Modeling the stability diagram and electrical conductance of a tunneling resonance system based on a carbon nanotube quantum dot

Abstract

Carbon nanotubes (CNTs) and CNT-based quantum dots (CNTQDs) hold significant promise for next-generation quantum and digital computing devices. This study presents a comprehensive theoretical analysis of the electronic properties of CNTs and CNTQDs, focusing on their stability diagrams and electrical conductance characteristics. Using the constant interaction model (CIM) and Green's function formalism within the Keldysh framework, we derive the quantized energy levels of the CNTQDs, evaluate their Coulomb blockade phenomena, and deduce the electrical conductance through the CNTQDs. This study illustrates how valley and spin degeneracy lead to the creation of four blockade diamonds as well as four conductance peaks per quantized energy level. The stability diagrams help in designing stable transistors and qubits that exhibit a voltage map that can be used for control in these devices.

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1. Introduction

While electronics industries have reached close to the end of the silicon age, scientists have been investigating suitable materials or new devices with higher computational powers that can replace silicon-based transistors in the last three decades. Researchers are pursuing two parallel paths: designing compact, high-efficiency transistors for digital computing using new materials and exploring quantum computing techniques with suitable materials that leverage quantum states, such as qubits, for advanced computation.^1–11 Carbon nanotubes have emerged as key dual-use materials, enabling the fabrication of digital devices such as new field transistors for digital computing and spin qubits, valley qubits, and charge qubits for quantum systems.^12–25

The exceptional properties of carbon nanotubes have attracted researchers to investigate useful applications for high-efficiency computational devices. The electronic properties of CNTs have been studied since they were discovered.^26,27 The transport properties, such as confining phenomena and the Coulomb blockade effect for electrons as well as ions,^28–32 valley degeneracy,^33,34 spintronics effects,^35,36 Fabry–Pérot interference of confined electrons,^37,38 and single-electron transport phenomena,^39,40 have been reported. CNTs exhibit induced superconductivity under many conditions, such as small-diameter CNTs,^41,42 doping with boron and nitrogen,^43,44 topological superconductivity under curvature effects,^45,46 and interfacing between CNTs and superconducting materials.⁴⁷ CNTs have been shown to be a high-efficiency resource for single-photon diodes and to enhance the efficiency of light absorption for better-performing photovoltaic devices.^48–51 Carbon nanotube quantum dots (CNTQDs) have attracted significant attention for quantum computing applications due to their ability to host spin, valley, and charge qubits.^16–25 Recently, qubits based on CNTs have been used in quantum communication experiments.^52,53 Using CNTs to design a device for a desired application and better efficiency requires an understanding of the electronic structures and how electrons can be transported in (to) them.

This paper shows how the energy dispersion relationships of different types of CNTs can be calculated, illustrating how CNTs can be electrically classified on the basis of chiral indices. The quantized energy levels and the additional charge energy at every sublevel of the CNTQDs are discussed. The confinement phenomenon of the CNTQDs was analyzed via a constant interaction model (CIM), and the electrical conductance was analyzed via quantum field theory to determine their stability. This would help in designing CNTQDs for various applications.

2. Graphene and carbon nanotube QDs

2.1. Graphene and carbon nanotube structure

A carbon nanotube is derived from graphene, a two-dimensional material of carbon atoms arranged in a hexagonal structure. The unit cell of graphene has a triangular shape and consists of two carbon atoms (see Fig. 1 (a)). The two basis vectors of the graphene unit cell are:^26,27where a₀ ≈ 1.42 Å is the carbon–carbon distance. The reciprocal lattice vectors of graphene b₁ & b₂ can be defined by using a_ib_j = 2πδ_ij (i, j = 1, 2) and are given by:

As illustrated in Fig. 1(b), the first Brillouin zone of graphene can be defined by constructing the perpendicular bisectors of the reciprocal lattice vectors. This first Brillouin zone has a hexagonal shape, with the corners defined as Dirac points. The positions of the Dirac points (K and K′) in reciprocal lattice space (momentum space) are as follows:

