Department of condensed matter physics
Přírodovědecká fakulta, Masarykova univerzita, Brno
Physical laboratory 4
LED spectra
Task and goals
Determine the relationship between voltammograms and LED spectra. Compare to a similar
relation between the voltage at the X-ray tube and the spectrum of the bremsstrahlung
X-ray radiation.
Measure the emission spectra of different colored LEDs for different currents, and determine
the wavelength by independent interference method. Measure the voltammetric characteristics
of different colored LEDs. Measure the X-ray bremsstrahlung spectra for different
voltages at the X-ray.
Verify the value of the ratio of fundamental physical constants hc/e.
Recommended procedure and tasks for measurements
The student has three weeks to make the measurement. Recommended Procedure:
• First and second week – spectra and voltammograms of LEDs. Spectra can be measured in
several ways: fiber spectrophotometer, Newtonian glass, and simple grating spectroscope.
Use at least two methods.
• Week three – the spectrum of bremsstrahlung x-ray as a function of voltage on the x-ray
tube.
The output of the practicum will be presented to the instructor in the form of graphs and
measured dependencies, including an estimate of uncertainties if possible and appropriate.
Voltage characteristics of LEDs
The first systematic measurement of Planck’s constant was made in 1912 by Robert Millikan,
best known for measuring the elementary charge, in which he observed the movement of charged
oil droplets in an electrostatic field. The Planck’s constant value of h = 6.57 × 10−34 : J, s was
determined by careful observation of the photoelectric effect on the surface of metals in a vacuum.
To approximate the value of Planck’s constant in this problem, the practitioner will use the
relationship between the characteristic voltage required to light a light-emitting diode (LED) and
the color of the light emitted. In this way, we can find the value of Planck’s constant with an error
of tens of percent.
Like other types of diodes, the LED is based on a PN transition between a semiconductor
type P-type and N-type. A depleted region – a layer of space charge – forms at the interface that
prevents majority electrons and holes from penetrating the interface. If we apply a voltage in
the permeable direction to the PN transition, it allows an additional electrostatic field to allow
charge carriers to more easily cross the depleted region, and PN current begins to flow through
the transition. Thus, in both regions (P and N type) dynamically increase the concentration of
LED spectra 2
minority carriers, which tend to recombine with the major carriers. For the production of LEDs,
semiconductors with a direct bandgap of suitable width (GaAs, Ga1−xAlxAs, GaP, GaN), which
allow radiative recombination in the visible wavelength range, or in the near IR or UV region.
vodivostnixpas
valencnixpas
Pxtyp Nxtyp
Eg
elektrony
diry
The width of the forbidden band is directly related to the photon energy of the emitted light
and to the the voltammetric characteristics of the diode, which brings a correlation between these
the two characteristics of the LED. We now discuss this relationship quantitatively and show how
it can be used to approximate the value of Planck’s constant.
An ideal diode has a voltammetric characteristic, i.e., the current dependence I flowing through
the diode on the voltage U applied to it, given by Shockley’s equation
I(U) = Is exp
eU
kBT
− 1 ; , (1)
where Is is the saturation current, e the elementary charge, T the temperature and kB Boltzmann
constant. The saturation current depends on the width of the forbidden band (a detailed discussion
can be found in e.g., the textbook [1]), which leads to an approximate equation
I(U) ≈ B exp −
Eg − eU
kBT
, (2)
where B is a constant determined by doping and transition geometry. In practice, a series of highly
luminous diodes with approximately the same parameters (e.g., a maximum operating current of
about 20 : mA) for which it is possible to can be expected to have approximately the same value
of the constant B.
Under this assumption, the preceding equation for the prescribed value of the current the
relationship between the voltage across the diode Uf and the width of the forbidden band
eUf |I = Eg + ∆E|I . (3)
The offset ∆E|I is universal for all diodes with the same value of B.
It remains to find the relation between the width of the forbidden band and the photon
wavelength emitted by the LEDs. In the case of a semiconductor with a direct bandgap, the
radiative recombination by electron hopping out of the conduction band to the valence band
while emitting a photon. We ignore the participation of recombination centers, which are essential
for the LED function are practically necessary for the function of the LEDs.
The energy of the emitted photons is then approximately equal to the width of the forbidden
band Eg, which determines the frequency and wavelength of the emitted radiation: hf = hc/λ =
Eg. When combined with the previous relationship between Uf and Eg, we get the relationship
Uf |I =
hc
e
λ−1
+ ∆U|I; . (4)
LED spectra 3
0
5
10
15
20
0 0.5 1 1.5 2 2.5 3 3.5
I(mA)
U (V)
Uf Uf
Obrázek 1: V-A characteristics of the red and blue LEDs with the voltage Uf indicated.
This session offers the possibility of experimentally determining the constant hc/e from the directive
dependence of Uf on λ−1. If we know the values of c and e from other experiments, the
value of Planck’s constant h can be calculated.
