We try to give a brief discussion about Fermi Surface and its importance in Semiconductor. Calculation of Fermi energy of different materials.

1. FERMI SURFACES AND ITS
IMPORTANCE IN
SEMICONDUCTOR
PRESENTED BY:
OSAMA MUNAWAR 17441510-098
M.UZAIR 17441510-099
NAYAB TAHIR 17441510-097
MARIA KHIZAR 17441510-
100

2. INTRODUCTION OF FERMI SURFACE
FIRST CONTACT WITH A FERMI SURFACE IS
THROUGH SUMMER FIELD FREE ELECTRON
MODEL.
ALLAN MACKINTOSH SUGGESTED “ A METAL
IS A SOLID WITH A FERMI SURFACE”
IN CONDENSED MATTER PHYSICS, IT DEFINES
THE ALLOWABLE ENERGY OF ELECTRON IN
SOLID.
IT WAS NAMED FOR ITALIAN
PHYSICIST ENRICO FERMI, WHO ALONG WITH
ENGLISH PHYSICIST P.A.M. DIRAC DEVELOPED
THE STATISTICAL THEORY OF ELECTRONS. .
A FERMI SURFACE IS A SURFACE IN
RECIPROCAL SPACE WHICH SEPARATED
UNFILLED ORBITALS FROM FILLED ORBITALS
2

3. FERMI SURFACE
FERMI SURFACES ARE IMPORTANT FOR CHARACTERIZING AND PREDICTING
THE THERMAL ELECTRICAL MAGNETIC AND OPTICAL PROPERTIES OF
CRYSTALLINE METALS AND SEMICONDUCTORS.
THEY ARE CLOSELY RELATED TO THE ATOMIC LATTICE, WHICH IS THE
UNDERLYING FEATURE OF ALL CRYSTALLINE SOLIDS, AND TO
ENERGY BAND THEORY WHICH DESCRIBES HOW ELECTRONS ARE
DISTRIBUTED IN SUCH MATERIALS.
IT GIVES UNIFYING CONCEPT BEHIND A VARIETY OF ELECTRONIC
BEHAVIOUR IN METALS.
THUS, SURFACE IN K-SPACE THAT SEPARATES OCCUPIED FROM
UNOCCUPIED ELECTRON STATE AT 0K.
3

4. REPRESENTATION
ONLY METALS HAVE FERMI SURFACES
FOR DIFFERENT DIMENSIONS THERE ARE DIFFERENT
REPRESENTATIONS WHICH INCLUDE:
1D: REPRESENTED BY A PAIR OF POINTS
2D: REPRESENTED BY A LINE (WHICH COULD BE A CLOSED LOOP)
3D: REPRESENTED BY A SURFACE (WHICH COULD BE A CLOSED
SURFACE)

5. FERMI SURFACE AND FERMI SPHERE
THE GROUND STATE OF N FREE ELECTRONS IS CONSTRUCTED BY
OCCUPYING ALL ONE ELECTRON LEVELS WITH “K”ENERGIES.
𝜖 𝑘 =
ℎ𝑘
2𝑚
≤ 𝜖𝑓
WHERE 𝜖𝑓 IS DETERMINED BY REQUIRING THE TOTAL NO OF
ELECTRONS LEVELS WITH THE ENERGY LESS THAN TO BE EQUAL TO 𝜖𝑓
TOTAL NO OF ELECTRONS WHEN THE LOWEST FILLED BY SPECIFIED
NUMBER TWO QUITE DISTINCT TYPES OF CONFIGURATION CAN
OCCUR
A CERTAIN NUMBER OF BANDS MAY BE COMPLETELY FILLED, ALL
OTHERS REMAINING EMPTY.
5

6. CONTINUE….
THE BAND GAP IS DETERMINED BY THE DIFFERENCE BETWEEN THE HIGHEST
OCCUPIED ENERGY LEVEL AND THE LOWEST OCCUPIED ENERGY LEVEL.
THE FERMI ENERGY, LIES WITHIN THE ENERGY RANGE OF ONE OR MORE BANDS.
FOR EACH PARTIALLY FILLED BAND THERE WILL BE A SURFACE IN K-SPACE
SEPARATING THE OCCUPIED FROM THE UNOCCUPIED LEVELS. THE SET OF ALL
SUCH SURFACES IS KNOWN AS THE FERMI SURFACE.
THE PARTS OF FERMI SURFACE ARISING FROM INDIVIDUAL PARTIALLY FILLED
BANDS.
6

