Experiments · Experiment 10

Fermi-Dirac Statistics: Thermionic Emission from a Metal

The Richardson-Dushman law, the work function, and the space-charge limit.

Objectives

On completing this experiment you should be able to:

Measure the current-voltage characteristic of a vacuum diode and identify the space-charge-limited and saturated regions.
Verify the Child-Langmuir three-halves power law.
Determine the filament temperature from its resistance.
Measure the saturation current as a function of temperature and obtain the work function from a Richardson plot.
Detect the Schottky effect and extrapolate to zero field.
Identify an unlabelled cathode from its work function.

Theory

At absolute zero the conduction electrons fill all states up to the Fermi energy. At a finite temperature the occupation is

f(E) = \frac{1}{\exp((E - \mu)/kT) + 1} ,

and for energies more than a few kT above mu this reduces to the Boltzmann factor exp(-(E - $\mu)/kT)$ . An electron escapes if the component of its kinetic energy normal to the surface exceeds the barrier. Integrating the Fermi-Dirac tail over the outward-moving states gives

J = A_G T^2 \exp(-\phi/kT), A_G = \frac{4 \pi m e k^2}{h^3} = 1.2017 \times 10^6\ \mathrm{A\,m^{-2}\,K^{-2}} ,

where phi = $E_{barrier}$ - mu is the work function. Real surfaces emit less than this because some electrons are reflected at the barrier, so the measured prefactor is $\lambda_R A_G$ with $\lambda_R$ below one.

The emitted electrons form a cloud in front of the cathode which repels further emission, so at low anode voltage the current is limited not by emission but by space charge, giving the Child-Langmuir law I proportional to $V^{3/2}$ .

Key relations

$f(E)=\frac{1}{e^{(E-\mu)/kT}+1}$
The Fermi-Dirac distribution.
$J=\lambda_R A_G T^{2}e^{-\phi/kT}$
Richardson-Dushman law.
$\ln\!\left(\frac{J}{T^{2}}\right)=\ln(\lambda_R A_G)-\frac{\phi}{k}\cdot\frac{1}{T}$
The Richardson plot.
$J=\frac{4}{9}\varepsilon_0\sqrt{\frac{2e}{m}}\,\frac{V^{3/2}}{d^{2}}$
Child-Langmuir space-charge law.

Procedure by part

Part A - Diode characteristic

At a fixed filament current, sweep the anode voltage and record the anode current.

Select the cathode and set the filament current. Record the filament resistance and deduce the temperature.
Sweep the anode voltage from 1 V to 300 V, recording the current.
Plot log I against log V and identify the region of gradient $\frac{3}{2}$ .
Identify the onset of saturation and comment on how it moves with filament temperature.

Part B - Richardson plot

With the anode voltage set well into saturation, measure the saturation current over a range of filament temperatures.

Fix the anode voltage at a value you have shown to be in saturation.
Step the filament current, recording the resistance, the deduced temperature and the anode current.
Plot $\ln(I/T^2)$ against $\frac{1}{T}$ . The gradient gives the work function and the intercept gives $\lambda_R A_G$ .
Repeat for at least three cathodes, including the unlabelled one.

Part C - The Schottky effect

Deep in saturation, the current still rises slowly with anode voltage because the applied field lowers the barrier.

Hold the filament temperature fixed and record the saturation current from 50 V to 300 V.
Plot ln I against the square root of the anode voltage.
Obtain the barrier lowering from the gradient and compare with $\sqrt{e^3 E/4}$ pi eps0).
Extrapolate to zero field to obtain the true zero-field work function, and compare with Part B.

Questions to answer in your report

Derive the Richardson-Dushman law from the Fermi-Dirac distribution, stating clearly where the approximation exp(-(E-mu)/kT) is justified. [10]
Explain why the electrons obey Fermi-Dirac statistics inside the metal but a Boltzmann distribution once emitted. Estimate the degeneracy parameter in each region. [8]
Compare your work function from the Richardson plot with the zero-field value obtained by extrapolating the Schottky plot. Which is the true work function, and what is the size of the correction? [6]