For higher currents flowing through a diode, its voltammogram is affected by the diode’s DC
resistance R
I(U) = Is exp
e(U − RI)
kBT
− 1 . (5)
From here, we can derive an approximate relation for the voltammetric characteristic for high
currents
I(U) =
0 forU < Uf
U−Uf
R forU ≥ Uf
, (6)
where the energy eUf can be approximately equal to the width of the forbidden band eUf ≈ Eg.
The energy of the emitted photons is approximately equal to the width of the forbidden band Eg,
which determines the frequency and wavelength of the emitted radiation: hf = hc/λ = Eg
Uf ≈
hc
e
λ−1
, (7)
where we can easily determine the Plack constant.
Úkoly
1. Determine the wavelengths of radiation of each LED in the series using diffraction grating.
Since the emission lines are very broad, we take the wavelength corresponding to the peak
2. Measure the voltammetric characteristics of the LED.
3. From the voltammograms of each LED, subtract Uf and construct a graph of the dependence
of Uf on λ−1, from which we can obtain the value of the constant hc/e.
Wavelength measurement by interference on Newton glasses
Theory
The wavelength of light is measured using the interference effect on a thin air gap between a
plane glass plate and a lens of radius R. When observed in reflected or transmitted light, we
see alternating light and dark circular bands of increasing radius r, called Newton’s rings. At the
LED spectra 4
0.0
0.5
1.0
400 450 500 550 600 650 700
I[a.u.]
λ [nm]
white
0.0
0.5
1.0
I[a.u.]
blue
0.0
0.5
1.0
I[a.u.]
pure green
0.0
0.5
1.0
I[a.u.]
green
0.0
0.5
1.0
I[a.u.]
yellow
0.0
0.5
1.0
I[a.u.]
red
Obrázek 2: Emission spectra of LEDs of different colors available in practice. For LEDs labeled
„white“ and „pure green“ intrinsic PN transition at wavelengths in the blue to UV region and the
resulting color is achieved by phosphorescence.
LED spectra 5
contact of a spherical lens with a plane glass plate, the lens and plate deform slightly, and the
point contact results in a flat circular contact which appears as a dark circular area in reflected
light and as a bright circular area in transmitted light, the so-called Hertzian spot, whose radius
a/2 depends on the attractive force. The situation in the plane of the section is shown in figure
3.
Obrázek 3: Schematic drawing of interfering rays on Newton’s glasses.
We assume that a plane monochromatic wave of wavelength lambda incident perpendicularly
on the plane refractive surface of the lens proceeds to the spherical refractive surface, where it is
partially reflected and reversed in phase. A part of the wave proceeds further through the air gap,
and at the air-plate interface, it is partially reflected without a change of phase and proceeds back
with the same phase. At points on a circle of radius r centered at the point where the lens touches
the plate, this is shown by three rays: the incoming „0“ and two reflected ones, „1“ and „2“. The
outgoing waves interfere, and the resulting intensity depends on the phase difference of the waves
or the path difference of the two reflected beams. According to the figure, the path difference ∆
of the rays „1“ and „2“ with respect to the phase change is equal to
∆ = 2∆r +
λ
2
(8)
The minimum of light intensity occurs on circles with radii rk for which the path difference is
equal to an odd multiple of λ/2, i.e.
∆ = (2k + 1)
λ
2
, resp.∆rk = k
λ
2
, eq2 (9)
where k = 1, 2, .... is the order of the minimum. The size of the air gap ∆r between the lens and
the plate at a distance r from the point of contact S is determined from geometry:
(R − ∆r − ∆a)2
+ r2
= R2
(10)
(R − ∆a)2
+
a
2
2
= R2
(11)
The height of the circular canopy ∆a formed by the deformation of the spherical surface of the
lens is determined from (11) assuming that 2R ≫ ∆a
∆a =
a2
8R
(12)
LED spectra 6
Assuming 2R ≫ ∆r + ∆a we get from the equation (10)
2R(∆r + ∆a) = r2
and using (12) to obtain for the radius of the circle r on which the size of the air gap ∆r
r2
= 2R∆r +
a2
4
(13)
If the size of the air gap satisfies the equation (??), we obtain the equation for the radii of the
circles rk with the minimum light intensity
r2
= λRk +
a2
4
(14)
The equation (14) shows that the square of the radius of the dark ring is a linear function of the
order of the minimum k. Plotting the dependence (14) on the graph gives the equation of the line
(Y = r2
k and X = k)
Y = A + BX (15)
and from the constants A, B, we can determine the wavelength and radius of the Hertzian spot:
λ =
B
R
a
2
=
√
A (16)
If we only want to determine the wavelength, we can determine it from the squared difference
of pairs of radii rk and rn by (14) as follows:
λ =
r2
k − r2
n
R(k − n)
(17)
Measurement progress
To measure the radii of Newton’s rings, we use a microscope with a top illumination and a
measuring eyepiece. Two lenses of the same radius of curvature are mounted in a metal fixture
with a cylindrical hole into which the microscope objective is inserted freely so that the interference
rings can be focused. In this case, the size of the air gap is twice that of the lens-planar plate
arrangement. The equation (14) for the radii of the circles rk with minimum intensity then
transitions to the relation
r2
=
1
2
λRk +
a2
4
(18)
and the relation (16) for determining the wavelength from the directive line to
λ =
2B
R
a
2
=
√
A. (19)
Similarly, the relation (17) for determining the wavelength from the squared difference of pairs of
radii rk and rn goes to:
λ = 2
r2
k − r2
n
R(k − n)
(20)
The equation (18) shows that the square of the radius of the dark ring is a linear function of the
order of the minimum of k. For illumination, we can use a sodium lamp or a luminescent diode.
