Free electron model

The free electron model , also known as the Sommerfeld model or the Drude-Sommerfeld model, is a simple quantum model of the behavior of valence electrons in a metal atom , developed by Arnold Sommerfeld on the basis of the classical Drude model taking into account the quantum mechanical Fermi-Dirac statistics. The metal electrons are considered in this model as a Fermi gas .

The difference between the Sommerfeld model and the Drude model is that not all valence electrons of the metal participate in kinetic processes, but only those that have energies within $k_{B}T$ ${\ displaystyle k_ {B} T}$ ${\ displaystyle k_ {B} T}$ from Fermi energy where $k_{B}$ ${\ displaystyle k_ {B}}$ $k_B$ Is the Boltzmann constant, T is the temperature. This limitation arises due to the Pauli principle, which forbids electrons to have the same quantum numbers. As a result, at finite temperatures, low-energy states are filled, which prevents electrons from changing their energy or direction of motion.

The density of states of a three-dimensional fermion gas is proportional to the square root of the kinetic energy of the particles.

Despite its simplicity, the model explains many different phenomena, among which:

Wiedemann-Franz law ;
temperature dependence of heat capacity;
electrical conductivity;
thermionic emission;
form of density of states of electrons;
range of binding energies.

Key Ideas and Assumptions

If in the Drude model the electrons of the metal were divided into bound and free, then in quantum mechanics, due to the principle of particle identity, the electrons are collectivized and belong to the whole solid. The skeletons of metal atoms form a periodic crystal lattice in which, according to the Bloch theorem, the states of electrons are characterized by a quasi-momentum . The energy spectrum of metal electrons is divided into zones, the most important of which is the partially filled conduction band formed by valence electrons.

The Sommerfeld model does not specify the dispersion law for electrons in the conduction band, assuming only that deviations from the parabolic dispersion law of free particles are insignificant. In the initial approximation, the theory neglects the electron-electron interaction, considering the electrons as an ideal gas. However, to explain kinetic processes, such as electrical and thermal conductivity, electron scattering on one another, on vibrations of the crystal lattice and defects, it must be taken into account. When considering these phenomena, it is important to know the energy distribution of particles. Therefore, to describe the kinetics of electrons, the Boltzmann equation is used . The electrostatic field inside the conductor is considered weak due to shielding.

Energy and wave function of a free electron

A plane wave moving along the x axis. Different colors correspond to different phases of the wave.

The Schrödinger equation for a free electron has the form ^[1] ^[2] ^[3]

-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t)=i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)

{\ displaystyle - {\ frac {\ hbar ^ {2}} {2m}} \ nabla ^ {2} \ Psi (\ mathbf {r}, t) = i \ hbar {\ frac {\ partial} {\ partial t}} \ Psi (\ mathbf {r}, t)}

Wave function $\Psi (\mathbf {r} ,t)$ ${\ displaystyle \ Psi (\ mathbf {r}, t)}$ can be divided into spatial and temporal parts. The solution to the time-dependent equation is

\Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )e^{-i\omega t}

{\ displaystyle \ Psi (\ mathbf {r}, t) = \ psi (\ mathbf {r}) e ^ {- i \ omega t}}

with energy

E=\hbar \omega

{\ displaystyle E = \ hbar \ omega}

The solution to the spatial, time-independent part will be

\psi _{\mathbf {k} }(\mathbf {r} )={\frac {1}{\sqrt {\Omega _{r}}}}e^{i\mathbf {k} \cdot \mathbf {r} }

{\ displaystyle \ psi _ {\ mathbf {k}} (\ mathbf {r}) = {\ frac {1} {\ sqrt {\ Omega _ {r}}}} e ^ {i \ mathbf {k} \ cdot \ mathbf {r}}}

with wave vector $\mathbf {k}$ ${\ displaystyle \ mathbf {k}}$ . $\Omega _{r}$ ${\ displaystyle \ Omega _ {r}}$ have the amount of space where the electron can be. The kinetic energy of an electron is given by the equation:

E={\frac {\hbar ^{2}k^{2}}{2m}}

{\ displaystyle E = {\ frac {\ hbar ^ {2} k ^ {2}} {2m}}}

The solution in the form of a plane wave of this Schrödinger equation will be

\Psi (\mathbf {r} ,t)={\frac {1}{\sqrt {\Omega _{r}}}}e^{i\mathbf {k} \cdot \mathbf {r} -i\omega t}

{\ displaystyle \ Psi (\ mathbf {r}, t) = {\ frac {1} {\ sqrt {\ Omega _ {r}}} e ^ {i \ mathbf {k} \ cdot \ mathbf {r} -i \ omega t}}

Solid state physics and condensed matter physics mainly deal with a time-independent solution $\psi _{\mathbf {k} }(\mathbf {r} )$ ${\ displaystyle \ psi _ {\ mathbf {k}} (\ mathbf {r})}$ .

Taking into account the periodicity of the crystal lattice by the Bloch theorem changes this function by

\Psi (\mathbf {r} ,t)={\frac {1}{\sqrt {\Omega _{r}}}}\phi (\mathbf {r} )e^{i\mathbf {k} \cdot \mathbf {r} -i\omega t}

{\ displaystyle \ Psi (\ mathbf {r}, t) = {\ frac {1} {\ sqrt {\ Omega _ {r}}}} phi (\ mathbf {r}) e ^ {i \ mathbf { k} \ cdot \ mathbf {r} -i \ omega t}}

,

Where $\phi (\mathbf {r} )$ ${\ displaystyle \ phi (\ mathbf {r})}$ - periodic function. The dependence of energy on the wave vector also changes. To account for these modifications, various model Hamiltonians are widely used, for example: the approximation of almost free electrons, the approximation of tight binding, and so on.

Fermi Energy

The Pauli principle forbids electrons to have wave functions with the same quantum numbers. For an electron described by a Bloch wave, the quantum numbers are quasi-momentum and spin. The ground state of an electron gas corresponds to a situation when all one-electron states with the lowest energy are filled to a certain energy $E_{F}$ ${\ displaystyle E_ {F}}$ called Fermi energy. For a parabolic zone, the energy is given as

E(\mathbf {k} )={\frac {\hbar ^{2}k^{2}}{2m}}

{\ displaystyle E (\ mathbf {k}) = {\ frac {\ hbar ^ {2} k ^ {2}} {2m}}}

,

such filling means that all states with a wave vector are less than $|\mathbf {k} |<k_{F}$ ${\ displaystyle | \ mathbf {k} | <k_ {F}}$ , $k_{F}$ ${\ displaystyle k_ {F}}$ , which is called the Fermi wave vector, are busy. Fermi vector equals

k_{F}=(3\pi ^{2}N_{e}/V)^{1/3}

{\ displaystyle k_ {F} = (3 \ pi ^ {2} N_ {e} / V) ^ {1/3}}

,

Where $N_{e}$ ${\ displaystyle N_ {e}}$ Is the total number of electrons in the system, and V is the total volume. Then the Fermi energy

E_{F}={\frac {\hbar ^{2}}{2m}}\left({\frac {3\pi ^{2}N_{e}}{V}}\right)^{2/3}

{\ displaystyle E_ {F} = {\ frac {\ hbar ^ {2}} {2m}} \ left ({\ frac {3 \ pi ^ {2} N_ {e}} {V}} \ right) ^ {2/3}}

In the approximation of almost free electrons $Z$ ${\ displaystyle Z}$ valent metal should be replaced $N_{e}$ ${\ displaystyle N_ {e}}$ on $N Z$ ${\ displaystyle NZ}$ where $N$ ${\ displaystyle N}$ - the total number of metal ions.

