The density of states in a two-dimensional sheet of graphene can be determined by considering the energy-momentum relationship of the low-energy electronic excitations. The energy of these excitations is given by E(k) = ħvk, where v is the velocity of the excitations and k is the wavevector.

Energy-Momentum Relationship

In graphene, the energy-momentum relationship follows a linear dispersion relation, which means that the energy of the excitations is directly proportional to their momentum. This is in contrast to the parabolic dispersion relation in most materials. The linear dispersion relation arises from the unique electronic band structure of graphene, which is characterized by two inequivalent points in the Brillouin zone called the Dirac points.

Density of States

The density of states (DOS) is a fundamental concept in condensed matter physics that describes the number of electronic states per unit energy interval. In two dimensions, the DOS is defined as the number of states per unit area per unit energy. It is denoted by g(E) and can be calculated using the relationship:

g(E) = (1 / A) * ∫d^2k * δ(E - E(k))

Where A is the area of the graphene sheet, ∫d^2k is the integration over the two-dimensional wavevector space, and δ(E - E(k)) is the Dirac delta function that enforces energy conservation.

Calculating the Density of States

To calculate the density of states, we need to evaluate the integral over the two-dimensional wavevector space. Since the energy-momentum relationship in graphene is linear, the integral can be simplified by converting it to an integral over energy:

∫d^2k = (1 / ħ^2v^2) * ∫dE

Substituting this into the expression for the density of states, we have:

g(E) = (1 / A) * (1 / ħ^2v^2) * ∫dE * δ(E - E(k))

The integral over energy can be evaluated by performing a change of variables:

∫dE = ∫(1 / v) * dE(k)

Substituting this into the expression for the density of states, we have:

g(E) = (1 / A) * (1 / ħ^2v^2) * ∫(1 / v) * dE(k) * δ(E - E(k))

Simplifying further, we get:

g(E) = (1 / A) * (1 / ħ^2v^3) * ∫dE(k) * δ(E - E(k))

The integral over the Dirac delta function can be evaluated by setting E = E(k):

g(E) = (1 / A) * (1 / ħ^2v^3)

Conclusion

In conclusion, the density of states in a two-dimensional sheet of graphene is inversely proportional to the cube of the velocity of the low-energy electronic excitations. This result arises from the linear dispersion relation in graphene, which gives rise to a unique band structure and electronic properties.