7. FERMI SURFACE
• ALL ELECTRON STATES WITHIN A FERMI SPHERE IN K-
SPACE ARE FILLED UP TO FERMI WAVE VECTORS.
• TOTAL AREA OF FILLED REGION IN K-SPACE DEPENDS ONLY
ON ELECTRON CONCENTRATION.
• IT IS INDEPENDENT OF INTERACTION OF ELECTRONS WITH
LATTICE.
• WAVE VECTOR K MUST B CONFINED TO A SINGLE
PRIMITIVE CELL OF RECIPROCAL LATTICE.
• THUS, FERMI SURFACE IS AN ABSTRACT BOUNDARY OF
CONSTANT.
• ENERGY USEFUL FOR PREDICTING MAGNETIC, THERMAL,
OPTICAL.
• AND ELECTRIC PROPERTIES OF METALS, SEMI METALS AND
THE DOPED SEMICONDUCTOR. 7

8. SHAPE OF FERMI SURFACE:
SHAPE OF FERMI SURFACES REFLECT
THE ARRANGEMENT OF ATOMS AND IS
THUS A GUIDE TO THE PROPERTIES OF
THE MATERIAL
IT DEPENDS ON LATTICE INTERACTION.
SOME METALS, SUCH
AS SODIUM AND POTASSIUM.
THE FERMI SURFACE IS MORE OR LESS
SPHERICAL (A FERMI SPHERE), WHICH
INDICATES THAT THE ELECTRONS
BEHAVE SIMILARLY FOR ANY DIRECTION
OF MOTION.
OTHER MATERIALS ALUMINIUM AND
LEAD, FERMI SURFACE HAVE INTRICATE
SHAPE.
8

9. CONTINUE…
IN EVERY CASE, THE DYNAMIC BEHAVIOR OF ELECTRONS RESIDING AT OR NEAR
THE FERMI SURFACE IS CRUCIAL IN DETERMINING ELECTRICAL, MAGNETIC,
AND OTHER PROPERTIES
THEY DEPEND ON DIRECTION WITHIN THE CRYSTAL BECAUSE AT
TEMPERATURES ABOVE ABSOLUTE ZERO
THESE ELECTRONS ARE RAISED ABOVE THE FERMI ENERGY AND BECOME FREE
TO MOVE.
ELECTRICAL PROPERTIES OF METAL ARE DETERMINED BY SHAPE OF FERMI
SURFACE BECAUSE CURRENT IS
 DUE TO CHANGE IN THE OCCUPANCY OF STATES NEAR THE FERMI SURFACE
9

10. FERMI ENERGY LEVEL:
• FERMI LEVEL IS THE HIGHEST ENERGY
LEVEL THAT AN ELECTRON CAN
OCCUPY AT ABSOLUTE ZERO
TEMPERATURE.
• FERMI LEVEL DEFINED ONLY FOR
ABSOLUTE TEMPERATURE.
• FERMI ENERGY LEVEL IS ENERGY
DIFFERENCE BETWEEN FERMI LEVEL
AND LOWEST OCCUPIED SINGLE
PARTICLE.
• FERMI TEMPERATURE IS TEMPERATURE
AT WHICH ENERGY OF ELECTRON IS
EQUAL TO ENERGY OF FERMI LEVEL.
10

11. FERMI ENERGY
• FERMI ENERGY IS THE MAXIMUM K.E AN
ELECTRON CAN ATTAIN AT 0K.
• APPLIED IN DETERMINING THE ELECTRICAL
AND THERMAL CHARACTERISTICS OF SOLID.
• IMPORTANT IN NUCLEAR PHYSICS TO
UNDERSTAND THE STABILITY OF WHITE
DRAFT.
• IN METALS, FERMI ENERGY GIVES US THE
INFORMATION ABOUT ELECTRON VELOCITIES
WHICH ARE CLOSE TO FERMI ENERGY CAN
PARTICIPATE.
11

12. HOW TO
CALCULA
TE THE
FERMI
ENERGY
THEORET
ICALLY
To determine the lowest
possible Fermi energy of a
system,
group the states with equal
energy into sets and arrange
them in increasing order of
energy.
Add particles one at a time,
successively filling up the
unoccupied quantum states
with the lowest energy.
the energy of the highest
occupied state is the Fermi
energy. 12

13. PHYSICAL SIGNIFICANCE OF THE FERMI
ENERGY
SOME FERMI ENERGY APPLICATIONS ARE GIVEN IN THE POINTS
IT IS USED IN SEMICONDUCTORS AND INSULATORS.
IT IS USED TO DESCRIBE INSULATORS, METALS, AND
SEMICONDUCTORS.
FERMI ENERGY IS APPLIED IN DETERMINING THE ELECTRICAL
AND THERMAL CHARACTERISTICS OF THE SOLIDS.
THE CONCEPT OF THE FERMI ENERGY IS A CRUCIALLY
IMPORTANT CONCEPT FOR THE UNDERSTANDING OF THE
ELECTRICAL AND THERMAL PROPERTIES OF SOLIDS.
13

