Next: Two dimensions, Bloch electrons
Up: magnetic_field
Previous: Introduction
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
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.
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 (remember that is the fractional filling
of the last Landau level). Physically, this
is because if one decreases below
, say,
then it is Landau level which
starts to be occupied, whereas if one increases above then it
is Landau level which starts to be depleted.
Fig. 2 is to be contrasted with the situation at a temperature where
, where
thermal fluctuations average over the oscillations.
Then one has only the smooth increase in
the energy proportional to . Since the magnetization is given by
, where is the energy per unit volume,
one then finds a small diamagnetic (i.e.
negative)
susceptibility,
,
where
, see Appendix A, AM p. 664, and Peierls,
Quantum Theory of Solids, pp. 144-149. This was first calculated by Landau in
1930.
Figure 3:
The energy of the two dimensional electron gas at
as a function of
where is the filling factor (the number of filled Landau
levels) and
is the field at which all the electrons are in a completely
filled lowest Landau level. For metals, is much greater than a
typical laboratory
field so in experimental situations the filling factor is large.
Note that the behavior is periodic apart from a
gentle overall decrease of the amplitude which is negligible for .
|
Note from Eq. (20)
that is a periodic function
of and hence, apart from the slowly varying factor of
, the energy, given by Eq. (25), is
an oscillatory function of . This is clearly seen in
Fig. 3.
The oscillation in the ground state energy give rise to oscillations in the
magnetization per electron , given by
|
(26) |
which, from Eqs. (5), (15), (17), (18),
(20) and (25) is given by
|
(27) |
where
|
(28) |
is the Bohr magneton.
For , which is generally the case of interest,
this is given, to a good approximation, by
|
(29) |
Since is periodic in , see Eq. (20),
it follows that is also periodic
in , see Fig. 4.
Figure 4:
The magnetization of the two dimensional electron gas
at
(in units of ) as a function of
where is the filling factor and
, obtained from Eq. (27).
Note that the magnetization is a periodic function of .
|
The main conclusion so far is that is a periodic function of with
period given by
|
(30) |
From Eq. (14) it follows that the density is related to the Fermi wave
vector by
|
(31) |
and so, using Eq. (5),
|
(32) |
where
|
(33) |
is the area of the Fermi surface. In Sec. III we shall show that
Eq. (32), which relates the periodicity to just fundamental constants
and the area of the Fermi surface,
is true quite generally, not just for free electrons.
In Sec. IV
we will also
discuss the implications of this result for de Haas-van Alphen experiments
which use oscillations in the magnetization to map out Fermi surfaces of
metals, see e.g. Ashcroft and Mermin, Ch. 12.
Figure 5:
The solid circles are the semi classical orbits of free electrons in two
dimensions in a magnetic field. Each circle corresponds to a particular Landau
level. The electrons follow orbits around these circles with a period
, where is the cyclotron frequency.
As the strength of the field is increased the
radii increase and the orbits ``pop out'' of the Fermi circle, denoted by a
dashed circle. This leads to oscillatory behavior in the energy and
magnetization provided is much less than the spacing of the Landau
levels,
.
|
It is interesting to see how Eq. (32) emerges from the semi-classical
picture. The semi classical orbits for free electrons
are circles, on
which the energy stays constant. Denoting by the magnitude of the
wave vector corresponding to Landau level , then equating
to
one gets
|
(34) |
and so the area in -space swept out by an electron in Landau level is
|
(35) |
which can also be written as
|
(36) |
Hence, in the semi-classical picture of free electrons is a magnetic field,
the allowed states in -space form circles whose radius increases with
, see Fig. 5. As the states go through the Fermi wave vector and
become de-populated this gives rise to oscillations in the energy. From
Eq. (36) the change in for one Landau level to go through the
Fermi surface (i.e.
let with , the area of the
Fermi surface) is given by
|
(37) |
in agreement with Eq. (32).
Next: Two dimensions, Bloch electrons
Up: magnetic_field
Previous: Introduction
Peter Young
2002-10-31