Carbon Nanotubes, Graphene, and Quantum Dots - Design Principles for High Performance
Upon completing this chapter, you will be able to explain:
Nanomaterials (Nanomaterials) are materials in which at least one dimension lies roughly in the 1-100 nm range. In this size regime, qualitatively different properties emerge compared with bulk materials. The origins are mainly the following two:
In a gold nanoparticle 10 nm in diameter, about 16% of all atoms are exposed at the surface (we compute this in Section 3.5). Whereas bulk gold (with a surface-atom fraction close to 0%) is chemically inert, gold nanoparticles catalyze CO oxidation at room temperature. Surface atoms are coordinatively unsaturated and highly reactive, and this difference gives rise to the catalytic, optical, and electronic properties of nanomaterials.
Nanomaterials are classified by the "number of directions confined to the nanoscale." For each confined direction the electrons lose a degree of freedom, changing the shape of the density of states (Density of States, DOS).
flowchart LR
A[Dimensional classification] --> B[0D
Quantum dots]
A --> C[1D
Nanotubes / nanowires]
A --> D[2D
Graphene / nanosheets]
B --> B1[Confined in 3 directions
discrete levels]
C --> C1[Confined in 2 directions
conduction along 1]
D --> D1[Confined in 1 direction
conduction in-plane]
style A fill:#f093fb
style B fill:#e3f2fd
style C fill:#e8f5e9
style D fill:#fff3e0
| Dimension | Confined directions | Representative examples | Density-of-states feature |
|---|---|---|---|
| 0D (zero-dimensional) | All 3 directions | Quantum dots, fullerenes | Discrete, delta-function-like |
| 1D (one-dimensional) | 2 directions | Carbon nanotubes, nanowires | 1/√E divergence (van Hove singularities) |
| 2D (two-dimensional) | 1 direction | Graphene, transition-metal dichalcogenides | Step-function-like (constant) |
| 3D (bulk) | None | Crystalline / polycrystalline solids | Proportional to √E |
For a sphere of radius r, the ratio of surface area S = 4πr² to volume V = (4/3)πr³ is given by:
That is, the specific surface area increases in inverse proportion to the radius. Reducing the particle size by a factor of 10 increases the surface area per unit volume tenfold, dramatically raising the fraction of atoms exposed at the surface. This geometric fact underlies the high catalytic activity, solubility, and reactivity of nanomaterials. Quantitative calculations are covered in the Python practice of Section 3.5.
A carbon nanotube (Carbon Nanotube, CNT) is a one-dimensional structure formed by rolling a graphene sheet into a cylinder. The direction of rolling is expressed by the chiral vector (chiral vector), specified as an integer pair (n, m) using the primitive translation vectors a₁, a₂ of graphene:
This (n, m) determines all of the CNT's geometry and electronic properties. The rolling direction gives three types:
The diameter d and chiral angle θ follow analytically from (n, m):
The most important property of CNTs is that whether a tube is metallic or semiconducting is fixed by its geometry alone. The rule can be stated as:
This rule follows from the band structure of graphene (Section 3.3). Graphene is a semimetal whose valence and conduction bands touch at the K points; in a CNT the circumferential wavenumber is quantized, so metallicity depends on whether the allowed wavenumber lines pass through a K point. Statistically, about 1/3 of randomly synthesized CNTs are metallic and 2/3 are semiconducting (we verify this in the Python practice of Section 3.5).
The band gap E_g of a semiconducting CNT is roughly inversely proportional to the diameter d, on the order of E_g ≈ 0.8 eV / d[nm]. A CNT 1 nm in diameter has about 0.8 eV, close to silicon (1.1 eV), making it promising as a transistor channel material. The ability to design the band gap by choosing the diameter is one of the attractions of CNTs.
CNTs exhibit outstanding properties owing to their strong sp² carbon-carbon bonds:
The dominant industrial synthesis method is chemical vapor deposition (Chemical Vapor Deposition, CVD). Hydrocarbon gases (methane, ethylene, etc.) are supplied over nanoparticles of a catalyst metal (Fe, Co, Ni) and decomposed and precipitated at 600-1000°C to grow CNTs.