Energy distribution of electrons

At a nonzero temperature, the electronic subsystem of the metal is not in the ground state, however, the difference will remain relatively small if $k_{B}T\ll E_{F}$ ${\ displaystyle k_ {B} T \ ll E_ {F}}$ that is usually performed. The probability that a one-electron state with energy E will be occupied is given by the Fermi function

f(E)={\frac {1}{e^{(E-E_{F})/k_{B}T}+1}}

{\ displaystyle f (E) = {\ frac {1} {e ^ {(E-E_ {F}) / k_ {B} T} +1}}}

,

Where $E_{F}$ ${\ displaystyle E_ {F}}$ - Fermi level. At absolute zero temperature $E_{F}=\mu$ ${\ displaystyle E_ {F} = \ mu}$ where $\mu$ ${\ displaystyle \ mu}$ - chemical potential .

Theory Predictions

The model allows you to correctly describe a number of properties of metals and their changes associated with temperature.

Heat capacity

When heated, energy is transferred to the metal electrons. However, electrons whose energy is less than the Fermi energy cannot change their state. To do this, they would have to go into a state with higher energy, which is already more likely to be occupied by another electron, and the Pauli principle prohibits this. Therefore, only electrons with energy close to the Fermi energy can get energy. There are few such electrons, approximately $N_{e}k_{B}T/E_{F}\ll N_{e}$ ${\ displaystyle N_ {e} k_ {B} T / E_ {F} \ ll N_ {e}}$ . Therefore, at high temperatures, the contribution of the electronic subsystem to the heat capacity of the metal is small compared to the contribution of the atoms of the crystal lattice.

The situation changes at low temperatures lower than the Debye temperature , when the heat capacity of the lattice is proportional $T^{3}$ ${\ displaystyle T ^ {3}}$ , while the heat capacity of the electronic subsystem is proportional $T$ ${\ displaystyle T}$ . Then the contribution of electrons to the heat capacity dominates, and the heat capacity of the metal, in contrast to dielectrics, is proportional to temperature.

Conductivity

The Sommerfeld model helped to overcome the problem of the Drude model with the mean free path of electrons. In the Drude model, the electric current density is given by the formula

\mathbf {j} =n{\frac {e^{2}\tau }{m}}\mathbf {E}

{\ displaystyle \ mathbf {j} = n {\ frac {e ^ {2} \ tau} {m}} \ mathbf {E}}

,

Where $n$ ${\ displaystyle n}$ Is the electron density, $\tau$ ${\ displaystyle \ tau}$ - relaxation time. If a $n$ ${\ displaystyle n}$ equal to the number of valence electrons in a solid, then to obtain real values of the conductivity of metals, the relaxation time, and therefore the mean free path of an electron, must be small, which contradicts the theory of an ideal gas. In the Sommerfeld Model $n$ ${\ displaystyle n}$ Is the fraction of electrons with energy close to the Fermi energy. It is proportional to a small value. $k_{B}T/E_{F}$ ${\ displaystyle k_ {B} T / E_ {F}}$ . Then the electrons that can be accelerated by the electric field in the metal are relatively small, but their path length is large.

Notes

↑ Albert Messiah. Quantum Mechanics. - Dover Publications, 1999. - ISBN 0-486-40924-4 .
↑ Stephen Gasiorowicz . Quantum Physics. - Wiley & Sons, 1974. - ISBN 0-471-29281-8 .
↑ Eugen Merzbacher. Quantum Mechanics. - 3rd. - Wiley & Sons, 2004 .-- ISBN 978-9971-5-1281-1 .

[Messiah-1] Albert Messiah. Quantum Mechanics. - Dover Publications, 1999. - ISBN 0-486-40924-4 .

[Gasiorowicz-2] Stephen Gasiorowicz . Quantum Physics. - Wiley & Sons, 1974. - ISBN 0-471-29281-8 .

[Merzbacher-3] Eugen Merzbacher. Quantum Mechanics. - 3rd. - Wiley & Sons, 2004 .-- ISBN 978-9971-5-1281-1 .