Nonequilibrium Kondo effect in a graphene-coupled quantum dot in the presence of a magnetic field

Background: Quantum dots connected to larger systems containing a continuum of states like charge reservoirs allow the theoretical study of many-body effects such as the Coulomb blockade and the Kondo effect. Results: Here, we analyze the nonequilibrium Kondo effect and transport phenomena in a quantum dot coupled to pure monolayer graphene electrodes under external magnetic fields for finite on-site Coulomb interaction. The system is described by the pseudogap Anderson Hamiltonian. We use the equation of motion technique to determine the retarded Green’s function of the quantum dot. An analytical formula for the Kondo temperature is derived for electron and hole doping of the graphene leads. The Kondo temperature vanishes in the vicinity of the particle–hole symmetry point and at the Dirac point. In the case of particle–hole asymmetry, the Kondo temperature has a finite value even at the Dirac point. The influence of the on-site Coulomb interaction and the magnetic field on the transport properties of the system shows a tendency similar to the previous results obtained for quantum dots connected to metallic electrodes. Most remarkably, we find that the Kondo resonance does not show up in the density of states and in the differential conductance for zero chemical potential due to the linear energy dispersion of graphene. An analytical method to calculate self-energies is also developed which can be useful in the study of graphene-based systems. Conclusion: Our graphene-based quantum dot system provides a platform for potential applications of nanoelectronics. Furthermore, we also propose an experimental setup for performing measurements in order to verify our model.


S1
The equation of motion for d σ nσ |d † σ r ω is: To determine the Green's function of the quantum dot, we need to calculate the new higher-order correlation functions that appear on the right-hand side of Eq. (S5). The equations of motion for these terms are expressed as: where the following notations are used: and To obtain an analytical formula for the Green's function of the quantum dot, we have to truncate the higher-order correlation functions that appear in Eqs.
(S6)-(S8) by using an approximation method. In order to do this, we apply the broadly used Lacroix decoupling scheme [2] that leads to close the infinite number of higher-order correlation functions.
Appendix B: Derivation of the self-energies at finite temperature In this section, we show a simple method to deduce self-energies Σ 3σ (ω) and Σ 4σ (ω) for finite temperatures. We introduce the self-energies by: and: with: and: where we introduced the notation ε =hv F k. Note that Σ (ω) explicitly depend on ω through ω 1σ and ω 2σ , respectively. Furthermore, by changing the variable β (ε − µ α ) = x where β = 1/T , then the Fermi function f (ε − µ α ) can be expressed as:

S5
where tanh(x/2) has the properties [5]: The following calculations will be based on the properties of function tanh(x/2) outlined in Eq.
Note that these results are valid as well at low temperatures. For absolute zero temperature we can substitute f α (ε k ) with the Heaviside function, i.e., f α (ε k ) = θ (µ α − ε k ), and using the method presented above the integrals can be calculated.

Appendix C: The verification of an analytical solution
In this section, we compare our analytical results for Σ 3σ (ω) presented in Appendix B with those of Z.-G. Zhu and J. Berakdar in Ref. [7]. In order to do this, we introduce the following integral: Z.-G. Zhu and J. Berakdar applied a contour integral method in complex plane and found that: where z = 1 2 + ω−µ 2πiT and ψ(z) is the digamma function. It can be shown that the integrating function in Eq. (S66) is not a holomorphic function, and thus the contour integral method can not be applied S13 for I(ω). As we shall see, their results differ from those obtained by numerical calculations (see Figure S1). Our analytical results, given by I α(+) 3σ (ω) in Apendix B, are in better agreement with the numerical calculations. Therefore, it can be verified that the relation (S67) does not accurately reproduce the case of absolute zero temperature.