theory and application of nanomaterials - lecture …ahrenkiel.sdsmt.edu/.../nano_702_103117.pdfs....

Outline Quantum Mechanical Description of Carbon Nanomaterials Forms of Carbon Graphene Carbon Nanotubes

Theory and Application of NanomaterialsLecture 12: Carbon Nanomaterials

S. Smith

SDSMT, Nano SE

FA17: 8/25-12/8/17

S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 1 / 22


Introduction

The carbon nano-tube (CNT) is one form which has been studied extensively. A single CNT canhave a tensile strength 25 times that of steel, may exhibit high carrier mobility orsuperconductivity, and many other extraordinary properties. The figure below illustrates thedensity-gradient ultracentrifugation method to sort carbon nanotubes by size and electrical type(images from the journal Nature Nanotechnology 1).

Figure: Carbon Nanotubes sorted by density gradient ultracentrifugation method. (left) variety of semiconducting CNTssorted by band-gap. (right) Semiconducting and metallic CNTs separated by weight after chemical-specific encapsulation. From

the journal Nature Nanotechnology1.

1Michael Arnold, et. al. Nature Nanotechnology 1 60 (2006).S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 2 / 22


Outline

1 Quantum Mechanical Description of Carbon NanomaterialsAtomic OrbitalsWavefunctions of Carbon

2 Forms of CarbonHybridizations of Carbon

3 GrapheneWavefunctions of GrapheneDispersion Relation in Graphene

4 Carbon NanotubesTypes of CNTsProperties of CNTsElectrical Properties of CNTs



Carbon Nanomaterials and atomic orbitals

Carbon is the 6th element in the periodic table. Due to the Pauli principle, the 4 lowest-energyorbitals which satisfy the Schrodinger equation are half-filled, leaving 4 states which can beshared with other elements to form covalent bonds. For this reason, carbon and it’s compoundsare ubiquitous in nature. The quantum mechanical origin of these 1/2-filled orbits follows fromanalysis of the Schrodinger equation for an electron in a spherically-symmetric potential. Thequantum mechanical derivation of the above mentioned orbitals is time-consuming, butstraightforward. We give an overview here: Given an electron in the spherically-symmetricCoulomb potential, Schrodinger’s equation takes the following form:

−~2

2µ

{1

r2

∂

∂r

(r2 ∂ψ

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂ψ

∂θ

)+

1

r2 sin2 θ

∂2ψ

∂φ2

}−

Ze2

rψ = Eψ

where µ is a reduced mass, taking into account the finite mass of the nuclei. This can berecognized to be another manifestation of the eigenvalue equation Hψ = Eψ, which is readilysolved in spherical coordinates by separation of variables, similar to the case in cartesiancoordinates treated earlier:

ψ(r , θ, φ) = R(r)Θ(θ)Φ(φ)



Solutions to Schrodingers Equation

Substitution of the above expression into Schrodinger’s equation will lead to three ordinarydifferential equations:

2

ρ

dR

dρ+

d2R

dρ2−

l(l + 1)

ρ2R +

(k

ρ−

1

4

)R = 0 ρ =

√8µ|E |~2

r k =Ze2

~

√µ

2|E |

sin θd

dθ

(sin θ

dΘ

dθ

)+ l(l + 1) sin2 θΘ−m2

l Θ = 0 where l = an integer and ml as given below

d2Φ

dφ2+ m2

l Φ = 0 with ml ∈ {−l ,−(l − 1), ..., 0, ..., l − 1, l}

The solutions of these equations are the so-called associated Laguerre polynomials Rn,l (r),

associated Legendre Polynomials P(ml )l (cos θ) and the complex exponentials Φml (φ) = e±imlφ.



Atomic Orbitals

The appropriately normalized product of these functions represents the (spatial) wavefunctionψn,l,ml

(r , θ, φ), and the quantum numbers n, l ,ml label the eigenstates and eignenergies,

En = −Z2e4µ2~2

1n2 , which miraculously agrees with the Bohr model. We note that the energy only

depends on n, the principal quantum number. The quantum number l , is often referred to inspectroscopic notation: n = 2, l = 0 −→ 2s, n = 2, l = 1 −→ 2p, etc. The wavefunction for agiven set of quantum numbers is termed an atomic orbital. As an example, the 2s and 2pzorbitals are calculated below using IDL:

Figure: Probability amplitude contours for the 2s and 2pz atomic orbitals of thehydrogen atom.



Linear Combination of Atomic Orbitals

We note here that the spin part of the wavefuntion is treated separately and would bean additional product to the ψn,l,ml (r , θ, φ) (the so-called spatial portion of thewavefunction) derived above.

