The de-Hass van-Alphen Effect
1930: Magnetization of bismuth
measured at 14.2 K as a function of
magnetic field oscillations in M/H or
magnetic susceptibility
Original dHvA data of Bismuth
Similar oscillations were seen in many other physical (non-magnetic) properties of
other metals, e.g., thermal conductivity, thermoelectric voltage, sound attenuation,
measured in the presence of a magnetic field
- Similar effects were seen in many other metals, like, silver, rhenium, etc.
Peltier effect in zinc
Typical dHvA
Results
Attenuation of sound -
- Magnetoaccoustic effect
Thermoelectric voltage and
thermal conductivity of Bismuth
de-Hass van-Alphen Effect explained
1952 : Onsager pointed out that the
oscillations were periodic in 1/B or 1/H
and the period was related to the
extremal cross sectional areas of Fermi
surface in the plane normal to the
magnetic field
Its explanation is based on Landau’s
theory for free electrons in a magnetic
field
de-Hass van-Alphen effect represents.
a definite failure of the semi-classical
description!
Notice the extremal sections
on the constant energy surface
Bloch electrons in uniform magnetic field
- de-Hass van-Alphen effect represents a failure of the semi-classical description!
- The failure arises whenever the semi-classical model predicts closed orbits for
electronic motion, projected on a plane perpendicular to the magnetic field, as under
this condition, the energies of motion perpendicular to B are quantized, as in the
case of free electrons
- To find these energy levels, in principle, the SE must be solved for an electron in a
periodic potential in the presence of a magnetic field, which is a difficult task, that
has been accomplished for free electrons in a magnetic field
- We have seen that Bloch electrons under the influence of a magnetic field move on
continuous paths perpendicular to B – This is not inconsistent with the discrete set of
k-vectors in reciprocal space, but recall that the discrete allowed values of k and the
use of kx, ky and kz arise essentially from three dimensional translational invariance
- In a magnetic field, Schrodinger equation is no longer translationally invariant in the
direction perpendicular to the magnetic field and kx and ky are no longer good
quantum numbers and in a way, the band model collapses!
Free electrons in uniform magnetic field
- Landau’s theory
The orbital energy levels of an electron in a cubical box of sides L along x, y and z, in
the presence of a magnetic field B along z are given by 2 quantum numbers, kz and ν:
Eν (kz) = ħ2kz2/2m + (ν + ½) ħωc - where ωc = eB/m is the cyclotron frequency
ν runs through all positive integers and kz = 2πnz/L (as in the absence of B)
- Each level is highly degenerate and the number of levels with energy for a given ν
and kz is 2eBL2/h, which for a field of 1 kG and L = 1cm is ~ 1010
- The high degeneracy reflects the fact that a classical electron, with a given energy
spirals about a line parallel to z-axis and can have arbitrary x- and y- coordinates
Explanation:
- In the absence of a force in B direction, the motion along z is unaffected by the
field and continues to be governed by ħ2kz2/2m
- But the energy of motion perpendicular to the field is quantized in steps of ħωc ,
where, ωc is the frequency of classical motion – Orbit Quantization!
The set of all levels with a given ν (and arbitrary kz) is collectively referred to as the
νth Landau level
Onsager’s generalization of Landau’s free
electron results
- Onsager’s argument generalizes the free electron results to Bloch electrons, which is
valid only for magnetic levels with high quantum numbers
- This is true for electrons on the Fermi surface (which participate in dHvA effect),
for which,
Ef ~ 1 eV and ħωc (= ħeB/m) ~ 10-4 eV (for B = 1Tesla) → Ef /ħωc ~ 104
Bohr’s correspondence principle: Energy levels with very high quantum numbers can
be accurately calculated by taking the difference in energy of two adjacent levels as
Planck’s constant times the frequency of classical motion at the energy of the levels
Since kz is a constant of semiclassical motion, this condition can be applied to levels
with a specified kz and quantum numbers ν and ν+1, so
if E ν(kz) is the energy of the νth level at a given kz , the correspondence principle
gives :
E ν+1(kz) - E ν(kz) = h/T(E ν(kz ),kz)
Bloch electron levels in a uniform magnetic field
From correspondence principle: E ν+1(kz) - E ν(kz) = h/T(E ν(kz ),kz)
From (6): T(E,kz) = (ħ2/eB) ∂A (E,kz) /∂E
So, [E ν+1(kz) - E ν(kz)] ∂A(E,kz)/∂E = 2πeB/ħ (8)
Note: E ν+1(kz) - E ν(kz) = ħωc ~ 10-4 eV
So for neighbouring Landau levels, the difference between energy levels is at least
10-4 times the energy of the level itself,
So we can write: ∂A (E,kz) /∂E as [A(E ν+1) – A(E ν)] / [E ν+1 - E ν] (9)
Placing (9) in (8)
A(Eν+1) – A(Eν) = 2πeB/ħ (10)
which states that the classical orbits of adjacent allowed energies (with the same kz)
enclose areas that differ by the fixed amount, given by
∆A = 2πeB/ħ (11)
and notice, that ∆A increases linearly with B!
