Vol.:(0123456789)1 3
Applied Physics A (2018) 124:381
https://doi.org/10.1007/s00339-018-1803-2
Electrical conductivity of high-purity germanium crystals at low temperature
Gang Yang1 · Kyler Kooi1 · Guojian Wang1 · Hao Mei1 · Yangyang Li1 · Dongming Mei1
Received: 3 September 2017 / Accepted: 18 April 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract
The temperature dependence of electrical conductivity of single-crystal and polycrystalline high-purity germanium (HPGe)
samples has been investigated in the temperature range from 7 to 100 K. The conductivity versus inverse of temperature
curves for three single-crystal samples consist of two distinct temperature ranges: a high-temperature range where the
conductivity increases to a maximum with decreasing temperature, and a low-temperature range where the conductivity
continues decreasing slowly with decreasing temperature. In contrast, the conductivity versus inverse of temperature curves
for three polycrystalline samples, in addition to a high- and a low-temperature range where a similar conductive behavior is
shown, have a medium-temperature range where the conductivity decreases dramatically with decreasing temperature. The
turning point temperature ( Tm
) which corresponds to the maximum values of the conductivity on the conductivity versus
inverse of temperature curves are higher for the polycrystalline samples than for the single-crystal samples. Additionally, the
net carrier concentrations of all samples have been calculated based on measured conductivity in the whole measurement
temperature range. The calculated results show that the ionized carrier concentration increases with increasing temperature
due to thermal excitation, but it reaches saturation around 40 K for the single-crystal samples and 70 K for the polycrystal-
line samples. All these differences between the single-crystal samples and the polycrystalline samples could be attributed
to trapping and scattering effects of the grain boundaries on the charge carriers. The relevant physical models have been
proposed to explain these differences in the conductive behaviors between two kinds of samples.
1 Introduction
Germanium, an important semiconductor material, has been
widely used in the optoelectronic industry. Single-crystal-
line germanium is used as window materials for infrared
optics and substrates for optoelectronic devices [1, 2]. HPGe
with an electrically active impurity concentration of only
109–1010 cm−3 can be used in the construction of large-vol-
ume, thick radiation detectors [2–5]. Since the conductiv-
ity behavior of HPGe crystals at low temperature plays an
important role in determining the purity level of such materi-
als used to fabricate the radiation detectors in the searches
for dark matter and measuring neutrinos properties, the
low-temperature measurement of germanium crystals was
carried out extensively in the 1950–1970s [6–8]. Hung and
Gliessman [6] carried out an experimental investigation of
the Hall coefficient and resistivity of HPGe crystals with an
impurity level of ~ 1014 cm−3 and doped germanium n-type
crystals at low temperatures. They found three anomalies in
both the Hall coefficient and resistivity versus inverse tem-
perature curves at low temperatures: (1) Hall curves went
through a maximum as the temperature decreased, (2) resis-
tivity approached a saturation value, and (3) the Hall curve
of low-resistivity samples became flat at low temperatures.
They proposed a modified band structure model to explain
these anomalies.
Frizsche [8] observed similar low-temperature anomalies
as reported in literature [8]: both the Hall coefficient and
electrical resistivity have a step maximum and a change in
slope of the log resistivity versus inverse temperature curve
at temperatures between 1.5 and 300 K. The lattice and
ionized impurity scatterings are regarded as the two main
mechanisms affecting the mobility of semiconductors. At
high temperatures, the lattice scattering obeys a T−3∕2 law,
while the ionized impurity scattering obeys T3∕2 law. Dunlap
[9] reported the resistivity, Hall coefficient, mobility, and
magneto-resistance of high-resistivity p-type germanium
* Gang Yang
[email protected]
