Band Gap of Germanium∗
Joel Ong†
(Dated: April 4, 2014)
I.
INTRODUCTION
Germanium is an intrinsic semiconductor. That is
to say, its Fermi energy µ lies between its valence and
conduction bands, and the band gap between its valence and conduction bands is small enough that conduction is possible at sufficiently high temperatures.
In the absence of dopants and other impurities, its
Fermi energy lies exactly in between the valence and
conduction bands; consequently, the number density
n of electrons and holes should be equal.
Electrons and holes are described by Fermi-Dirac
statistics. The law of mass action then demands that
at sufficiently high temperatures,
]
[
√
∆E
,
(1)
ne nh = nc ∝ exp −
2kT
where nc is the total carrier concentration, k is Boltzmann’s constant and ∆E is the band gap between the
conduction and valence bands.
The conductivity of a material can be roughly described using the Drude model of electrical conductivity; in sum, the current density passing through a
material is proportional to the carrier concentration
and applied electric field. Therefore, we must have
that the conductivity σ is proportional
[ ∆E ] to the carrier
concentration, and so σ ∝ exp − 2kT
.
A.
Objectives
In this experiment, we examine the temperature dependence of the conductivity of a germanium sample,
and use it to estimate the band gap of germanium.
II.
EXPERIMENTAL PROCEDURE
We were provided with a sample of germanium of
dimensions 20 × 10 × 1 mm3 , which was attached to a
heating unit (Figure 1). The circuit onto which these
were mounted was capable of passing a constant current through the germanium sample while measuring
both the temperature T and the potential difference
V across the sample simultaneously.
By maintaining a constant current I through the
germanium sample, our measurements of the potential
V across the sample should be inversely proportional
to the conductivity σ. We recorded various values of
V for constant I, at different values of T . This was
achieved by letting the heating coil heat up the germanium sample while the computer was left running
∗
†
Lab Summary for 8 Credit Hours
Matric Number: A0098750U; Lab Partner: Kho Zhe Wei
to collect data. We collected data in this manner over
the range of temperatures 25◦ C to 131◦ C.
FIG. 1: Experimental equipment: germanium
sample, heating unit and sensors on a single chip
III.
RESULTS AND ANALYSIS
We carried out the above procedure for I =
5.0(5) mA. Given the above relationship between the
conductivity, band gap
and temperature, we
]
[ ∆Eenergy
. Therefore, we should in
then have V ∝ exp 2kT
principle be able to measure the band gap by measuring V at different temperatures while keeping the current I constant, and fitting the resulting curve against
the two-parameter model
[ ]
B
V = A exp
.
(2)
T
Such a model should yield ∆E = 2kB.
4
V/V
3
2
1
T/K
320
340
360
380
400
420
FIG. 2: Fitted curve with experimental data for
I = 5.0(5) mA (with error bars for instrument
uncertainty)
We performed this curve-fitting procedure on the
data collected for this value of I (Figure 2 shows a plot
of the fitted curve). The fitting procedure returned
2
parameter estimates of A = 4.5(9) × 10−6 V and B =
4120(6) K−1 to 95% confidence, with a correlation
coefficient of r2 = 0.9992. This gave us an estimate
of the band gap energy ∆E = 0.71(1) eV (to within a
95% confidence interval).
However, we observe that the computer obtained
samples of V against T at irregular intervals of temperature. Upon closer inspection, we found that values of V and T were recorded at regular intervals of
time, at ∆t = 1 s. Therefore, we can improve the
accuracy of our regression estimate by taking samples while the germanium sample was cooling, so as
to obtain more data points at smaller temperature intervals.
We recorded the same data with the germanium
sample cooling down instead of heating up, and performed the same curve-fitting procedure. The curve
fitting procedure returned parameter estimates of A =
2.25(4) × 10−6 V and B = 4321(6) K−1 to 95% confidence, with a correlation coefficient of r2 = 0.99997,
which is much higher than that obtained from taking
readings while heating the germanium sample.
Using these parameter estimates we obtain an estimate for the band gap energy as ∆E = 0.745(1) eV,
to within a 95% confidence interval. We see that the
95% confidence interval returned from performing the
fitting procedure on the cooling curve is an order of
magnitude smaller than that returned by the heating
curve; moreover, we see that the plot of our experimental data is qualitatively smoother than the heating
curve (Figure 3).
V/V
TABLE I: Parameter estimates with 95% confidence
intervals for curves obtained from different values of
I, using data collected while the sample was either
being heated up or cooled down
I/mA
Heating
3.0(5)
5.0(5)
9.0(5)
Cooling
3.0(5)
5.0(5)
9.0(5)
B/K−1 ∆E/eV
A/V
0.0000025(5) 4150(6) 0.72(1)
0.0000045(9) 4120(6) 0.71(1)
0.000008(1) 4150(6) 0.71(1)
r2
0.999172
0.999199
0.999231
0.00000126(3) 4354(7) 0.750(1) 0.999966
0.00000225(4) 4321(6) 0.745(1) 0.999974
0.00000400(7) 4329(5) 0.7458(9) 0.999978
the sample was cooling (on average, 0.747(1) eV), as
compared to those estimates calculated from data collected while the sample was being heated up (on average, 0.71(1) eV).
IV.
A.
DISCUSSION
Negative I
In principle, we should have obtained the same results as above with the current flowing in the opposite direction (i.e. for negative I). However, when we
performed the experiment in such a configuration, we
observe a discontinuity at T = 310 K (Figure 4).