Landau levels for Bloch electrons – Landau Tubes
(11) ∆A = 2πeB/ħ
In other words, at large ν, the area enclosed by a semi-
classical orbit at an allowed energy and kz is given by
A(E ν(kz ),kz) = (ν + λ) ∆A (12)
where, λ is independent of ν
Notice: Each Landau state ν is represented by
an orbit in reciprocal space in the plane
perpendicular to B, instead of a point (kx ,ky)!
As ‘A’ is independent of kz, the electrons located on
a set of orbits satisfying (11) for a given ν, form a
tubular structure of cross-sectional area (ν + λ) ∆A,
called Landau Tubes
The cross section of the tubes is determined by the nature of the CES in zero field!
For circular orbits (parabolic bands with scalar m*): Landau tubes are circular cylinders
The Landau tube/cylinder depicts the set of all orbits satisfying (12) for a given ν,
but notice that it is not a constant energy surface, since the total energy depends on kz!
Landau tube features
The energy of electron on the νth Landau tube is
E = (ħ2/2m*)(kx2 + ky2 + kz2 ) = (ħ2/2m*)(k┴2 + kz2 ) Landau cylinders in RS
enclosed by a zero-field CES
= ( ħ2 kz2 /2m*) + (ν + ½) ħωc (13)
In the presence of B, all the k- states between the
tubes condense on the tubes with no change in the
total number of states!
In effect, the field moves all the states with zero-field
energies between
(ħ2 kz2 /2m*) + ν ħωc to (ħ2 kz2 /2m*) + (ν+1) ħωc
to Landau tubes of energy ħ2 kz2 /2m*) + (ν+1/2) ħωc
Radius of the νth Landau cylinder (k┴): ( ħ2 k┴2 /2m*) = (ν + ½) ħωc (14)
No. of states/length of Landau cylinder is equal to the No. states between neighbouring
cylinders of unit length
No. of states/length of cylinder = 2. (△A.1)/(8π3/V) = (2πeB/ħ)/(4π3/V)
= (Vm*ωc /2π2 ħ) (15)
Plot of energy along Landau tubes and their filling
E(kz,ν)
Notice that the filled length kz(max) of each tube
depends on ν!
Ef
At T= 0
kz(max,ν)
Ef = (ħ2/2m*) kz2(max,ν) + (ν + ½) ħωc (16)
Landau levels fill according to FD statistics
and the total No. of electrons is given by
kz
∞ ∞
N = (Vm*ωc /2π2 ħ) ∑ ∫ dkz /{1+ exp (E – μ)/kT}
ν = 0 -∞
where, E = E (kz ,ν)
and μ(T) is the chemical potential,
which is a complicated function of B Fermi sphere
and can be determined from the
above expression for N
Explanation of the oscillatory phenomenon
- The separation between the tubes is very small compared to kf (but large wrt 2π/L)
and expands with increase of B
- The contribution to g(E)dE from the Landau levels associated with orbits on the
νth tube will be the number of such levels (of different kz ) between E and E+dE
- This, in turn, is proportional to the area on the wall of the tube contained between
two constant energy surfaces (in the absence of B)
Notice from Fig. A:
When the orbits of energy
Eν on the tube are NOT
extremal wrt the constant
energy surface, the total
number of states on the
tubes (or, dkz for different
values of kz , which contribute
to g(E)dE is much smaller,
as compared to Fig. B,
when the orbits on a
tube are extremal wrt A B
the constant energy surface
Oscillatory phenomenon in dHvA effect
Most electronic properties of metals depend on the density of electron states on the
Fermi surface, g(Ef ),
which will become large, whenever the magnetic field causes an extremal orbit on
the Fermi surface and contribute to the physical phenomenon under consideration
As the tubes expand continuously with increase in B (∆A = 2πeB/ħ), this will happen
whenever the value of the magnetic field is such that an extremal orbit on the
Fermi surface (Ae) satisfies the orbit quantization condition (12), i.e.,
Ae(Ef) = (ν + λ) ∆A = (ν + λ) (2πeB/ħ)
→ 1/B = (ν + λ) (2πe/ħ) / Ae(Ef) (17)
This will happen at regularly spaced intervals in 1/B (as ∆ν = ±1)
∆(1/B) = (2πe/ħ) / Ae(Ef) (18)
Notice, that period of oscillation in 1/B gives extremal area of Fermi surface
Thus by measuring Ae(Ef) in different orientations, the Fermi surface can be
constructed!
