值（相应电导率ρ≡1/δ)。

绝缘体如熔融石英和玻璃具有很低电导率，在10-18 到10-8 S/cm;and conductors such as aluminum and silver have high conductivities, typically from 104 to 106 S/cm. Semiconductors have conductivities between those of insulators and those of conductors. The conductivity of a semiconductor is generally sensitive to temperature, illumination（照射）, magnetic field, and minute amount of impurity atoms. This sensitivity in conductivity makes the semiconductor one of the most important materials for electronic applications.导体如铝和银有高的电导率，典型值从104到106S/cm;而半导体具有的电导率介乎于两者之间。

关于拓扑超导的英文演讲Topological superconductivity is a fascinating topic in the field of condensed matter physics that has garnered significant attention in recent years. In this speech, I will provide an overview of the concept, its potential applications, and the ongoing research in this exciting field.Firstly, let's understand what topological superconductivity is. Superconductivity is a quantum phenomenon that occurs at very low temperatures, where certain materials can conduct electricity without any resistance. This property is due to the formation of Cooper pairs, which are pairs of electrons with opposite spins. Topological superconductivity refers to a special class of superconductors where the Cooper pairs exhibit an additional quantum property known as non-Abelian statistics.Non-Abelian statistics means that the quantum wavefunction of the system is not invariant under the exchange of particles. This unique characteristic holds the potential for storing and manipulating quantum information, making topological superconductors a promising platform for developing quantum computers. Unlike conventional superconductors, which are described by Abelian statistics, the non-Abelian nature of topological superconductivity provides protection against certain types of local perturbations and disturbances, making them more stable against noise.The study of topological superconductivity is closely connected to the field of topological insulators. Topological insulators are materials that have a unique electronic band structure that results in conducting surface states while remaining insulating in the bulk. This distinct behavior arises due to the nontrivial topology of the electron wavefunctions. By introducing superconductivity into topological insulators, researchers have been able to realize topological superconductivity.One of the most exciting prospects of topological superconductivity is its potential for hosting Majorana fermions. Majorana fermions are hypothesized particles that are their own antiparticles, meaning they can annihilate and reappear as their own particle. Majorana fermions have distinct properties that make them attractive for quantumcomputing, as they are expected to have a higher resistance to decoherence. Decoherence is a phenomenon that can disrupt quantum states and is a major challenge in quantum computing.Numerous experimental efforts have been dedicated to the search for evidence of Majorana fermions in topological superconductors. One of the most notable experiments is the creation of a hybrid structure called a topological superconductor nanowire. This nanowire, made of materials with strong spin-orbit coupling and proximity-induced superconductivity, exhibits the predicted signatures of Majorana fermions. These experimental advancements have sparked great excitement and sparked further research in the field of topological superconductivity.Apart from quantum computing, topological superconductivity also has potential applications in other areas, such as topological quantum computation and fault-tolerant quantum memories. Researchers are actively exploring the possibilities of using the unique properties of topological superconductors to create new technologies that can revolutionize various fields.In conclusion, topological superconductivity is a captivating area of research with great potential for quantum technologies. Its non-Abelian nature and the possible existence of Majorana fermions make it a promising platform for quantum computing and other applications. Continued experimental efforts and theoretical investigations are crucial in unraveling the mysteries and realizing the full potential of topological superconductivity. The future of this field holds exciting possibilities that could shape the future of quantum technology.。