99 Measuring Planck's constant by means of LED's

Historical burdens on physics
99 Measuring Planck’s constant by means of LED’s
Subject:
Planck’s constant can be measured by using light emitting diodes. The voltage
applied to the LED is increased until the diode begins to emit light. The
corresponding threshold voltage U0 multiplied with the elementary charge
is, so it is said, equal to the band gap energy and thus equal to the energy
of the emitted photons. The experiment is carried out with various LED’s
which emit light with different frequencies.
Deficiencies:
There is no threshold voltage for the light that is emitted by the diode. The
light intensity is proportional to the electric current in the diode. The electric
current I as a function of the applied voltage U is in good approximation
given by:
⎛
I = I S· exp
eU ⎞
⎝
⎜
ηkT ⎠
⎟ = I S· exp ⎛ U ⎞
⎝
⎜
U T ⎠
⎟ . ! (1)
Here, k is the Boltzmann constant, T the absolute temperature and e the
elementary charge. η is called the non-ideality factor whose value is between
one and two. It would be equal to one if all the electron-hole pairs
would recombine radiatively. η has no significance for the following considerations
as long as it has the same value for all the diodes that are compared.
IS is the saturation current. It depends on the temperature and on the
band gap energy Eg . The following proportionality holds:
⎛
I S A · exp − E g ⎞
⎝
⎜
ηkT ⎠
⎟ ,
where A is the pn contact surface area. Apart from
U T = ηkT
e
there is no characteristic voltage in equation (1). However, UT has nothing
to do with the band gap [1, 2].
There is not a minimum voltage for which the diode begins to emit, since it
emits always – but with an intensity that depends on the applied voltage. It
even emits when U = 0 V, namely the thermal radiation. When the voltage
increases, the intensity of the emitted light increases exponentially,
whereas its spectral distribution does not change. It may surprise that the
diode emits photons whose energy is roughly equal to the band gap energy,
even though the energy eU supplied to the electron-hole pairs is smaller.
Actually the diode would cool down a little when working with small applied
voltages. It works as a Peltier element. Since this effect is small, it is covered
by the unavoidable dissipative heat.

Fig. 1. Characteristic of one and the same diode represented with the current axis scaled
differently. The curves have the same shape and can be made to coincide by a horizontal
displacement.
The procedure that is applied to get a “threshold voltage” U0 is based on an
illusion. Fig. 1 shows three times the same exponential characteristic, the
difference between the three representations consisting only in the choice
of the axis of ordinates. Each time that the scale of the vertical current axis
is changed by a factor of 100, the graph is displaced horizontally by
kT
e · ln102 = 0,119 Volt
(We have admitted that η = 1.)
The “threshold voltage” which one would read from the graph changes by
the same amount.
Origin:
The experiment was introduced as a simple and inexpensive experiment for
the physics lab at the high school and the university. The incorrect interpretation
has a certain plausibility. Apparently, it was overlooked that a threshold
voltage cannot be defined for an exponential function in principle.
Disposal:
Planck’s constant can be determined by means of several diodes with different
band gaps, Fig. 2.

Fig. 2. Characteristics of two diodes, that emit light of different frequencies. The curves can
be made to coincide by a horizontal displacement.
But there is a condition: the pn contact surface area must be the same for
all of the diodes. If this is the case the corresponding characteristics are
distinguished only in the factor [3]
⎛
exp − E g ⎞
⎝
⎜
ηkT ⎠
⎟ .
The band gap energy Eg is related to the average frequency of the emitted
light by
Eg = h · f.
Thus, the horizontal distance between the two curves 1 and 2 is
(Eg1 – Eg2)/e.
We now choose an arbitrary value I0 of the current and read the corresponding
voltages Ui . We get
U1 – U2 = (Eg1 – Eg2)/e
or
e(U1 – U2) = Eg1 – Eg2 = h(f1 – f2),
and thus
h = e(U 1 − U 2 )
f 1 − f 2
.
Notice that neither
eU1 = hf1
nor
eU2 = hf2
is valid separately.

When plotting eUi over the frequency of the emitted light, one gets a
straight line whose slope is equal to Planck’s constant, fig. 3.
Fig. 3. For two (or more) diodes the voltage at I0 multiplied by e is plotted over the frequency
of the emitted light. The slope of the straight line is equal to the Planck constant.
(Sometimes tangents at the points of equal current are drawn and the voltage
is read where they cut the axis of abscissas. Obviously the value is the
same as when reading the voltage directly as in Fig. 2, but it may give the
illusion that this section represent something like a threshold voltage.)
Whether the straight line in Fig. 3 runs through the origin or not, depends
only on the arbitrary choice of the value of the current I0.
Instead of reading the voltage values for a given current, one often uses
another procedure: One chooses that voltage where the diode visibly begins
to emit light. Since one automatically compares the light intensity with
the ambient light, a voltage value can quite reliably be determined. By this
procedure a current value is defined with sufficient precision that is the
same for all diodes. This explains why the procedure gives satisfying results.
However, the fact that the straight line obtained in this way often
passes through the origin is pure coincidence.
[1] Herrmann, F. und Schätzle, D.: Question # 53. Measuring Planck’s constant
by means of an LED, Am. J. Phys. 64, 1996, S. 1448
[2] Morehouse, R.: Answer to Question # 53. Measuring Planck’s constant
by means of an LED, Am. J. Phys. 66, 1998, S. 12
[3] Würfel, P.: Physics of Solar Cells, Wiley-VCH, Weinheim 2009
Friedrich Herrmann and Peter Würfel, Karlsruhe Institute of Technology