| Krishna Ray|

THE CONCEPTS OF FERMI ENERGY

Fermi Level for Intrinsic, p-type and n-type materials

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Fermi Level for Intrinsic, p-type and n-type materials

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1. History

Before the introduction of Fermi-Dirac statistics in 1926, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena. It was difficult to understand, for example, why electrons in a metal can move freely to conduct electric current, while their contribution in the same metal to the specific heat was negligible, as if there were considerably fewer electrons. It was also difficult to understand why the emission currents, generated by applying high electric fields to metals at room temperature, were almost independent of temperature.

The difficulty encountered by the electronic theory of metals at that time was due to considering that electrons were (according to classical statistics theory) all equivalent. In other words it was believed that each electron contributed to the specific heat an amount of the order of the Boltzmann constant k. This statistical problem remained unsolved until the discovery of F-D statistics.

F-D statistics was first published in 1926 by Enrico Fermi and Paul Dirac. F-D statistics was applied in 1926 by Fowler to describe the collapse of a star to a white dwarf. In 1927 Sommerfeld applied it to electrons in metals and in 1928 Fowler and Nordheim applied it to field electron emission from metals. Fermi-Dirac statistics continues to be an important part of physics.

2. Introduction to Fermi Level

The concepts of Fermi energy and Fermi level are of paramount importance in theories of electrical conductivity, thermal conductivity, and a wide array of thermodynamical and chemical phenomena. The Fermi level was named after the famous physicist Enrico Fermi, who had a significant hand in developing the atom bomb. In thermodynamics and solid-state physics, the Fermi level is the most energetic state occupied by an electron when the system is in the lowest energy configuration (i.e. at absolute zero). It is a measure of the energy of the least tightly held electrons within a solid. It is important in determining the electrical and thermal properties of solids. The value of the Fermi level at absolute zero

3. The Concept of Fermi Level

(−273.15 °C) is called the Fermi energy and is a constant for each solid.

Fermi level is the term used to describe the top of the collection of electron energy levels at absolute zero temperature. It is the energy level at which an average of 50% of available quantum states are filled by an electron. The Fermi Level relates the probable location of electrons in a band diagram. In a band diagram of a substance (usually a semiconductor) the

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Fermi Level tells the average location of electron. For metals the Fermi Level lies in the conduction band and in insulators, the Fermi level lies between, and far from the bands and for semiconductors the Fermi Level lies in the band gap.

Figure 1: The energy arrangement in atoms. Figure 2: Fermi Energy Level for Different Materials

The Fermi level changes as the solid is warmed and as electrons are added to or withdrawn from the solid. In doped semiconductors, p-type and n-type, the Fermi level is shifted by the impurities, illustrated by their band gaps. Each of the many distinct energies with which an electron can be held within a solid is called an energy level. According to the laws of quantum mechanics

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, each energy level can accommodate only a limited number of electrons. The Fermi level is any energy level having the probability that it is exactly half filled with electrons. Levels of lower energy than the Fermi level tend to be entirely filled with electrons, whereas energy levels higher than the Fermi tend to be empty.

In

Formation of Fermi Level

quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons are fermions) obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. The states are labeled by a set of quantum numbers. In a system containing many fermions (like electrons in a metal) each fermion will have a different set of quantum numbers. Considering electrons as fermions, at absolute zero they pack into the lowest available energy states and build up a "Fermi sea" of electron energy states. The Fermi level is the surface of that sea at absolute zero where no electrons will have enough energy to rise above the surface. The concept of the Fermi energy is a crucially important concept for the understanding of the electrical and thermal properties of solids. Both ordinary electrical and

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thermal processes involve energies of a small fraction of an electron volt. But the Fermi energies of metals are on the order of electron volts. This implies that the vast majority of the electrons cannot receive energy from those processes because there are no available energy states for them to go to within a fraction of an electron volt of their present energy. Limited to a tiny depth of energy, these interactions are limited to "ripples on the Fermi sea".

5. Semiconductor and its Purity 5.1 Intrinsic Material

An intrinsic semiconductor, also called an undoped semiconductor or i-type semiconductor, is a perfect semiconductor crystal with no impurities or lattice defects pure. The number of charge carriers is therefore determined by the properties of the material itself. In intrinsic semiconductors the number of excited electrons and the number of holes are equal: n = p. The conductivity of intrinsic semiconductors can be due to crystal defects or to thermal excitation.

