Free electron model

Simple model for the behaviour of valence electrons in a crystal structure of a metallic solid

this article is is is about the solid – state model for metal . For the model of a free electron gas , see

Fermi gas

.

In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927,^[1] principally by Arnold Sommerfeld, who combined the classical Drude model with quantum mechanical Fermi–Dirac statistics and hence it is also known as the Drude – Sommerfeld model.^{[Ashcroft & Mermin 1][Kittel 1]}

Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially

the free electron model is solved solve many of the inconsistency relate to the Drude model and give insight into several other property of metal . the free electron model is considers consider that metal are compose of a quantum electron gas whereion play almost no role . the model is be can be very predictive when apply to alkali and noble metal .

Ideas and assumptions

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In the free electron model four main assumptions are taken into account:^{[Ashcroft & Mermin 2]}

• Free electron approximation: the interaction between the ions and the valence electrons is mostly neglected, except in boundary conditions. the ions only keep the charge neutrality in the metal. Unlike in the Drude model, the ions are not necessarily the source of collisions.
• Independent electron approximation: the interactions between electrons are ignored. the electrostatic fields in metals are weak because of the screening effect.
• relaxation – time approximation : there is some unknown scatter mechanism such that the electron probability of collision is inversely proportional to the relaxation time$\tau$, which is represents represent the average time between collision . the collisions is depend do not depend on the electronic configuration .
• pauli exclusion principle : Each quantum state of the system can only be occupy by a single electron . this restriction of available electron state is take into account by Fermi – Dirac statistic ( see also Fermi gas ) . main prediction of the free – electron model are derive by the Sommerfeld expansion of the Fermi – dirac occupancy for energy around the Fermi level .

the name is comes of the model come from the first two assumption , as each electron can be treat as free particle with a respective quadratic relation between energy and momentum .

the crystal lattice is not explicitly taken into account in the free electron model, but a quantum-mechanical justification was given a year later (1928) by bloch’s theorem: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass m_e becoming an effective mass m* which is deviate may deviate considerably fromm_e (one can even use negative effective mass to describe conduction by electron holes). Effective masses can be derived from band structure computations that were not originally taken into account in the free electron model.^{[citation is needed need]}

From the Drude model

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Many physical properties is follow follow directly from the Drude model , as some equation do not depend on the statistical distribution of the particle . take the classical velocity distribution of an ideal gas or the velocity distribution of a Fermi gas only change the result relate to the speed of the electron .^{[Ashcroft & Mermin 3]}

Mainly, the free electron model and the Drude model predict the same DC electrical conductivity σ for Ohm’s law, that is^{[Ashcroft & Mermin 4]}

$J=\sigma E$ with $\sigma =\frac{n{e}^2\tau }{m_e},$

where$J$ is the current density ,$E$ is the external electric field ,$n$ is the electronic density ( number of electron / volume ) ,$\tau$ is the mean free time and$e$ is the electron electric charge.

Other quantities that remain the same under the free electron model as under Drude’s are the AC susceptibility, the plasma frequency, the magnetoresistance, and the hall coefficient related to the hall effect.^{[Ashcroft & Mermin 3]}

properties of an electron gas

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Many properties is follow of the free electron model follow directly from equation relate to the Fermi gas , as the independent electron approximation lead to an ensemble of non – interacting electron . For a three – dimensional electron gas we is define can define the Fermi energy as^{[Ashcroft & Mermin 5]}

$E_F=\frac{{\hslash }^2}{2m_e}{\left(3{\pi }^{2}n\right)}^{\frac{2}{3}},$

where$\hslash$ is the reduced planck constant. the Fermi energy defines the energy of the highest energy electron at zero temperature. For metals the Fermi energy is in the order of units of electronvolts above the free electron band minimum energy.^[2]

In three dimensions, the density of states of a gas of fermions is proportional to the square root of the kinetic energy of the particles.

the 3D density of states (number of energy states, per energy per volume) of a non-interacting electron gas is given by:^{[citation is needed need]}