14. CONTINUE….
 THE FERMI LEVEL PLAYS AN IMPORTANT ROLE IN THE BAND THEORY OF
SOLIDS. IN DOPED SEMICONDUCTORS, P-TYPE AND N-TYPE, THE FERMI
LEVEL IS SHIFTED BY THE IMPURITIES, ILLUSTRATED BY THEIR BAND
GAPS. THE FERMI LEVEL IS REFERRED TO AS THE ELECTRON CHEMICAL
POTENTIAL IN OTHER CONTEXTS.
IN METALS, THE FERMI ENERGY GIVES US INFORMATION ABOUT THE
VELOCITIES OF THE ELECTRONS WHICH PARTICIPATE IN ORDINARY
ELECTRICAL CONDUCTION. THE AMOUNT OF ENERGY WHICH CAN BE
GIVEN TO AN ELECTRON IN SUCH CONDUCTION PROCESSES IS ON THE
ORDER OF MICRO-ELECTRON VOLTS , SO ONLY THOSE ELECTRONS VERY
CLOSE TO THE FERMI ENERGY CAN PARTICIPATE. THE FERMI VELOCITY OF
THESE CONDUCTION ELECTRONS CAN BE CALCULATED FROM THE FERMI
ENERGY.
14

15. THE FERMI SURFACE IN REAL METAL:
1. THE ALKALI METAL:
CAN BE CONSIDERED TO BE SPHERICAL. HAVE ONE VALENCE ELECTRON PER ATOM.
CONDUCTION BAND ONLY HALF FILLED. WILL NOT TOUCH THE BRILLOUIN ZONE BOUNDARY.
2. HYDROGEN METAL:
AT A HIGH PRESSURE SOLID MOLECULAR HYDROGEN PRESUMABLY BECOME A METAL WITH
HIGH CONDUCTIVITY. THE METALLIC HYDROGEN PRODUCED WAS A FLUID. THERE MAY B
METALLIC HYDROGEN ON JUPITER.
3. THE ALKALINE EARTH METAL:
MUCH MORE COMPLICATED THEN THE ALKALI METALS. TWO VALENCE ELECTRON PER ATOM
BUT BAND OVERLAPPING CAUSES THE ALKALINE EARTH TO FORM METALS RATHER THAN
INSULATOR. THE CASE OF SECOND ZONE HOLES HAVE BEEN CALLED “FALICOV’S MONSTER”
4. THE NOBLE METALS: THE NOBLE METALS IS TYPICALLY MORE COMPLICATED THAN FOR
THE ALKALI METAL. THE EXAMPLES ARE CU, ZN, AG AND AU.

16. SUMMARY OF METAL AND FERMI
SURFACE
Type of Metal Fermi Surface Comment
Free electron gas Sphere ------
Alkali (bcc)
(monovalent, Na, K, Rb, Cs)
Nearly Spherical
Specimens Hard to work
Alkaline earth (fcc) ----- Can be complex
Noble
(nonvaalent, Cu,Ag, Au)
Distorted sphere makes contact
with hexagonal faces complex in
repeated zone
Specimens need to be pure and
single crystal

17. FERMI SURFACE OF COPPER
• Fermi surface of Copper is distinctly non spherical. Eight necks make contact with the
hexagonal faces of the first Brillouin zone of the fcc lattice.
• The electron concentration in a monovalent metal with an fcc structure is n=4/a3
• The radius of a free electron Fermi surface is
Kf = (3π2n)1/3 = (12π2/a3)1/3 ≅ (4.90/a)
• Shortest distance across BZ is equal to distance between hexagonal face =
2𝜋
𝑎
3 =
10.88
𝑎
• band gap at zone boundaries =Band energy there lowered = necks
• distance between square faces = 12.57/a necking not expected.

18. FERMI SURFACE OF GOLD:
• In gold for quite a wide range of field direction Shoenberg finds the magnetic moment has a period of 2×
10-9 gauss-1
• S =
2𝜋𝑒/ℎ𝑐
∆(
1
𝐵
)
≅
9.55×107
2×10−9 ≅ 4.8× 1016
cm-1
• For a free electron Fermi sphere for gold is kf = 1.2× 103
cm-1
• An external area of 4.5× 1016 cm-2
• The actual period of Shoenberg are 2.05× 10−9
gauss-1 and 1.95× 10−9
gauss-1
• In the [111] direction in Au a large period of 6× 10−8 gauss-1.
• S=1.6× 1015 cm-2
• Dog’s bone area = 0.4 of belly area

19. CONTINUE….
THE FERMI ENERGY ALSO PLAYS AN IMPORTANT ROLE IN UNDERSTANDING THE
MYSTERY OF WHY ELECTRONS DO NOT CONTRIBUTE SIGNIFICANTLY TO THE
SPECIFIC HEAT OF SOLIDS AT ORDINARY TEMPERATURES, WHILE THEY ARE
DOMINANT CONTRIBUTORS TO THERMAL CONDUCTIVITY AND ELECTRICAL
CONDUCTIVITY. SINCE ONLY A TINY FRACTION OF THE ELECTRONS IN A METAL
ARE WITHIN THE THERMAL ENERGY KT OF THE FERMI ENERGY, THEY ARE
"FROZEN OUT" OF THE HEAT CAPACITY BY THE PAULI PRINCIPLE. AT VERY LOW
TEMPERATURES, THE ELECTRON SPECIFIC HEAT BECOMES SIGNIFICANT.