Fig: The carrier distribution of an intrinsic material. Here, Ei = EF

5.2 Extrinsic Material

, is the Fermi level of the intrinsic material.

It’s possible to create carriers in semiconductors by doping or purposely introducing impurities into the crystal. By doping a semiconductor can be altered so that it has a predominance of either electrons or holes. Therefore, there are two types of doped semiconductors, n-type (mostly electrons) and p-type (mostly holes). When a crystal is doped the carrier concentrations n0 and p0 is different from the intrinsic carrier concentration ni

Fig: Semiconductor doping by introducing Acceptor and Donor states.

. That is why the carrier distribution also differs from that of intrinsic.

Acceptor state

Donor state

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5.2.1 N-Type Material

Extrinsic semiconductors with a larger electron concentration than hole concentration are known as n-type semiconductors. The phrase 'n-type' comes from the negative charge of the electron. In n-type semiconductors, electrons are the majority carriers and holes are the minority carriers. N-type semiconductors are created by doping an intrinsic semiconductor with donor impurities. In an n-type semiconductor, the Fermi energy level is greater than that of the intrinsic semiconductor and lies closer to the conduction band than the valence band.

Fig: The carrier distribution of a n-type material. Here, EF

5.2.2 P-Type Material

, is closed to the conduction band.

As opposed to n-type semiconductors, p-type semiconductors have a larger hole concentration than electron concentration. The phrase 'p-type' refers to the positive charge of the hole. In p-type semiconductors, holes are the majority carriers and electrons are the minority carriers. P-type semiconductors are created by doping an intrinsic semiconductor with acceptor impurities. P-type semiconductors have Fermi energy levels below the intrinsic Fermi energy level. The Fermi energy level lies closer to the valence band than the conduction band in a p-type semiconductor.

Fig: The carrier distribution of a n-type material. Here, EF

6. Fermi-Dirac Distribution Function

, is closed to the valance band.

The Fermi function is a probability distribution function. The distribution of electron in a semiconductor in a semiconductor is governed by the law of Fermi-Dirac statistics. The result is a distribution function F(E), which yields the probability F that an energy state E is occupied

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by an electron. The Fermi energy level, EF

, is the energy at which the probability of occupancy is exactly 𝟏𝟏 𝟐𝟐� for temperatures greater than zero. The Fermi function is given by,

Here, EF is the Fermi energy of the electrons in the solid, k (=1.38062´10-23 J K-1

In a band diagram, the position of the Fermi level determines which carrier dominates. If the semiconductor contains more electrons than holes then it’s an n-type material and the Fermi level is positioned above mid gap. If holes are more abundant than electrons then it’s a p-type material and E

) is the Boltzmann’s constant and T is the temperature (kT is the thermal energy, 25 meV at room temperature).

F is positioned below mid gap. When the electron and hole concentrations are approximately equal or in an intrinsic material, EF

• The probability an energy state is occupied: 𝑓𝑓(𝐸𝐸) = 1

1+𝑒𝑒(𝐸𝐸−𝐸𝐸𝐹𝐹)

𝑘𝑘𝑘𝑘�

is positioned at mid gap. The Fermi function, or level, also varies with temperature and carrier concentration.

• The probability an energy state is empty: [1- f (E)] • The Fermi function at E = EF : f (E) = 1 2� • The Fermi function is simply a mathematical 0 < f (E) < 1

function and has no units

An energy level may contain several sublevels, all with the same energy. Each sublevel is called a "state," and can be occupied by exactly one electron. Suppose there are N allowed states at energy E. Then the probability of finding an occupied state at energy E is N×F(E). In continuous-band theory we represent N as a density of states. The density of states reveals how the allowed sublevels are spread out across energy bands in a specific material. In a semiconductor, not every energy level is allowed. For example, there are no allowed states within the forbidden gap. To make use of the Fermi function, we need another function that has units of states per energy level per volume. In a solid with numerous atoms, a large number of states appear at energy levels very close to each other. We approximate these states as a

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continuous "band" and imagine that an "energy level" is a vanishingly small energy interval of width dE. The density of states, N(E) is the fraction of all allowed states that lie within E and E+dE. The Fermi function tells us the probability that a state is occupied. The density of states complements the Fermi function by telling us how many states actually exist in a particular material.

