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Physics 2

Ohm's law from the structure of matter: the Drude model

Current density from the carriers, conductivity derived from electron collisions with the lattice, the numbers for copper, Joule heating, resistance of conductors of any shape, temperature dependence, limits of Ohm's law and the Hall effect.

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This page goes deeper into the lesson on electric current: it derives Ohm’s law from the microscopic properties of matter, that is, from the free electrons of a metal colliding with the crystal lattice. This is the Drude model (1900). From a simple idea we obtain the conductivity in terms of atomic quantities, the numerical value of the time between collisions in copper, Joule heating seen as energy given to the lattice, the temperature dependence, the resistance of conductors of any shape, the limits of validity of the law and the Hall effect, which allows the number and sign of the carriers to be measured.

Current density from the carriers

Theorem. If a conductor contains n carriers per unit volume, each with charge q and mean (drift) velocity \vec v_d, the current density is

    \[ \vec J = n\,q\,\vec v_d \]

Proof. Consider a cross-section of area A perpendicular to \vec v_d. In a time dt it is crossed by all the carriers lying within a distance v_d\,dt upstream, that is, those contained in the cylinder of volume A\,v_d\,dt. There are n\,A\,v_d\,dt of them, and they carry the charge dQ = q\,n\,A\,v_d\,dt. The current is I = dQ/dt = nqv_dA, and the density J = I/A = nqv_d. ∎

Segment of conductor with the charge carriers
In a time d t the section A is crossed by all the carriers, in orange, contained in the cylinder of length v d times d t. The field E pushes positive carriers in its direction and the current density J has the same direction.

For electrons q = -e: \vec v_d is opposite to \vec J, but the product q\vec v_d has the direction of the conventional current.

The Drude model

There are four assumptions, all simple:

  1. in the metal some of the electrons (one or two per atom) are free to move, like a gas, among the fixed ions of the lattice;
  2. between collisions an electron moves under the action of external forces only, here -e\vec E;
  3. collisions are instantaneous, and after each collision the velocity has a random direction: on average, just after a collision, the velocity is zero;
  4. the probability of a collision in an interval dt is dt/\tau, independent of the electron’s history. \tau is called the relaxation time or mean time between collisions.
Path of a free electron in a metal
The motion is a sequence of stretches interrupted by collisions, in almost random directions, because the thermal speed is enormously larger than the speed due to the field. The field E adds a small push in the opposite direction to each stretch, and after many collisions the electron has moved by v d times delta t to the left.

Theorem. In the Drude model the drift velocity and the conductivity are

    \[ \vec v_d = -\frac{e\tau}{m}\vec E,\qquad \vec J = \sigma\vec E,\quad \sigma = \frac{n\,e^2\,\tau}{m} \]

Proof. Look at one electron at a given instant. A time t_0 has passed since its last collision, and during that time the field has accelerated it with \vec a = -e\vec E/m. Its velocity is therefore \vec v = \vec v_0 - \frac{e\vec E}{m}t_0, where \vec v_0 is the velocity just after the collision.

Average over all the electrons. By assumption 3, the average of \vec v_0 is zero. By assumption 4, the time elapsed since the last collision has an exponential distribution with mean \tau (the probability that no collision has occurred in the last t_0 seconds is e^{-t_0/\tau}). Hence

    \[ \vec v_d = \langle\vec v\rangle = 0 - \frac{e\vec E}{m}\langle t_0\rangle = -\frac{e\tau}{m}\vec E \]

Substituting into \vec J = n(-e)\vec v_d: \vec J = \frac{ne^2\tau}{m}\vec E. ∎

Another way to see it. On average, the force of the field and the effect of collisions balance: collisions behave like viscous friction, which every second destroys a fraction 1/\tau of the ordered momentum. The equation of the average motion is

    \[ m\frac{d\vec v_d}{dt} = -e\vec E - \frac{m}{\tau}\vec v_d \]

At steady state d\vec v_d/dt = 0 and the same result follows. The transient lasts a few \tau, about 10^{-13} s: for every practical purpose the current follows the field instantly.

Velocity component along the field as a function of time
Between two collisions the velocity grows linearly because of the field; at each collision, marked by the grey lines, it restarts from a random value. The average over many collisions is the drift velocity, the dashed red line.