1 Department of Physics, The University of South Dakota,
Vermillion, SD 57069, USA
G. Yang et al.
1 3
381 Page 2 of 8
single crystals. Dunlap found that the mobility obeys a T−2.0
law in the temperature range of 78–400 K. Prince [10] found
that the mobility of high-resistivity n-type germanium obeys
a 9.1 × 108 T−2.3 law, whereas high-resistivity p-type germa-
nium obeys 3.5 × 107 T−1.6 law. Wichner et al. [11] reported
Hall measurements on 58 samples at low temperature (to
~ 7 K), and found that HPGe was well compensated, indi-
cating that the individual acceptor or donor concentrations
were 3–10 times the net concentration at 77 K. Addition-
ally, Haller et al. [3–5] have published a series of articles on
the low-temperature electrical properties of detector-grade
HPGe, and have established the profile of impurity distri-
bution in HPGe crystals. In recent years, Yang et al. have
also reported the low-temperature electrical properties of
detector-grade HPGe zone-refined and single crystals using
the van der Pauw–Hall measurement technique [12–14].
In our lab, we grow high-purity germanium (HPGe) crys-
tals and use them to fabricate germanium detectors. Because
HPGe detectors usually operate at 77 K or below, the low-
temperature measurement of electrical properties of HPGe
crystals are very important and inevitable. While a lot of
research has been carried out on low-temperature electrical
properties of single germanium crystals, most of them only
include germanium crystals with impurity levels of more
than ~ 1013 cm−3. It is well known that the grain bound-
ary should have scattering and trapping effects on charge
carriers in polycrystalline semiconductors, which have
been identified in polycrystalline silicon by several authors
[15–18]. However, very little investigation of influences of
grain boundary on conductivity behaviors involves poly-
crystalline germanium crystals. As a result, in the present
work, we have investigated the electrical behaviors of both
polycrystalline and single HPGe crystals at temperatures
from 7 to 100 K. The differences in the low-temperature
electrical behavior between the two kinds of crystals have
been emphasized.
2 Experimental details
Low-temperature measurements were made on two kinds
of samples: polycrystalline samples cut from zone-refined
HPGe ingots and single-crystalline samples cut from HPGE
single crystals grown by Czochralski (Cz) pulling. Three
single-crystalline samples and three polycrystalline samples
were denoted as SCGe-1, SCGe-2 and SCGe-3, and ZRGe-
1, ZRGe-2 and ZRGe-3, respectively. The samples were cut
into 1.5 cm2 sheet samples with thicknesses of ~ 1 mm. After
cutting and squaring, the samples were then polished, rinsed
with deionized water, and etched to reduce surface effects.
The etching solution consisted of nitric acid and hydro-
fluoric acid with a mixture ratio of 3:1. After etching was
completed, samples were rinsed again with deionized water
and dried with nitrogen gas. Preliminary measurements
were conducted at 77 K using van der Pauw–Hall measure-
ment method. Two gold wires with a diameter of 0.5 mm
were soldered to both side surfaces of a sample using spark
welding method to form an Au/Ge eutectic alloy at about
360 °C, which is a stable contact at low temperature, and
used as current and potential. The low-temperature measure-
ment system consisted of an Optistat CFV2 cryostat (Oxford
Instrument, UK), FT/IR-6000 (Jasco Inc., Japan) with liq-
uid helium cylinder, Temperature Controller with Heater,
Keithley 2400-C Source Meter (Tektronix Company, USA),
and Vacuum Pump (Pfeiffer Vacuum GmbH, Germany), as
shown in Fig. 1. Because the temperature below 7K was not
stable, the dependence of conductivity on temperature of all
samples was measured in the temperature range of 7–100 K.
To observe the difference of microstructures between poly-
crystalline samples and single-crystalline samples, after the
low-temperature measurement was finished, two representa-
tive samples were further polished using alumina powders
of decreasing particle sizes as polishing media to remove
scratches and create a mirror surface, then etched in a solu-
tion of nitric acid, hydrofluoric acid, acetic acid, and iodine.
The observation of microstructures was carried out under
optical microscopy (Olympus Inc, Japan).