CVD can control the diameter to some degree, but making the chirality ((n,m)) uniform remains difficult. Obtaining only semiconducting tubes at high purity requires post-synthesis separation and purification (density-gradient ultracentrifugation, gel chromatography, etc.). Direct synthesis of single-chirality CNTs is still an active research topic.
Graphene (Graphene) is a two-dimensional material one atom thick in which carbon atoms form a honeycomb lattice (honeycomb lattice) through sp² bonding. In 2004, Geim and Novoselov isolated it by mechanical exfoliation using adhesive tape, work that led to the 2010 Nobel Prize in Physics.
The honeycomb lattice has two carbon atoms per unit cell (the A and B sublattices). Solving it in the tight-binding approximation (tight-binding) shows that the valence and conduction bands touch at the six K points (and K' points) of the Brillouin zone.
The decisive feature of graphene is that near the K points the energy disperses linearly with wavenumber:
This touching point is called the Dirac point (Dirac point). Whereas ordinary semiconductors have parabolic dispersion E ∝ k², graphene's linear dispersion makes electrons behave as if they were massless relativistic particles (Dirac fermions). The Fermi velocity v_F is about 1/300 of the speed of light.
Because graphene's valence and conduction bands touch at a single point, it is a semimetal (semimetal) with a band gap that is exactly zero. This yields high mobility, but is a weakness for transistors since no off state can be created. Research is underway to open a gap by forming nanoribbons, bilayers, or through interactions with the substrate.
| Method | Quality | Area / quantity | Main applications |
|---|---|---|---|
| Mechanical exfoliation (Scotch-tape method) | Highest quality, minimal defects | Tiny flakes only | Basic research, property measurement |
| CVD (growth on Cu foil) | High quality, large area | Roll-to-roll capable | Transparent electrodes, flexible devices |
| SiC thermal decomposition (epitaxial growth) | High quality | Wafer scale | High-frequency electronic devices |
| Reduced graphene oxide (rGO) | More defects, low cost | Mass production, solution process | Conductive inks, composites, electrodes |
A quantum dot (Quantum Dot, QD) is a semiconductor nanocrystal roughly 2-10 nm in diameter. Because the motion of electrons and holes is confined in all three directions to the crystal size, the energy levels become discrete and the band gap varies with particle size. This phenomenon is called the quantum confinement effect (Quantum Confinement Effect).
Confinement becomes pronounced when the crystal radius falls to about the exciton Bohr radius (exciton Bohr radius) or below. In CdSe the exciton Bohr radius is about 5.6 nm, and once the particle size drops below this the band gap widens markedly.
The most basic picture is the quantum-mechanical "particle in a box (particle in a box)." The ground-state energy of a particle confined in a three-dimensional infinite well of side L increases as E ∝ 1/L². In other words, the smaller the box, the higher the ground level, so the smaller the particle, the wider the band gap.
The Brus equation (Brus equation) quantifies this intuition for semiconductor nanocrystals. The effective band gap of a spherical nanocrystal of radius R is:
The right-hand side has three terms:
As the particle shrinks, the second term (1/R²) grows faster than the third term (1/R), so the net band gap widens and the emission wavelength shifts toward shorter wavelengths (toward blue). We compute this relationship in Python in Section 3.5.
Because the band gap is set by particle size, quantum dots have the striking feature that the emission color can be tuned continuously simply by changing the particle size at fixed composition. In CdSe quantum dots, changing the size from about 2 nm to 6 nm shifts the emission color from blue to red.
Many high-performance quantum dots contain cadmium, as in CdSe and CdTe, raising issues of toxicity and environmental regulation (RoHS, etc.). Cadmium-free alternatives such as InP-based dots, carbon dots, and perovskite quantum dots are being studied, but they can fall short of Cd-based dots in emission efficiency and stability, so materials development continues.
In this section we actually compute the three themes covered above. All code is self-contained using only NumPy, with execution results shown alongside.