7. The Distribution Function and the Fermi Energy To begin to understand the meaning of the distribution function and the Fcrmi energy, we can plot the distribution function versus energy.

• E < EF [At T = 0 K] Semiconductor energy level (E) is lower than Fermi energy level E

𝑓𝑓(𝐸𝐸) = 11+0

= 1

F

𝑓𝑓(𝐸𝐸) =1

1 + 𝑒𝑒(𝐸𝐸−𝐸𝐸𝐹𝐹)

𝑘𝑘𝑘𝑘�=

11 + 𝑒𝑒−∝

∴ At T = 0 K, the lower states of EF

• E > EF [At T = 0 K]

are always filled.

Semiconductor energy level (E) is higher than Fermi energy level E

𝑓𝑓(𝐸𝐸) = 11+∝

= 0

F

𝑓𝑓(𝐸𝐸) =1

1 + 𝑒𝑒(𝐸𝐸−𝐸𝐸𝐹𝐹)

𝑘𝑘𝑘𝑘�=

11 + 𝑒𝑒+∝

∴ At T = 0 K, the upper states of EF are always empty.

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Fig: Fermi Level, Fermi Function and Electron Occupancy of Localized Energy States at T=0K

• E = EFSemiconductor energy level (E) and Fermi energy level (E

[At any Temperature] F

𝑓𝑓(𝐸𝐸) = 11+1

= 12

) are same at any temperature.

𝑓𝑓(𝐸𝐸) =1

1 + 𝑒𝑒(𝐸𝐸−𝐸𝐸𝐹𝐹)

𝑘𝑘𝑘𝑘�=

11 + 𝑒𝑒0

∴ At any temperature, at Fermi Level the probability of finding electron is 50%.

Fig: Fermi Level, Fermi Function and Electron Occupancy of Localized Energy States at any Temperature

Empty States

Field states

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7.1 Temperature dependence of the Distribution Function Throughout nature, particles seek to occupy the lowest energy state possible. Therefore electrons in a solid will tend to fill the lowest energy states first. Electrons fill up the available states like water filling a bucket, from the bottom up. At T=0, every low-energy state is occupied, right up to the Fermi level, but no states are filled at energies greater than E F

Figure 1: Illustration of the Fermi function for zero temperature. All electrons are stacked neatly below the Fermi level.

.

For T > 0, some electrons can be excited into higher-energy states. This is similar to a bucket of hot water. Most of the water molecules stick around the bottom of the bucket. The Fermi level is like the water line. A fraction of water molecules are excited and drift above the water line as vapor, just as electrons can sometimes drift above the Fermi level.

Figure: Illustration of the Fermi function for temperatures above zero. Some electrons drift above the Fermi level. Their density

at higher energies is proportional to the Fermi function.

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The plot of the distribution function versus energy shows a rectangular shaped distribution at T=0K. Therefore, at T = 0 K, every available energy state up to EF

is filled and with electrons and all states above EF is empty.

Figure: Fermi-Dirac Distribution Function

At temperature higher than 0 K, some probability exists for status above the Fermi Level to be filled. Thus, at T = 300 K, there is some probability f(E) that status above EF are filled and there is a corresponding probability [1- f (E)] that states below EF

7.2 Material dependence of the Distribution Function

are empty.

One feature that is very important about the Fermi-Dirac distribution is that it is symmetric about the chemical potential. F(E) is the probability of occupancy of an available state at E. Thus if there is no available state at E, there is no possibility of finding an electron there. We can best visualize the relation between f(E) and band structure by turning the f(E) vs. E diagram on its side so that the E-scale corresponds to the energies of the band diagram. For intrinsic material, the concentration of holes in the valance band is equal to the concentration of electrons in the conduction band. Therefore, the Fermi level EF must lie at the middle of the band gap in the intrinsic material. Since f(E) is symmetrical above EF, the electron probability “tail” of f(E) extending into the conduction band is symmetrical with the hole probability tail [1- f (E)] in the valence band. The distribution has values within the band gap between EV and EC but there is no energy status available, and electrons occupancy results from f(E) in this range.

