Harrison's method. Construction of Fermi surfaces. FERMI SURFACE. The boundary between the filled and empty states. BASIC Of HARRISON’S METHOD. Importance of Fermi surfaces. HOW INFORMATION CAN BE EXTRACTED FROM FERMI SURFACE? Kondo Insulators. Galvano-magnetic Measurements. A complete explanation by Syed Hammad Ali Gillani.
3. FERMI
SURFACE
• The surface of constant energy
in k-space inside which all the
states are occupied by valence
electrons.
• The boundary between the
filled and empty states.
• Strictly Valid at Absolute Zero.
• (wikipedia) 3
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4
BASIC Of HARRISON’S METHOD:
• It is based on weak pseudo-potential method 𝑉𝑒𝑓𝑓(r).
• Its effect to introduce a small perturbation in motion of free electrons.
• This potential Cause the energy discontinuities as well as distortions in
F.S at zone boundaries.
• The 𝑉𝑒𝑓𝑓(r) is made arbitrarily low, the discontinuities and the
distortions in F.S can be removed.
• 𝑉𝑒𝑓𝑓(r) is made arbitrarily low so that we describe the F.S of any radius
from center of 1st Brillioun zone i.e. a F.S consisting of any number of
electrons.
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5
CONSTRUCTION:
The reciprocal lattice points of a square lattice are determined, and free-
electron sphere of radius appropriate to the electron concentration is
drawn around each point.
Any point in k space that lies within at least one sphere corresponds to an
occupied state in the first zone.
Points within at least two spheres correspond to occupied states in the
second zone, and similarly for points in three or more spheres.
In other words, the darker the shading, the higher the zone number
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2-D SQUARE LATTICE:
We assume that there is only one
atom per square lattice. Each atom
has p electrons. Then we get
2
𝐿2
(2𝜋)2
𝜋𝐾𝑓
2
=
𝐿2
𝑎2
𝑝 = 𝑁
Or 𝐾𝑓 =
2𝑝
𝜋
𝜋
𝑎
So 𝜀𝑓 =
ℏ2
2𝑚
𝑘𝑓
2
= 𝐸0
2𝑝
𝜋
Table;
𝐾𝑓
𝜋
𝑎
vs p and
𝜀𝑓
𝐸0
vs p
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8
1. p=1
• 𝐾𝑓 =
2
𝜋
𝜋
𝑎
= 0.797885
𝜋
𝑎
• The radius 𝐾𝑓 of Fermi circle is shorter
then the length Γ𝑋.
• The Fermi circle does not contact the
zone boundary of 1st Brillioun Zone.
• The 2-D Fermi circle corresponding to
p=1 electron per atom.
• Each Brillouin Zone is separated by
straight lines.
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9
2. p=2
• 𝐾𝑓 =
4
𝜋
𝜋
𝑎
= 1.12838
𝜋
𝑎
• The radius 𝐾𝑓 of Fermi circle is longer
then the length Γ𝑋.
• The Fermi circle does contact the zone
boundary of 1st Brillioun Zone around
point X.
• A part of the Fermi circle passes the
region of 2nd Brillouin Zone.
• The construction is shown in figure.
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10
• This Hole like Fermi Spheres.
• The Fermi Surface in the 1st Brillouin
Zone. It is periodic zone scheme.
• P=2
• The shaded areas contains electrons
forming a hole-like Fermi surface at
each corner of the Brillouin Zone- the
Point M.
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11
• This Electron like Fermi Spheres.
• The Fermi Surface in the 2nd Brillouin
Zone. It is periodic zone scheme.
• P=2
• The shaded areas green forms an
electron like Fermi surfaces.
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12
3. p=3
• 𝐾𝑓 =
6
𝜋
𝜋
𝑎
= 1.38198
𝜋
𝑎
• The radius 𝐾𝑓 of Fermi circle is longer then
the length Γ𝑋 But slightly shorter than
𝚪𝑴.
• The Fermi circle passes through the region
of 2nd Brilloiun zone and a very small part
of 1st Brillouin zone.
• The 1st zone is almost full.
• ThE 2-D F.C corresponding to p=3.
• The Harrisons construction of the Fermi
Surface is shown:
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13
ELECTRON LIKE FERMI SURFACE
The 2nd zone:
• The Fermi surface in the 2nd
zone.
• P=3
• The shaded area contains
electrons.