From the (n, m) indices we determine the diameter, chiral angle, geometric type, and metallic/semiconducting character. We confirm the rule (n − m) mod 3 = 0 for metallic tubes, and that enumerating all 0 ≤ m ≤ n ≤ 20 gives a metallic fraction of about 1/3.
# ===================================
# Example 1: CNT chirality classification
# ===================================
import numpy as np
a = 0.246 # graphene lattice constant [nm]
def cnt_diameter(n, m):
"""Compute the CNT diameter from chiral indices (n, m) [nm]"""
return a * np.sqrt(n**2 + n*m + m**2) / np.pi
def chiral_angle(n, m):
"""Compute the chiral angle [degree]"""
return np.degrees(np.arctan(np.sqrt(3)*m / (2*n + m)))
def cnt_type(n, m):
"""Determine metallic vs. semiconducting character"""
if (n - m) % 3 == 0:
return "metallic"
return "semiconducting"
def cnt_class(n, m):
"""Geometric classification (armchair / zigzag / chiral)"""
if m == 0:
return "zigzag"
if n == m:
return "armchair"
return "chiral"
# Classify representative (n, m)
examples = [(5,5), (9,0), (10,0), (7,3), (6,4), (10,10), (8,4), (11,7)]
print("(n, m) type geometry d [nm] theta [deg]")
print("-" * 58)
for n, m in examples:
print(f"({n:2d},{m:2d}) {cnt_type(n,m):14s} {cnt_class(n,m):9s} "
f"{cnt_diameter(n,m):5.3f} {chiral_angle(n,m):5.2f}")
# Enumerate all (n, m) with 0 <= m <= n <= 20 and tally the metallic fraction
total = 0
metal = 0
for n in range(1, 21):
for m in range(0, n + 1):
total += 1
if (n - m) % 3 == 0:
metal += 1
print()
print(f"Enumerated (n,m) with 0<=m<=n<=20: {total}")
print(f"Metallic count: {metal} ({100*metal/total:.1f}%)")
# Execution result:
# (n, m) type geometry d [nm] theta [deg]
# ----------------------------------------------------------
# ( 5, 5) metallic armchair 0.678 30.00
# ( 9, 0) metallic zigzag 0.705 0.00
# (10, 0) semiconducting zigzag 0.783 0.00
# ( 7, 3) semiconducting chiral 0.696 17.00
# ( 6, 4) semiconducting chiral 0.683 23.41
# (10,10) metallic armchair 1.356 30.00
# ( 8, 4) semiconducting chiral 0.829 19.11
# (11, 7) semiconducting chiral 1.231 22.69
#
# Enumerated (n,m) with 0<=m<=n<=20: 230
# Metallic count: 83 (36.1%)
The armchair tubes (5,5) and (10,10) are always metallic (n − m = 0), and the diameter of (10,10) is twice that of (5,5). Over the full enumeration, 83 of 230 tubes are metallic (36.1%), in good agreement with the theoretical "about 1/3 metallic."
For CdSe quantum dots, we vary the radius from 1.5 to 4.0 nm and compute the band gap and emission wavelength. We confirm that smaller particles have a wider gap and emit at shorter (bluer) wavelengths.