The dispersion relation of bulk graphene is given by:^54,55

where t ≈ 2.7 eV is the nearest-neighbor hopping term of graphene. In this context, the (+) symbol represents the conduction band, whereas the (−) symbol corresponds to the valence band. These bands meet at the Dirac points. As a result, graphene is considered a zero-bandgap semiconductor. Near the Dirac points, the dispersion relation can be approximated by a linear equation:⁴⁵where . A carbon nanotube (CNT) is formed by rolling a graphene sheet into a concentric cylindrical shape (Fig. 1(c)). The structural characteristics of CNTs, which play crucial roles in determining their electronic properties, could be derived from the geometric structure of graphene. The circumference of the CNT is defined by the chiral vector, which is expressed in terms of the basis vectors of graphene as follows:

C_h = na₁ + ma₂ (6)

Fig. 1 (a) Hexagonal monolayer graphene and lattice vectors a₁ and a₂. (b) Graphene Brillouin zone with reciprocal lattice vectors b₁ & b₂. (c) The graphene sheet and geometrical parameters of the CNTs.

Here, (n,m) are known as the chiral indices. The circumference of a carbon nanotube is given by , where d is the diameter of the nanotube. The chiral vector C_h defines the first vector of the nanotube's unit cell. The shortest vector perpendicular to C_h and parallel to the tube axis is called the translational vector T. This vector represents the second vector of the nanotube unit cell. Mathematically, it is expressed in terms of the graphene basis vectors of a₁ and a₂ as follows:

where N_R = gcd(2m + n,2n + m) is the greatest common divisor of 2m + n and 2n + m, which is obtained from the condition C_h·T = 0. The length of the translational vector is:^26,27

The angle between a₁ and C_h is defined as the chiral angle and is given by:

This angle is used to categorize the nanotubes into three different types:

■ θ = π/6 Armchair CNT with (n = m)

■ θ = 0 Zigzag CNTs with (n ≠ 0, m = 0)

■ 0 < θ < π/6 Chiral CNTs with (n ≠ m)

The total number of graphene unit cells (graphene unit cells) per CNT unit cell is as follows:

This means that there are 2N_C carbon atoms in the primitive cell of a carbon nanotube. Since the vectors C_h and T define the Bravais lattice of a carbon nanotube, the reciprocal vectors of a nanotube (K₁ and K₂) can be defined via the following relationships.

C_h·K₁ = 2π & T·K₁ = 0, C_h·K₂ = 2π & T·K₂ = 2π. (11)

This leads to

where K₁ and K₂ represent the reciprocal vectors of a nanotube along its circumference (tube axis).

2.2. Carbon nanotube energy band structure

Since a carbon nanotube has a cylindrical shape, electrons are confined along the tube's circumference, leading to quantized energy levels in two directions. In contrast, the electron energy along the tube's axis depends on its length: it is continuous for an infinitely long nanotube or quantized for a finite-length nanotube. Thus, the wave vector k in the nanotube can be decomposed into circumferential and axial components accordingly.

k = k_⊥K₁ + k_∥K₂. (13)

Here, k_⊥ represents the circumferential component of the wave vector, which is perpendicular to the tube axis (along the K₁ direction), whereas k_∥ is the axial component, which is aligned with the tube axis (along the K₂ direction). Since a nanotube is formed by rolling a graphene sheet, traversing an integer number of chiral vectors C_h leads back to the same point in space. This periodic boundary condition enables the electron wave function to be expressed as u_j(k,r + C_h) = u_j(k,r), which means that e′^ikC_h = 1. This leads to the following allowed values of k_⊥:

In the case of an infinite nanotube, the allowed values of another component, the parallel component k_∥, form a continuous spectrum but remain confined within a specific range k_∥∈[−π/|T|,π/|T|]. The pair (k_⊥,k_∥) defines a series of cutting lines that represent the allowed values of the wave vector in the Brillouin zone. The number, length, orientation, and spacing of these lines, the separation between two adjacent lines (2/d), depend on the chiral indices (n,m) of the carbon nanotube. The energy dispersion of a carbon nanotube can be derived from the energy dispersion of graphene by applying appropriate boundary conditions: one around the nanotube's circumference and the other along its axis. The general expression for the energy dispersion of a carbon nanotube is

This equation defines N pairs of energy dispersion curves that result from the cross sections of the two-dimensional energy dispersion surface of 2D graphene, where the cross sections are cut from the lines of υK₁ + k_∥(K₂/|K₂|), and they represent the allowed values of the wave vector k_∥. The space between every two successive lines is Δk_∥ = 2/d. For a particular armchair nanotube, where (n = m), the expression of the energy dispersion of carbon nanotubes can be simplified to:

with Δk_∥ = 2π/3a₀n and υ = 0, 1,…2n. In the case of the zigzag nanotube, where (n ≠ 0,m = 0), the energy dispersion of the carbon nanotubes can be simplified to:

The energy dispersion relationships for various types of carbon nanotubes are plotted as a function of k_∥ in Fig. 2. This figure illustrates the differences in electronic properties based on nanotube chirality. In armchair nanotubes, the allowed wavevector lines (cutting lines) intersect at the Dirac points. Consequently, armchair nanotubes exhibit no band gap and are thus classified as metallic. In contrast, zigzag and chiral nanotubes may either possess a band gap or not, depending on their chiral indices. Those exhibiting a band gap are classified as semiconductors, whereas those without a gap are considered metallic. This classification prompts the following question: What determines whether a nanotube is metallic or semiconducting?

Fig. 2 The band structure of carbon nanotubes. (a) Band structure of carbon armchair nanotubes (n,m) = (7,7), which is plotted using eqn (16). (a) and (c) Band structure of carbon zigzag nanotubes, where (n,m) = (6,0) for (b), which is a metal tube, and (n,m) = (7,0) for (c), which is a semiconductor tube; both graphs are plotted using eqn (17). (a) and (c) Band structures of carbon chiral nanotubes, where (n,m) = (7,3) in (b), which is a semiconductor tube (7 × 2 + 3 = 17 is not a multiple of 3), and (n,m) = (6,3) for (c), which is a metal tube (6 × 2 + 3 = 15 is a multiple of 3), (d) and (e), are plotted using eqn (15).

The electronic behavior of a carbon nanotube with chiral indices (n,m) is governed by the positions of the cutting lines relative to the Dirac points in the Brillouin zone of graphene. If one or more cutting lines pass through a Dirac point, the conduction and valence bands touch, resulting in a zero-band gap and metallic behavior. Conversely, if none of the cutting lines intersect the Dirac points, the nanotube exhibits a band gap and behaves as a semiconductor. The variance between the metal and semiconductor tubes of zigzag and chiral tubes depends on the nanotube indices. Moreover, the Dirac points in a metal nanotube must be included in one (or more) of the allowed wave vector lines.^26,54 Hence, these facts could be used to derive an expression that helps to form the classification criterion. Fig. 3 shows the cutting lines on the graphene sheet for a semiconductor and a metal nanotube. In metal nanotubes, the projection of the vector that links the center of the graphene Brillouin zone is XK, with the Dirac point K₁ being YK. This vector, YK, must be an integer value of K₁ if the allowed wave vectors (cutting lines) pass through Dirac points. The projection of XK can be calculated from the vector geometry as:

Fig. 3 (a) Nanotube energy dispersion for semiconductor tubes, where none of the energy-crossing lines pass through Dirac points. (b) Energy dispersion of a metal tube where the energy-crossing line passes through Dirac points.