3 Results
3.1 Temperature dependence of electrical conductivity
The influences of temperature on electrical conductivity for
three single-crystal samples and three polycrystalline sam-
ples are shown in Fig. 2a, b, respectively. The conductivity
versus inverse of temperature curves of the three single-
crystal samples consist of two distinct temperature ranges:
the high-temperature range and low-temperature range. In
the high-temperature range, as the temperature decreases the
conductivity of each sample increases until it reaches a turn-
ing point ( Tm
) where the conductivity is the maximum. After
Tm
, the samples enter the low-temperature range where the
conductivity steadily decreases with decreasing temperature.
The conductivity versus inverse of temperature curves
for the three polycrystalline samples shown in Fig. 2b are
divided into three distinct temperature ranges. Similar to
the single-crystal samples, the polycrystalline samples have
high- and low-temperature ranges, but in between these two
ranges is a distinct medium-temperature range. The high-
temperature range again ends at Tm
, but as the temperature
continues to decrease into the medium-temperature range
the conductivity decreases dramatically until the low-tem-
perature range, where the conductivity continues to steadily
Electrical conductivity of high-purity germanium crystals at low temperature
1 3
Page 3 of 8 381
decrease with decreasing temperature, but at a slower rate.
The reasons for these temperature ranges can be understood
using the following analysis.
As we know, the conductivity of semiconductors depends
on the concentration and mobility of charge carriers, which
can be interpreted by Eq. (1) [5]:
where is mobility, q is charge of electron ( 1.6 × 10−19
coulombs), and ||NA− N
D|| represents the carrier concentra-
tion. Both mobility and carrier concentration are dependent
on temperature. According to Matthiessen’s rule [19], the
mobility of free charge carriers can be written as:
where 1∕L , 1∕
I , and 1∕
D correspond to the scattering of
charge carriers due to lattice vibration, impurity ions and
defects which are dislocations in the single-crystal sam-
ples or grain boundaries in the polycrystalline samples,
respectively.
(1) = q||NA − ND||,
(2)1
=
1
L
+1
I
+1
D
,
Figure 3 is the enlarged section of Fig. 2 to show the con-
ductivity of all investigated samples in the high-temperature
range, indicating an obvious increase in the conductivity
with decreasing temperature for both single-crystal samples
and polycrystalline samples. First, in the high-temperature
range, lattice scattering is a dominant scattering, so the
mobility is limited by lattice scattering due to thermal vibra-
tion of the lattice as presented by:
Thus, in the high-temperature range, L increases with
decreasing temperature. Therefore, the total mobility in
Eq. (2) increases as the temperature decreases. Second, the
charge carrier concentration in the high-temperature range
is not expected to decrease significantly as the temperature
decreases. Consequently, the conductivity increases with
decreasing temperature in the high-temperature range. Simi-
lar results have been reported by Hung and Gliessman [8]
and Frizsche [10].
In the low-temperature range, the lattice scattering in the
single-crystal samples and grain boundary scattering in the
(3)L∞T
−3∕2.
Fig. 1 Schematic drawing
diagram of low-temperature
measurement setup
G. Yang et al.
1 3
381 Page 4 of 8
polycrystalline samples become very weak. Therefore, the
ion impurity scattering becomes the dominant influence on
mobility in both single-crystal samples and polycrystalline
samples. This is because the electrons move more slowly in
the low-temperature range, so the decreased momentum of
the electrons allows them to be deflected by Coulomb force
from the impurity ions. This impact on mobility is repre-
sented by the following relationship:
Since I decreases with decreasing temperature, the total
mobility in Eq. (2) decreases. Meanwhile, the carrier con-
centration also decreases gradually with decreasing tempera-
ture. Therefore, both single-crystal samples and polycrystal-
line samples exhibit a gradual decrease in conductivity with
further decreasing temperature.
(4)I∝ T
3∕2.
