J. Phys. Chem. SoMs
THE
Pergamon
DRIFT
Press 1960. Vol. 16. pp. 207-219.
MOBILITY
IN GERMANIUM
OF
Printed in Great Britain.
ELECTRONS
AT LOW
AND
HOLES
TEMPERATURES
E. G. S. PAIGE
Royal Radar Establishment,
(Rem&d
St. Andrew’s Road, Malvem,
Worm.,
England
18 January 1960; rewised 21 March 1960)
Abstract-The
drift mobility of electrons and hoIes has been measured in the temperature range
from 20°K to 300°K in samples of germanium containing impurity concentrations from 7 x 101s
cm-s to 4 x 101s cm-s. Conductivity measurements were also made. Below about lOOoK the observed
minority carrier mobility is less than the mobility calculated from the effects of scattering by phonons
and ionized and neutral impurity atoms. The discrepancy, which is greater than a factor of 2 in some
circumstances, has been attributed to electron-hole scattering. It is proposed that the unexpectedly
large effect of electron-hole scattering is due to a drag exerted on the minority carriers by the
majority carriers when an electric field is applied.
Qualitative observations on the drift mobility of electrons have been made below 20°K. There is
no evidence that electrons remain localized about the same minimum in k space for the duration of a
transit time (3 psec). An extreme example of conductivity modulation of the injected distribution of
carriers has been observed to occur when impact ionization is taking place.
1. INTRODUCTION
effect is complicated by the nature of the band
structure of germanium. (51s)This serves to emphafirst measurement
of the transit time of
size the advantage of the drift experiment as a
minority carriers injected into ge~~ium
was perdirect method of dete~i~ng
mobility.
formed by HAYNESand SHOCKLEY in 1949.(l) A
Another difference between the Hall effect and
more sophisticated version of the experiment was
the drift mobility experiment is that the majority
presented in 1951.(s) In this, which is now called
carrier mobility is determined from the Hall effect
the conventional drift mobility experiment, the
while the minority carrier mobility is determined
drift mobilities of both holes in n-type and elecfrom the drift experiment. It is, of course, the
trons in p-type material were measured at room
majority carrier mobility which is relevant to the
temperature.
The first measurements
of drift
conductivity.
mobility below room temperature were reported
Normally the mobility of a particular type of
by LAWRENCEf3). His observations were confined
charge carrier is assumed to be independent of
to the drift mobility of holes in the temperature
whether that charge carrier is a minority or majorrange IOO-360”K, where he found that the temity carrier. However, a distinction does become
perature dependence of the lattice mobility folscattering is conlowed a T-s*3 law. PRINCE(*) confirmed LAWRENCE’S necessary when carrier-carrier
sidered. A majority carrier is scattered by lattice
observations on holes and obtained data on the drift
defects and by other majority carriers. As DEBYE
mobility of electrons in the temperature range
and cONWEK,L(7)have pointed out the scattering of,
150-370°K
besides measuring the variation of
mobility with resistivity at room temperature.
say, an electron by other electrons does not affect
the total momentum of the free electrons but only
In general there was disagreement between both
alters the distribution of momentum though this in
the value and the temperature dependence of the
turn can affect the mobility if the momentum remobility deduced from the IIall effect and from the
laxation time is energy dependent. Scattering of a
drift mobility experiment.
Subsequent
experimajority carrier by other majority carriers is therements have confirmed the drift mobility results
fore a second-order effect. In contrast, scattering
and have shown that the interpretation of the Hall
THE
207
208
E.
G.
S.
between charge carriers of opposite sign can produce a change in the total momentum of both types
of carrier. As a result the mobility of minority
carriers can be significantly affected by scattering
by majority carriers, i.e. in the drift mobility experiment the effect of carrier-carrier scattering can
be important.
For convenience the following notation will be
introduced. The mobility of minority electrons, of
minority holes and of minority carriers in general
will be denoted by F~, ph and pc respectively. The
mobility of majority electrons, of majority holes
and of majority carriers in general will be denoted
by pg, f~# and pq respectively.
