chiral m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Three major categories of nanotube structures can be identified based on the values of m and n

Transcription

1 zigzag armchair Three major categories of nanotube structures can be identified based on the values of m and n m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Nature 391, 59, (1998) chiral

2 J. Tersoff, APL, 74, 1, (99) a) Graphite Valence(π) and Conduction (π*) states touch at 6 Fermi points Carbon nanotube: Quantization from the confinement of electrons in the circumferential direction b) (3,3) CNT; allowed energy states of CNT cuts pass through Fermi point metallic c) (4,) CNT; no cut pass through a K point semiconducting In general, for a chiral tubule, we have the following results: n - m = 3q metallic, no gap n - m 3q semiconductor with gap E gap 4hv = 3d F CNT circumference = nλ F.46 n + nm + m d CNT = π nm

3 .5eV

4 Cross-section view of the vibration modes Determination of the tube diameter from A 1g Raman vibration frequency Symmetric stretch Asymmetric stretch One can then guess a set of (m,n) from Figs and n + nm + m d CNT = π nm

5 A SWCNT CMOS device 1. Two p-type CNT FETs in series. Potassium bombardment on the unprotected one results in a p n conversion 3. CMOS CNT FET with gain (V out /V in ) > 1

6 Introduction to Nanotechnology Chapter 9 Quantum Wells, Wires and Dots Lecture ChiiDong Chen Institute of Physics, Academia Sinica chiidong@phys.sinica.edu.tw

7 Size effect: For a nano-meter cube, the surface to volume ratio increases with decreasing size For an FCC cubic: N surface = 1 n N volume = 8n 3 + 6n + 3n n = number of atoms along edges d = na, a = lattice constant For GaAs, a=0.565 nm 1E+7 1E+6 N volume 1E+5 N surface 1E+4 1E+3 1E+ 1E size d in nm 60 N surface /N volume in% size d in nm 60

8 Charge motion in a conductor or semiconductor with periodic crystal potential: Resistance arises from scattering with phonons and defects Force on an electron F=eE Momentum P=mv=F t; t=τ v = τee/m τ = average scattering time mean free path l = v F τ Ohm s law: current density j = nev = ne τe m j σe σ = ne τ m 1 = σ m ne τ ρ mobility: v τe µ = E m potassium Scattering time: For T>>Θ D : ρ ~ T = + τ τ L τ i For T<< Θ D : ρ ~ T 3 For T<<< Θ D : ρ ~ T 5 Barron, R. F., Cryogenic Systems, nd Edition, Oxford University Press, New York, Residual resistivity: 1% atomic impurity = 1µΩ-cm

9 Types of defects: 1. mission atoms vacancies. extra atoms interstitial atoms 3. a vacancy interstitial atom pair Frenkel defect in semiconductors: doping level of ~10 18 donors/cm ~10 3 conduction electrons in (100nm) 3 cube 10 cubes share one electron

10 Sec Dimensionality Example: a D Cu film: 10cm 10cm 3.6nm 0% of atoms are in unit cells at the surface confinement of electron in vertical direction Length scales for electron motion: mean free path; Fermi wavelength Relevant scale: Fermi wavelength

11 Sec Fermi Gas and Density of States Classical description: Momentum p = mv Kinetic Energy E = mv / = p /m Quantum description: p x = hk x All conduction electrons are equally spread out in the k space (reciprocal space) Available space in k dn ( E) de

12 Confined electron wavefunction in a infinite square well n = kf π a a m = π h 1 E n 1 c.f. 1D in Table A Let L=a, do not need to consider spin E n = π h ma n ψ n = cos(nπx/a) n = 1,3, 5, even parity ψ n (-x) = ψ n (x) ψ n = sin(nπx/a) n =,4, 6, odd parity ψ n (-x) = -ψ n (x) x = -a/ x =0 x = a/ Probability of finding an electron at a particular value of x = Ψ n (x)

13 Energy levels for a 1D parabolic potential well Fig V ( x) = kx En ψ 1 = n hω 0 n ( ) ( ) n ω 0 x = H x e = αx k m Hermite polynomials H n (x) n=

14 Degeneracy: Energy of a D infinite rectangular square π h ma n = n n x + y = ( ) E E 0 n eq. 9.9 Degeneracy (including spin states) : n degeneracy n x, n y n x, n y n x, n y n x, n y 1 4 0,1 1,0 4 0,, ,3 3,0 a ,4 0,5 4,0 5,0,3 3,

15 N(E) and D(E) in 1D, D and 3D

16 Measurement of electronic density of state at the Fermi level: D(EF) 1. heat capacity at low temperature C el = π D(E F )k B T/3. Pauli susceptibility χ el = µ B D(E F ) note: 1. χ M/H. no temperature dependence 3. Spectrum of e-beam induced X-ray Other methods: Photoemission spectroscopy, Seebeck effect, tunneling effect

17 Excitons: Rydberg series Radius of an exciton: a eff = (ε/ε 0 ) / (m*/m 0 ) nm In semiconductors, large ε, screening effect reduced e-h interation a eff >> lattice spacing Mott-Wannier exciton For GaAs: ε/ε 0 = 13., m*/m 0 =0.067 a eff = 10.4 nm d >> a eff : no confinement d > a eff : weak confinement d < a eff : strong confinement Increasing e-h interaction blue shift in optical absorption

18 Single Electron Tunneling Capacitance of a dielectric disk : C = 8ε 0 ε r r Capacitance of a dielectric sphere : C = 4ε 0 ε r r For a GaAs sphere, C = r farad for radius r in nm p. 801 C 7aF for ε r =10

19 pa Ct zf Cd 1.8 Vt af Rt 4.40 GOhm Rd 7.0 MOhm Cg 0.01 af Vp Cg = Vg V Vg Ct + Cd

20 Asymmetric SET (simulations) Coulomb Staircase R 1 = 50 kω, R = 1 MΩ, C 1 =0.fF, C =0.15fF, C g =16aF, E C =.54K, T=0mK appears when R 1 C 1 R C V g =0 V SD =e/c Σ =0.4mV appears when R 1 R e/(rc) large I SD (na) 4E C /e e/c small I SD (pa) slope = (C g /C Σ )/R slope = (C g /C Σ )/R 1 V SD (mv) V g (mv) e/c g

21 ϕ = i e C eff Π i k Π x x 1 x = + C C 1 0 C eff = + C 0 4CC0

22 IV characteristics for 1D array with C=100aF, R=0kΩ I (na) # of islands # of islands Vb (mv) Vth in mv e/c=0.8mv 10

23 APL, 70, 859 (97)

24 Laser: light amplification by stimulated emission of light Monochromatic, coherence Requires: 1. Atoms with discrete energy levels for laser emission transition. Population inversion Helium-Neon Neodymium -YAG Quantum dot laser : Quantum dots = atoms 1-5 mm 4-60 µm 1 lasing

25 Superconductivity A vortex core ξ λ type II type I enclose one flux quantum H ~ exp(r/λ) h Φ0 = = e 15 Tm = λ at H C 1, ξ at H C

26 Josephson effect j = j c sin( φ) S I S P. 160 J1=tip to Pb particle J=Pb particle to other Pb particles for Pb = 1.5±0.1meV