Ohm’s law is therefore the consequence of one fact: collisions make the electron “forget” the velocity it has gained, and the average velocity stays proportional to the field instead of growing without limit. In a conductor without collisions (\tau \to \infty) the current would grow indefinitely under a constant field: that is what happens in the vacuum of a cathode-ray tube, not in a wire.

The numbers for copper

The time between collisions can be obtained from the measured resistivity. For copper \rho = 1.7\cdot 10^{-8} Ω·m and n = 8.5\cdot 10^{28} m^{-3} (one free electron per atom):

    \[ \tau = \frac{m}{ne^2\rho} = \frac{9.11\cdot 10^{-31}}{8.5\cdot 10^{28}\cdot(1.60\cdot 10^{-19})^2\cdot 1.7\cdot 10^{-8}} \approx 2.5\cdot 10^{-14}\ \text{s} \]

Three comparisons show the scales.

Drift versus thermal motion. In a 1 mm² wire carrying 1 A the field is E = \rho J = 0.017 V/m and the drift velocity v_d = e\tau E/m \approx 7\cdot 10^{-5} m/s, the same value found in the lesson from J = nev_d. The electrons, on the other hand, move randomly at speeds of order 10^6 m/s: the drift is a tiny correction to a very fast disordered motion.

The mean free path. With the “classical” thermal speed \sqrt{3kT/m} \approx 1.2\cdot 10^5 m/s at 300 K, the distance between collisions would be \lambda = v\tau \approx 3 nm, about ten interatomic distances: consistent with Drude’s picture of electrons hitting the ions. But quantum mechanics says that conduction electrons move at the Fermi velocity, 1.6\cdot 10^6 m/s in copper, and then \lambda \approx 39 nm, about 150 interatomic distances.

The role of the ions. An electron thus travels past hundreds of atoms without colliding: a perfectly regular lattice does not scatter electrons (a result of quantum mechanics, linked to the wave nature of the electron). Collisions happen with the imperfections of the lattice: thermal vibrations of the ions (phonons), impurities, defects, grain boundaries. The Drude model has the wrong picture of collisions but the right structure of the formula: with the right effective mass and relaxation time, \sigma = ne^2\tau/m remains valid in the quantum theory too.

Joule heating seen from the lattice

Theorem. In an ohmic conductor the power dissipated per unit volume is

    \[ p = \vec J\cdot\vec E = \sigma E^2 = \frac{J^2}{\sigma} \]

and integrated over a wire it gives P = RI^2.

Proof. On each carrier the field does work at a rate q\vec E\cdot\vec v. On average, per unit volume, p = nq\vec E\cdot\vec v_d = \vec J\cdot\vec E. At steady state the average kinetic energy of the electrons does not grow: all this energy is given to the lattice in collisions, and becomes thermal agitation of the ions, that is, heat. In a wire of cross-section A and length L with uniform field, P = pAL = (JA)(EL) = IV = RI^2. ∎

The resistance of a conductor of any shape

From the local law \vec J = \sigma\vec E the resistance of an object follows by integration. For a uniform wire, V = EL and I = JA give R = \rho L/A. If the cross-section changes along the current path, divide the conductor into slices of thickness dl in series, each with resistance dR = \rho\,dl/A(l):

    \[ R = \int\frac{\rho\,dl}{A(l)} \]

Example: leakage through the insulation of a coaxial cable. Between the inner conductor (radius a) and the braid (radius b) there is an insulator of very large resistivity \rho, of length L. The leakage current flows radially, through cylindrical shells of radius r, thickness dr and area 2\pi rL, all in series:

    \[ R = \int_a^b\frac{\rho\,dr}{2\pi rL} = \frac{\rho}{2\pi L}\ln\frac{b}{a} \]

The resistance decreases with length, unlike a wire: a longer cable has more surface through which to leak current. With \rho = 10^{13} Ω·m (polyethylene), b/a = 3.5 and L = 100 m: R \approx 2\cdot 10^{10} Ω.

Radial resistance of the insulation of a coaxial cable
On the left the cross-section: the leakage current, in green, flows from the inner conductor of radius a to the braid of radius b through shells of thickness d r. On the right the side view, with the length L of the cable.

