Two-Dimensional Debye Solid

An atom confined to a surface may be thought of as an object “living” in a two-dimensional world. There are a variety of ways to look at such an atom. Suppose that the atoms adsorbed on the surface are not independent but undergo collective oscillations as do the atoms in a Debye crystal. Unlike the atoms in a Debye crystal, however, there are only two dimensions in which these collective vibrations can occur.

a) Derive an expression for the number of normal modes between ω and ω + dω and, by thinking carefully about the total number of vibrational frequencies for N atoms confined to a surface, rewrite it in terms of N and ω_D, the maximum vibration frequency allowed due to the discreteness of the atoms.

b) Obtain an integral expression for the energy E for the two-  dimensional Debye crystal. Use this to determine the limiting form of the temperature dependence of the heat capacity (analogous to the Debye τ³ law) as τ → 0 for the two-dimensional Debye crystal up to dimensionless integrals.

SOLUTION

a) The number of normal modes in the 2D solid within the interval dk of a wave vector k may be written

(1)

In the 2D solid there are only two independent polarizations of the excitations, one longitudinal and one transverse. Therefore,

(2)

where ⟨v⟩ is the average velocity of sound. To find the Debye frequency ω_D, we use the standard assumption that the integral of (2) from 0 to a certain cut-off frequency ω_D is equal to the total number of vibrational modes; i.e.,

(3)

Therefore,

(4)

We can express ⟨v⟩ through ω_D:

(5)

Then (2) becomes

(6)

b) The free energy then becomes

(6)

Defining θ_D = hω_D, and introducing a new variable x = hω/τ, we can rewrite (7) in the form

(8)

Integrating (8) by parts, we obtain

(9)

where the 2D Debye function D₂(z) is

(10)

The energy is given by

(11)

The specific heat C (C_P ≈ C_V at low temperatures) is

(12)

At low temperatures, τ << θ_D, we can extend the upper limit of integration to infinity:

Therefore, at τ << θ_D,

(13)

where

and  is the Riemann  function. Note that the specific heat in 2D is

Note also that you can solve a somewhat different problem: When atoms are confined to the surface but still have three degrees of freedom, the results will, of course, be different.