**Fundamental Quanta of Resistance**

*A very simple account of a quantum phenomenon which connects Solid State Physics to the universal constants.*

In 1826 Georg Simon Ohm published the well-known, basic law of linear electric circuits,V=RI, where V is the potential difference between the ends of a metallic wire, R the resistance and I the intensity of the current. One can also write the same law as I=GV, where G=1/R is called the conductance. The resistance R depends on the material properties of the wire as well as on the geometry. For a cylindrical wire, one can write

where L is the length of the wire, A the area its cross-section and ρ the resistivity, which is a property of the material ; equivalently, one may write

where

is the conductivity. Ohm’s discovery was the beginning of electrical engineering and more generally of the theory of electron transport in matter. This is a very rich subject, since many interesting and useful phenomena and transport mechanisms have been discovered, and quantum transport is even now an important active area of research.

The old, classical explanation of the resistance R depicts the electrons that carry the current are like marbles. These marbles are accelerated by the field but scatter against the hard ion cores of the metal and therefore the current experiences some kind of friction, similar to a stream that hits the stones and the irregularities of the ground. Otherwise the electrons were supposed to move in the electric field similar to the raindrops that fall from the clouds but do not exceed a limiting speed since are held back by the air resistance. In 1904 the latter idea inspired the classical Drude theory of electric and optical properties of metals which had some success in the case of Alkali metals.

However, it is evident that the above classical expressions for R and G must fail when the length L of the wire goes to 0, because then the resistance should vanish (and the conductance should diverge); if that were true, a short enough wire should support an arbitrarily large current.

Indeed, the old classical picture is wrong in principle. Atoms in solids form regular periodic lattices; Quantum Theory shows that electrons propagate as waves that can diffract unhindered through a perfect periodic potential. On the other hand, the electrons do scatter against impurities, lattice defects, walls and phonons (i.e. quantized lattice waves). For each metal, the electrons have a (temperature dependent) mean free path λ between two scatterings. For example, at room temperature, λ =22 nm for Al and 55 nm for Cu [1] (1 nm is the billionth of a meter). Scattering events dissipate energy and produce heat (Joule effect). Electric stoves and lamps with a Tungsten filament use the Joule effect. A detailed quantitative theory of such dissipative processes is very far from trivial.

**Quantum Transport in Nanoscopic devices**

For nanoscopic objects (i.e. conductors smaller than the mean free path λ ) the description simplifies because we can do without the theory of dissipation . Instead, we must fully account for the quantum nature of electrons: in fact, one speaks about **quantum transport**. The electron wave functions can be found by using quantum mechanics for particles moving in an external potential. We also say that when all lengths are small compared to the electron mean free path, the transport is **ballistic**. This occurs, for example, in experiments with Carbon Nanotubes (CNT), Graphene, nanowires and single electron transistors.

In such conditions, the theory of transport requires quantum Green’s function techniques [2]; however a very simple argument is enough to show that conductance and resistance must be quantized. Since electrons are Fermions, no two of them can occupy the same quantum state. Therefore the electrons in metals cannot all be on the lowest level. They must form a conduction band and fill the lowest available quantum states up to a maximum energy, which is called the Fermi energy. Let’s imagine the electrons that under the influence of a potential difference V flow from a nanoscopic conductor (which I will call the conducting channel) to an electrode (see the Figure below). Then, eV must be the energy difference between the Fermi energy in the conducting channel and in the electrode .The electrons available for conduction are those whose energy is below the Fermi level of the conducting channel (left in the Figure) and above the Fermi level of the positive electrode (right).

Let us say that

is the average time an electron takes to jump. The frequency

is related to the current i by the obvious formula

On the other hand, the electron energy makes a jump eV and in Quantum Mechanics there is a frequency

associated to eV, where h is Planck’s constant. But there is nothing else than the current in this problem, so it is natural to conclude that

Therefore,

which implies

But we must multiply the current by 2 in order to account for both spin directions. Therefore we find a fundamental **quantum of conductance**

and therefore a **quantum of resistance**

It turns out that an approximate value is R=12900 Ohm. In macroscopic conductors the resistance can be much less than that, because they may have a large number of alternative conducting channels. In other terms, G is the conductance that we should measure if we could make a conductor consisting of a single conducting channel. In reality any nanoscopic conductor behaves like a bunch of a number M conducting channels due to transverse degrees of freedom. The number M of conducting channels is of order of

where

is the wave vector of an electron at the Fermi level,

is the wave length of an electron at the Fermi level, and W is the width of the conductor. Typically the wave length is comparable to a lattice parameter, so M can easily be of the order of 10 in nano wires.

B.J. Van Wees et al. [3] performed an experiment using a negative potential applied across the sample, the so called gate potential. By increasing the size of the gate potential, the conducting channels can be switched off one after the other, thereby reducing M, and so the quantization becomes evident.

The staircase results in the above Figure show clearly that the conductance drops when the gate potential becomes more negative and is indeed quantized. One can see clearly that there is a conductance quantum G.

**General significance. Why is the resistance quantum important?**

The resistance quantum has nothing to do with macroscopic circuits and electric stoves, where energy is dissipated into heat by mechanisms that are not operating in nanoscopic conductors. Instead, the resistance quantum R has a special meaning for nanoscopic conductors where quantum effects dominate. In particular, it shows up in the Quantum Hall Effect, discovered by Klaus von Klitzing, which does not lend itself to a simple explanation like the present one, but allows to measure R very accurately. This is important for fundamental Physics because it is related to the fine structure constant α. In cgs units,

where c is the speed of light; numerically,

This is a pure number, which measures the strength of the electromagnetic interaction and appears to be the same in all the universe within the experimental accuracy, according to the astronomers. Therefore if there are physicists in other planetary systems or in other galaxies, they must find the same numerical value for α. Since the value of c is accurately known, improving the value of R improves the knowledge of α, which enters all the formulas of Quantum Electrodynamics.

[1] See for instance Yuko Hanaoka et al, Materials transactions 43,7 (2002).

[2] Michele Cini, Physical Review B22, 5887 (1980)

[3] B.J. van Wees, et al., Physical Review Letters 60, 848 (1988)