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The most important concern we have to determine conductivity is not any of these steps by themselves. It is the combination of how many electrons are free to carry current (due to promotion by the field and temperature) and how fast they are moving. In short, to determine the electrical conductivity of a metal, we must determine the number density of free electrons and the velocity of the free electrons under the applied electric field. In a metal, the number density of states depends on $\sqrt{E}$. At 0 K, we fill this will the appropriate number density of bonding electrons. Then, we promote electrons using Fermi-Dirac statistics because thermal promotion typically dominates promotion by the electric field. Using a convolution integral, we obtain a picture of electron density as a function of energy and temperature $\rho_E(E,T)$ as shown here. Those electrons above the Fermi energy are free to carry current.

The thermal energy that is applied to promote electrons from occupied states is on the order of $k_BT$ (this is the dashed line in the picture). The energy of an electric field that is applied in a metal in typical cases (room temperature without high fields) is below $k_BT$. Therefore, in most cases, more electrons are free due to thermal promotion than are free due to the electric field.

The most important concern we have to determine conductivity is not any of these steps by themselves. It is the combination of how many electrons are free to carry current (due to promotion by the field and temperature) and how fast they are moving. In short, to determine the electrical conductivity of a metal, we must determine the number density of free electrons and the velocity of the free electrons under the applied electric field. In a metal, the number density of states depends on $\sqrt{E}$. At 0 K, we fill this with the appropriate number density of bonding electrons following Pauli exclusion principles. Then, we promote electrons using Fermi-Dirac statistics because electrons are fermions. This happens regardless of whether a field is applied or not. Using a convolution integral, we obtain a picture of electron density as a function of energy and temperature $\rho_E(E,T)$ as shown here. Those electrons above the Fermi energy are free to carry current.

The thermal energy that is applied to promote electrons from occupied states is on the order of $k_BT$ (this is the dashed line in the picture). When the energy of an electric field that is applied in a metal is below $k_BT$, more electrons are free due to thermal promotion than are free due to promotion by the electric field.

In conclusion, the initial shape of the Fermi surface has nothing to say about electrical conduction. The perturbation that occurs in the shape by the electrical field has nothing to say about electrical conduction. Finally, the promotion of electrons above the Fermi energy, whether by an applied field or by thermal means, is only the first (and required) step to determine electrical conduction. The most important concern we have to determine conductivity is not any of these steps by themselves. It is the combination of how many electrons are free to carry current (due to promotion by the field and temperature) and how fast they are moving.

To determine the electrical conductivity of a metal, we must determine the number density of free electrons and the velocity of the free electrons under the applied electric field. In a metal, the number density of states depends on $\sqrt{E}$. At 0 K, we fill this will the appropriate number density of bonding electrons. Then, we promote electrons using Fermi-Dirac statistics because thermal promotion dominates promotion by the electric field. Using a convolution integral, we obtain a picture of electron density as a function of energy and temperature $\rho_E(E,T)$ as shown here.

The thermal energy that is applied to promote electrons from occupied states is on the order of $k_BT$. The energy of an electric field that is applied in a metal in typical cases (room temperature without high fields) is below $k_BT$. Therefore, in most cases, more electrons are free due to thermal promotion than are free due to the electric field.

An electric field perturbs the unbound electrons by causing them to move throughout the lattice. This perturbation is independent of whether the free electron has only an infinitesimal energy above the Fermi energy or is at the limit of $k_B T$ or above the Fermi energy.

In conclusion, the initial shape of the Fermi surface has nothing by itself to say about electrical conduction. The perturbation that occurs in the shape by the electrical field has nothing to say by itself about electrical conduction. Finally, the promotion of electrons above the Fermi energy, whether by an applied field or by thermal means, is only a first (and required) step to determine electrical conduction. 

The most important concern we have to determine conductivity is not any of these steps by themselves. It is the combination of how many electrons are free to carry current (due to promotion by the field and temperature) and how fast they are moving. In short, to determine the electrical conductivity of a metal, we must determine the number density of free electrons and the velocity of the free electrons under the applied electric field. In a metal, the number density of states depends on $\sqrt{E}$. At 0 K, we fill this will the appropriate number density of bonding electrons. Then, we promote electrons using Fermi-Dirac statistics because thermal promotion typically dominates promotion by the electric field. Using a convolution integral, we obtain a picture of electron density as a function of energy and temperature $\rho_E(E,T)$ as shown here. Those electrons above the Fermi energy are free to carry current.

The thermal energy that is applied to promote electrons from occupied states is on the order of $k_BT$ (this is the dashed line in the picture). The energy of an electric field that is applied in a metal in typical cases (room temperature without high fields) is below $k_BT$. Therefore, in most cases, more electrons are free due to thermal promotion than are free due to the electric field.

