Three major categories of nanotube m = n “Armchair” structures can be identified based m = 0 or n = 0 “Zigzag” on the values of m and n m ≠ n “Chiral” Nature 391, 59, (1998)
chiral
zigzag armchair J. Tersoff, APL, 74, 2122, (99) http://www.ece.eng.wayne.edu/~jchoi/06012004.pdf
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 circumference = nλ b) (3,3) CNT; allowed energy states of CNT cuts pass through Fermi point → metallic F c) (4,2) CNT; no cut pass through a K point → semiconducting 4hvF Egap = 3dCNT In general, for a chiral tubule, we have the following results: 2.46 n2 + nm + m2 n - m = 3q metallic, no gap d = nm n - m ≠ 3q semiconductor with gap CNT 2π ≈2.5eV Cross-section view of the vibration modes
Determination of the tube diameter
from A1g Raman vibration frequency
Symmetric stretch Asymmetric stretch
One can then “guess” a set of (m,n) from
2.46 n2 + nm + m2 d = nm Figs. 5-19 and 5-20 CNT 2π A SWCNT CMOS device
1. Two p-type CNT FETs in series 2. Potassium bombardment on the unprotected one results in a p→n conversion
3. CMOS CNT FET with gain ≡ (Vout/Vin) > 1 Introduction to Nanotechnology
Chapter 9 Quantum Wells, Wires and Dots Lecture 2
ChiiDong Chen Institute of Physics, Academia Sinica [email protected] 02 27896766 Size effect: For a nano-meter cube, the surface to volume ratio increases with decreasing size
2 For an FCC cubic: Nsurface = 12 n n = number of atoms along edges 3 2 d = na, a = lattice constant Nvolume = 8n + 6n + 3n
For GaAs, a=0.565 nm 1E+7 N 1E+6 volume
1E+5 Nsurface 1E+4 80 in% 60 1E+3
volume 40 1E+2 /N 20
surface 0 1E+1 N 0 10 20 30 40 50 60 0 5 10 15 20 25 30 35 40 45 50 55 60 size d in nm size d in nm 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 = vF τ
Ohm’s law: ρ 2 σ 2 ne τE ne τ σ1 m current density j = nev = j ≡ σE →→= ≡ = 2 m m ne τ v τe mobility: µ ≡ = E m potassium
τ1 1 1 Scattering time: = τ + L τ i For T>>ΘD: ρ ~ T
3 For T<< ΘD : ρ ~ T
5 For T<<< ΘD : ρ ~ T Barron, R. F., Cryogenic Systems, 2nd Edition, Oxford University Press, New York, 1985 http://www.electronics-cooling.com/html/2001_august_a3.html Residual resistivity: 1% atomic impurity = 1µΩ-cm Types of defects: 1. mission atoms ≡ vacancies 2. extra atoms ≡ interstitial atoms 3. a vacancy – interstitial atom pair ≡ Frenkel defect in semiconductors: doping level of 1014~1018 donors/cm3 → 10-1~103 conduction electrons in (100nm)3 cube
10 cubes share one electron Sec. 9.3.2 Dimensionality Example: a 2D Cu film: 10cm×10cm×3.6nm 20% 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 Sec. 9.3.3 Fermi Gas and Density of States Classical description: Momentum p = mv Kinetic Energy E = mv2/2 = p2/2m Quantum description: px = h kx All conduction electrons are equally spread out in the k space (reciprocal space) Available space in k
dN(E) dE Confined electron wavefunction in a infinite square well
1 2k a 2m 2 1 π F 2 n = = 2 En 2 a π h c.f. 1D in Table A2 Let L=a, do not need to consider spin