The (2l + 1)-fold degenerate atomic orbitals of the hydrogen atom are a starting pointfor the description of higher elements and molecules. It is an often-used approach totake the wavefunction of higher elements and molecules as linear combinations of atomicorbitals derived from the hydrogen atom, and this is precisely the approach taken for thevalence electrons of carbon:

ψ(r) = φs(r) +∑i

λφpi (r)

where the functions φs,pi are the orbitals obtained from the Schrodinger equation for thespherically-symmetric potential of the hydrogen atom and λ is termed the couplingconstant.



Filling of Atomic Orbitals of Carbon

The Pauli principal states that electrons have a 4th quantum number S, which describes theelectron spin, whose allowed values are ±1/2, and that no two electrons may share the samequantum state. As a result, each of the 2l + 1 orbitals for the valence electrons in carbon canhold at most 2 electrons each. The filling of the first few states in carbon is shown below infigure 3, where the filled 1s states are shown in light-blue:

Figure: Atomic orbitals of Carbon. 2 electrons fill the 1s-level, with spin quantum numberS = ±1/2, shown in light-blue (designating “filled”). The remaining three 2px , 2py , 2pz orbitalsand the 2s-orbital each have a single electron, shown with S = +1/2, allowing a covalent bondwith each of the 4 empty electron levels.



Hybridization of Carbon Based Materials

In carbon-based materials, the above linear combinations of the s and p orbitals can take avariety of forms, summarized in the table below:

Table: Hybridization of carbon valence electrons

hybridization sp sp2 sp3

basis functions s, p s, px , py s, px , py , pzcoupling constant λ 1

√2

√3

bond angle 180◦ 120◦ 109.3◦

geometry linear trigonal tetrahedralexamples graphite diamond

The sp2 hybridization is stable in the layered materials such as graphite, but also in the singlesheet-form of graphene. Other stable carbon-based clusters, such as C60 also incorporate carbonin this way. The quantum mechanical description of carbon nanotubes, for instance, can beunderstood by considering the properties of graphene, where three of the four valence electronsare shared in covalent bonds of the hexagonal lattice (sp2 hybridization), and the 4th electronremains in the pz orbital and contributes a valence electron to the two dimensional crystal.



Wavefunctions in Graphene

The wave function of the bonding electrons in graphene take the form:

1√

3(ψ(2s) +

√2ψ(2pi ) where i = x , y

Following the calculation of Wallace2, we use the tight-binding approximation, which states thatthe remaining pz atomic orbitals are mixed by plane wave phase factors, essentially replacing theBloch function by the atomic orbitals of choice. We assume that only the pz orbitals describe theelecrons at each site in the unit cell of figure 4, as the other 3 electrons participate in the strongcovalent bonds described by the sp2 hybridization. The problem then amounts to the mixing ofthe wavefunctions for the two nearest neighbor atoms in the unit cell shown in figure 4:

ψ = ϕ1 + λϕ2

where ϕ1 and ϕ2 are given by the wavefunctions:

ϕ1 =∑A

e ik·rAX (r− rA) (1)

ϕ2 =∑B

e ik·rBX (r− rB) (2)

2P.R. Wallace, ”Band Theory of Graphite”, Phys. Rev. 71 (9) 622 (1947).S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 10 / 22


Wavefunctions in Graphene (continued)

where X (r− rA(B)) are given by the pz atomic orbitals, assumed to be oriented out of the plane,the positions of the A and B atoms are as shown in figure 4, and the sums are over allequivalent nearest neighbors (dashed lines).

Figure: 2D hexagonal structure of graphene, a carbon atom resides at every vertex in thehoneycomb lattice. Two atoms A an B reside in the rhombohedral unit cell, bounded by theprimitive lattice vectors shown in light blue.



Energy Eigenvalues

We will assume there is no overlap between the two nearest-neighbor wavefunctions:∫X (r− rA)X (r− rB)d3r = 0

thus we can write the Schrodinger’s equation Hψ = Eψ in a 2x2 matrix form:(H11 H12

H21 H22

)(ϕ1

λϕ2

)= E

(ϕ1

λϕ2

)upon setting the determinant |H − EI | = 0, we see that λ falls out and we obtain thecharacteristic equation:

E2 − (H11 + H22)E − (H12H21 − H11H22) = 0

Which yields the new eigenenergies:

E± =1

2{(H11 + H22 ±

√(H11 − H22)2 + 4H12H21} = H11 ± |H12|

where the last form applies if we make the logical assumption (by symmetry) that H11 = H22

and H12 = H21, where these terms describe the interaction of A(B) atoms with equivalent A(B)atoms, and the interaction of A(B) atoms with B(A) atoms, respectively.