In summary, the Fermi surface is the boundary between occupied and free electrons. The number density of free electrons is obtained by an integral over the convolution of the density of states and the Fermi-Dirac function. The electric field perturbs the free electrons (those above the Fermi energy, whether by thermal or electrical means) by causing them to move throughout the lattice. This perturbation happens independently of whether the free electron has only an infinitesimal energy above the Fermi energy or is at the limit of $k_B T$ or above the Fermi energy.

Rather than considering a Fermi sphere, you may be better to consider a basic muffin-tin potential system. The electrons that you say are inside the Fermi sphere are localized in the muffin potentials. The electrons that are free are delocalized throughout the entire set of potentials. Electrons are free because they are thermally promoted above the Fermi energy.

An electric field perturbs the bound electrons by shifting them relative to their centers. This is equivalent to polarization in dielectrics.

An electric field perturbs the unbound electrons by causing them to move throughout the lattice. This perturbation is independent of whether the free electron is just an infinitesimal energy above the Fermi energy or is at the limit of $k_B T$ or beyond the Fermi energy.

The two key points to consider when an electric field is applied to a material with metallic bounds are

• Bound electrons cannot move, they can only shift relative to their centers (the nucleus)
• ALL free electrons can move, regardless of their energy above the Fermi energy

Also ...

• Electrons follow a Pauli exclusion principle in the bound state. In the unbound state, we have no need to consider the details of defined quantum numbers and exclusion because the electrons choose from an infinite number of possible energy states.

• Electrons do not change their kinetic energy in the bound state. They only shift their positions relative to the nucleus.

• Electrons gain kinetic energy in the unbound state. When they would not scatter (from the electron clouds), they would hypothetically reach an infinite velocity.

As for the Fermi sphere paradigm, one confusion may be to believe that the electric field shifts only the electrons at the exact boundary of the surface of the sphere. The electrons at the sphere surface are already perturbed by thermal energy. This is where the $k_BT$ factor arises.

The Fermi sphere is the $E(k)$ boundary between the occupied (bonding) and unoccupied (non-bonding) states in a metal at zero Kelvin 1. In a metal, conduction is due primarily if not exclusively to the motion of free electrons. Free electrons are those that are not in occupied (bonding) states. At 0 K with no electric field, all electrons are in occupied states. Therefore, the metal carries no electrical current.

Let's perturb the metal in one of two ways.

• Put an electric field on the metal. This can distort the Fermi surface. Such a distortion is NOT the cause for the conduction of current. The distortion is analogous to how the shape of the Fermi surface is different along different crystallographic orientations. All that is being changed is the position of the Fermi energy. Nothing is said about the motion of the free electrons.

• Put an electric field on the metal. This promotes electrons from occupied to non-bonding (initially unoccupied) band states. This action is independent of the above change in the shape of the surface. This promotion is NOT the root cause for the conduction of current. It is however a step toward that result.

• Put an electric field on the material. This applies a force to the free electrons (those in non-bonding states). The free electrons move (accelerate). This is electrical current.

• Put the material at a temperature above 0 K. This promotes electrons from occupied to non-bonding (initially unoccupied) band states. Those free electrons are just as free to move as are the electrons that were promoted by the electric field.

In conclusion, the initial shape of the Fermi surface has nothing to say about electrical conduction. The perturbation that occurs in the shape by the electrical field has nothing to say about electrical conduction. Finally, the promotion of electrons above the Fermi energy, whether by an applied field or by thermal means, is only the first (and required) step to determine electrical conduction. The most important concern we have to determine conductivity is not any of these steps by themselves. It is the combination of how many electrons are free to carry current (due to promotion by the field and temperature) and how fast they are moving.

To determine the electrical conductivity of a metal, we must determine the number density of free electrons and the velocity of the free electrons under the applied electric field. In a metal, the number density of states depends on $\sqrt{E}$. At 0 K, we fill this will the appropriate number density of bonding electrons. Then, we promote electrons using Fermi-Dirac statistics because thermal promotion dominates promotion by the electric field. Using a convolution integral, we obtain a picture of electron density as a function of energy and temperature $\rho_E(E,T)$ as shown here.

The thermal energy that is applied to promote electrons from occupied states is on the order of $k_BT$. The energy of an electric field that is applied in a metal in typical cases (room temperature without high fields) is below $k_BT$. Therefore, in most cases, more electrons are free due to thermal promotion than are free due to the electric field.

Alternative insights can also be obtained using a basic muffin-tin potential system. The electrons that are bound are localized in the muffin potentials. The electrons that are free are delocalized throughout the entire set of potentials above the Fermi energy. Electrons are free primarily because they are thermally promoted above the Fermi energy.

An electric field perturbs the unbound electrons by causing them to move throughout the lattice. This perturbation is independent of whether the free electron has only an infinitesimal energy above the Fermi energy or is at the limit of $k_B T$ or above the Fermi energy.