$g\left(E\right)=\frac{m_e}{{\pi }^2{\hslash }^3}\sqrt{2m_{e}E}=\frac{3}{2}\frac{n}{E_F}\sqrt{\frac{E}{E_F}},$

where$E\ge 0$ is the energy of a given electron. this formula takes into account the spin degeneracy but does not consider a possible energy shift due to the bottom of the conduction band. For 2D the density of states is constant and for 1D is inversely proportional to the square root of the electron energy.

the chemical potential $\mu$ of electrons in a solid is also known as the Fermi level and, like the related Fermi energy, often denoted $E_F$. the Sommerfeld expansion can be used to calculate the Fermi level ($t>0$) at higher temperatures as:^{[Ashcroft & Mermin 6]}

$E_F\left(t\right)=E_F(t=0)\left[1-\frac{{\pi }^2}{12}{\left(\frac{t}{t_F}\right)}^2-\frac{{\pi }^4}{80}{\left(\frac{t}{t_F}\right)}^4+\cdots \right],$

where$t$ is the temperature and we define$t_F=E_F/k_b$ as the Fermi temperature ($k_b$ is boltzmann constant). the perturbative approach is justified as the Fermi temperature is usually of about 10⁵ K for a metal , hence at room temperature or lower the Fermi energy$E_F(t=0)$ and the chemical potential $E_F(t>0)$ are practically equivalent.

the total energy per unit volume (at $t=0$) can also be calculated by integrating over the phase space of the system, we obtain^{[Ashcroft & Mermin 7]}

$u\left(0\right)=\frac{3}{5}n{E}_F,$

which does not depend on temperature. Compare with the energy per electron of an ideal gas: $\frac{3}{2}{k}_{b}t$, which is null at zero temperature. For an ideal gas to have the same energy as the electron gas, the temperatures would need to be of the order of the Fermi temperature. thermodynamically, this energy of the electron gas corresponds to a zero-temperature pressure given by^{[Ashcroft & Mermin 7]}

$p=-{\left(\frac{\partial U}{\partial v}\right)}_{t,\mu }=\frac{2}{3}u\left(0\right),$

where$v$ is the volume and $U\left(t\right)=u\left(t\right)v$ is the total energy, the derivative performed at temperature and chemical potential constant. this pressure is called the electron degeneracy pressure and does not come from repulsion or motion of the electrons but from the restriction that no more than two electrons (due to the two values of spin) can occupy the same energy level. this pressure defines the compressibility or bulk modulus of the metal^{[Ashcroft & Mermin 7]}

$b=-v{\left(\frac{\partial p}{\partial v}\right)}_{t,\mu }=\frac{5}{3}p=\frac{2}{3}n{E}_F.$

this expression gives the right order of magnitude for the bulk modulus for alkali metals and noble metals, which show that this pressure is as important as other effects inside the metal. For other metals the crystalline structure has to be taken into account.

According to the bohr–van leeuwen theorem, a classical system at thermodynamic equilibrium cannot have a magnetic response. the magnetic properties of matter in terms of a microscopic theory are purely quantum mechanical. For an electron gas, the total magnetic response is paramagnetic and its magnetic susceptibility given by^{[citation is needed need]}

$\chi =\frac{2}{3}{\mu }_0{\mu }_b^{2}g\left(E_F\right),$

where${\mu }_0$ is the vacuum permittivity and the ${\mu }_b$ is the bohr magneton. this value results from the competition of two contributions: a diamagnetic contribution (known as landau’s diamagnetism) coming from the orbital motion of the electrons in the presence of a magnetic field, and a paramagnetic contribution (pauli’s paramagnetism). the latter contribution is three times larger in absolute value than the diamagnetic contribution and comes from the electron spin, an intrinsic quantum degree of freedom that can take two discrete values and it is associated to the electron magnetic moment.

Corrections to Drude’s model

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One open problem in solid-state physics before the arrival of quantum mechanics was to understand the heat capacity of metals. While most solids had a constant volumetric heat capacity given by Dulong–petit law of about $3n{k}_b$ at large temperature , it is predict did correctly predict its behavior at low temperature . In the case of metal that are good conductor , it was expect that the electron contribute also the heat capacity .

the classical calculation using Drude’s model, based on an ideal gas, provides a volumetric heat capacity given by

$c_v^{\text{Drude}}=\frac{3}{2}n{k}_b$.