2 2 π h 2 En = 2 n 2ma
ψn = cos(nπx/a) n = 1,3, 5,… even parity
ψn (-x) = ψn (x)
ψn = sin(nπx/a) n = 2,4, 6,… odd parity
ψn (-x) = -ψn (x)
Probability of finding an electron x = -a/2 x =0 x = a/2 2 at a particular value of x = Ψn (x) Energy levels for a 1D parabolic potential well
1 2 1 k V (x) = kx En = n − hω0 ω = 2 2 0 m αx2 Fig. 9.14 − 2 ψ nn()xHxe= ()
Hermite polynomials Hn(x) n= 0 1 2 3 4 5 6 7 8 9 10 11 Degeneracy:
Energy of a 2D infinite rectangular square
π 2 2 ∞ E = h ()n2 + n2 = E n2 n 2 x y 0 2ma eq. 9.9
Degeneracy (including spin states) :
nndegeneracy x, ny nx, ny nx, ny nx, ny 1 4 0,1 1,0 2 4 0,2 2,0 3 4 0,3 3,0 4 4 0,4 4,0 a 5 8 0,5 5,0 2,3 3,2 N(E) and D(E) in 1D, 2D and 3D Measurement of electronic density of state at the Fermi level: D(EF)
2 2 1. heat capacity at low temperature Cel = π D(EF)kB T/3
2 note: 2. Pauli susceptibility χel = µB D(EF) 1. χ≡M/H 2. no temperature dependence
3. Spectrum of e-beam induced X-ray
Other methods: Photoemission spectroscopy, Seebeck effect, tunneling effect Excitons: Radius of an exciton: aeff = 0.0529 (ε/ε0) / (m*/m0) nm Rydberg series In semiconductors, large ε, → screening effect → reduced e-h interation
→ aeff >> lattice spacing Mott-Wannier exciton
For GaAs: ε/ε0 = 13.2, m*/m0 =0.067 → aeff = 10.4 nm
Increasing d >> a : no confinement eff e-h interaction d > a : weak confinement eff blue shift
d < aeff: strong confinement in optical absorption Single Electron Tunneling
Capacitance of a dielectric disk : C = 8ε0εr r Capacitance of a dielectric sphere : C = 4ε0εr r For a GaAs sphere, C = 1.47 × 10-18 r farad for radius r in nm
p. 801
C≈7aF for εr=10 40 30 20 10 0 pA -10 -20 -30 -40 -300 -200 -100 0 100 200 300 Vt Ct 350.0 zF Cd1.8 aF Cg0.01 aF Vp C = g Rt 4.40 Rd 7.20 GOhm MOhm Vg3.87 V Vg Ct + Cd Asymmetric SET (simulations)
Coulomb Staircase
R1= 50 kΩ, R2= 1 MΩ, C1=0.2fF, C2=0.15fF, Cg=16aF, EC=2.54K, T=20mK appears when R C ≠R C appears when R1≠R2 1 1 2 2 VSD=e/CΣ=0.42mV
Vg=0
e/(RC)large 2 1
R R (nA) / /
) ) Σ
(pA) Σ SD
C C /
I / g
SD g I
C C
4E /e ( ( C
= = = e/C small e e
p p o o
l l
s s
V (mV) e/C VSD (mV) g g e i−k −ϕi = Π Ceff
2 Π ≡ x − x −1 2 Ceff = C0 + 4CC0 x =1+ C0 2C IV characteristics for 1D array with C=100aF, R=20kΩ
# of islands
80 6 7 8 9 3 4 5 60 2 40 1 7 20 6 0 5 I (nA) 4 -20 3 2 -40 Vth in mV 1 0 -60 1 2 3 4 5 6 7 8 9 10 -80 # of islands -25 -20 -15 -10 -5 0 5 10 15 20 25 Vb (mV)
e/C=0.8mV APL, 70, 859 (97) Laser: light amplification by stimulated emission of light
Monochromatic, coherence
Requires: 1. Atoms with discrete energy levels for laser emission transition 2. Population inversion Helium-Neon Quantum dot laser : Quantum dots = atoms Neodymium -YAG
4-60 µm m m 5 1-
12× lasing Superconductivity A vortex core
ξ λ
H ~ exp(r/λ)
enclose one flux quantum type I h Φ = = 2.0678×10−15Tm2 0 2e type II
= λ at HC1, ξ at HC2 • Josephson effect SIS
j = jc sin(∆φ)
P. 12260
J1=tip to Pb particle J2=Pb particle to other Pb particles
∆ for Pb = 1.25±0.1meV