YK has an integer value of K₁ only if 2n + m (or n − m) is a multiple of 3, which means that the nanotube is a metal tube only if 2n + m is an integral multiple of 3.²⁶ Armchair nanotubes (n = m) satisfy this condition, and they are classified as metal tubes (see Fig. 2(a), where 6 × 2 = 12 is a multiple of 3). The zigzag nanotube is a metal only if n is a multiple of 3 (Fig. 2(b)). The chiral tubes vary between the metal and the semiconductor depending on the chiral indices (Fig. 2(d and e)).

The plotted graphs of the energy dispersion for the zigzag and chiral nanotubes show a distinct value of the energy at Dirac points K and K′. The difference between these distinct values of energy is known as the mismatch energy. This mismatch in energy is attributed to curvature-induced symmetry breaking and circumferential confinement. This energy can be estimated as the difference between the values of the valence energy bands at the Dirac points δ = |ε⁺_i(K) − ε⁺_i(K′)|. The magnitude of this energy is small compared with the quantized energies and the Coulomb energy, but it plays an important role in the stability of valley qubits by maintaining coherence for a long time, facilitating valley polarization and controlling valley currents in the valleytronic region.^54,55

A simplified form of energy depression of carbon nanotubes can be derived if k_x and k_y are replaced by their values in terms of k_⊥ and k_∥.^45–60 The simplified form is . In this form, the circumference component k_⊥ could approach k_⊥ = K_⊥ + κ_⊥, where k_⊥ is the projection of k on K₁. The confinement along the tube circumference imposes discrete values of momentum, which means that C_h(k_⊥ + κ_⊥) = 2πυ. The geometric calculation gives . In the case of a semiconductor tube, the Dirac points are missed, and n − m = ±1. Hence, the energy dispersion relation takes the following form:

The energy gap of semiconductor nanotubes is related to the chiral indices and can be calculated by ε_g = ε⁺(k_∥ = 0) − ε⁻(k_∥ = 0). If the last derived equation is used, the energy gap will be ε_g = 4ħv_F/3d = 0.72/d(nm) eV.

2.3. Carbon nanotube quantum dots

In the case of a finite length “L” nanotube with M unit cells (M = L/|T|), the electrons would also be confined along the tube axis. This would make this tube a 0D material (quantum dot), wherein the allowed values of momentum k_∥, in this case, would be converted from continuous values to quantized values. The allowed values of k_∥ would be

This condition imposes that the tube ends with hard-wall boundaries. In general, the tube could end with different boundaries (suppose right R and left L boundaries). The shape of the tube boundary adds a phase shift to the quantization condition. The modified condition is:⁶¹

with
where N_s is the number of graphene unit cells along the nanotube axis, r = ±1 represents the branch index (r = 1 right-going branch and r = −1 left branch), p is the parity index (r = 1 for even parity with a symmetric wavefunction, while r = −1 for odd parity with an antisymmetric wavefunction), θ_r is the phase shift of the branch which could be determined by the boundary condition, and φ_r = arg(S(k)), where is the graphene tight-binding structure factor.⁶¹ In the next discussion, we will consider the first form of the quantization condition. The quantization energy of the carbon nanotube quantum dot (CNTQD) levels can be obtained from eqn (16) after replacing the continuous k_∥ with the corresponding quantized values derived from the preceding equations. A simplified form of the CNTQD energy level would result from eqn (19) after considering quantized k_∥,^45,60 which leads to