EF

f(E)

E

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Figure: Fermi-Dirac Distribution Function in intrinsic semiconductor.

For an extrinsic semiconductor the situation is slightly more complicated. At absolute zero in an n-type semiconductor, the chemical potential must lie in the centre of the gap between the donor level and the bottom of the conduction band. At low temperatures in such a semiconductor there are more conduction electrons than there are holes. If the donor level is more than half full, the chemical potential must lie somewhere between the donor levels and the conduction band. At higher temperatures, when the donor level is completely depleted of electrons, and the contribution from intrinsic electrons to the overall electrical conductivity becomes more substantial, the chemical potential tends towards that for an intrinsic semiconductor. In n-type material there is a high concentration of electron in the conduction band compared with the hole concentration in the valence band. Thus in n-type material the distribution function f(E) must be above its intrinsic position on the energy scale. Since, f(E) retains in its shape for a particular temperature, the larger concentration of electrons at EC in n-type material implies a correspondingly smaller hole concentration at EV. We obtain that the value of f(E) for each energy level in conduction band increases as EF moves closer to EC. Thus energy difference (EC- EF) gives a measure of ni

Figure: Fermi-Dirac Distribution Function in n-type semiconductor.

For

.

p-type semiconductors the behavior is similar, but the other way around. For p-type material the Fermi level lies near the valence band such that the [1- f (E)] tail below EV is larger than the f(E) tail above EC. The value of (EF - EC) indicates how strongly p-type the material is.

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Figure: Fermi-Dirac Distribution Function in n-type semiconductor.

8. Distribution Function in calculating Electron and hole concentration at Equilibrium

In a solid semiconductor at thermal equilibrium, every mobile electron leaves behind a hole in the valence band. Since holes are also mobile, we need to account for the density of "hole states" that remain in the valence band. Because a hole is an unoccupied state, the probability of a mobile hole existing at energy E is 1-F(E).

Figure 4:

Figure: The density of mobile electrons is shown in the conduction band. The corresponding density of mobile holes is shown in

the valence band.

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The concentration of electrons in the conduction band is

𝑛𝑛0 = � 𝑓𝑓(𝐸𝐸)𝑁𝑁(𝐸𝐸)𝑑𝑑𝐸𝐸∝

𝐸𝐸𝑐𝑐… … … … … … … … … . (𝑖𝑖)

Here, N(E) is the density of states in the energy range dE. Now,

𝑁𝑁(𝐸𝐸) ∝ 𝐸𝐸12�

∴ 𝑁𝑁(𝐸𝐸) ↑ → 𝐸𝐸 ↑ ∴ 𝑓𝑓(𝐸𝐸) ↓ → 𝐸𝐸 ↑

f(E)N(E) decreases rapidly above EC

The concentration of holes in the valance band is

, and very few electrons occupy energy states far above the conduction band edge.

𝑝𝑝0 = � [1 − 𝑓𝑓(𝐸𝐸)]𝑁𝑁(𝐸𝐸)𝑑𝑑𝐸𝐸… … … … … … … … … . (𝑖𝑖𝑖𝑖)𝐸𝐸𝑣𝑣

0

Figure: Tthe density of states. For an intrinsic semiconductor the Fermi energy is in the middle of the band gap and n=p.

By solving eqn

And, 𝑝𝑝0 = 𝑁𝑁𝑣𝑣𝑒𝑒−(𝐸𝐸𝑓𝑓−𝐸𝐸𝑣𝑣) 𝑘𝑘𝑘𝑘⁄

Here,

(i) and (ii), 𝑛𝑛0 = 𝑁𝑁𝑐𝑐𝑒𝑒−(𝐸𝐸𝑐𝑐−𝐸𝐸𝑓𝑓) 𝑘𝑘𝑘𝑘⁄

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𝑁𝑁𝑐𝑐 = 2 �2𝜋𝜋𝑚𝑚𝑛𝑛

∗ 𝑘𝑘𝑘𝑘ℎ2 �

32�

And,

𝑁𝑁𝑣𝑣 = 2 �2𝜋𝜋𝑚𝑚𝑝𝑝

∗𝑘𝑘𝑘𝑘ℎ2 �

32�

Here, 𝑚𝑚𝑛𝑛

∗= effective mass of electron and, 𝑚𝑚𝑝𝑝

∗= effective mass of hole

Figure: Density of states, occupancy factors and carrier contributions for different positions of the Fermi energy.

