Lecture1CASGS-纳米科学概论,低维体系量子力学

1、,纳米结构物理学 课程内容,纳米科学概论, 低维体系量子力学 固体物理, 表面/界面科学及材料生长简介 纳米结构常用分析与制备方法 纳米线(管,带,杆) 团簇与晶粒 磁性纳米结构及自旋电子学,1 nm = 10-9 m = 10-3 m = 10 ,纳米结构 (Nanostructures): material systems with length scale of 1-100 nm in at least one dimension,2-D: quantum wells, thin films, 2-D electron gas 1-D: quantum wires, nanowires,

2、 nanotubes, nanorods 0-D: quantum dots, macro-molecules, clusters, nano-crystallites,Between individual atoms/molecules and macroscopic bulk materials: Mesoscopic structures (介观结构), with distinct properties not available from atoms or bulk crystals,类型,材料性质随体系尺度的变化：量变到质变,Quantum confinement: quantiza

3、tion and reduced dimensionality of electronic states Quantum coherence and de-coherence Surface/interface states Metastability, adjustable size and shape Properties tunable High speed, compact density and efficiency,Unique properties of nanostructures:,Two approaches in our understanding and exploit

4、ation of material world: from the bottom up and from the top down,The bottom-up approach: Atoms, simple molecules (well-understood sub-nm world) Macro-molecules, polymers clusters, crystallites, nanowires, bio-molecules,The top-down approach: Bulk crystals Discrete devices Integrated circuits LSI VL

5、SI ULSI ( 0.1-0.05 m) ? Shrinking and shrinking into deep sub-0.1-m,两种途径在纳米尺度相会,For up-to-date Edition visit ,半导体工业路线图,Bottom-up approach can deal with systems consisting of 104 atoms quite accurately,纳米研究的目标,Search for new physical phenomena existing at nanoscales Fabricate nano-devices with novel

6、functions Search for processes to fabricate nanostructures with high accuracy and low cost Explore new experimental and theoretical tools to study nanostructures,Nanoscience & nanotechnology:,Multi-disciplinary and rapid-developing,现状与未来: 一个学术界，政府和产业部门高度重视的战略性研究领域,Quantum mechanics of low-dimensiona

7、l systems,Time-independent Schrdinger equation:,Free particle with V(r) = 0, plane wave:,(r , t) = A exp(ikr - iEt/),Energy and momentum of the particle: E = = 2k2/(2m) = 2(kx2 + ky2 + kz2)/(2m) = (k) p = k de Broglie wavelength: = h/p Probability of finding the particle at r : P(r , t) = |(r , t)|2

8、,For a free particle, the probability is the same everywhere,Potential well, quantization and bound states,1D potential well of infinite depth:,V(x),0 a x,n,n,Confined, discrete energy levels, with n = 1, 2, 3,Ground-state (n =1) energy = h2/(8ma2), zero-point or confinement energy,Potential wells o

9、f finite depth:,For negative E, only a certain number of E values are allowed. The particle remains confined, but not completely within the well.,For E above zero, any values are allowed, the probability of finding particle does not approach zero away from the well: The particle is free,Quantum well

10、: particle confined by a 1-D potential well, but free in other 2-D, quantum states labeled by n, kx and ky:,Each n represents a branch or subband,Quantum wire: particle confined by 2-D potential wells, free only in 1-D (1-D free particle), quantum states labeled by n1, n2 and kz:,Quantum dot: partic

11、le confined by potential wells in 3-D, quantum states labeled n1, n2 and n3:,All discrete levels, like in atom,Density of states (DOS): N(E),N(E)E = number of states with energies of E to E + E Plays a important role in many physical processes: conductivity, light emission, magnetism, chemical react

12、ivity A measurable quantity to characterize a physical system, e.g. to determine the dimensionality,1-D: plane wave (x) = A exp(ikx), with periodic boundary conditions:,(L) = (0) and,(L later),k and only take values:, n = 0, 1, 2,k,0,1-D k-space & allowed states,Dispersion relation (k) for 1-D syste

13、m,Count states in k-space: Allowed states are separated by a spacing 2/L,DOS in k-space N(k):,(2-fold spin degeneracy),n1D(k) = N1D(k)/L = 1/ Independent of L!,DOS in energy n1D(E):,n1D(E)E = n1D(E)k = 2n1D(k)k,n1D(E) = 2n1D(k)/(d/dk) = =,(k branches),n1D(E) diverges as E- when E 0, van Hove singula

14、rity,For a unit length:,DOS for a 2-D system:,n2D(E) =,It is a constant!,DOS for a 3-D system:,n3D(E) =,3-D k-space,DOS of a quantum well: sum up all branches, each has a 2-D DOS,Dispersion relation:,n2D(E) =,Multi-step function of step size g0 = m/2,DOS of a quantum wire: superposition of a series