Every quantized level can be occupied by two electrons with different spin directions. However, graphene shows two inequivalent valleys at the Dirac points because of the time-reversal symmetry. The electrons could be in either the K valley or the K′ valley, which means that these valleys add a second degree of freedom. In other words, every quantum level consists of two sublevels at the K and K′ valleys (valley sublevels), and every valley sublevel can be occupied by two electrons owing to spin degeneracy. This results in a total of four electrons that can occupy each quantum level.^{54–56,58–61} These electrons interact with each other via Coulomb interactions. If the constant interaction model (CIM) is used to describe the CNTQDs, the chemical potential of the CNTQDs can be determined as follows:

where l = 0, 1, 2, and 3 ensures the inclusion of the valley and spin degeneracy of every level p. U_c is the Coulomb energy unit U = e²/C_∑ (where C_∑ = 2πε₀L/ln(b/a) is the total capacitance of the CNTQD when the tube is considered a cylindrical capacitor; L is the tube length). The valley sublevel mismatch δ has been added, where δ_l,2 is a Kronecker delta function that ensures that δ is added after every valley sublevel is filled. A scheme of the energy level of a CNTQD with its sublevels is shown in Fig. 4, where the four sublevels differ.⁵⁵ The energy required to add an electron to every sublevel is given bywhere E_add is the addition energy, Δε = E_p+1,l − E_p,l is the separation between the quantized levels, and E_c is the charge energy, which is equal to the Coulomb energy unit U_c.

Fig. 4 Energy levels of the CNTQDs. (a) Scheme illustrating the CNT valleys and how to generate a new degree of freedom in CNTQDs with mismatch energy. (b) Energy levels of the CNTQDs and the additional charge energy required to enable an electron to occupy every level.

3. Blockade phenomena and stability diagram

The single electron transistor based on CNTQDs consists of a CNTQD that is coupled to conducting electrodes from the left and right, which form the source and drain. A voltage (V) must be applied to the electrodes to create a difference in their chemical potential, which is known as a bias window. The difference in chemical potential forces electrons to flow from a higher electrode chemical potential (which acts as a source) to a lower electrode chemical potential (which acts as a drain). However, electrons can flow only if one (or more) energy level is located within the bias window. The energy of the dot level can be replaced or modified by applying another voltage through a third electrode that must be connected to the CNTQD, which is called the gate electrode with voltage V_g. Moreover, single-electron transport phenomena can be observed if the thermal energy kT is smaller than the separation between energy levels in the QDs.^58–63 The variations in the thermal energy, quantum energy levels, and Coulomb interaction lead to a difference between the two systems. The first is when the thermal energy is greater than the quantum energy and less than the Coulomb energy. This regime is classified as a metallic Coulomb blockade regime. The second is where the thermal energy is smaller than both the Coulomb energy and the quantum energy. This regime is known as the quantum Coulomb blockade regime. In CNT QDs, the energy of the quantum level is greater than the thermal energy, and the SET based on CNT QDs can be categorized into a quantum Coulomb blockade regime. In the quantum-Coulomb blockade regime, an electron can be transported through a QD if it has sufficient energy to occupy the quantum energy level and overcome the Coulomb repulsive interaction. This sufficient energy can be provided to electrons by applied voltages. In the constant energy model, the QD is considered a capacitor, and the Coulomb energy is treated as the charging energy of this capacitor. We have three electrodes in the SET: the source, drain, and gate. The total capacity C_∑ is divided by the capacity of the source C_s, the capacity of the drain C_d, and the capacity of the gate C_g (C_∑ = C_d + C_s + C_g). The capacities of these capacitors C_s, C_d, and C_g, depend on the spatial arrangement of the electrodes relative to the QD. These capacities could also be defined from the experimental data of any Coulomb diamond by defining the intersection points.

Electron transport from the source (or drain) to (from) the QD that has N electrons when the source (or drain) chemical potential aligns with the QD level E_p,l. This statement leads us to write:⁶³

where is an asymmetric parameter and where C_∑ is the total capacity of the CNTQDs. These equations represent four straight equations that intersect with each other and define the borders between the values of the applied voltage, which can provide electrons with the energy required for transport, and those that cannot. Moreover, from the intersection points of these lines, the operation conditions of the CNTQD devices could be extracted. The intersection points are:⁶³

From these points, the maximum source–drain voltage that can be applied, the positions of conductance peaks, and the voltage range where Coulomb blockade would be observed at zero value of source–drain voltage could be given by:⁶²

