1. 1
Temperature Dependence of
Semiconductor Conductivity
Nikolas Guillemaud
EE 145L
Professor: Nobuhiko P Kobayashi
Submitted: 12/02/14
Abstract:
We investigate a sample of semiconductor crystal by measuring its electrical
resistance  R over a range of temperatures  T from 165.0 360.0K T K  in 10K
steps. Utilizing the temperature dependence of conductivity, exp
2
gE
kT

 
  
 
, to produce
a linear fit to the intrinsic temperature region of the semiconductor provided an
estimation for the bandgap energy 1
6.29 10gE x eV
 . With a percent difference of 4.7% ,
our result is consistent with the expected 1
6.6 10gE x eV
 for germanium.

2. 2
1. Introduction
1.1.Conductivity of a Semiconductor
The conductivity of a semiconductor can be expressed as
( )n pq n p    , (1)
where q is the electric charge, n and p are the electron and hole mobilities,
while n and p are the electron and hole densities. In the case of a doped
semiconductor the majority carrier has a much higher density than the minority
carrier. This allows conductivity to be written as a function of the majority carrier
mobility and density, disregarding the minority contribution. As we see from Eq.
1, conductivity depends upon carrier concentration and mobility; both of these
parameters are temperature dependent. To better illustrate the dependence we
can recast Eq. 1 as
       n pq T n T T p T      . (2)
1.2.Temperature Dependence of Carrier Mobility in a Semiconductor
The carrier mobility is influenced by temperature due to carrier scattering. The
two types of scattering are impurity scattering and lattice scattering. Impurity
scattering is due to ionized crystal defects interacting with the charge carriers.
This is dominant only at low temperatures where the carriers are less energetic
and are moving more slowly. As the temperature decreases the interaction time,
between the ion and carrier, increases and results in a higher probability of
carriers being scattered. Thus, for impurity scattering, decreasing the

3. 3
temperature also decreases carrier mobility. This experiment does not probe
temperatures cold enough for this impurity scattering to be observable.
Therefore, the carrier mobility we are concerned with is influenced primarily by
lattice scattering.
Lattice scattering, we recall, occurs at higher temperatures and will dominate
the carrier mobility in this experiment. This type of scattering is the result of ions
within the crystal gaining energy and vibrating with increasing amplitude as
temperature increases. As temperature increases, this form of scattering will
decrease the carrier mobility. The carrier mobility in this temperature regime
follows
3
2
T

 . (3)
1.3.Temperature Dependence of Carrier Concentration in a Semiconductor
Also dependent on temperature is the charge carrier density. The intrinsic
carrier concentration in is modeled by
 
3
2 3
24
2
2
2
gE
kT
i n p
kT
n m m e
h

 
  
  
  
 
. (4)
In this equation, nm and pm are the effective electron and hole masses
respectively, gE is the bandgap energy, and T the temperature. When
calculating the total carrier density it is important to remember that the entire
crystal itself must be neutrally charged. The number of positive and negative
charges must be equal. Thus, expressing the carrier concentrations as

4. 4
     
 
 
2
i
D A
n T
n T N T N T
n T
 
   and      
 
 
2
i
A D
n T
p T N T N T
p T
 
   (5)
ensures the required charge neutrality for n-type and p-type semiconductors. In
Eq. 5,  DN T
and  AN T
are the concentrations of ionized donor and acceptor
ions respectively. These are especially helpful in the case of compensation
doping.
The carrier concentration of a doped semiconductor will vary according to
temperature in three distinct ways. These ranges of temperature are defined by
the ion saturation temperature ST and the intrinsic temperature iT .
The saturation temperature is the level at which the majority of the donor or
acceptor atoms have been ionized. Colder than ST  ST T , the carrier
concentration increases with T as dopant ionization occurs. Hence, it is known
as the ionization region. Increasing the temperature above ST , in the extrinsic
region  S iT T T  , the carrier concentrations remain constant. This is a result of
nearly all the dopant ionization having occurred at ST , thus in this region, dn N
and the concentration is dopant dependent not temperature dependent. Note that
in this region, if we refer to Eq. 5,  D DN T N
 ,  A AN T N
 , and
 i D An T N N  .
The ionization of dopants requires much less thermal energy than that of a
charge carrier traversing the entire bandgap. Thermal excitation across the
bandgap occurs at iT . At this temperature the bonds of the host element

5. 5
(Germanium) loosen and carriers gain enough energy to cross from the valence
band to the conduction band. This intrinsic region  iT T has its carrier
concentration dominated by those thermally excited across the bandgap
 i dn N . In this region  in n T because in becomes dominant in Eq. 5 at
these temperatures.
1.4.Temperature Dependence of Conductivity for a Semiconductor
Equation 1 modeled conductivity using two temperature dependent functions,
carrier mobility and carrier concentration. For this experiment we are interested in
the extrinsic and intrinsic temperature ranges. At these temperatures it is
expected that lattice vibration will be the dominant mechanism for carrier
scattering in the semiconducting crystal. As such, conductivity can be expressed
as
2
gE
kT
e
 
  
 
 . (6)
2. Procedure
We measured the resistance  R of a germanium sample over a temperature
range  T from 165.0K to 360.0K in 10K steps. This range corresponds to
162.4 189.6o o
F T F   . The sample was approximately 0.5mm thick, 1mm wide,
and 3mm in length. The equipment consisted of a cryostat with liquid argon
cooling and a digital multimeter (DMM) for direct resistance measurements. This
equipment was setup by the lab instructor.