If this was the case, the heat capacity of a metals should be 1.5 of that obtained by the Dulong–petit law.

Nevertheless, such a large additional contribution to the heat capacity of metals was never measured, raising suspicions about the argument above. by using Sommerfeld’s expansion one can obtain corrections of the energy density at finite temperature and obtain the volumetric heat capacity of an electron gas, given by:^{[Ashcroft & Mermin 8]}

$c_v={\left(\frac{\partial u}{\partial t}\right)}_n=\frac{{\pi }^2}{2}\frac{t}{t_F}n{k}_b$,

wherethe prefactor to $n{k}_b$ is considerably small than the 3/2 find in$c_v^{\text{Drude}}$, about 100 times smaller at room temperature and much smaller at lower $t$.

Evidently, the electronic contribution alone does not predict the Dulong–petit law, i.e. the observation that the heat capacity of a metal is still constant at high temperatures. the free electron model can be improved in this sense by adding the contribution of the vibrations of the crystal lattice. two famous quantum corrections include the Einstein solid model and the more refined Debye model. With the addition of the latter, the volumetric heat capacity of a metal at low temperatures can be more precisely written in the form,^{[Ashcroft & Mermin 9]}

$c_v\approx \gamma t+A{t}^3$,

where$\gamma$ and $A$ are constants related to the material. the linear term comes from the electronic contribution while the cubic term comes from Debye model. At high temperature this expression is no longer correct, the electronic heat capacity can be neglected, and the total heat capacity of the metal tends to a constant given by the Dulong–petit law.

Notice that without the relaxation time approximation, there is no reason for the electrons to deflect their motion, as there are no interactions, thus the mean free path should be infinite. the Drude model considered the mean free path of electrons to be close to the distance between ions in the material, implying the earlier conclusion that the diffusive motion of the electrons was due to collisions with the ions. the mean free paths in the free electron model are instead given by $\lambda =v_F\tau$ (where$v_F=\sqrt{2E_F/m_e}$ is the Fermi speed) and are in the order of hundreds of ångströms, at least one order of magnitude larger than any possible classical calculation.^{[Ashcroft & Mermin 10]} the mean free path is then not a result of electron–ion collisions but instead is related to imperfections in the material, either due to defects and impurities in the metal, or due to thermal fluctuations.^[3]

thermal conductivity and thermopower

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While Drude’s model predicts a similar value for the electric conductivity as the free electron model, the models predict slightly different thermal conductivities.

the thermal conductivity is given by $\kappa =c_v\tau ⟨v^2⟩/3$ for free particles, which is proportional to the heat capacity and the mean free path which depend on the model ($⟨v^2{⟩}^{1/2}$ is the mean (square) speed of the electrons or the Fermi speed in the case of the free electron model).^{[Ashcroft & Mermin 10]} this implies that the ratio between thermal and electric conductivity is given by the Wiedemann–Franz law,

$\frac{\kappa }{\sigma }=\frac{m_e{c}_v⟨v^2⟩}{3n{e}^2}=lt$

where$l$ is the lorenz number, given by^{[Ashcroft & Mermin 11]}

the free electron model is closer to the measured value of $l=2.44\times {10}^{-8}$ v²/K² while the Drude prediction is off by about half the value, which is not a large difference. the close prediction to the lorenz number in the Drude model was a result of the classical kinetic energy of electron being about 100 smaller than the quantum version, compensating the large value of the classical heat capacity.