Bandstructure of GrapheneThese two interaction integrals are calculated by Wallace as:

H11 =1

N

∑A,A′

eik·(rA−r′A)

∫X∗(r− rA)HX (r− r′A)d3r = Eo − 2γ′o

(cos(ky a) + 2 cos(kx a

√3/2) cos(ky a/2)

)(3)

H12 =1

N

∑A,B

e ik·(rA−rB )∫

X∗(r− rA)HX (r− rB )d3r = −γo(e ikx a/

√3 + 2 cos(ky a/2)e−ikx a/(2

√3))

(4)

if we include only the nearest neighbors A-B, we set H11 = 0, and have:

E± = ±|H12| = ±γo√

1 + 4 cos2(ky a/2) + 4 cos(ky a/2) cos(kx a√

3/2)

often touted as the energy band diagram for graphene and plotted in figure 5.



Carbon NanotubesCarbon Nanotubes (CNT) can be imagined to have been formed by rolling-up a graphene sheet whose edges are commensuratewith forming a tube. Each CNT has a unique Chiral Vector:

Ch = na1 + ma2

where a1 and a2 are primitive translation vectors which translate to equivalent points in the hexagonal lattice, as shown in figure

2 (see also: Saito et. al.3). The vector describes all the possible ways of rolling-up a graphene sheet

Figure: A carbon atom resides at every vertex in the honeycomb lattice. Shown are theprimitive translational vectors a1 and a2 , and the chiral vector Ch, which is an integralcombination of translations a1 and a2, indexed by the integers n and m.

3Saito, R., Dresselhaus, G., and Dresselhaus, M. Physical Properties of Carbon Nanotubes(London: Imperial College Press) (1998).



Classification of CNTs

Carbon nanotubes are often classified simply by these indeces: (n,m), from which one canderive several important properties of the CNT. We may classify all the possible combinations of(n,m) and what type of CNT they form:

n 6= 0,m = 0 zig zagn = m arm chair

n,m 6= 0 chiral

Figure 7 below is a visualization of the these three classifications of CNTs:

Figure: Visualizations of 3 classifications for CNTs. From left to right: zig zag (n 6= 0, m = 0),arm chair (n = m), and chiral (n,m 6= 0). The red lines at the terminus illustrate the origin ofthe naming convention.



Properties of CNTs

Having Ch one can calculate the diameter and the chiral angle θc :

D =|Ch|π

=a

π

√n2 + nm + m2 and θc = tan−1

{ √3m

m + 2n

}

Typical diameters may be in the range D ∼ 0.6− 2.0nm. The electronic properties of the CNTare typically understood in terms of the band structure of graphite, where the appropriateboundary conditions are applied. Calculations of γo in the dispersion relation for graphenetypically give values in the 2-3eV range, and the properties of the CNT will vary according tothe size and chirality. For instance, the bandgap

Eg =2dccγo

D∼ 0.7-0.9 eV

for D=1nm, dcc=0.142nm (carbon-carbon bondlength) and a=0.242nm (lattice parameter).



Conductivity of CNTs

Of particular interest is the condition for metallic CNTs, which can be derived by consideringthe Brillouin zone of graphene, as shown if figure 8. If the points of high symmetry, wherein theband-gap in graphene goes to zero, the so-called K -points, are such that an integral number ofchiral vectors (which describe the diameter of the CNT) is equal to an integral number ofwavelengths, the CNT will be metallic. This is expressed by the equation:

K · Ch = 2πq −→ n −m = 3q

This and an equivalent condition can be found by taking the dot product using the explicitforms for Ch and K. Based on this condition, where q is an integer, roughly 1/3 of all CNTsshould be metallic.

Figure: Real space and reciprocal space illustrations of primitive lattice vectors {a1, a2} andreciprocal lattice vectors {b1, b2}. Points of high symmetry K and K ′ are shown in thereciprocal space lattice.



Density of states of a CNT

A related quantity which also can be derived from the dispersion relations is the density of statesof the CNT. One can expand E(k) in a Taylor series around the K ′-point, as shown in figure 9:

Figure: Zoom-in of energy dispersion relation E(k) around the K ′-point. Dashed line showsthe Taylor expansion of the energy dispersion relation around the K ′-point.

expanding the term H12 of equation (7) in a Taylor series about the K ′-point yields:

H12 ≈∂H12

∂kx

∣∣∣∣k=K′(kx − K ′x ) +

∂H12

∂ky

∣∣∣∣k=K′(ky − K ′y ) = −iγoa

√3

2

[(kx − K ′x ) + i(ky − K ′y )

]