V/V
4
0.0
- 0.5
3
- 1.0
2
- 1.5
- 2.0
1
T/K
320
340
360
380
400
FIG. 3: Fitted curve for cooling germanium sample
with experimental data for I = 5.0(5) mA (with
error bars for instrument uncertainty)
However, we see that this parameter estimate differs
significantly from the one we obtained when heating
the germanium sample.
To verify that this is a real observation and not a
consequence of an error in our analysis, we decided
to repeat the procedure outlined above for I = 3 mA
and I = 9 mA; for each value of I we collected data
both while the germanium sample was heated up, and
while it was cooling down. We then performed the
same fitting procedure as outlined above on all the
data sets so obtained. We show our results below in
Table I.
We see that the estimates for the band gap energy
∆E are persistently higher for all data collected while
- 2.5
T/K
320
340
360
380
400
420
440
FIG. 4: Data collected while the germanium sample
was cooling, with fitted curve, for I = −5.0(5) mA
(with error bars for instrument uncertainty)
This discontinuity appears irrespective of whether
the data was collected while the germanium sample
was being heated up or cooled down. However, truncating data collected for temperatures below this discontinuity and then performing the same regression
procedure as above returns the estimates for the band
gap, for both the heating and cooling curves, that are
consistent with the ones in Table I; this indicates that
data collected at temperatures below this discontinuity are inaccurate when I is negative.
This fact leads us to hypothesise that this observed
discontinuity is an artifact of the instrumentation used
to measure the voltage and current across the ger-
3
manium sample. The fact that the change is induced abruptly at a particular temperature, as well as
the predictable behaviour of the discontinuity (nearly
doubling the measured value of V ) indicates that a
likely cause is that the combination of negative current and temperature change induces a change in the
gain of an amplifier circuit used as part of the ammeter. However, given that the internal workings of
the equipment provided was essentially a closed black
box, there was little that we could do to verify this
conjecture or investigate this phenomenon.
B.
Discrepancy with Literature Value
All of our estimates differ from the literature value
of 0.67 eV [1], with percentage discrepancies on the
order of 6% for estimates obtained from the heating
curves, and 12% for estimates collected from the cooling curves.
However, we find also that the band gap energy
varies with temperature, even in the literature, instead of being a constant as we have earlier supposed.
In particular, at 0 K, the band gap of germanium is
0.74 eV, which is quite close to our measured values.
Phenomenologically, this temperature dependence can
be modelled as
∆E(T ) = ∆E(0) −
αT 2
.
T −β
(3)
Moreover, the literature values for these fitting parameters α and β are small enough that within the
range of temperatures that we have examined, the
variation of the band gap is approximately linear with
temperature [2]. Consequently, the fitting procedure
outlined above returns us not the actual band gap
energy, but rather the band gap energy evaluated
at 0[ K, since
the additional linear term falls out as
]
αT
≈ constant. Compared to that value, our
exp 2kT
parameter estimates exhibit far smaller discrepancies
than when compared to the literature value for the
room-temperature band gap.
Aside from this, there must of course exist impurities within the germanium sample with which we
were provided, whose electrons must also contribute to
the conductance of the material. We therefore expect
some degree of deviation from the literature value, alternative model notwithstanding. However, we were
not able to accurately characterise the purity of the
provided germanium sample; consequently, we were
unable to draw any quantitative conclusions as to the
[1] C. Kittel and P. McEuen, Introduction to solid state
physics, Vol. 8 (Wiley New York, 1986).
[2] P. Lautenschlager, P. Allen, and M. Cardona, Physical
Review B 31, 2163 (1985).
significance of any contribution from impurity carriers.
C.
Discrepancy between heating and cooling
curves
We see that we have larger estimates for the band
gap energy when using data collected while the sample
was being cooled down compared to data collected
when it was being heated up. We can attribute this
to the fact that all of the physical description outlined
above (including Equation 1) applies only to systems
in thermal equilibrium.
Manifestly, when the germanium sample is being
heated up rapidly by an external heating unit, it is
not in thermal equilibrium with itself, since the side
facing the heating unit is hotter than the side facing away (given that the heating unit was at a much
higher temperature than the germanium sample, and
we turned the heating unit off before equilibrium with
it could be achieved in order to prevent heat damage
to the sample).
Conversely, for the data collected when the germanium sample was cooling down, the entire process of
data collection took much, much longer, on the order
of 12 minutes. During this time, the germanium sample was not in thermal equilibrium with the ambient
air (as it was being cooled down). However, the relaxation time for the sample to come into equilibrium
with itself is much smaller than the time taken for it to
equilibrate to room temperature. Therefore, on sufficiently coarse time scales, we would expect that the
cooling process is quasistatic, in that at each point in
time the germanium sample is approximately in equilibrium with itself.
Consequently, the analysis above should in principle
hold more accurately for when the germanium sampled is being slowly cooled down that when it is being
heated up; this should mostly account for the discrepancy in the results returned by the two different experimental methodologies. Therefore, we should privilege
the values returned by cooling the germanium sample
as being the more accurate of the two methodologies;
this yields an average estimate for the band gap energy of 0.747(1) eV.
V.
CONCLUSION
Using the temperature dependence of the conductance of a germanium sample with the current flowing
through it held constant, we have determined that the
band gap of germanium has a value of 0.747(1) eV.