Using eqn (24)–(27), the stability diagram of the CNTQDs can be plotted for any dot. Fig. 5(a) shows the stability diagram of metallic CNTQDs (where ε_g = 0) which has E_c = 4.5 meV and ΔE = 6.8 ± 2.5 meV (data are taken from ref. 64). The gray diamonds in Fig. 5(a) represent the diamonds of the quantum-Coulomb blockade, where the applied voltages cannot provide electrons for transport through the CNTQDs. The other diamonds represent the values of the applied voltages that enable electron transport. As shown, the quantum-Coulomb diamond size depends on the CNTQD level. The size of the diamonds depends on the quantum dot chirality indices. For smaller quantum dots (QDs) with reduced n and m values, the Coulomb diamond is larger, requiring higher voltages to control CNTQD devices. This results in improved controllability and reduced sensitivity to environmental fluctuations. By leveraging the stability diagram, these devices can be designed for applications such as single-electron transistors, spin qubits, or charge qubits, depending on operational requirements.

Fig. 5 Stability diagram and the conductance of metallic CNTQDs. (a) Stability diagram of CNTQDs, plotted by using eqn (24)–(27). The CNTQDs used in this study are fabricated from a metallic tube (i.e., ε_g = 0 and the mismatch energy is assumed to be δ = 0; the tube is connected to the source and drain i.e., ζ = 0 also) and it has E_c = 4.5 × 10⁻³ eV and ΔE = 6.8 ± 2.5 meV (data taken from ref. 64). (b) Conductance of the tunneling resonance system based on CNTQDs, which is plotted using eqn (42), where the tunneling rates are assumed to be Γ^L/R = 1 × 10⁻³ eV.

Finally, it should be noted that altering the configuration of the connected electrodes leads to corresponding changes in the capacitances C_s, C_d, and C_g. Consequently, the stability diagram of CNTQDs is affected, as the range of applied voltages in which the Coulomb blockade effect is observed and the range where it is not both vary accordingly. As evident from eqn (29), the maximum source–drain voltage is independent of the system capacitances; therefore, modifying the electrode configuration does not influence the source–drain voltage range, which depends solely on the quantum dot size. Eqn (31) shows that the range of gate voltage over which the blockade effect occurs is proportional to C_∑/eC_g. Hence, decreasing the gate capacitance extends this range. Such an extended gate-voltage range enables more precise control over the occupation of quantum dot energy levels during the operation of CNTDQ-based devices. Furthermore, the capacitances of the source, drain, and gate electrodes can be extracted from the experimental data by identifying the intersection points of any Coulomb diamonds in the stability diagram. These intersection coordinates can then be substituted into the corresponding equations (eqn (28)) to form a solvable set, allowing the determination of C_s, C_d, and C_g.

4. Conductance in CNTQDs

Electron transport over short distances in nanodevices preserves the quantum information encoded in the electron's wavefunction phase factor, a process known as quantum transport. Moreover, the transport phenomenon itself occurs when a physical system transitions from equilibrium to nonequilibrium conditions, such as when a bias voltage is applied. To study electron transport in quantum devices while accounting for its quantum nature, the Keldysh formalism for nonequilibrium systems is commonly employed. In CNTQDs, electron transport under applied voltages is treated as quantum transport, and this formalism is used to analyze electrical conductance.

