A metal is characterised by the presence of conduction electrons. The highest occupied level of these electrons is called the Fermi level. In the free electron model, these electrons move throughout the metal even at absolute zero. However, the conductivity is zero in the absence of an electric field, because of random motion of electrons. When an electric field is applied, because of a drift motion, there is a finite current. The current density is proportional to the electric field for small field and the proportionality parameter is conductivity. It is well known Ohm’s law.

The reciprocal of conductivity is resistivity. Resistivity in a metal is a function of temperature; it increases with temperature due primarily to electron-phonon interaction. However, due to the presence of small amount of impurities and defects, the resistivity is finite even at absolute zero. Thus both resistivity and conductivity are finite at absolute zero.

However, for superconductors and semiconductors the situation is different. For a superconductor, below some characteristic temperature known as the superconducting transition temperature, resistivity is zero and current becomes perennial. On the other hand in an intrinsic semiconductor, at absolute zero, there is no current, and a semiconductor behaves like an insulator. Impurity doping and/or finite temperature give rise to a finite conductivity, but the conductivity is not as good as in a metal.

Not generally. Many metals transition to superconductors near 0K. Otherwise there is a finite number of states below the Fermi energy, so the Pauli exclusion will keep some electrons in the conduction band.

Certainly, the metals would stay conducting at zero temperature. External electric field tries to increase the energy of the electron and there is enough unoccupied states above the Fermi level. You forgot this factor in your considerations. In metals, only those electrons conduct which are close to the Fermi surface. 

0k does not mean zero energy. So by understanding the Pauli Exclusion principal even at 0k there are many moving electrons although with energy not above Fermi level.

However, based solid state Physics, the resistance of metals is based on the imperfection in their structure so the higher the perfection the lower the resistance at  0k

As I understand, classically, there are two contributions in the resistivity/conductivity.

1)      The number or charge carrier,

2)      The interaction (scattering) between the electrons and the ions (which in the semiconductor theory appears in the “mobility”)

Only the electrons at the energy where there exist both occupied states and free states contribute to the resistivity/conductivity.

When all states of a band are occupied, there is no scattering possible since we have no empty state where the electrons which occupy the band (at the same energy) can go. The conductivity is zero. It is an insulator or a semiconductor at low temperature.

When the temperature increases, some electrons of the valence band can go into the conduction band creating both free states in the valence band and occupied states in the conduction band (intrinsic conductivity). The electrons in the valence states can now be scattered in the few states which have been released (conduction by holes), simultaneously the electrons in the conduction band can be scattered in the many free bands (conduction by holes). The temperature increases the number of scatterers in both the conduction and valence bands thus increases the conductivity.

The conduction by impurities (extrinsic) is less sensitive to temperature since when all electrons of the impurities states (localized, not in a band) are transferred in the valence band, the number of scatterers no more increase.

The number of scatterers is the main effect in semiconductors.

In metals there are many electrons and many free states. There is also a (smaller) effect due to the number of scatterers with temperature. Near kF there is a blurring of the Fermi sphere which increases with temperature. However at very low temperatures, there are less “vibrations” of the ions. The “mobility” increases when the temperature decreases, the electrons are less scattered, the mean free path increases.

In solid state silicon just below the melting temperature is a semi conductor. Just above it is a metal. The mobility is nearly identical. What changes is the number of scatterers.

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It is well known that in general conductivity of metals increases with decreasing temperature. YES Although there are less electrons that are thermally excited above the Fermi energy to contribute to conduction, the decrease in thermal vibrations that scatters conducting electrons has a larger effect, thus conductivity increases. THERE ARE TWO EFFECTS: THE NUMBER OF ELECTRONS WHICH CAN BE SCATTERED IS LOWER (NUMBER) AND THE VIBRATIONS OF THE IONS ARE SMALLER (MOBILITY)

Similarly, electrical conductivity of semiconductors increases with increased temperature because there is more thermal energy to excite a greater number of electrons from the valence band to the conduction band. YES However, as more electrons populate the conduction band the influence of thermal scattering becomes dominant and conductivity begins to decrease like in a metal. I DO NOT KNOW THIS EFFECT, IF IT EXISTS IT IS ONLY AT VERY HIGH TEMPERATURE, MORE PROBABLY TEMPERATURE MAY CHANGE THE STRUCTURE INTO A METALLIC STATE.

If electrons need to be excited above the Fermi energy in order to contribute to conduction, would a metal at 0 K become an insulator since there is no thermal energy to excite electrons?

