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INSTITUTE OF PHYSICS PUBLISHING
Eur. J. Phys. 28 (2007) 85–91
EUROPEAN JOURNAL OF PHYSICS
doi:10.1088/0143-0807/28/1/009
Estimating p–n diode bulk parameters,
bandgap energy and absolute zero by a
simple experiment
R O Ocaya and F B Dejene
Department of Physics, University of the Free State, Private Bag X13, Phuthaditjhaba 9866,
South Africa
E-mail: [email protected]
Received 19 September 2006, in final form 31 October 2006
Published 27 November 2006
Online at stacks.iop.org/EJP/28/85
Abstract
This paper presents a straightforward but interesting experimental method
for p–n diode characterization. The method differs substantially from many
approaches in diode characterization by offering much tighter control over the
temperature and current variables. The method allows the determination of
important diode constants such as temperature coefficient of voltage, bandgap
and indicates the existence of a unique temperature, hypothesized to be absolute
zero, at which the voltage–temperature characteristics of the diode converge
under different constant diode currents.
1. Introduction
The current I flowing through a p–n diode at applied voltage V and junction temperature T is
given by the well-known Shockley equation:
I = I0 [exp(qV /nkT ) − 1]
(1)
where I0 is the reverse saturation current that is known to depend on temperature, k is
Boltzmann’s constant, q is the electronic charge, n is a diode-dependent ideality factor
(typically 1.9 ± 0.02 for silicon diodes) and T is the absolute temperature. More appropriately
equation (1) has the form
Eg
[exp(qV /nkT ) − 1]
(2)
I = In exp −
kT
where In is a constant that depends on the geometry and doping densities of the material
of the diode. Eg is the bandgap of the semiconductor [1–3]. The form of In is described
in section 2. When the p–n diode is introduced to students the reverse saturation current
is not normally written as a function of temperature. This rather innocent simplification
c 2007 IOP Publishing Ltd Printed in the UK
0143-0807/07/010085+07$30.00 85
86
R O Ocaya and F B Dejene
leads to an understatement of a useful property of the diode: its voltage–temperature response
under constant current. As this paper shall show, by maintaining diode current constant and
varying diode temperature, it is possible to obtain a set of plots of V versus T for different
constant currents that indicate a temperature convergence near or at absolute zero. This
approach differs substantially from most taken in the evaluation of diodes by undergraduates
that do little beyond attaining the ‘accepted shape’ of the I–V characteristic with questionable
temperature constancy. Furthermore, this experiment allows the deduction of the bandgap of
the semiconductor as an added bonus.
1.1. A note on bandgap
The electrical conductivity of all materials, whether insulators, semiconductors or conductors,
can be explained in terms of the difference in energy between the valence band and the
conduction band; this difference or bandgap is large in insulators and relatively small in
semiconductors. In good conductors there is an overlap of the valence and conduction bands.
The variation of the bandgap of silicon with temperature has been studied widely, resulting in
a number of experiment-based approximations. According to [2]
Eg (T ) = Eg (0) −
αT 2
β +T
(3)
where Eg (0) is the bandgap at 0 K. The fitting parameters are α = 0.473 meV K−1 and
β = 636 K. Another, earlier approximation of the variation of bandgap with temperature is
given in [4]. The literature states that Eg (0) = 1.17 eV. It may be argued that the 0 K bandgap
is itself an estimate because although 0 K has been closely approached, it has not yet been
reached. It is also known that doping density and impurities affect bandgap. For example,
[5] presents a detailed theoretical study of many-body effects in heavily doped silicon and
gallium arsenide at 0 K. They find that heavy doping generally leads to bandgap narrowing.
Similarly, [6] quantifies the bandgap narrowing as approximately 40 meV for a one order of
magnitude change in doping concentration. The bandgaps were determined by both electrical
measurements and spectroscopy.
