Question

For a three-dimensional (3D) solid, we have
found the low-temperature heat capacity to be proportional to T3. How does
it depend on the temperature for the two-dimensional (2D) graphene? Hint:
In order to give a correct answer to this question, you have to know a curious
fact about the phonon dispersion in graphene. We have seen that the acoustic
phonon branches in both one dimension and 3D have a dispersion with
?(k) ∝ k. For graphene, this is not so: Graphene has three acoustic branches
and one of them has a dispersion for which ? ∝ k2. For very low temperatures,
this is the important branch and you should base your calculation on
this dispersion only. The reason for this unusual behavior is that graphene
may be 2D, but it exists in a 3D world. It, therefore, has two “normal” in-plane
acoustic phonon branches, one longitudinal and one transverse. In addition
to these, it has a phonon branch that corresponds to a “flexing” motion out of
the plane and this is the one with the unusual dispersion.

Answer 1

According to Debye Theory for specific heat,the motion of the atom is coupled to each other therefore they cannot vibrate independently contradicting with Einstein theory which assumed that atoms vibrate with same fundamental frequency. Therefore in solid large number of frequency modes are possible. These modes are have discrete energies. Using the dispersion relation for phonon as given,

where, omega,v_s and k are the angular frequency, velocity of sound wave and magnitude of wave vector for phonon.

That implies,

Now the 2D density of states is given by,

We notice that for the given dispersion relation in 2D, the density of states becomes independent of energy.

Thus thermal energy is

here, w_D is Debye frequency.

Substitute,

x_D = T_D/T

Specific heat Cv is defined as

Case:For low temperature, T<<T_D which implies x_D tends to infinity. Therefore the integration becomes indefinite integral and hence a constant.

Thus C_phonon is proportional to T.

The specific heat of the electron is T for all the temperature and dimensions.

There the complete specific heat of the graphene(as conductor) with the given dispersion relation at low temperature is

C = C_electron + C_phonon = AT+BT, A and B are constants.

Note: The given dispersion relation is used for massive particles like electron magnon,etc.