1. 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”

2. 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 !!

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

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

5. From E-k to Density of StatesΣ1
dNs = D(E)dE =  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

6. 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

7. 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)

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

9. Quantizing k along confinement direction
Source: Dragica Vasileska, ASU

10. 2-D  step function DOS E E ~ q(E-Ec) ky kx DOS 2D ħ2(kx2 + ky2)
2mc
(Bottommost subband only, included in Ec)

11. 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

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

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

14. 1-D
~ 1/(E-Ec)1/2 E E kx
DOS
E =
2mc
ħ2kx2

15. 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)

16. 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))

17. 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

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

19. 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

20. Semi-metallic bands

21. Recap: Graphene bandstructureky
(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

22. 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)

23. Quantizing k along circumference

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

25. Choosing the Chiral vector
Different Circumference vectors give different tubes

26. Graphene subbands  CNT bandsrc
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

27. 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)
, 1993.

28. 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

29. 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

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

31. 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 = kW/√ns kx

32. 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