1.2. Experimental details
The gist of the experiment is the variation of diode temperature while measuring diode voltage
under constant current. The method used by the authors is the controlled heating of an
electrically insulated p–n diode in a water bath. The diode temperature is monitored on an
appropriately mounted thermometer as the water is heated from near zero to boiling point. The
experimental set-up with the 1N4148 diode is described in detail in [7]. The availability of a
current source capable of outputting any constant current from 0 to 1500 µA may be the only
hurdle, but one that can nonetheless be overcome. The bias current should be at maximum
1.5 mA to prevent self-heating as well as to keep the diode within its exponential region of the
I–V characteristic. Suitable current sources are described in [7, 8].
1.3. The theory: V –T response, I constant
After a rather messy differentiation of equation (2) with respect to T under the assumption that
I is constant, one gets
dV
qV
q
qV
T
− 1 = 0.
(4)
exp
− V + Eg exp
n
nkT
dT
nkT
Absolute zero and bandgap by a p–n diode experiment
87
Figure 1. Temperature dependence of I0 under the constant diode current for the experimental
current extremes. The plots show a Pearson’s coefficient of linearity, R 2 ≈ 1.
Subject to the approximation
qV
qV
− 1 ≈ exp
,
exp
nkT
nkT
equation (4) reduces to
V =T
dV n
+ Eg
dT q
≡ mT + C,
where
m=
dV
dT
and
C=
n
Eg .
q
(5)
By substituting equation (5) into Shockley’s equation and solving for m it can be shown that
dV
≈ η ln I + ρ
dT
where η = nk/q and ρ = nk ln In /q are constants. This shows that m depends only on
the forward diode current. Therefore if the diode current is constant, then the temperature
coefficient is essentially constant. This result is indicated by experiment [7].
m=
2. Results and analysis
The experiment was carried out for diode currents of 100, 400, 800 and 1500 µA. For each
current, the diode temperature was varied from 0 to 95 ◦ C (boiling point in the locality). The
diode voltage was recorded at suitable intervals.
2.1. Determination of the reverse saturation current
At a given diode current, I0 can be calculated at the point (T , V ) using equation (1). Thus for
each constant diode current a new data table can be built up for the points (T , I0 ). Figure 1
shows the combined plot of the natural logarithm of I0 versus the reciprocal of absolute T for
88
R O Ocaya and F B Dejene
the four currents mentioned above. Figure 1 implies that
ln I0 (T ) =
a
+ b.
T
(6)
A linear fit of figure 1 gives a = 7506 ± 77 A K, b = 5.69 ± 0.24 A. It is easy to show
that equation (2) can then be written as
qV
7506
exp
−1 .
(7)
I (V , T ) = 295.9 exp −
T
nkT
A quick check shows that equation (7) reproduces all the experimentally obtained I, V , T
data with n = 1.90. Note that in a standard form, written in terms of current densities [9], the
total current density is
Eg
qV
exp
−1
(8)
JTOT = J∞ exp −
kT
nkT
where J∞ is temperature independent and is given by
J∞ =
2qD
Nc Nv
×
l
Nh
(9)
where D is the diffusion constant, l is the length, and Nc , Nv and Nh are carrier concentrations.
2.2. Calculating the bandgap
Comparing equations (7) and (8) implies that
Eg
7506
=
.
kT
T
(10)
This gives Eg = 1.04 ± 0.02 eV for the diode, placing the value within the range of silicon
(1.03–1.17) eV for 0 < T 600 K [2]. In addition, In = 295.9 = J∞ γ , where γ is a
determinable constant that depends on the cross-sectional area of the diode junction.
2.3. Predicted responses
The diode voltage at a given forward current and temperature can be calculated by
rearranging the terms of equation (7) for V . Using the Microsoft Excel spreadsheet program,
equation (7) was solved for varying temperature and different constant currents. This gives rise
to figure 2, which predicts the voltage response of the diode for some temperatures between
11 K and 600 K. It is interesting to note that at or near absolute zero the diode goes from being
almost ideal with a forward drop of about 1.23 V to being Ohmic (purely resistive), where
the forward current is directly proportional to the diode voltage. There are two models that
explain the transition to Ohmic behaviour in an extrinsic semiconductor. One model is based
on the classical approach of the Drudes law [10]; the other model is quantum mechanical, and
assumes a free-electron Fermi gas [5].
