Lecture1CASGS纳米科学概论低维体系量子力学

1、纳米结构物理学纳米结构物理学 课程内容课程内容1. 纳米科学概论, 低维体系量子力学 2. 固体物理, 表面/界面科学及材料生长简介3. 纳米结构常用分析与制备方法 4. 纳米线(管,带,杆)5. 团簇与晶粒6. 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

2、: quantum wires, nanowires, nanotubes, nanorods0-d: quantum dots, macro-molecules, clusters, nano-crystallitesbetween individual atoms/molecules and macroscopic bulk materials: mesoscopic structures (介观结构介观结构), with distinct properties not available from atoms or bulk crystals类类型型材料性质随体系尺度的变化：量变到质变量

3、变到质变quantum confinement: quantization and reduced dimensionality of electronic statesquantum coherence and de-coherencesurface/interface statesmetastability, adjustable size and shape properties tunablehigh speed, compact density and efficiencyunique properties of nanostructures:two approaches in ou

4、r understanding and exploitation 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

5、 integrated circuits lsi vlsi 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 http:/半导体工业路线图半导体工业路线图bottom-up approach can deal with systems consisting of 104 atoms quite accurately纳米研究的目标纳米研究的目标search for new physical phenomena existing at na

6、noscalesfabricate nano-devices with novel functionssearch 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现状与未来现状与未来: 一个学术界，政府和产业部

7、门高度一个学术界，政府和产业部门高度重视的战略性研究领域重视的战略性研究领域 quantum mechanics of low-dimensional systems time-independent schrdinger equation:)()()()(222rrrrevmfree 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 br

8、oglie wavelength: = h/pprobability of finding the particle at r : p(r , t) = |(r , t)|2for a free particle, the probability is the same everywhere potential well, quantization and bound states 1d potential well of infinite depth: otherwise ,ax0 if 0,)(xv v(x) 0 a xnn otherwise , 0ax0for x),sin()(nan

9、axn)(22222222nnkmanmknconfined, discrete energy levels, with n = 1, 2, 3ground-state (n =1) energy = h2/(8ma2), zero-point or confinement energy potential wells of finite depth: otherwise , 0axa- if ,0v-)(xvfor negative e, only a certain number of e values are allowed. the particle remains confined,

10、 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 freequantum well: particle confined by a 1-d potential well, but free in other 2-d, quantum states labeled by n, kx and ky: )(2

11、),(222222yxyxkkanmkkneach n represents a branch or subbandquantum 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: )(2),(222222212221zzkbnanmknnquantum dot: particle confined by potential wells in 3-d, quantum states label

12、ed n1, n2 and n3:)(2),(22322222122321cnbnanmnnnall discrete levels, like in atomdensity 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 reactivity a measurable quantity to

13、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 lxxxx0(l later) k and only take values:,2lnknmkknn2)(22 , n = 0, 1, 2, kl6l4l20l2l4l61-d k-space & allowed states dispersion relation (k) for 1-

14、d systemcount states in k-space: allowed states are separated by a spacing 2/l dos in k-space n(k): klkknd22)(1(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)kn1d(e) = 2n1d(k)/(d/dk) = = em21)/(22mk(k branches)n1d(e) diverges as e- w

15、hen e 0, van hove singularityfor a unit length:dos for a 2-d system:n2d(e) = 2mit is a constant! dos for a 3-d system:n3d(e) = mem232 3-d k-space dos of a quantum well: sum up all branches, each has a 2-d dos dispersion relation: )(2),(222222yxyxkkanmkknn2d(e) = nnem)(2multi-step function of step si

16、ze g0 = m/2 dos of a quantum wire: superposition of a series of individual 1d dos functionsn(e) = 2, 12, 12, 1)(21nnnnnneemenergy gap due to confinementdos 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 throu

17、gh a barrier of v0 e v0 a exp(ikx) b exp(-ikx) c exp(ikx) region i barrier region iiaedefine: /2mek / )(20evmtunneling probability: 2222222224)(sinh)(4kakkactfor a thick or tall barrier, a 1)(22exp)(16)2exp()(16020022222evmaveveakktfor an irregular shaped barrier,badxexvmt)(22exp(a & b are class

18、ical 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 regions separated by an arbitrary potential barrier: / )(21iuemk/ )(22iiuemk a exp(ik1z) b exp(-ik1z) c exp(ik2z) region

19、i barrier region ii uii ui21221ackktttransmission and reflection coefficients, t and r:2abrt + r = 1 for same e, t21(e) = t12(e) transport between two 1deg with fermi level difference: i - ii = eviiievuiuii current due to electrons from region i to ii:0212)()(),(2dkktkvkefeiii(form of current densit

20、y j = nqv, dk/2 counts states in 1d) fermi distribution function: / )exp(1/1),(kteefstep function at low tcurrent due to electrons from region ii to i: iiiiiiudeetefhei)(),(212for coherent transport, t21 = t12 = t, the net current:iiiiiiiudeetefefheiii)(),(),(2iiideethe)(2(f step function at low t)f

21、or small bias v, t(e) a constant, )(2)()(22thvetheiiii landauer formula of conductance:)(2/2thevigquantum 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! trans2 2nhegfor a sys

22、tem with ntrans transmitted states (modes) : classical case: a perfect wire has no resistance (superconductor), or it increases with length2d 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 & mosef反相层反相层低维电子系统制备与输运实验further conf

23、inement to 2deg 1deg (q-wire) 0d (qd)quantum point-contact量子触点conductance through a short wire or constriction (quantum point contact) between two leads of 2deg quantized conductance as a function of gate voltage vgntrans can be changed by varying split-gate bias vg classical effect in transport thr

24、ough nanoparticles: coulomb blockade coupling of qd to external worldweak coupling: the number of electrons located at the qd is well defined coulomb repulsion energy between electrons in a qd of size a: (nm) (ev) 44. 1402aaeerrcthe discrete nature of electron charge becomes strongly evident when ec

25、 kbt. for r 5, t = 300 k, this occurs at a 10 nmcoulomb blockade: one electron located on a qd creates an energy barrier to stop the further transfer of electrons onto the qdclassical effect in transport through nanoparticles: coulomb blockade furthermore, the charging energy can stop any electron j

26、umping on a qdelectrostatic energy stored in this capacitor is: cqe22capacitance for observing coulomb 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:04(1)2racal for typical semiconductors, r 10, a 2.7 nm at rt energy dia

27、gram of a double-junction qd structure with coulomb blockade in equilibrium under an applied biasexperimental (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

28、qd current is nearly a constant single electron transistor (set)third electrode - gate - to adjust qd potential independentlyanother version of setvg = v0 + v1 cos(2ft)i = ef, set can be used as a current standardapplication example of set:参考文献参考文献1. p. moriarty, nanostructured materials, rep. prog.

29、 phys. 64, 297 (2001).2. g. timp (ed), nanotechnology (springer, new york, 1999). 3. hari singh nalwa (ed), nanostructured materials and nanotechnology (academic press, london, 2002).4. for 2003 international technology roadmap for semiconductors (itrs), see website http:/ the royal society, nanosci

30、ence and nanotechnologies: opportunities and uncertainties, .uk/finalreport.htm (july 2004).6. 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

31、 press, new york, 1998). 8. a. shik, quantum wells: physics and electronics of two-dimensional systems (world scientific, 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 nanostru