Measurement of Fermi Surface (Chap 15 A&M)
The shape of Fermi surface is intimately involved with the transport and optical
properties of metals and in general, all physical properties, hence, any information
about the size and shape of their Fermi surfaces is of special importance in
understanding various solid state phenomena
An experimentally measured Fermi surface also provides a target for first principles
band structure calculations
It often provides fitting parameters in the phenomenological crystal potential, which
can be used to calculate different properties
On the other hand, band structure information from theoretical calculations can be
used with dHvA data to suitably construct Fermi surfaces
De-Hass van-Alphen effect is the most powerful technique for the measurement of
Fermi surfaces!
Fermi surface measurements- Alkali metals
K, Na, Cs, Rb: FCC reciprocal lattice (BCC direct lattice)
kf = ( 3π2 n )1/3 with n = 2/a3
→ kf = 0.620 (2π/a) < Γ – N distance [ 0.707 (2π/a) ]
So, the free electron sphere is contained well within the FBZ
- For alkali metals, such as K, Na, Rb, Cs, the value of kf can be easily
estimated
- De-Hass van-Alphen measurements of Fermi surfaces of all these metals
have been performed by working at a constant magnetic field and
observing the magnetic susceptibility at different orientations of the crystal
- The fractional deviations of (∆kf /kf) for these Fermi surfaces, measured
along different orientations, were ~ 10-3, which confirmed that the Fermi
surfaces were nearly spherical and were contained well within the first BZ,
as expected for the half filled band of monovalent metals
Fermi surface measurements- Noble metals
Cu, Ag, Au: BCC reciprocal lattice
- The Fermi surface of the partly filled free electron bands of Cu, Ag, Au is a sphere
entirely contained within the the first BZ, approaching the surface of the zone, most
closely in the <111> direction and making contact with the hexagonal faces
- De-Hass van-Alphen measurements in
all the three noble metals reveal that their
Fermi surfaces are closely related to the
free electron sphere
- Interestingly, the dHvA measurement data
with B along <111> direction, contains two
periods, determined by the two extremal
orbits corresponding to belly (maximum)
and neck (minimum) orbits. The ratio of
the two periods has been used to Ag
determine the ratio of the maximal to
minimal cross-sections of
27 (Cu), 51 (Ag) and 29 (Au)
Cyclotron resonance in metals
The cyclotron resonance does not measure Fermi surface geometry !
By observing the absorption of rf signal at resonance frequencies of the electric field
with the orbital motion of Bloch electrons in a magnetic field, it measures the cyclotron
effective mass (∂A/ ∂E), leading to the determination of effective mass tensor
- In semiconductors, electrons/hole exist in pockets of ellipsoidal/spherical constant
energy surface, so for a given orientation of crystal wrt B, all electrons/holes precess
at a frequency determined by the effective mass tensor describing that pocket
- In metals, electrons only within the skin depth can absorb energy, so the preferred
geometry of experiment is to place the magnetic field parallel to the surface
- If the electron orbiting near the surface, enters the skin depth and experiences the
same phase of the electric field every time, then it can resonantly absorb energy from
the field
- This requires, T = nTE
which implies, ωE = n ωc → (1/B) = n.2πe / ħ2ωE (∂A/∂E) as the resonance
condition
For a certain rf frequency, this gives uniformly spaced resonant peaks due to a given
ωc, which are spaced uniformly in (1/B), if they originate from the same period
Cyclotron resonance in metals (cont’d)
Analysis and the interpretation of data from cyclotron resonance in metals is
complicated because the identification of orbits contributing to resonance is often
difficult for the following reasons:
- T(Ef,kz) can depend on kz (unlike ellipsoidal/spherical Fermi surfaces), and several
orbits of electrons on the Fermi surface with different T (~ ∂A/ ∂E) may give rise to
almost a continuum of resonances
- Although extremal orbits dominate the determination of resonant frequencies, but
contributions from other orbits can often result in a complicated averaging of T
- The detailed dependence of energy loss peaks can thus have a complicated
structure, as several extremal periods could be involved and resonance peaks due
to a range of T’s are seen for a given B, some of which are spaced uniformly in
(1/B) if they originate from the same period
So, in general, one may not be measuring the extremal values of T(Ef,kz)
unambiguously, but, rather some complicated average over the Fermi surface