# ===================================
# Example 2: Brus equation (quantum dots)
# ===================================
import numpy as np
# Physical constants (SI units)
h = 6.62607015e-34 # Planck constant [J s]
me0 = 9.1093837015e-31 # electron rest mass [kg]
e = 1.602176634e-19 # elementary charge [C]
eps0 = 8.8541878128e-12 # vacuum permittivity [F/m]
c = 2.99792458e8 # speed of light [m/s]
# CdSe parameters
Eg_bulk = 1.74 # bulk band gap [eV]
m_e = 0.13 * me0 # electron effective mass
m_h = 0.45 * me0 # hole effective mass
eps_r = 10.6 # relative permittivity
def brus_gap_eV(R_nm):
"""Compute the effective band gap with the Brus equation [eV]"""
R = R_nm * 1e-9
confinement = (h**2 / (8 * R**2)) * (1/m_e + 1/m_h) / e # quantum confinement term [eV]
coulomb = 1.8 * e / (4 * np.pi * eps0 * eps_r * R) # Coulomb term [eV]
return Eg_bulk + confinement - coulomb
def emission_nm(Eg_eV):
"""Compute the emission wavelength from the band gap [nm]"""
return h * c / (Eg_eV * e) * 1e9
print("CdSe quantum dot: Brus equation")
print(f"{'radius [nm]':>11} {'diameter [nm]':>13} {'E_gap [eV]':>11} {'lambda [nm]':>12}")
print("-" * 50)
for R in [1.5, 2.0, 2.5, 3.0, 3.5, 4.0]:
Eg = brus_gap_eV(R)
lam = emission_nm(Eg)
print(f"{R:11.1f} {2*R:13.1f} {Eg:11.3f} {lam:12.1f}")
print()
print(f"Bulk CdSe gap {Eg_bulk} eV -> lambda {emission_nm(Eg_bulk):.1f} nm")
# Execution result:
# CdSe quantum dot: Brus equation
# radius [nm] diameter [nm] E_gap [eV] lambda [nm]
# --------------------------------------------------
# 1.5 3.0 3.234 383.4
# 2.0 4.0 2.550 486.3
# 2.5 5.0 2.239 553.8
# 3.0 6.0 2.073 598.2
# 3.5 7.0 1.974 627.9
# 4.0 8.0 1.912 648.5
#
# Bulk CdSe gap 1.74 eV -> lambda 712.6 nm
At a radius of 1.5 nm the emission wavelength is 383 nm (violet), and at 4.0 nm it is 649 nm (red), so particle size alone can cover the entire visible range. Compared with bulk CdSe (712 nm, near-infrared), nanocrystallization greatly widens the band gap. This monotonic size-emission relationship is the basis of color design in displays.
Because the Brus equation is based on the effective-mass approximation, it tends to overestimate the band gap in the strong-confinement regime where the radius drops below 1 nm. This is why the calculation above starts at a radius of 1.5 nm. Smaller particles require tight-binding or first-principles methods.
For a spherical nanoparticle, we treat atoms within a shell one atomic layer thick from the surface as "surface atoms" and compute their fraction as a function of particle size. We confirm that the surface-atom fraction rises sharply as the particle shrinks.
# ===================================
# Example 3: Surface-atom fraction
# ===================================
import numpy as np
# Shell model: atoms within one atomic diameter of the surface are "surface" atoms.
# F_surface = 1 - ((R - t)/R)^3, t = surface-shell thickness [nm]
r_atom = 0.144 # metallic radius of gold [nm]
t = 2 * r_atom # surface-shell thickness (one atomic diameter) [nm]
def surface_fraction(D_nm):
"""Surface-atom fraction of a spherical nanoparticle of diameter D"""
R = D_nm / 2.0
if R <= t:
return 1.0
return 1.0 - ((R - t) / R)**3
def n_total(D_nm):
"""Approximate total atom count (assuming fcc with packing fraction 0.74)"""
R = D_nm / 2.0
return 0.74 * (R / r_atom)**3
print("Gold nanoparticle: surface-atom fraction (r_atom = 0.144 nm)")
print(f"{'diameter [nm]':>13} {'~N_total':>10} {'surface fraction':>17}")
print("-" * 44)
for D in [1, 2, 5, 10, 20, 50, 100]:
print(f"{D:13d} {n_total(D):10.0f} {surface_fraction(D)*100:16.1f}%")
# Execution result:
# Gold nanoparticle: surface-atom fraction (r_atom = 0.144 nm)
# diameter [nm] ~N_total surface fraction
# --------------------------------------------
# 1 31 92.4%
# 2 248 63.9%
# 5 3872 30.7%
# 10 30978 16.3%
# 20 247825 8.4%
# 50 3872258 3.4%
# 100 30978063 1.7%
At a diameter of 1 nm, more than 90% of all atoms are exposed at the surface, whereas at 100 nm only 1.7% are. This sharp rise in the surface-atom fraction underlies size-dependent properties of gold nanoparticles such as catalytic activity, melting-point depression, and plasmonic coloration. The relation S/V = 3/r from Section 3.1.3 is seen to act in the same way at the level of atom counts.