15、of individual 1D DOS functions,n(E) =,Energy gap due to confinement,DOS of a quantum dot: Summation of a set of -functions (as in atoms and molecules),Quantum tunneling: A particle can be reflected by or tunnel through a barrier of V0 E,V0,A exp(ikx),B exp(-ikx),C exp(ikx),Region I Barrier Region II

16、,a,E,Define:,Tunneling probability:,For a thick or tall barrier, a 1,For an irregular shaped barrier,(a & b are classical turning points),Coherent quantum transport in 1-D channel,When phase coherence is maintained, electrons should be treated as pure waves 1D electron transportation between two reg

17、ions separated by an arbitrary potential barrier:,A exp(ik1z),B exp(-ik1z),C exp(ik2z),Region I Barrier Region II UII,UI,Transmission and reflection coefficients, T and R:,T + R = 1,For same E, T21(E) = T12(E),Transport between two 1DEG with Fermi level difference: I - II = eV,Current due to electro

18、ns from region I to II:,(Form of current density J = nqv, dk/2 counts states in 1D),Fermi distribution function:,step function at low T,Current due to electrons from region II to I:,For coherent transport, T21 = T12 = T, the net current:,(f step function at low T),For small bias V, T(E) a constant,L

19、andauer formula of conductance:,Quantum conductance unit: G0 = 2e2/h = 7.75 S,Quantum resistance unit: R0 =h/2e2 = 12.9 k,For a perfect quantum wire T = 1, its conductance is G = 2e2/h, independent of its length!,For a system with Ntrans transmitted states (modes) :,Classical case: a perfect wire ha

20、s no resistance (superconductor), or it increases with length,2D electron gas (2DEG),低维电子系统制备与输运实验,Double hetero-junction quantum well e.g.,AlGaAs-GaAs-AlGaAs,Single hetero-junction & MOS,EF,反相层,低维电子系统制备与输运实验,Further confinement to 2DEG 1DEG (Q-wire) 0D (QD),Quantum point-contact,量子触点,Conductance th

21、rough a short wire or constriction (quantum point contact) between two leads of 2DEG,Quantized conductance as a function of gate voltage Vg,Ntrans can be changed by varying split-gate bias Vg,Classical effect in transport through nanoparticles: Coulomb blockade,Coupling of QD to external world,Weak

22、coupling: the number of electrons located at the QD is well defined,Coulomb repulsion energy between electrons in a QD of size a:,The discrete nature of electron charge becomes strongly evident when EC kBT. For r 5, T = 300 K, this occurs at a 10 nm,Coulomb blockade: one electron located on a QD cre

23、ates an energy barrier to stop the further transfer of electrons onto the QD,Classical effect in transport through nanoparticles: Coulomb blockade,Furthermore, the charging energy can stop any electron jumping on a QD,Electrostatic energy stored in this capacitor is:,Capacitance for observing Coulom

24、b blockade at RT:,C 3 10-18 F,Spherical QD of radius a at a distance l (a) above a ground plane, the capacitance of this system:,For typical semiconductors, r 10, a 2.7 nm at RT,Energy diagram of a double-junction QD structure with Coulomb blockade,In equilibrium Under an applied bias,Experimental (

25、A) and theoretical (B and C) I-V curves of a STM tip/10-nm In island/AlOx film/Al substrate,When e/2C Va 3e/2C, maximum occupation number of QD is n = 1 one electron at a time jump through QD current is nearly a constant,Single electron transistor (SET),Third electrode - gate - to adjust QD potentia

26、l independently,Another version of SET,VG = V0 + V1 cos(2ft),I = ef, SET can be used as a current standard,Application example of SET:,参考文献,1. P. Moriarty, Nanostructured materials, Rep. Prog. Phys. 64, 297 (2001). 2. G. Timp (ed), Nanotechnology (Springer, New York, 1999). 3. Hari Singh Nalwa (ed),

27、 Nanostructured materials and nanotechnology (Academic Press, London, 2002). 4. For 2003 International Technology Roadmap for Semiconductors (ITRS), see website . 5. The Royal Society, Nanoscience and nanotechnologies: opportunities and uncertainties, ..uk/finalReport.htm (July 2004). 6.

28、D.J. Griffiths, Introduction to quantum mechanics (Prentice Hall, New Jersey, 1995). 7. J.H. Davis, The physics of low-dimensional semiconductors: an introduction (Cambridge University Press, New York, 1998). 8. A. Shik, Quantum wells: physics and electronics of two-dimensional systems (World Scient

29、ific, Singapore, 1997). 9. K. Barnham, D. Vvedensky (eds.),Low-dimensional semiconductor structures: Fundamentals and device applications (Cambridge University Press, New York, 2001).,10. D.K. Ferry, S.M. Goodnick, Transport in nanostructures (Cambridge University Press, New York, 1997). 11. T. Ando et al., Mesos