the dependency between electron density and fermi energy for a 2D free electron
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Answer:
It is easier to consider first the case of two dimensions, where the electrons are confined to the - plane in a perpendicular field. Hence there is no motion in the direction so the term in Eq. (2) involving does not occur. You showed in Qu. 3 of Homework 3 that the density of states in two dimensions is a constant,
(6)
in the absence of a field. Here we give the total density of states, rather than the density of states per unit area, so a factor of  appears. In a field, the allowed energy values are discrete with a constant spacing between levels of . Now the number of zero field states in an interval  is given by
(7) (8) (9)
the degeneracy of a Landau level. In other words, the field bunches the states into discrete levels, but the total number of states in a region much larger than the Landau level spacing is unchanged by the field. The density of states, both with and without a magnetic field is shown in Fig. 1.
Figure: The dashed line shows the density of states of the two dimensional free electron gas in the absence of a magnetic field. It has the constant value . In the presence of a magnetic field the energy levels are bunched into discrete values  where , and , where  is the cyclotron frequency. Hence the density of states is a set of delta functions, shown by the vertical lines. The weight of the delta function is equal to the zero field density of states times , so the energy levels are just shifted locally, the total number of states in a region comprising a multiple of  being unchanged.
At zero temperature, as we increase the magnetic field the number of occupied Landau levels will change, since the number of electrons is fixed but the degeneracy of each Landau level changes with . This will lead to an oscillatory behavior of the energy as a function of the magnetic field, that we discuss below. At finite temperature these oscillations will be washed out if , because then many Landau levels will be partially filled. In order to see the oscillations, we therefore need
(10)
which requires large fields and very low temperatures, of order a few K. At higher temperatures, there is a smooth change in the energy with  which leads to a small diamagnetic response, see Appendix A, Ashcroft and Mermin (AM) p. 664, and Peierls, Quantum Theory of Solids, pp. 144-149.
It is also important to realize that if there are impurities, then the electron states will have a finite lifetime, , which will broaden the levels by an amount . This will also wash out the oscillations if the level broadening is greater than the Landau level splitting. Hence, to observe the oscillations we also need a second condition,
(11)
which requires very clean samples.
Let us now determine the change in energy of our two-dimensional model at  as a function of field. Let  be the number of electrons per unit area. Hence, for  the Fermi energy is determined from
(12)
or
(13)
which gives
(14)
Hence the ground state energy per electron in zero field is
(15)
using Eq. (14).
In the presence of the field the levels,  will be fully occupied with electrons and level  will be partially occupied with  electrons, where .
Counting up electrons one has
(16)
where
(17)
is called the filling factor of the Landau levels. Note that  and  take a continuous range of values, whereas  is an integer. From Eq. (4) we have
(18)
where the last equality is from Eq. (14), in which
(19)
is the field required to put all the electrons in the lowest Landau level. Consequently
(20)
where  means the largest integer less than or equal to . Note that  is the fractional filling of the last Landau level.
The energy per electron is then just obtained by summing the energies of each Landau level, i.e.
(21) (22) (23)
Eliminating  in favor of  using Eq. (18) one finds
(24)
which, from Eqs. (3), (15) and (18)-(20), can be conveniently written as
(25)
Eq. (25) is our main result. It gives the field dependence of the ground state energy (per electron) of free electrons in two-dimensions.
Figure: The energy of the two dimensional electron gas at  according to Eq. (25), as a function of  where  is the field at which all the electrons are in a completely filled lowest Landau level. Note that the overall size of the energy change varies as  in addition to the oscillations. The oscillations get closer together for small . In fact, apart from the smooth variation, the data is a periodic function of , see Figs. 3 and 4.
A sketch of the of energy as a function of , according to Eq. (25), is shown in Fig. 2. One clearly sees oscillations whose amplitude increases smoothly which , actually as . The period of the oscillations gets smaller with decreasing , since the energy is actually a periodic function of not  (apart from the  variation in amplitude). This is seen in Fig. 3 below. Although the energy is a continuous function of  the derivative is discontinuous when the field is such that the Landau levels are completely filled
Electron density is directly proportional to the square root of Fermi energy.
- Electron density is defined as the number of electrons per unit volume in space.
- Fermi energy is defined as the energy difference at zero temperature between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions.
- Electron density is directly proportional to the square root of Fermi energy.