Graphene nanoribbon (GNR) is a promising alternative to carbon nanotube (CNT) to overcome the chirality challenge as a nanoscale device channel. Due to the one-dimensional behavior of plane GNR, the carrier statistic study is attractive. Research works have been done on carrier statistic study of GNR especially in the parabolic part of the band structure using Boltzmann approximation (nondegenerate regime). Based on the quantum confinement effect, we have improved the fundamental study in degenerate regime for both the parabolic and nonparabolic parts of GNR band energy. Our results demonstrate that the band energy of GNR near to the minimum band energy is parabolic. In this part of the band structure, the Fermi-Dirac integrals are sufficient for the carrier concentration study. The Fermi energy showed the temperature-dependent behavior similar to any other one-dimensional device in nondegenerate regime. However in the degenerate regime, the normalized Fermi energy with respect to the band edge is a function of carrier concentration. The numerical solution of Fermi-Dirac integrals for nonparabolic region, which is away from the minimum energy band structure of GNR, is also presented.

Single layer of graphite which is also known as graphene has been discovered as a material with attractive low-dimensional physics, and possible applications in electronics [

Armchair GNRs with dimer lines

Zigzag GNRs with dimer lines

The band energy throughout the entire Brillouin zone of graphene is given by [

The band structure of GNR near the minimum energy is parabolic.

By taking the derivatives of energy

Carrier concentration is an essential parameter for semiconductor. The numbers of electrons/cm^{3} and holes/cm^{3} with energies between

Therefore the total carrier concentration in a band can be obtained by simply integrating the Fermi-Dirac distribution function over energy band as follows [

Energy band diagram showing degenerate and nondegenerate regions.

The simplified form of the occupancy factors is a Maxwell-Boltzmann-type function that also describes, for example, the energy distribution of molecules in a high temperature. The simplified occupancy factors lead directly to the nondegenerate relationships. In closed-form relationships we find limited usage in device analysis since the nondegenerate relationships are only valid for an intrinsic and low-doped semiconductor [

Cross-section of a rectangular one-dimensional GNR with

In nondegenerate limit condition, the carrier concentration of GNR can be expressed as

For degenerate GNR we have

Comparison of Fermi-Dirac integral and Fermi −1/2, 0, 1/2 for Q3D bulk, Q2D, and Q1D devices, respectively [

Also shown in Figure

As illustrated in Figure

Comparison of the GNR Fermi-Dirac integral in degenerate and nondegenerate regimes in non-parabolic region. Also shown is the comparison with general equation (

The modeling of carrier statistic in a both parabolic and non-parabolic region is presented. The one-dimensional GNR approaches degeneracy at relatively lower values of carrier concentration as compared to 2D and 3D structures. The Fermi energy with respect to the band edge of the GNR is a function of temperature but is independent of the carrier concentration in the nondegenerate regime. In the strongly degenerate regime, the Fermi energy is a function of carrier concentration but it is independent of temperature.

The authors would like to acknowledge the financial support from FRGS grant of the Ministry of Higher Education (MOHE), Malaysia. Mohammad Taghi Ahmadi is thankful to the Research Management Centre (RMC) of Universiti Teknologi Malaysia (UTM) for providing excellent research environment in which to complete this work.