The unique conductive behavior of the three polycrys-
talline samples in the medium-temperature range is attrib-
uted to trapping and scattering effects of grain boundaries
on charge carriers. In addition to the scattering by ion
impurities, the grain boundary is a dominant scattering
source that significantly lowers mobility. Meanwhile, there
is also a significant reduction of charge carrier concen-
tration with decreasing temperature. Therefore, the con-
ductivity of the polycrystalline samples reduces dramati-
cally. The medium-temperature range ends when the grain
boundary scattering is inhibited by the very low-temper-
ature range and only the ion impurity scattering remains.
Both will be further analyzed in Sect. 4.
Another interesting phenomenon highlighted by Fig. 3
is the difference in Tm
temperature between the polycrys-
talline and single-crystal samples. Tm
of the ZRGe-1,
ZRGe-2 and ZRGe-3 samples (80, 70, and 60 K, respec-
tively) are much higher than Tm
of the SCGe-1, SCGe-2
and SCGe-3 samples (20, 30, and 40 K, respectively). This
phenomenon can be also attributed to the trapping and
scattering effects of the grain boundary on charge carriers
(a detailed explanation of this trapping effect is given in
Sect. 4). Figure 4a, b is optical microscopy photos of the
microstructures of samples ZRGe-1 and SCGe-1, showing
that sample SCGe-1 is a single crystal (with a dislocation
density of 4280 cm−2) and the sample ZRGe-1 is polycrys-
talline with strip-shaped grains. In the polycrystalline sam-
ples, because of trapping and scattering effects of grain
boundaries, the conductive behaviors of the polycrystal-
line samples start becoming worse at a higher temperature
compared to the single-crystal samples.
Fig. 2 Conductivity as a function of temperature inverse of all inves-
tigated samples in the temperature range of 7–100 K, a single-crystal-
line samples and b polycrystalline samples
Fig. 3 Enlarged section of Fig. 2 to show the conductivity of all
investigated samples in the high temperature range
Electrical conductivity of high-purity germanium crystals at low temperature
1 3
Page 5 of 8 381
3.2 Temperature dependence of net carrier concentration
In our lab, we usually measure the electrical properties
such as carrier concentration, mobility and resistivity at
77 K using van der Pauw–Hall measurement system. The
electrical properties of all investigated samples at 77 K are
shown in Table 1. Now, we are exploring the possibility
of using germanium detectors at much lower temperature
than 77 K, even to 4.2 K. We have another low-temperature
measurement system, which can conduct measurement of
conductivity but not mobility at very low temperature since
it cannot apply magnetic field to produce Hall electric field.
Thus, we estimated the temperature dependence of carrier
concentration using measured conductivity in combination
with mobility model [6, 20]. From Eq. (1), we can get the
net carrier concentration ||NA− N
D|| as follows:
where can be calculated from Eq. (2). For the simplifi-
cation in calculation, we ignored the term 1∕D in Eq. (2).
However, a further investigation of D will be needed in the
future. Land
I can be calculated by Eqs. (6) [6] and (7)
[20], respectively.
A in Eq. (6) is the value of L at room temperature, which
is 3600 cm2/V s. Ns in Eq. (7) is the concentration of Cou-
lomb scattering centers, which is assumed equal to the car-
rier concentration. By substituting Eqs. (6) and (7) for L
and I in Eq. (2), we can solve for total mobility as below:
We set up a relationship between ||NA− N
D|| and conduc-
tivity by substituting Eq. (8) for in Eq. (5) as below:
(5)||NA − ND
|| =
× q,
(6)L = A ×
(
T
300
)−3
2
,
(7)I=
8.25 × 1017 × T
3
2
Ns× ln
1 + 2.57 × 108 × (N
s)−
2
3 × T2
.
(8)
=8.25 × 10
17× T
3
2
4.41 × 1010+ N
s× ln
(
1 + 2.57 × 108×
(
Ns
)−2
3 × T2
).