We consider the Anderson model, where a QD is coupled symmetrically to noninteracting electrodes. The corresponding Hamiltonian of this system can be written as.^65–67

with,where H_α; α = L, R are the electrode Hamiltonians, H_T is the tunnelling Hamiltonian, and H_D is the CNTQD Hamiltonian. Here, c_αkσ (c^†_αkσ) destroys (creates) a conduction electron with momentum k and spins σ at electrode α, and d_nσ (d^†_nσ) destroys (creates) an electron with spin σ in the QD. where ε_αkσ is the kinetic energy of electrons in the electrode, V_kσα,nσ represents the matrix elements of the tunnelled electrons, ε_dσ is the energy dot level, and U_nn′ is the interaction potential.^67–72 In the constant interaction model where the electrons are assumed to interact with the same energy (U_nn′ = U), the electron–electron interaction Hamiltonian is:^55,59where (d^†_nσd_nσ)² = n_nσ² = n_nσ = 1, the number of electrons in the CNTQD is , and U = e²/C_∑. The Coulomb interaction can be added to the dot-level energy as . The linear conductance through the CNTQD can be given by the Landauer–Büttiker formula:^66–69where f_α(ε) is the Fermi distribution function, Γ^L(ε)(Γ^R(ε)) is the tunneling rate from the left (right) electrode, which behaves as a source (drain), and A(ε) is a spectral function that has the following form:where is the real part (imaginary part) of self-energies, which must be defined. The self-energies can be extracted from the equilibrium self-energies of the causal Green function (GF) after applying the Langreth rules. The causal Green's function is obtained by modeling the tunneling Hamiltonian as perturbed, with the dot Hamiltonian acting as the unperturbed foundation. This leads to

Considering the first-order corrections and then applying Wick's theory, one could gate the tunneling self-energy . The next steps involve applying Langreth's rules and conducting a Fourier transformation into energy space, which yields the following tunnelling self-energy expressions for each dot level “l”.

where ∑⁻⁺_T and ∑^R/A_T are the lesser, retarded, and advanced tunnelling self-energies, respectively. Under the wide-band approximation, the real part could be neglected compared with other energies, and the tunnelling rate's imaginary part “Γ_li(ε)” could be approximated to its value at the Fermi level of electrodes and considered constant for all dot levels (Γ^α_ll(ε) = Γ^α). Hence, after the self-energies in the spectral function are substituted, the conductance expression is as follows: one can find the following expressions for conductance:

The plot of the derived expression of conductance is shown in Fig. 5(b), where eqn (42) is plotted for V = 0. The CNTQDs present peaks of conductance, which means that the conductivity of the CNTQDs is quantified. Starting from the ground state, when an electron passes through a CNTQD, the energy required to occupy the ground state at the lowest energy during pass is ε₁. Hence, the first peak would be obtained. With increasing gate voltage, electrons can pass through the second sublevel, and the second peak will appear at V_g = C_∑[ε₁ + U]/eC_g. The third peak will appear because of the first level of the second valley, where this peak would appear at V_g = C_∑[ε₁ + 2U]/eC_g. The fourth peak is obtained when an electron passes through the second valley sublevel in the second valley, and this peak appears at V_g = C_∑[ε₁ + 3U]/eC_g. The same discussion could be held for higher quantized levels and their spins and valley sublevels. As a result, the conductance peaks in the CNTQD group into sets of four peaks, which has been reported in many works.

5. Conclusion

This study provides a detailed theoretical framework for understanding the electronic properties of carbon nanotube quantum dots (CNTQDs). By employing the constant interaction model (CIM) and Green's function formalism, we derived the quantized energy levels of CNTQDs and analyzed their stability diagrams under Coulomb blockade regimes. This study illustrates that spin and valley degeneracies lead to four distinct conductance peaks per quantized energy level, a key feature of quantum transport in CNTQDs. Furthermore, we investigated how the chiral indices (n,m) affect the type of carbon nanotube and energy levels of CNTQDs, which could be considered as design parameters when carbon nanotube-based devices are fabricated. Stability diagrams revealing diamond-shaped blockade regions offer insights into the control of single-electron tunneling in CNTQD-based devices. These overview studies provide researchers with the necessary information about the electronic structures of CNTs and CNTQDs, which will help in designing and controlling CNT-based devices.

Conflicts of interest

There are no conflicts to declare.

Data availability

All data supporting the findings of this study are included within the main article. No additional datasets were generated or analyzed.

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