n-type

Intrinsic

p-type

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9. Fermi Level Positioning

9.1 Fermi Level Positioning for Intrinsic Material In an intrinsic semiconductor the density of electrons equals that of holes

∴ 𝒏𝒏𝟎𝟎 = 𝒑𝒑𝟎𝟎 = 𝒏𝒏𝒊𝒊 ∴ 𝑵𝑵𝒄𝒄𝒆𝒆−(𝑬𝑬𝒄𝒄−𝑬𝑬𝒇𝒇) 𝒌𝒌𝒌𝒌⁄ = 𝑵𝑵𝒗𝒗𝒆𝒆−(𝑬𝑬𝒇𝒇−𝑬𝑬𝒗𝒗) 𝒌𝒌𝒌𝒌⁄

This yields a Fermi level of,

∴ 𝑬𝑬𝒇𝒇𝒊𝒊 = 𝑬𝑬𝒊𝒊 =𝟏𝟏𝟐𝟐

(𝑬𝑬𝒄𝒄 + 𝑬𝑬𝒗𝒗) +𝟏𝟏𝟐𝟐𝒌𝒌𝒌𝒌 𝐥𝐥𝐥𝐥

𝑵𝑵𝒗𝒗

𝑵𝑵𝒄𝒄

where Ei is called intrinsic (Fermi) level. This level is in the middle of the bandgap, displaced from it by a term which can usually by neglected. Now, from the above eqn

If the electron and hole effective masses are equal so that 𝑚𝑚𝑝𝑝∗ = 𝑚𝑚𝑛𝑛

∗ , then the intrinsic Fermi level is exactly in the center of the band gap. If 𝑚𝑚𝑝𝑝

∗ > 𝑚𝑚𝑛𝑛∗ , the intrinsic Fermi level is slightly

above the center, and if 𝑚𝑚𝑝𝑝∗ < 𝑚𝑚𝑛𝑛

∗ , it is slightly below the center of the band gap. The density of states function is directly related to the carrier effective mass; thus a larger effective mass means a larger density of states function. The intrinsic Fermi level must shift away from the band with the larger density of states in order to maintain equal numbers of electrons and holes. The distribution (with respect to energy) of electrons in the conduction band is given by the density of allowed quantum states times the probability that a state is occupied by an electron. This statement is written in equation form as,

𝒏𝒏(𝑬𝑬) = 𝒎𝒎𝒄𝒄(𝑬𝑬)𝒇𝒇𝑭𝑭(𝑬𝑬)

,

∴ 𝑬𝑬𝒇𝒇𝒊𝒊 =𝟏𝟏𝟐𝟐

(𝑬𝑬𝒄𝒄 + 𝑬𝑬𝒗𝒗) +𝟑𝟑𝟒𝟒𝒌𝒌𝒌𝒌 𝐥𝐥𝐥𝐥

𝑚𝑚𝑝𝑝∗

𝑚𝑚𝑛𝑛∗

But,

𝟏𝟏𝟐𝟐

(𝑬𝑬𝒄𝒄 + 𝑬𝑬𝒗𝒗) = 𝑬𝑬𝒎𝒎𝒊𝒊𝒎𝒎𝒎𝒎𝒎𝒎𝒑𝒑

∴ 𝑬𝑬𝒇𝒇𝒊𝒊 − 𝑬𝑬𝒎𝒎𝒊𝒊𝒎𝒎𝒎𝒎𝒎𝒎𝒑𝒑 =𝟑𝟑𝟒𝟒𝒌𝒌𝒌𝒌 𝐥𝐥𝐥𝐥

𝑚𝑚𝑝𝑝∗

𝑚𝑚𝑛𝑛∗

where 𝑓𝑓𝐹𝐹(𝐸𝐸)is the Fermi-Dirac probability function and 𝑔𝑔𝑐𝑐(𝐸𝐸) is the density of quantum states in the conduction band.