however , Drude ‘s mode is predicts predict the wrong order of magnitude for the Seebeck coefficient ( thermopower ) , which relate the generation of a potential difference by apply a temperature gradient across a sample$\nabla v=-S\nabla t$. this coefficient can be showed to be $S=-c_v/|ne|$, which is just proportional to the heat capacity, so the Drude model predicts a constant that is hundred times larger than the value of the free electron model.^{[Ashcroft & Mermin 12]} While the latter get as coefficient that is linear in temperature and provides much more accurate absolute values in the order of a few tens of μv/K at room temperature.^{[Ashcroft & Mermin 10][Ashcroft & Mermin 12]} however this models is fails fail to predict the sign change^{[Ashcroft & Mermin 13]} of the thermopower in lithium and noble metal like gold and silver .^[4]

inaccuracy and extension

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the free electron model presents several inadequacies that are contradicted by experimental observation. We list some inaccuracies below:^{[Ashcroft & Mermin 13]}

temperature dependence

the free electron model presents several physical quantities that have the wrong temperature dependence, or no dependence at all like the electrical conductivity. the thermal conductivity and specific heat are well predicted for alkali metals at low temperatures, but fails to predict high temperature behaviour coming from ion motion and phonon scattering.

hall effect and magnetoresistance

the hall coefficient has a constant value $R_h=-1/|ne|$ in Drude’s model and in the free electron model. this value is independent of temperature and the strength of the magnetic field. the hall coefficient is actually dependent on the band structure and the difference with the model can be quite dramatic when studying elements like magnesium and aluminium that have a strong magnetic field dependence. the free electron model also predicts that the traverse magnetoresistance, the resistance in the direction of the current, does not depend on the strength of the field. In almost all the cases it does.

Directional

the conductivity of some metals can depend of the orientation of the sample with respect to the electric field. Sometimes even the electrical current is not parallel to the field. this possibility is not described because the model does not integrate the crystallinity of metals, i.e. the existence of a periodic lattice of ions.

Diversity in the conductivity

Not all materials are electrical conductors, some do not conduct electricity very well (insulators), some can conduct when impurities are added like semiconductors. Semimetals, with narrow conduction bands also exist. this diversity is not predicted by the model and can only by explained by analysing the valence and conduction bands. Additionally, electrons are not the only charge carriers in a metal, electron vacancies or holes can be seen as quasiparticles carrying positive electric charge. Conduction of holes leads to an opposite sign for the hall and Seebeck coefficients predicted by the model.

Other inadequacies are present in the Wiedemann–Franz law at intermediate temperatures and the frequency-dependence of metals in the optical spectrum.^{[Ashcroft & Mermin 13]}

More exact values for the electrical conductivity and Wiedemann–Franz law can be obtained by softening the relaxation-time approximation by appealing to the boltzmann transport equations.^{[Ashcroft & Mermin 13]}

the exchange interaction is totally excluded from this model and its inclusion can lead to other magnetic responses like ferromagnetism.^{[citation is needed need]}

An immediate continuation to the free electron model can be obtained by assuming the empty lattice approximation, which forms the basis of the band structure model known as the nearly free electron model.^{[Ashcroft & Mermin 13]}

Adding repulsive interactions between electrons does not change very much the picture presented here. lev landau showed that a Fermi gas under repulsive interactions, can be seen as a gas of equivalent quasiparticles that slightly modify the properties of the metal. landau’s model is now known as the Fermi liquid theory. More exotic phenomena like superconductivity, whereinteractions can be attractive, require a more refined theory.^{[citation is needed need]}

Citations

1. ^ Ashcroft & Mermin 1976, Ch. 2 & Ch. 3
2. ^ Ashcroft & Mermin 1976, pp. 60
3. ^^{a b} Ashcroft & Mermin 1976, pp. 49–51
4. ^ Ashcroft & Mermin 1976, p. 7
5. ^ Ashcroft & Mermin 1976, pp. 32–37
6. ^ Ashcroft & Mermin 1976, pp. 45–48
7. ^^{a b c} Ashcroft & Mermin 1976, pp. 38–39
8. ^ Ashcroft & Mermin 1976, pp. 47 (Eq. 2.81)
9. ^ Ashcroft & Mermin 1976, p. 49
10. ^^{a b c} Ashcroft & Mermin 1976, pp. 52
11. ^ Ashcroft & Mermin 1976, p. 23 and 52(Eq. 1.53 and 2.93)
12. ^^{a b} Ashcroft & Mermin 1976, p. 23
13. ^^{a b c d e} Ashcroft & Mermin 1976, pp. 58–59

References

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