Subbands So far, we saw how to calculate bands for solids Boundary conditions we used were artificial periodic bcs Now we’ll see what happens if you have real boundaries Quantization of bands along the confinement direction  Subbands If we confine all 3 directions, the levels are fully quantized, as in an atom Can then calculate “density of states” and “number of modes”

Material to transport - New Ohm’s Law !! Resistor Waveguide L > 1 mm Semiclassical Transport/ “Meso”scopics (Complicated!) L ~ 100s nm Ballistic QM “Nano” (Cleaner and thus Simpler !) R = (h/2q2MT) source drain L ~ 10 nm Drift-Diffusion (“Macro/Micro”) (Obsolete !) R = rL/A Resistance of two elements in series can be smaller than the sum !!

Where are the states? E dE dE dk k x k For 1D parabolic bands, DOS peaks at edges ~1/(E-Ec)

Increasing Dimensions dE dE dE dk # k points increases with diameter squared D ~ q(E-Ec) 2pkdk . k k In higher dimensions, DOS has complex shapes

From E-k to Density of States Σ1 dNs = D(E)dE = 2  2 (dk/[2p/L]) for each dimension k For E = Ec + ħ2k2/2mc D = (Wmc/2p2ħ3)[2mc(E-Ec)]1/2 in 3-D = (Smc/2pħ2)q(E-Ec) in 2-D = (mcL/pħ)/√2mc[E-Ec] in 1-D

3-D DOS E E ~ (E-Ec)1/2 ky kz Ec kx DOS E = + Ec ħ2(kx2 + ky2 + kz2) Hard to draw a 3-D paraboloid ! ~ (E-Ec)1/2 ky kz Ec kx DOS E = + Ec ħ2(kx2 + ky2 + kz2) 2mc

Quasi 2D  subbands E ~ Spq(E-Ec-p2ez) ky kx DOS Quasi-2D d ħ2(kx2 + ky2) 2mc Subband bottoms quantized due to confinement along z ez ≈ ħ2p2/2mcd2 (like modes in a waveguide)

Thin Films E1 = + + Ec ħ2(kx-k0);2 2ml ħ2(ky2+kz2) 2mt ml=0.91m0, mt=0.19m0

Quantizing k along confinement direction Source: Dragica Vasileska, ASU

2-D  step function DOS E E ~ q(E-Ec) ky kx DOS 2D ħ2(kx2 + ky2) 2mc (Bottommost subband only, included in Ec)

Quasi 1D E E kx DOS Quasi-1D ħ2kx2 E = + Ec + p2ez + q2ey 2mc (p,q=0,1,2,3,…) E = + Ec + p2ez + q2ey 2mc ħ2kx2 Confinement in 2 directions, 1D subbands

Silicon Nanowire Wang, Rahman, Ghosh, Klimeck, Lundstrom, APL ‘05

Carbon Nanotube (12,0) (9,0) Kienle, Cerda, Ghosh, JAP ‘06

1-D ~ 1/(E-Ec)1/2 E E kx DOS E = 2mc ħ2kx2

General Results NT(E) = Saq(E-Ea) As each mode starts, you get a new step function D(E) = dNT(E)/dE = Sad(E-Ea)

Separable Problems = ∫dE’Dx(E’)Dy(E-E’) Convolve individual DOS to sum over its modes e(m,n) = ex(n)+ey(m) D(E) = Sm,nd(E-ex(n)–ey(m)) = ∫dE’Dx(E’)Dy(E-E’) = Sm,n∫dE’d(E-ey(n)–E’)d(E’-ex(m))

Evolution of DOS 1D wire Quasi 0-D Quantum Dot E Quasi 1D wire 2D well 0-D artifical molecule (quantum dot) 3D Bulk Solid Ec DOS

From graphite to nanotube Animation: Dr. Shigeo Maruyama Real boundary conditions (periodic) along circumference

Graphene BZ K1 = (p/a)x + (p/b)y K2 = (p/a)x - (p/b)y 6 BZ vertices, each shared by 3 unit cells (0,2p/3b) K2 K1 K1 + K2 (p/a,p/3b) (-p/a, p/3b) K1 K2 K1 + K2 Can translate three BZs (each 1/3rd valley) using RLVs (-p/a,-p/3b) (p/a,-p/3b) (0,-2p/3b) Only 2 distinct BZ valleys K1 = (p/a)x + (p/b)y K2 = (p/a)x - (p/b)y

Semi-metallic bands

Recap: Graphene bandstructure ky (0,2p/3b) kx (0,-2p/3b) E(k) = ±t √[1 + 4cos(3kxa0/2)cos(kya0√3/2) + 4cos2(kya0√3/2)] Periodic boundary conditions quantize ky If ky goes through above BZ points  metallic, else semiconducting

Recap: Graphene Bandstructure E(k) = ±|h0|, h0 = -t(1 + 2eikxacoskyb) (0,2p/3b) At kx=0, ky = ±2p/3b, E = 0 (Same for other BZs) The question is whether these BZ points are included in the set of quantized k’s (0,-2p/3b)

Quantizing k along circumference

Rolling up graphene rc = ma1 + na2 gives the circumference vector

Choosing the Chiral vector Different Circumference vectors give different tubes

Graphene subbands  CNT bands rc k.rc = m(kxa + kyb) + n(kxa-kyb) = 2pn BZ (0,±2p/3b) is included in these k points if (m-n) = 3n Then we have a metallic nanotube Otherwise semiconducting

Chirality controls metallicity SWNT as molecular interconnects: Cylindrical boundary conditions define a tube: Chiral indices (n,m) determine the band structure‡: |n-m| = 0,3,6,… , metallic; otherwise semiconducting. Reference ‡ J.W. Mintmire et al., J. Phys. Chem. Sol. 54(12) 1835-1840, 1993.

Linearize around minima E(k) = ±|h0|, h0 = -t(1 + 2eikxacoskyb) Linearize around BZ points (kx = 0,ky = ±2p/3b) by Taylor expansion -iat(-1) (-at). h0 ≈ ∂h0/∂kx|0kx + ∂h0/∂ky|2p/3b(ky-2p/3b) Zigzag tube (m,0), m(kxa+kyb) = 2pn E = √[En2 + (takx)2], En = takn, kn = (2p/3b).(3n/2m – 1) D(E) = Sn (2L/pat).[E/√(E2 – En2)] = L/pħv

Nanotubes are quasi 1-D D(E) = Sn (2L/pat).[E/√(E2 – En2)] E DOS Zigzag MetallicTube (n = 3m), allowed ks include BZ Looks like our quasi-1D results

Nanotubes are Quasi 1-D E Gap DOS Zigzag SemiconductingTube (n ≠ 3m)

Mode Counting E W L kx Modes confined along W DE = ħ2p2/2mcW2 M = Int √(EF – EC)/DE = Int(kFW/p) = Int(2W/lF) Gmax = (2q2/h)M ns = (mc/pħ2)(EF-EC) RminW = 16.28 kW/√ns kx

Summary Confinement in a solid breaks its bands into subbands Counting subbands gives us the density of states DOS Each subband mode carries a fixed current