Upon completing this chapter, you can explain the following:
Is the carbon nanotube with chiral indices (12, 6) metallic or semiconducting? Answer using the rule.
Correct answer: metallic
Explanation:
The rule is "metallic if (n − m) is a multiple of 3."
n − m = 12 − 6 = 6 = 3 × 2, which is a multiple of 3. Therefore (12, 6) is judged metallic.
Note that even when n = 2m, only the value of n − m matters for the determination.
Classify the following nanomaterials as 0D, 1D, or 2D: (a) graphene, (b) CdSe quantum dot, (c) single-walled carbon nanotube.
Correct answer:
Corresponding to the number of confined directions (3, 1, 2), the number of freely moving directions is 0, 2, 1.
Two quantum dots made of the same CdSe are given: A (radius 2.0 nm) and B (radius 3.5 nm). Which emits at the shorter (bluer) wavelength? State the reason.
Correct answer: A (radius 2.0 nm)
Explanation:
The quantum confinement term scales as 1/R², so the smaller the radius, the wider the band gap. A wider band gap means higher-energy emitted photons and thus shorter wavelength.
In the calculation of Section 3.5.2, the emission wavelength was 486 nm at radius 2.0 nm and 628 nm at radius 3.5 nm. Therefore the smaller dot A emits at the shorter wavelength (blue-green) and B at the longer wavelength (red).
Compute the diameter of the armchair (10, 10) CNT, using the lattice constant a = 0.246 nm and the formula d = (a/π)·√(n² + nm + m²).
Answer:
Substitute n = m = 10.
n² + nm + m² = 100 + 100 + 100 = 300
√300 ≈ 17.32
d = (0.246 / π) × 17.32 = 0.0783 × 17.32 ≈ 1.356 nm
This matches the code output of Section 3.5.1 ((10,10) → 1.356 nm). For reference, (5,5) gives √75 ≈ 8.66 and d ≈ 0.678 nm, exactly half the diameter.
A gold nanoparticle 10 nm in diameter has a surface-atom fraction of about 16%, and one 2 nm in diameter about 64%. When used as a catalyst, explain from the standpoint of surface effects why smaller particles tend to be advantageous. Also give one drawback of making them too small.
Sample answer:
Why smaller particles are advantageous for catalysis:
Catalytic reactions proceed on the atoms exposed at the surface. The higher the surface-atom fraction, the more atoms of the same mass of gold can participate in the reaction, raising the activity per unit mass. Surface atoms also have low coordination numbers (coordinatively unsaturated) and readily adsorb and activate reactant molecules, so they act as active sites. By the relation S/V = 3/r, the smaller the particle, the larger the specific surface area.
Drawback of making them too small (any one):
For a QD display that uses a blue backlight (450 nm), design CdSe quantum dots that emit green (about 530 nm) and red (about 630 nm). Referring to the results of Section 3.5.2, estimate the approximate radius required for each, and also state two advantages of using quantum dots in a display compared with organic phosphors.
Answer:
Radius estimates (interpolated from the table in 3.5.2):
| Target emission wavelength | Nearby points in table | Estimated radius |
|---|---|---|
| Green ~530 nm | 2.0 nm→486 nm, 2.5 nm→554 nm | about 2.3-2.4 nm |
| Red ~630 nm | 3.5 nm→628 nm | about 3.5 nm |
Green is obtained with dots of radius about 2.3 nm and red with radius about 3.5 nm. The scheme is that the green and red dots absorb the blue backlight and re-emit in their respective colors.
Advantages over organic phosphors (any two):
Note: In practice, cadmium-free versions (e.g., InP-based) and synthesis control to narrow the size distribution are challenges.
In Chapter 3 we learned the fundamentals of nanomaterials (size effects and dimensionality), the structures and properties of carbon nanotubes, graphene, and quantum dots, and how to compute their properties in Python. In Chapter 4, we address the design principles for integrating these nanomaterials and advanced materials into real devices and systems.