Since we use the microscope to determine the radii of the Newtonian rings in parts of the eyepiece
scale, we must first determine the microscope magnification Z = y′/y using a test slide, where
y is the distance of the indentations on the test slide in mathrmm and y′ is the distance of the
indentations in parts. We determine the actual size of the Newton rings in µm as rk = r′
k/Z. To
make the measurements, place the fixture on the microscope stage, focus the interference rings,
LED spectra 7
and gently move the fixture or microscope stage to place the rings in the center of the field of
view. The size of the rings is determined by the two extreme positions on the ring, the difference
of which determines the diameter of the ring. Proceed from the smallest to the largest ring as far
as the scale of the eyepiece will allow.
Remark:
To determine the wavelength of light, we need to know the radius of curvature of the refractive
surface of the lens R. If its value is unknown or is not given with sufficient accuracy, but we have
a source of monochromatic radiation of a known wavelength, e.g., sodium lamp with wavelength
λ = 589,30,mathrmnm, we can determine the radius of the lens R from the equations (18) or
(20) by measuring the radii of the Newton rings.
Úkoly
1. Assemble the preparation with the lens and plate, insert it into the microscope objective,
focus the interference fringes, and place the center of the rings in the center of the microscope
field of view. Check the function of the measuring eyepiece and, if necessary, focus the scale
with the eyepiece lens.
2. Illuminate the specimen with the LED (powered via a 4.5 V battery regulation resistor) and
measure the diameters of all rings in the scale range. Determine the wavelength of the LED.
3. From the results of measurements 1 and 2, determine the diameter a of the Hertzian spot.
Bremsstrahlung radiation
Theory
intenzita
vlnová délka
Obrázek 4: Spectrum of X-rays as the sum of the continuous and characteristic spectra.
The incident electron builds up in the anti-cathode material an X-ray spectrum which has
continuous and line component, Fig. 4.
Bremsstrahlung radiation
The braking of the incident electron in the material produces the continuous component of the
X-ray spectrum. anticathodes – bremsstrahlung radiation. The photon energy of the bremsstrahlung
radiation is maximal when the entire kinetic energy of the incident electron is converted to photon
energy. For the minimum wavelength (edge) of a continuous spectrum, the following holds
λmin =
hc
eU
(21)
LED spectra 8
where U is the accelerating voltage in the rtg lamp. The maximum intensity is the bremsstrahlung
radiation for wavelengths from approximately 1.5 λmin to 1.8 λmin (the value also depends on the
type of X-ray tube).
Measurement progress
rtg svazek
analyzátor
2θ(λ)
I(λ)
detektor
Obrázek 5: Scheme of an energy-sensitive measurement with a crystal analyzer.
The spectrum is measured by diffracting the collimated radiation on a single crystal (analyzer);
see Fig. 5. At a given angle θ between the incident radiation and the crystallographic plane, the
a diffraction for the wavelength spectrum for which the Bragg diffraction condition (cubic crystals)
2a sin θ = λ
√
N, N = h2
+ k2
+ l2
. (22)
The dependence of the diffracted intensity on the angle θ measured by rotating the crystal with
can be converted to a wavelength dependence of the intensity. In doing so, it is necessary to
the effect of the superposition of higher orders of diffraction on the analyzer must be taken into
account.
We will therefore measure the spectrum as the intensity dependence of the Bragg angle of the
analyzer, which we then convert to wavelength. We will measure the dependencies for the series
of voltages on the X-ray and the current flowing through it. We analyze the dependencies of the
minimum wavelength and maximum intensity of the continuous spectrum and the maximum of
the characteristic spectrum. We perform measurements with an inserted nickel filter and analyze
its effect on the spectrum.
Experimental equipment
X-ray source with copper, molybdenum, and tungsten anti-cathode, goniometer, LiF analytical
single crystal (lattice constant 4.028 Å, surface plane (001)), X-ray ionization detector (dead time
τ = 90 µs), control computer.
Reference
[1] S.M. Sze: Physics of semiconductor devices, John Wiley and Sons Inc., New York (1981).
[2] R.A. Millikan, Phys. Rev. 7, 355 (1916)