Density of states of a CNT (continued)from which we can write the magnitude |H12| and invert this to give ky as a function of energy ε:

|H12| = γoa

√3

2

√(kx − K ′x )2 + (ky − K ′y )2 −→ ky = ±

2

γo√

3a

√ε2 − ε2

q +K ′y where ε2q = (

γo√

3a

2∆kx )2

While we derived this expression with a chosen coordinate system, in general, the energy dispersion relation will always posessradial symmetry about the K -points, and thus we can write:

E±(k−K′) ≈ ±γoa√

3

2

√∆k2⊥ + ∆k2

|| −→ ∆k|| = ±2

γo√

3a

√ε2 − ε2

q where ε2q = (

γo√

3a

2∆k⊥)2

Figure: (left): Allowed CNT k-states in 1st Brilloin zone of graphene, represented by parallellines which satisfy k · Ch = 2πq. (right): zoom in of CNT k-state referenced to K ′-point.



Density of states of a CNT (continued)2

Which is a more general re-statement of the derived expressions, where the expansion is referenced to the chiral vector Ch . Asseen in the figure above, the allowed k-states in CNTs are derived from the dispersion relation of graphene by requiring theboundary condition k · Ch = 2πq. Thus specific “slices” of the graphene dispersion relation apply to a given CNT with chiralvector Ch and indeces (n,m). Combining this requirement with the previously stated relation (you can show it in homework)

K′ · Ch = 2π3

(n − m) yields the following expression for the ∆k⊥ shown in the figure, and thus the condition on εq whichfollows:

∆k⊥ =

∣∣∣∣∣(k− K′) ·Ch

|Ch|

∣∣∣∣∣ =2π

3|Ch||3q − n + m| =

2

3D|3q − n + m| −→ εq =

γoa√3D|3q − n + m|

from which we see that the transverse momentum ∆k⊥, and likewise the energy εq , are quantized4, thus we will havesub-bands appearing for various values of the integer q. Since the density of states is the derivative of the total number of statesat energy ε, and the total number of states at energy ε is just the total volume in k-space divided by the volume of the unit cell(in k-space), we can now calculate the density of states: the volume of all filled states up to energy ε in a 1-D k-space is just2kF , where kF is the Fermi wavevector. We can compute the number of states N(ε) by dividing this volume by l , the equivalentlength of the 1st Brillouin zone in a CNT formed by rolling-up a graphene sheet, and readily find the density of states by takingthe derivative dN(ε)/dε:

N(E) = 22k

l=

4

l

(2

γo√

3a

√ε2 − ε2

q + K ′||

)−→

dN(ε)

dε=

4

l

2

γo√

3a

|ε|√ε2 − ε2

q

4C. T. White, J. W. Mintmire, Nature 394 29 (1998).S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 20 / 22


Density of states of a CNT (continued)3

Since there are a number of sub-bands, we must sum over each:

D(ε) =4

l

2

γo√

3a

∑q

g(ε, εq) where g(ε, εq) =

|ε|√ε2 − ε2

q

|ε| > εq

0 |ε| < εq

and εq =|3q − n + m|γoa√

3D

where l = (4π/√

3)|Ch|/a2 is the equivalent length of the first Brillouin zone5 for the CNT and the other parameters are asdefined previously. We sketch this relation in figure 11 for a semiconducting CNT. We note the sharp spikes when ε = εq ,which are known as van Hove singularities, in the density of states.

Figure: Typical density of states for (semiconducting) carbon nanotube as a function of biasvoltage (energy).

5J. W. Mintmire and C. T. White, Phys. Rev. Letters, 81 2506 (1998)S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 21 / 22


Applications of CNTsThe superior strength to weight ratio of CNTs is of interest in applications where high tensile strength (100X steel in theory and50X steel in practice), and lightweight (less than 1/6 that of steel) are required, such as for aircraft or spacecraft. Though wedid not discuss it here, CNTs may be switched between a semiconducting to a metallic state by the application of strain, as thebandgap in solids depends on the lattice spacing a and can be modified by applying isostatic pressure. Thus CNTs could be usedas strain gauges, in replacement of capacitive or piezo-resistive sensors currently used. With a little imagination, you can comeup with your own applications, here’s one crazy idea: The concept of a hot-carrier solar cell, where energy normally lost to heatcould be retrieved and converted to useful electrical energy, depends on an energy selective contact. The above density of statescould provide such a contact, extracting the hot-carriers before they cool to the band edge (see below).

Figure: An idea to utilize the density of states of the CNT: an energy selective contact in ahot-carrier solar cell.


theory and application of nanomaterials - lecture …ahrenkiel.sdsmt.edu/.../nano_702_103117.pdfs....

Documents