Temperature dependence

In metals n does not depend on temperature; \tau does. Collisions with lattice vibrations are more frequent the more the ions vibrate, and the mean square amplitude of the vibrations is proportional to T (above a few tens of kelvin). So 1/\tau \propto T and the resistivity grows linearly. Collisions with impurities, instead, do not depend on T. The two causes add up (Matthiessen’s rule):

    \[ \rho(T) = \rho_0 + \rho_{phonons}(T) \approx \rho_0 + \rho_{20}\,\alpha\,(T - T_{20}) \]

where \rho_0 is the residual resistivity at very low temperature, which measures the purity of the metal. For copper \alpha \approx 0.0039 K^{-1}.

In semiconductors the story is the opposite: the number of carriers grows exponentially with temperature, n \propto e^{-E_g/(2kT)}, because thermal agitation frees electrons from the bonds. The increase of n outweighs the decrease of \tau, and the resistivity decreases rapidly as T rises. This is the reason for NTC thermistors.

Resistivity as a function of temperature
On the left a metal: at low temperature the residual resistivity rho zero remains, then the resistivity grows almost linearly. On the right a semiconductor: the resistivity decreases rapidly as the temperature rises.

When Ohm’s law does not hold

Proportionality between J and E requires \tau and n not to depend on the field, and the field to vary slowly compared with \tau. Outside these conditions:

  • strong fields in semiconductors: the drift velocity saturates (about 10^5 m/s in silicon), because the electrons give energy to the lattice as fast as they receive it;
  • high frequencies: with an alternating field of angular frequency \omega, the same equation of average motion gives \sigma(\omega) = \dfrac{\sigma_0}{1 + j\omega\tau} (with the phasor convention of electrical engineering). For metals \omega\tau \ll 1 up to about 10^{12} Hz: the DC Ohm’s law holds for all of electronics, but not for light: at optical frequencies the free-electron model, extended, explains why metals reflect visible light;
  • non-linear devices: diodes, transistors, gas discharges, where n depends on the applied voltage;
  • conductors shorter than the mean free path: in nanowires electrons pass without collisions (ballistic conduction) and the conductance is quantised: each conduction channel contributes G_0 = 2e^2/h \approx 77 µS, and a conductor with a single channel has a resistance of about 12.9 kΩ with no collisions at all;
  • superconductors: below the critical temperature the resistance is exactly zero.

The Hall effect: counting the carriers

In a strip carrying current and placed in a perpendicular magnetic field \vec B, the Lorentz force q\vec v_d\times\vec B pushes the carriers towards one edge. Charge accumulates there until the transverse electric field E_H that forms balances the magnetic force: qE_H = qv_dB.

Theorem. In a strip of thickness t (in the direction of \vec B), the Hall voltage is

    \[ V_H = \frac{I\,B}{n\,q\,t} \]

and its sign depends on the sign of the carriers.

Proof. With width w, V_H = E_Hw = v_dBw. From the current, I = nqv_d\,wt, so v_d = \frac{I}{nqwt}. Substituting, V_H = \frac{IB}{nqt}. If the carriers are negative, for the same current direction they move the opposite way, but their charge is also opposite: the Lorentz force pushes them onto the same edge as positive carriers, and the voltage changes sign. ∎

Example. A copper strip 0.1 mm thick carrying 1 A in a 1 T field: V_H = \frac{1\cdot 1}{8.5\cdot 10^{28}\cdot 1.6\cdot 10^{-19}\cdot 10^{-4}} \approx 0.7 µV. In semiconductors n is a million times smaller and V_H becomes tens of millivolts: this is the principle of Hall sensors, used to measure magnetic fields and currents without contact.

Hall effect in a conducting strip
The current I flows to the right, the electrons move to the left with velocity v d, the magnetic field B goes into the page. The Lorentz force F pushes the electrons towards the upper edge, which becomes negatively charged, while the lower one stays positive: the Hall voltage is measured between the two edges.

Historically the Hall effect showed that in some metals, such as zinc, the carriers behave like positive charges (holes): a fact the Drude model could not explain and one of the proofs of the quantum theory of solids.

Exercises on resistance, DC circuits and the RC circuit are on the exercise page linked at the top.

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