TO MY OPINION, NO. EVEN AT ZERO K THERE ARE AT THE SAME ENERGY STATES OCCUPIED BY ELECTRONS AND FREE STATES. THE FIRST PEAKS OF THE STRUCTURE FACTOR ARE INFINITELY SMALL, THUS THE MOBILITY INCREASES VERY MUCH. THE CONDUCTVITY OF A METAL INCREASES.

HOWEVER THERE ARE ALSO OTHER EFFECTS THAT I DO NOT KNOW (LIKE PHONON DRAG) .

Let me know if I am thinking about this correctly. Thanks

Thank you all for your contributions. I understand this question may have a very intuitive answer (of course metals are conductive at low temperature), but it has been on my mind as of lately.

Igor Goliney, your statement does well at answering my question. To reiterate what you said, an applied external electric field will increase the energy of electrons near the Fermi surface to excite them to unoccupied states above the Fermi level which allows the electrons to contribute to conduction. However, would this material no longer be at 0K? Hypothetically speaking, without an applied electric field, would a metal at 0K have 0 conductivity because the number of electrons above the Fermi level would be 0?

Gado Abubakar Abubakar, you say that at 0K the electrons will be below the Fermi level, I agree because there is no thermal energy to excite electrons above the Fermi level. Based on that, wouldn't you say that a metal at 0K by definition would not be a conductor because the concentration of conducting electrons is 0?

Alan Ahlberg Elliot, let us simplify the question and exclude superconducting metals from the argument at this time. By definition, isn't the Fermi level of a metal the highest occupied state at 0K? Therefore there would be no electrons above it at 0K. 

A system conducting an electric current is not in equilibrium. In a way, conducting  electrons are always hot. The distribution is no longer Fermi-Dirac. Therefore, the notion of temperature is not well defined. Usually, the thermal equilibrium establishes faster than the concentrational. Therefore we may consider the current at a set temperature. This would hardly be so at zero temperature. At zero temperature the system is supposed to be coupled with a bath that sucks out every possible energy from it. This hardly works with a bias. Zero temperature is such a weird notion. Actually it is unattainable. 

In metals the Fermi level is in the conduction band at 0K, therefore it is positive always. Moreover the number of carriers (electrons) does not change with temperature, the only parameter that changes is the mobility, which reduces at high temperatures, thus the conductivity decreases

A metal is characterised by the presence of conduction electrons. The highest occupied level of these electrons is called the Fermi level. In the free electron model, these electrons move throughout the metal even at absolute zero. However, the conductivity is zero in the absence of an electric field, because of random motion of electrons. When an electric field is applied, because of a drift motion, there is a finite current. The current density is proportional to the electric field for small field and the proportionality parameter is conductivity. It is well known Ohm’s law.

The reciprocal of conductivity is resistivity. Resistivity in a metal is a function of temperature; it increases with temperature due primarily to electron-phonon interaction. However, due to the presence of small amount of impurities and defects, the resistivity is finite even at absolute zero. Thus both resistivity and conductivity are finite at absolute zero.

However, for superconductors and semiconductors the situation is different. For a superconductor, below some characteristic temperature known as the superconducting transition temperature, resistivity is zero and current becomes perennial. On the other hand in an intrinsic semiconductor, at absolute zero, there is no current, and a semiconductor behaves like an insulator. Impurity doping and/or finite temperature give rise to a finite conductivity, but the conductivity is not as good as in a metal.

Shmuel E Schacham, you said that the number of charge carriers does not change with temperature (for a metal). It was my understanding that only electrons above the Fermi level would contribute to conduction, therefore only electrons near the Fermi level have enough energy at room temperature to be excited above the Fermi level. So that would make it seem like charge carrier concentration is temperature dependent, but electron mobility is the dominating factor. The electrons below the Fermi level are responsible for metallic bonding. Please correct my thinking if I am wrong.

Igor Goliney, yes, it is not a very practical question. Thank you for your response.

In addition see the simple picture clarifying that no сontradictions occures between that all electrons are below the Fermi surface and still conductivity is high at low T in metals.

At T=0 all electron states below Fermi sphere are occupied symmetrically, but if one apply electric field, electrons acquires additional impulse towards the field direction and Fermi surface slightly moves (red arrow). In this case the nonequillibrium Fermi-surface doesnt match an initial. So no termal excitation required in the metals to reach good conductivity