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Similarly, the distribution (with respect to energy) of holes in the valence bend is the density of allowed quantum states in the valence band multiplied by the probability that a state is not occupied by an electron. We may express this as,

𝒑𝒑(𝑬𝑬) = 𝒎𝒎𝒗𝒗(𝑬𝑬)[𝟏𝟏 − 𝒇𝒇𝑭𝑭(𝑬𝑬)]

Figure : (a) Density of states functions, Fermi-Dirac probability function, and areas representing electron and hole

concentrations for the case when EF

Figure(a) shows a plot of the density of states function in the conduction band 𝒎𝒎𝒄𝒄(𝑬𝑬), the density of states function in the valence band 𝒎𝒎𝒗𝒗(𝑬𝑬), and the Fermi-Dirdc probability function for

is near the midgap energy; (b) expanded view near the cunduction band energy; and (c) expanded view near the valence band energy.

T >0K when Ef is approximately halfway between Ec and Ev,. If we assume, for the moment, that the electron and hole effective masses are equal, then 𝒎𝒎𝒄𝒄(𝑬𝑬)and 𝒎𝒎𝒗𝒗(𝑬𝑬) are symmetrical functions about the midgap energy. The function, fF(E) for E > EF is symmetrical to the function 1 - fF(E) for E < EF about the energy E = EF. This also means that the function fF(E) for E = EF + dE is equal to the function 1 -fF(E)fo r E = EF - d E .

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9.2 Fermi Level Positioning for Extrinsic Material

The Fermi energy level changes as dopant atoms are added. In general, both donors and acceptors are present in a piece of a semiconductor although one outnumbers the other one. When EF > EFi , the electron concentration is larger than the hole concentration, and when EF < EFi

Figure: Density of states functions, Fermi-Dirac probability function and areas representing electron and hole concentrations

for the case when E

. the hole concentration is larger than the electron concentration..

F is (a) above and (b) below the intrinsic Fermi energy.

An extrinsic semiconductor in thermal equilibrium does not contain an intrinsic carrier concentration, although some thermally generated carriers are present. The intrinsic electron and hole carrier concentrations are modified by the donor or acceptor impurities. The Fermi energy level in a semiconductor changes as the electron and hole concentrations change and, again, the Fermi energy changes as donor or acceptor impurities are added. The Fermi energy may vary through the bandgap energy casing significant change in the carrier concentration of electrons and holes. The change in the Fermi level is actually a function of the donor or acceptor impurity concentrations that are added to the semiconductor. Electron and hole concentrations change by orders of magnitude from the intrinsic carrier concentration as the Fermi energy changes by a few tenths of an electron-volt.

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References Books-

• Fundamentals of classical and statistical thermodynamics By Bimalendu Narayan Roy • System Integration By Kurt Hoffmann • On the Theory of Quantum Mechanics• Solid State Electronics Devices (Sixth edition) By Ben G. Streetman , SanjayKumar Banerjee

By Dirac, P. A. M. (1926)

• Semiconductor Physics And Devices (3rd edition) By J. Neamen • Physics and Technology of Semiconductor Devices By A.S. Grove (1967) • Mobility of holes and electrons in high electric fields By E.J. Ryder (1953) • Introduction to Solid State Physics By C. Kittel (1986)

Websites-

• http://ecee.colorado.edu/~bart/book/book/chapter2/ ( Semiconductor Fundamentals) • http://people.seas.harvard.edu/~jones/es154/lectures/lecture_2/fermi_level/fermi_level.html • http://jas.eng.buffalo.edu/education/semicon/fermi/bandAndLevel/fermi.html • http://cnx.org/content/m13458/latest/ • http://kottan-labs.bgsu.edu/teaching/workshop2001/chapter6.htm • http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html • http://jas.eng.buffalo.edu/education/semicon/fermi/functionAndStates/functionAndStates.html • http://en.wikipedia.org/wiki/Fermi-Dirac_statistics • http://www.ece.utep.edu/courses/ee3329/ee3329/Studyguide/ToC/fundamentals/Carriers/fermi.

html • http://en.wikipedia.org/wiki/Fermi_energy • http://en.wikipedia.org/wiki/Fermi_level • http://people.seas.harvard.edu/~jones/es154/lectures/lecture_2/fermi_level/fermi_level.html • http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_5.htm