Figure 3 shows the predicted V –T response for a current of 100 µA for temperatures
between 11 K and 600 K in steps of 20 K. The spreadsheet program was not able to calculate
the diode voltage for lower temperatures. Note that above about 450 K the response becomes
nonlinear. In the linear portion of the graph the predicted gradient is −2.41 mV K−1 , a value
that is found to agree with the literature value [7, 11].
Absolute zero and bandgap by a p–n diode experiment
89
Figure 2. Predicted behaviour of the 1N4148 diode at different temperatures.
Figure 3. V –T behaviour of the 1N4148 diode for a 100 µA forward current showing a gradient
of −2.41 mV K−1 .
2.4. Convergence of the V –T characteristic
It is hypothesized in this paper that there is a temperature at which the V –T characteristic
converges. In order to explore the idea further the different V –T behaviours at different
constant currents were plotted on the same graph and then extrapolated backwards using
Excel as shown in figure 4. By using the 100, 400 and 800 µA curves, it can be calculated
under the assumptions of continued linearity to 0 K that the estimate of absolute zero is 0 K.
It is known that near 0 K practical cryogenic diode sensors
exhibit a sudden rise in the magnitude of the gradient of the V–T characteristic. It is thought
that this upturn arises from carrier freeze-out, the influence of impurity conduction and increase
in Ohmic behaviour [12]. However, the fact that the extrapolated curves themselves appear to
converge under the assumptions of continued constancy of gradient is in itself interesting.
2.4.1. Silicon diodes below 10 K.
Equation (3) is a commonly used approximation for
the variation of bandgap with temperature. Figure 5 shows the effect of the temperatureinduced bandgap variation on the V –T characteristic at the 100 µA diode current. It is clear
from the plot that even though the approximation represented by equation (3) describes a
2.4.2. The effect of bandgap variation.
90
R O Ocaya and F B Dejene
Figure 4. Extrapolated plots of V –T , I = constant for (a) 11–600 K, (b) extended plot for 0–10 K
showing convergence at 0 K to VD = 1.228 V.
Figure 5. Graph showing the effect on the V –T characteristic of a temperature-induced variation
of bandgap, Eg (T ), at the 100 µA bias current.
parabolic variation of Eg with T, the overall effect on the plot is simply an increase in the
temperature coefficient of voltage (steeper graph). The linearity is maintained.
3. Discussion and conclusions
Undergraduate experiments that characterize the I–V response of the p–n diode tend to
understate the role that temperature plays in the reverse current of the diode. With current,
voltage and temperature tightly controlled, this paper shows that some interesting temperaturerelated facts emerge about the simple p–n diode. By writing the reverse saturation current I0
as a function of temperature the paper highlights a more experimentally interesting form of
Shockley’s equation. The equation is found to correctly predict the temperature coefficient of
forward voltage at different currents and reproduces the familiar I–V behaviour at different
temperatures. In addition, it suggests the convergence of the V –T characteristic at 0 K. The
equations also show that near absolute zero the diode exhibits near zero resistance and near
ideal behaviour with a mean diode voltage of 1.23 ± 0.01 V; the experimentally determined
Absolute zero and bandgap by a p–n diode experiment
91
bandgap energy is Eg = 1.04 ± 0.02 eV. At high temperatures, Ohmic behaviour is predicted
as expected for diodes as shown in figure 2 for some temperatures between 11 K and 600 K.
The paper rationalizes the constants in the equations in terms of the bulk properties of the
material of the diode and bandgap energy. Finally, it may be one of very few experiments to
indicate absolute zero by convergence implications without the use of gas laws. In this regard
it makes a significant contribution particularly for undergraduate teaching.
References
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