By appropriately doping a sample of graphene, it is possible to increase the Fermi temperature of the conduction electrons in order to simulate the electrical conductivity of metals.
Introduction
As seen in the previous issue, the study of the semiconductor/semimetal/metal distinction can be carried out in two distinct “environments”. The first (temperature domain) involves the numerical determination of the chemical potential µ as a function of the absolute temperature T. In the second (chemical potential domain) the variable T is incorporated into a dimensionless quantity x, free to vary from −∞ a +∞ (this is the range of variation of the chemical potential of fermions). In this schematization, semimetals present themselves as an interface between semiconductors and metals since x = 0, as illustrated in the logarithmic scale graph produced in the previous issue and which we propose again in Figure 1.
N-type graphene
Being a semimetal, graphene has a zero bandgap and, therefore, electrons exist in the conduction band regardless of the temperature T. In this way, they simulate the behavior of metals. However, they differ from the latter due to the mechanism of electrical conductivity, since electrons and holes are involved (bipolar mechanism). It follows that, if we want to best simulate the behavior of metals, we must appropriately dope the graphene in order to create a device in which the dominant contribution to conductivity comes from electrons. The final result is therefore a sample of n-type graphene.
We remind that the effects of doping with donor atoms in a semiconductor with bandgap εg consists in the generation of an extremely dense level spectrum centered in −εd, as illustrated in Figure 2, where we have drawn a single level for graphical needs.
The extreme proximity of −εd to the levels of the conduction band favors the transition of the corresponding electrons to the conduction band.
What happens if εg = 0 as in the case of graphene? The answer is very simple: the excess electrons (whose charge is compensated by the related ionized atoms) are “added” to the electrons already present in the conduction band. In this way, with a suitable density of dopant atoms, it is possible to achieve an electronic concentration of the same order of magnitude as that of a metal.
The Sommerfeld functional equation
An increase in the electronic concentration in the conduction band corresponds to an increase in the Fermi energy εF . In fact, as we already know that:
We see that εF is proportional to n2/3, with n = Ne/V, and being V the volume of the sample considered. For a given sample of graphene with a concentration of impurities such as to have n = 4 · 1021 cm−3, from the equation (1) we obtain ε = 4.6 eV and, therefore, a Fermi temperature T = 53378.7 K (we assumed the effective mass of the electron in graphene m∗e = 0.2me).
It follows that, at room temperature, the electron gas is strongly degenerate. In Figure 3 we report the trend of the corresponding Fermi-Dirac distribution function at various temperatures, having assumed µ(T) = εF . This last position is justified by the fact that we considered temperatures T ≪ TF for which the chemical potential differs slightly from εF = µ (0).
To obtain a more rigorous trend, it is necessary to solve the Sommerfeld functional equation1. This new expression refers to a functional equation in µ(T) valid for T ≪ TF deriving from the well-known Sommerfeld expansion of the Fermi-Dirac integral expressed in physical rather than dimensionless units. The result is the following:
From the corresponding plot of Figure 4 we see that, in the range T ≪ TF, it deviates from the Fermi energy by a negligible amount.
Inserting equation (2) into the expression of the Fermi-Dirac distribution function, we have:
It follows that the “jump” of the function towards zero does not occur at εF, but at a slightly different value, as we see from the graph in Figure 5. Please note that the temperatures 3600, 10600 K are theoretical and never experimentally reachable.
Conclusion
The chemical potential is, therefore, a fundamental physical quantity in the semiconductor/semimetal/metal classification. More precisely, it is its dependence on temperature that determines the behavior of that state of aggregation of matter classified as a “solid state”. In the specific case of n-type graphene, it can be deduced from the arguments carried out in this issue that the conductivity is with the ob- vious meaning of the symbols σ = enµe. Since n does not vary appreciably with the temperature, there is a behavior similar to that of metals.
References
1 Colozzo M. Sommerfeld Functional Equation.
2 Landau L.D., Lifshitz E.M. Statistical Physics, Third Edition, Part 1: Volume 5 . Editori Riuniti.
3 Kittel C. Kroemer H. Termodinamica statistica.
4 Scientific notes on power electronics.
The post Scientific Notes on Power Electronics: the Fermi temperature of electrons in doped graphene appeared first on Power Electronics News.
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