Fig. 4 Optical microscopy images of microstructures of representa-
tive samples, a SCGe-1 and b ZRGe-1
Table 1 Measured electrical
properties of samples
investigated
Sample Type Carrier concentration at
77 K (cm−3)
Mobility at 77 K
(cm2/V s)
Resistivity at 77 K
(Ω·cm)
Activation
energy
(eV)
SCGe-1 P 4.80 × 109 5.00 × 104
2.60 × 104 0.0004
SCGe-2 P 1.94 × 1011 4.53 × 104
7.09 × 102 0.0026
SCGe-3 P 2.86 × 1011
4.71 × 104
4.60 × 102 0.0024
ZRGe-1 P 7.49 × 1010
4.21 × 104
1.98 × 103 0.0662
ZRGe-2 P 1.32 × 1011
3.80 × 104
1.24 × 103 0.0271
ZRGe-3 P 1.19 × 1012
4.74 × 104
1.11 × 102 0.0129
G. Yang et al.
1 3
381 Page 6 of 8
By solving Eq. (9), we calculate the net carrier concentra-
tion ||NA− N
D|| as a function of temperature from 7 to 100 K,
which is shown in Fig. 5a, b. For all three single-crystalline
samples, as temperatures increases from 7 K, the net car-
rier concentration increases rapidly, then reaches the maxi-
mum at ~ 40 K, beyond which it remains constant. Similarly,
the net carrier concentration versus temperature curves of
all three polycrystalline samples have the same changing
trends, but the temperature corresponding to the maximum
net carrier concentration is at ~ 70 K. These results show
that excitation from both donor and acceptor states reach
(9) =0.139 × |
|NA− N
D|| × T
3
2
4.41 × 1010 + |
|N
A− N
D||× ln(1 + 2.57 × 10
8 × ||N
A− N
D||
−2
3 × T2
.
saturation for the single-crystalline samples at 40 K, and for
the polycrystalline samples at 70 K. This is to be expected
from the difference between their activation energies in
Table 1, which shows that the activation energies of the three
single-crystalline samples are 0.0026, 0.0024 and 0.0004 eV,
respectively, while for the three polycrystalline samples,
they are 0.0662, 0.0129 and 0.0271 eV, respectively. Since
existence of grain boundaries in polycrystalline samples
results in an increase in the potential barrier of conductivity
at grain boundary (detailed description will be provided in
Sect. 4). As a result, the activation energies and, therefore,
saturation temperatures of the three polycrystalline samples
are higher than the saturation temperatures of three single-
crystal samples.
Additionally, we found that the calculated carrier concen-
trations match our measured results using van der Pauw–Hall
measurement system well at 77 K in Table 1, and the carrier
concentration versus temperature curves closely resemble
experimental results of high-purity germanium crystals with
dislocation reported by Haller et al. [3]. This suggests that
the calculated carrier concentration is an accurate approxi-
mation of the temperature dependence.
4 Analysis
Since the contribution of intrinsic electrons and holes to the
conductivity is negligible in the measurement temperature
range of 7–100 K, the only contribution to the conductivity
of the investigated samples comes from the ionization of
impurity atoms in the samples. The above results can be
explained by the band structure of impurity semiconductors.
The investigated samples have four impurity atoms, such as
boron (B), phosphorus (P), aluminum (Al) and gallium (Ga),
which have been identified by photothermal ionization spec-
troscopy (PTIS) [13]. Therefore, we proposed a modified
schematic band structure for the investigated single-crystal
samples based on Hung and Gliessman’s energy level dia-
gram [6], as shown in Fig. 6. P atoms occupy the donor band
while B, Ga, and Al atoms are in the acceptor band. The
electrons in the donor states are excited to the conductivity
band due to thermal vibration energy. The movement of the
electrons from the valence band to the acceptor band due
to thermal excitation can be regarded as the movement of
holes in the acceptor band to the valence band. NA
and ND
are the concentration of donor and acceptors, respectively.