• The Fermi surface is electron
like.
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14
2-D HEXAGONAL LATTICE METAL:
• Assume one atom per area
3𝑎2
2
. Each atom has p-electrons.
Then we get: 2
𝐴
(2𝜋)2 𝜋𝐾𝑓
2
=
𝐴
3𝑎2
2
𝑝
• The 𝑘𝑓 is calculated as: 𝑘𝑓 =
2
3
1
4 𝑎
𝜋𝑝
• The Fermi energy is given as: 𝜀𝑓 =
ℏ2
2𝑚
𝑘𝑓
2
= 𝐸0
4𝑃
3
1
2𝜋
• The reciprocal lattice vector is: 𝐺1 =
4𝜋
3𝑎
• The ratio of 𝑘𝑓 to the zone boundary of 1st Brillouin zone 𝐺
2 is:
𝑘𝑓
𝐺1
2
= 3
1
4
𝑝
𝜋
= 0.742515 𝑝
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15
1. p=1
• For p=1, this ratio is smaller than 1,
while for p>2, the ratio is larger than
1.
• For p=1, There is a Fermi circle inside
the 1st Brillouin zone.
• We calculate:
𝑘𝑓
𝐺1
3
=
1
2
3
3
4
𝑝
𝜋
• Table is shown in figure:
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17
2. p=2
•
𝑘𝑓
𝐺1
2
= 1.05008
•
𝑘𝑓
𝐺1
3
= 0.909392
• The radius 𝑘𝑓 of Fermi circle is a little
longer than the shortest length 𝚪𝑴 of 1st
Brillouin zone.
• The Fermi circle does not contact the zone
corner-the K-point of the 1st Brillouin
zone.
• The F.C passes the region of 1st and 2nd
Brillouin zone.
• The construction is shown in Figure:
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18
• Hole Like Fermi Sphere-The 1st
Brilloiun zone.
• The Fermi surface in the 1st zone.
• P=2
• The shaded area represents occupied
electron states forming a hole-like
Fermi Surface.
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19
• Electron Like Fermi Sphere-The 2nd
Brilloiun zone.
• The Fermi surface in the 2nd zone.
• P=2
• The shaded area represents occupied
electron states forming a electron-
like Fermi Surface.
• (Chambers 1966)
20. The electrical and thermal properties of
metals are largely determined by one small
group of electrons: the conduction electrons
which lie at and near the Fermi surface.
Fermi surface is the surface
in reciprocal space which separates
occupied from unoccupied electron
states at zero temperature.
FERMI SURFACES:
IMPORTANCE:
21. • Those electrons are free to move through the
metal. These electrons fill a certain region in
‘wave-vector space', and the Fermi surface is
the surface bounding this region.
HOW INFORMATION
CAN BE EXTRACTED
FROM FERMI SURFCES:
21
Just as a tightly-packed crowd of people.
so the conduction electrons can only move, in
wave-vector space, if the electrons near the
Fermi surface move.
(Suzuki 2012)
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22
Kondo Insulators:
• Hybridization.
• Torque magneto-metry.
• Samarium hexa-boride (SmB6)
• The observed oscillation patterns reveal
two Fermi surfaces on the (100) surface plane
and one Fermi surface on the (101) surface
plane.
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23
Galvano-magnetic
Measurements
• The topology, shape and some details of the
Fermi surface in silver are discussed on the
basis of galvano-magnetic measurements.
• The axes of these cylinders are parallel to the
principal rational directions [001], [110], and
[111].* It is most natural to assume that the
open Fermi surface in silver, gold, and copper
is a surface formed by "spheres," joined in the
[111] direction by ''necks.'‘
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24
REFERENCES:
1. wikipedia "https://en.wikipedia.org/wiki/Fermi_surface."
2. Chambers, R. J. S. P. (1966). "The Fermi Surface." 163-191.
3. Suzuki, M. S. (2012). "Construction of Fermi surface Masatsugu Sei
Suzuki and Itsuko S. Suzuki Department of Physics, SUNY at Binghamton
(Date: March 28, 2012).“
4. Li, G., et al. (2014). "Two-dimensional Fermi surfaces in Kondo insulator
SmB6." 346(6214): 1208-1212.
5. Alekseevskii, N. and Y. P. J. S. P. J. GAlDUKOV (1962). "Fermi surface of
Silver." 15(1).
