Quantized Electrical Conductance of Carbon nanotubes(CNTs)

One of the main factors that impacts the efficiency of solar cells is due to the scattering of electrons inside the cells. Carbon nanotubes (CNTs) have proposed to use in solar cells due to their ballistic transport properties: electrons travel through CNTs without experiencing scattering or experiencing negligible scattering. The conductance of CNTs is shown to be quantized and has a multiple amounts of G0 which has the conductance 2e/h = 1 12.9 kΩ . This paper mainly focuses on the quantized electrical conductance of CNTs through the analysis of the theory and the experimental results. What are Carbon nanotubes? Carbon nanotubes (CNTs) can be thought as rolled up graphene sheets in to cylindrical shapes. The diameter of a CNT is about a few nanometers, and the ratio between its diameter and its length can reach as high as 1: 132,000,000 [1]. CNTs include two types: single-walled nanotubes (SWNT) and multiple-walled nanotubes (MWNT). The couple integer indices (n,m) represents the numbers of unit vectors (a1, a2) along two direction of a crystal lattices of a graphite sheet [1]. The circumference of the nano-tubes is represented by the magnitudes of the chiral vector, Cmn = na1 ����⃗ +ma2 ����⃗ , and the nanotube diameter d = a�(n2 + nm + m2)/π (where a=0.246 nm); the chiral angle is θ = tan−1 √3m/(2n + 7) and is measured from the chiral vector and the zigzag axis (See Figure 1) [2]. Figure 1.Left: A tube has a couple of integer indices (n,0) and is called zigzag tube, while a tube with indices (n,n) is called a armchair tube. Right: A single-walled armchair nanotube which has a couple integer indices (n,n) is shown above. The electrical properties of a CNT changes with its geometrical structures, depending on the ways a graphene sheet is rolled up. CNTs have the electrical properties of metal when (n − m) = 3z; CNTs become semiconducting tubes when (n −m) = 3z ± 1 (where z is integer) [2]. The electrical properties of a material are determined by its energy bands that can be below or above the Fermi level. Fermi level describes the top collection of electrons energy level at absolute zero temperature (See Figure 2) [3]. For semiconductors, the Fermi level is between the band gap of conduction band and the valences band. For metallic conductors, the band gap is zero and the Fermi level is in the overlapped area of conduction band and the valences band. Then, a portion of valence electrons can travel across the material. Figure 2. Schematic diagram of conduction band and valance band of insulator, semiconductor, and conductor are shown above. The energy gap between the conduction and valence band determines the conductivity of a material. The energy of electrons is along the vertical direction. By the same token, the band structures of a CNT determine its conductivity. The conductivity of a CNT depends on how a graphene sheet is rolled up. When a nanotube is made by rolling up a graphene sheets around the y-axis, the nanotube behaves like metal with a Fermi velocity that is similar to metals due to the Fermi level is in the overlapped region of the two bands of the tube (See Figure 3) [4]. When a graphene sheet is rolled up around the x-axis, the semiconducting nanotube is constructed because the Fermi level is between the two bands. Figure 3. For a metallic nanotube, the Fermi level is in the overlapped region between the conduction and valence band. For a semiconducting nanotube, the two bands do not across the Fermi level. The theory of quantized electrical conductance of CNTs When the transport of electrons in a medium experiences negligible or no scattering due to atoms or impurities, this phenomenon is called ballistic transport [4]. When the electronic mean free path of a wire is larger than the length of the wire, the electron transport in the wire is ballistic [5]. More surprisingly, the conductance in the wire is quantized. The wire acts like an electron waveguide and each conduction channel (or transverse waveguide mode) has an amount conductance of G0 = 2e2 h , where G0 is called “conductance quantum” and has the value of 1 12.9 kΩ , given e is the electron charge and h is the Planck’s constant [5]. The following derivation from reference [5] shows how G0 is obtained. Consider a one dimensional wire that connects “adiabatically” (no heat is being lost from the system) to two electrochemical potential reservoirs μ1 and μ2 (See Figure 4).Two assumptions also need to be taken into account: {1} there are no reflections of electrons between the reservoirs; {2} the wire is very narrow in order for the lowest transverse modes (corresponding to the lowest level of energy) in the wire to be below the Fermi energy. Fermi energy is the energy of highest occupied quantum state in a system of fermions (including electrons) at absolute zero temperature. Figure 4. A one dimensional wire is connected to two reservoir adiabatically with electrochemical potential μ1 and μ2. The current I is the same as the current density in one dimension. Hence, the current density is J = e ∗ v(μ1 − μ2) dn dε (1) where dn dε is the density of states and v is the electron velocity. The density of state for an electron including spin degeneracy (spins up and spins down) is