ΔEA and ΔE
D are the respective acceptor and donor energy
gaps from the valence and conduction bands, which are
Fig. 5 Net carrier concentration as a function of temperature of all
investigated samples in the temperature range of 7–100 K, a single-
crystalline samples and b polycrystalline samples
Electrical conductivity of high-purity germanium crystals at low temperature
1 3
Page 7 of 8 381
much smaller than band gap between the conduction band
and valence band. Thus, electrons’ leaps from the donor
band to conduction band and from valence band to acceptor
band require less energy, and are much easier than the leap
from valence band to conduction band. For polycrystalline
samples, based on Lee and Cheng’s energy-band diagram
for n-type polycrystalline silicon [15], a modified schematic
band structure for the polycrystalline samples is proposed
in Fig. 7a, b. The valence, conduction, donor, and acceptor
bands bend upwards at the potential barriers produced at the
grain boundaries, indicating an increase in activation energy,
which can be verified by experimental results in Table 1.
There are two different models for interpreting the electri-
cal conductivity: the dopant segregation model and grain
boundary carrier trapping model [15–18]. The grain bound-
ary carrier trapping model has been widely accepted [15].
According to the grain boundary carrier trapping model,
there are many trapping states at the grain boundary due
to its incomplete atomic bonding. When the charge carri-
ers move to the grain boundaries, they will be trapped and
thereby the potential barriers are created at the grain bounda-
ries. The potential barriers further impede the motion of
charge carriers from one crystallite to the other and limit the
conductivity of polycrystalline samples [16]. Additionally,
Fig. 6 Schematic energy level
diagram of the single-crystalline
samples
CONDUCTION BAND
VALENCE BAND
DONOR STATES (ND)
ACCEPTOR STATES (NA)
(P)
(Al, B, Ga)
= −
Fig. 7 Schematic grain bound-
ary diagram (a) and energy
level diagram (b) of polycrystal-
line samples
G. Yang et al.
1 3
381 Page 8 of 8
the grain boundaries also have the scattering effect on mov-
ing charge carriers and reduce the mobility of charge carri-
ers. Lu et al. [21] indicated that the grain boundary scatter-
ing effects are essential factor at low temperature. It is the
grain boundary trapping and scattering effects that results in
the following differences of conductive behaviors between
the single-crystal samples and polycrystalline samples: (a)
the conductivity versus inverse of temperature curves of the
polycrystalline samples have a medium-temperature range
with a rapid decrease in conductivity, (b) the turning points
( Tm
) are higher for the polycrystalline samples than for the
single-crystal samples and (c) the saturation temperatures of
carrier concentration are higher for the crystalline samples
than for the single-crystal samples.
5 Conclusion
The conductivity measurements of both single-crystalline
and polycrystalline p-type HPGe crystals have been carried
out in the temperature range of 7–100 K. The net carrier
concentrations of all the samples have been calculated in
the measurement temperature range using measured con-
ductivities and existing mobility models. As the tempera-
ture decreases, the conductivity of the single-crystal samples
initially increases to a maximum then decreases gradually,
while the conductivity of the polycrystalline samples ini-
tially increases to a maximum, then decreases dramatically,
and finally continues a slow decrease. The turning points
( Tm
) on the conductivities versus the inverse of temperature
are higher for the polycrystalline samples than for the single-
crystal samples. As the temperature increases, the calculated
ionized carrier concentration increases until saturation. Sim-
ilarly, the saturation temperatures on carrier concentration
versus temperature are higher for the polycrystalline samples
than for the single-crystal samples. These differences in the
conduction behaviors between two kinds of samples could
be attributed to the trapping and scattering effects of grain
boundaries in the polycrystalline samples.
Acknowledgements The authors would like to thank the members of
the crystal growth group at The University of South Dakota. This work
was supported by DOE DE-FG02-10ER46709, NSF OISE-1743790,
NSF OIA-1738632 and the state of South Dakota.
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