6. 6
The cryostat was first used to cool and then heat the sample as we measured
electrical resistance. The sample was mounted within a vacuum chamber inside
the cryostat and soldered to electrical probes, seen here in Fig. 1.
Figure 1: Semiconductor sample mounted within the cryostat.
The highly pressurized gas undergoes Joule-Thompson expansion through
serpentine microchannels in the sample mount. Once cooled by this refrigerating
thermodynamic process, the sample can then be precisely heated with wires
which are embedded within the stage.
3. Results and Analysis
Using the resistance data  R and the sample dimensions we determined the
resistivity   and the conductivity   . The data can be seen here in Table 1.

7. 7
Table 1: Collected temperature and resistance data. The linearized data
values  ln and  1
T
are also included.
The relationship in Eq. 6 made it useful to find values for the natural log of
conductivity  ln and the inverse of the temperature  1
T
. Taking the natural
log of Eq. 6 linearizes the exponential dependence and generates
1
ln
2
gE
T kT

 
  
 
. (7)
In this linear relationship,  ln is the dependent variable,  1
T
the independent
variable, and thus
2
gE
k
 
 
 
corresponds to the slope. For the linear fit we
T (K) R (kΩ) ρ (Ωm) σ (Ωm)-1
ln(σ) (Ωm)-1
1/T (K-1
)
165.0 87.0 1.45E+01 6.90E-02 -2.67 6.06E-03
180.3 87.2 1.45E+01 6.88E-02 -2.68 5.55E-03
185.0 85.2 1.42E+01 7.04E-02 -2.65 5.41E-03
195.0 81.6 1.36E+01 7.35E-02 -2.61 5.13E-03
205.0 77.5 1.29E+01 7.74E-02 -2.56 4.88E-03
215.0 72.8 1.21E+01 8.24E-02 -2.50 4.65E-03
225.0 67.0 1.12E+01 8.96E-02 -2.41 4.44E-03
235.0 60.2 1.00E+01 9.97E-02 -2.31 4.26E-03
245.0 51.8 8.63E+00 1.16E-01 -2.16 4.08E-03
255.0 41.5 6.92E+00 1.45E-01 -1.93 3.92E-03
265.0 29.8 4.97E+00 2.01E-01 -1.60 3.77E-03
275.0 19.3 3.22E+00 3.11E-01 -1.17 3.64E-03
285.0 12.1 2.02E+00 4.96E-01 -0.70 3.51E-03
295.0 7.33 1.22E+00 8.19E-01 -0.20 3.39E-03
305.0 5.01 8.35E-01 1.20E+00 0.18 3.28E-03
315.0 3.31 5.52E-01 1.81E+00 0.59 3.17E-03
325.0 2.36 3.93E-01 2.54E+00 0.93 3.08E-03
335.0 1.80 3.00E-01 3.33E+00 1.20 2.99E-03
345.0 1.43 2.38E-01 4.20E+00 1.43 2.90E-03
355.0 1.16 1.93E-01 5.17E+00 1.64 2.82E-03
360.0 1.05 1.75E-01 5.71E+00 1.74 2.78E-03

8. 8
included only the data where 265.0 335.0K T K  , this is the intrinsic region.
These specific data points were chosen for the fit so that only thermal carrier
excitation is responsible for the change in conductivity. Looking at the resulting fit
3652 12.15y x   we determined 19 1
1.01 10 6.29 10gE x J x eV 
  .
Figure 2: Natural logarithm of sample conductivity as a function of inverse
temperature. The full data set is plotted in blue while the intrinsic region is
overlaid with red data points and the extrinsic region with green points.
4. Conclusion
The sample of germanium had an electrical conductivity that responded to
temperature variation as expected. It was apparent that as temperature increased,
beyond the extrinsic region, the carrier concentration became temperature dependent
and dominated the conductivity of the sample.
y = -3652x + 12.15
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0060
ln(σ)
1/T
ln(σ) vs. 1/T
Data
Intrinsic
Extrinsic
Linear (Intrinsic)

9. 9
The plot in Fig. 2 illustrates the transition from the extrinsic (green) to the intrinsic
(red) temperature regions. The shape hints to the drastic shift from a temperature
independent carrier concentration in the extrinsic region, to a temperature dependent
concentration in the intrinsic region. The cryostat did not reach cold enough
temperatures for the ionization region to be observed. For the intrinsic region, it was
decided that the last three data should be left off of the linear fit. This is due to what
appeared to be the effects of lattice scattering decreasing carrier mobility and therefore
diminishing conductivity at the highest temperatures. By omitting these data we were
able to fit only the data for which the increase of intrinsic carrier concentration
dominated the increase in conductivity.
The linear fit to the intrinsic temperature region provided an estimation for the
bandgap energy 1
6.29 10gE x eV
 . Using a percent difference calculation
theoretical - experimental
% difference = 100
theoretical
x
we found that our experimentally determined bandgap energy was within 4.7% of the
referenced value for germanium 1
6.6 10gE x eV
 .(2)
This suggests that our empirically
derived value was subject to negligible experimental error.
Further analysis could be done with more information on the sample; knowing the
type of doping (p, n, or compensated) would assist with the calculation of dopant
concentration. It would also be helpful to characterize the ionization temperature range
and calculate the dopant ionization energy. Additionally, values for carrier mobility and
the effective masses of holes and electrons, for this specific sample, are needed for the
determination of dopant type and concentration.

10. 10
References
1. “Lab 5: Temperature Dependence of Semiconductor Conductivity”, N.
Kobayashi, 2014
UCSC, Baskin School of Engineering, EE Department
EE-145L: Properties of Materials Laboratory
2. “Electronic Materials and Devices” S.O. Kasap, 3rd
edition, 2006