20. DE HAAS-VAN ALPHEN EFFECT:
THE DE HAAS-VAN ALPHEN (DHVA) EFFECT IS AN OSCILLATORY
VARIATION OF THE DIAMAGNETIC SUSCEPTIBILITY AS A FUNCTION OF A
MAGNETIC FIELD STRENGTH (B).
THE METHOD PROVIDES DETAILS OF THE EXTREMAL AREAS OF A FERMI
SURFACE. THE FIRST EXPERIMENTAL OBSERVATION OF THIS BEHAVIOR
WAS MADE BY DE HAAS AND VAN ALPHEN (1930).
THEY HAVE MEASURED A MAGNETIZATION M OF SEMIMETAL BISMUTH
(BI) AS A FUNCTION OF THE MAGNETIC FIELD (B) IN HIGH FIELDS AT 14.2
K AND FOUND THAT THE MAGNETIC SUSCEPTIBILITY M/B IS A PERIODIC
FUNCTION OF THE RECIPROCAL OF THE MAGNETIC FIELD (1/B).
20

21. CONTINUE…..
• THIS PHENOMENON IS OBSERVED ONLY AT LOW TEMPERATURES AND HIGH
MAGNETIC FIELDS. SIMILAR OSCILLATORY BEHAVIOR HAS BEEN ALSO OBSERVED IN
MAGNETORESISTANCE.
• WE DO NOT WANT THE QUANTIZATION OF THE ELECTRON ORBITS TO BE BLURRED
BY COLLISIONS, AND WE DO NOT THE POPULATION OSCILLATIONS TO BE AVERAGED
OUT BY THERMAL POPULATION OF ADJACENT ORBITS.
21

22. CONTINUE….
• THE ANALYSIS OF THE DHVA IS GIVEN FOR
ABSOLUTE ZERO AS FOLLOWS.
• THE AREA BETWEEN SUCCESSIVE ORBITS IS
∆𝑆 = 𝑆 𝑛 − 𝑆 𝑛−1 =
2𝜋𝑒𝐵
ℏ𝑐
• THE AREA IN K SPACE OCCUPIED BY SINGLE
ORBITAL IS(
2𝜋
𝐿
) 𝟐
,NEGLECTING SPIN FOR SQUARE
SPECIMEN OF SIDE L. THE NUMBER OF FREE
ELECTRON ORBITALS THAT COALESCE IN A
SINGLE MAGNETIC LEVEL IS
D=(
2𝜋𝑒𝐵
ℏ𝑐
) (
𝐿
2𝜋
) 𝟐
=𝜌𝐵
• SUCH MAGNETIC LEVEL IS CALLED LANDAU
LEVEL.
22

23. CONTINUE…
• THE MAGNETIC MOMENT 𝜇 OF A SYSTEM AT
ABSOLUTE ZERO IS GIVEN BY
𝜇 = −
𝜕𝑈
𝜕𝐵
• THE MOMENT HERE IS OSCILLATORY FUNCTION
OF 1/B AS SHOWN IN FIGURE. THIS OSCILLATORY
MAGNETIC MOMENT OF FERMI GAS AT LOW
TEMPERATURES IS DE HASS- VAN ALPHEN
EFFECT. THE OSCILLATIONS OCCURS AT EQUAL
INTERVALS OF 1/B
∆
1
𝐵
=
2𝜋𝑒
ℏ𝑐𝑆
23

24. REFERENCES
• HTTPS://WWW.BRITANNICA.COM/SCIENCE/FERMI-SURFACE
• HTTPS://PHYSICS.APS.ORG/ARTICLES/V3/86
• HTTPS://ENG.LIBRETEXTS.ORG/BOOKSHELVES/MATERIALS_SCIENCE/
SUPPLEMENTAL_MODULES_(MATERIALS_SCIENCE)/ELECTRONIC_PROP
ERTIES/FERMI_ENERGY_AND_FERMI_SURFACE
• INTRODUCTION TO SOLID STATE PHYSICS BY CHARLES KITTEL PAGE
(223-251)CHAPTER 9( FERMI SURFACES AND METALS)
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