In previous measurements of the drift mobility a
distinction has not been made between pc and pa
for a particular type of carrier. This has been
permissible because the drift mobility experiments
have been carried out in such temperature ranges
and on such material that the effect of carriercarrier scattering was small. An exception is the
work of PRINCE@) on heavily doped germanium at
room temperature. He has taken the scattering by
majority carriers into account by assuming that
their effect is identical to scattering by ionized impurity atoms. Another experiment in which the
effects of electron-hole
scattering was important
was the measurement of the conductivity of germanium at high temperature
by MORIN and
MAITAW. They incorporated the effect of electronhole scattering into their analysis by assuming that
the ionized impurity scattering formula applied,
with the modification that the mass of the scattered
particle is replaced by the reduced mass of the
electron and hole. PRINCE and MORIN and MAITA
were able to get reasonably good agreement between theory and experiment. However, in both
experiments
electron-hole
scattering
played a
minor role in reducing the mobility since the effects
of nhonon scattering were large.
In the experimeit
to be-described
we have
deliberately set out to make detailed measurements
of pc in a temperature and concentration range
where it is reasonable to expect, if the behaviour of
electron-hole
scattering is similar to ionized impurity scattering, that electron-hole scattering will
be comparable in magnitude with ionized impurity
and phonon scattering. The measurements have
been made in the temperature range 20”-300°K on
e+ and p-type germanium containing impurity
PAIGE
concentrations varying from 7 x 1012cm-sto 5 x 101s
cm-3. Conductivity measurements have been made
simultaneously on some samples. An examination
of the observations reveals that electron-hole scattering has an appreciable effect on pLcand dominates
the mobility in the more impure specimens. The
results show that electron-hole scattering lowers
the mobility more than the amount predicted by
the ionized impurity scattering formula as used by
PRINCE or as modified by MORIN and MAITA. As
explained in Section 4 this is principally due to the
neglect of the drag which can be exerted on the
minority carriers by the majority carriers. The detailed analysis of the experimental results is not
attempted in this paper, but has been carried out
by MCLEAN and PAIGE@) and will be presented in a
subsequent paper.
These authors have derived
expressions for both IQ and pV incorporating
carrier-carrier scattering. The treatment, based on
the Boltzmann transport equation, is applicable
_.
only in certain ranges of temperature and free
carrier concentration
but the experimental observations presented in this paper fall within these
ranges.
In Section 5 some qualitative observations of
pe below 20°K are discussed. The attempt is
described-and
reasons for its failure given-to
observe the transport of electrons which have remained in the same energy minimum in momenturn space during the transit time. Finally an
extreme conductivity modulation effect which can
occur at very low temperatures is described.
2. EXPERUvXENTAL TECHNIQUE!3
The method of measuring the minority carrier
mobility is based on the techniques developed by
HAYNESand SEIOCKLEU@).
There are some small
but important differences to facilitate measurements at low temperatures. These are
(i)
the use of alloy junctions
to the specimen,
for all contacts
(ii)
the use of an I/zjunction for the collector
instead of the conventional pn junction,
(iii)
the pulsing of the sweep field, the emitter
current and the collector bias,
and
(iv)
the illumination of the specimen
light in the absorption edge.
with
DRIFT
MOBILITY
OF
ELECTRONS
The reasons for these modifications are as follows. The
use of alloyed junctions enabled contacts to be made to
the specimen which were reproducible and mechanically
reliable. The replacement of the conventional pn junction collector by an Zh junction was made at the suggestion of Dr. A. F. GIBSON of this laboratory. Its advantages were that it maintained its efficiency and had a 20
MC/S bandwidth over the complete temperature range.
The use of a low resistance collector necessitated pulsing
of the collector bias to minimize the power dissipation.
Illumination of the specimen by light in the absorption
edge quenches trapping effects by saturation of the
traps.c3J It was necessary to illuminate all samples at low
temperatures to prevent trapping affecting the mobility.
2.1 Specimen fabrication
and mounting
Filaments of germanium were cut perpendicular to the
growth axis of ingots specially selected for long lifetime
and uniformity of resistivity. Germanium doped with
arsenic or gallium was used ranging in resistivity from
intrinsic to about 1 Q cm. Typical dimensions of the
filaments were 13 x0.8 x 0.8 mm3. The specimens were
ground and then etched in Ha02 at 60°C for at least 2 hr
to remove surface damage caused by sawing and grinding(“) and hence to keep the surface recombination
velocity low. A large area non-injecting contact was
alloyed to one end of the filament. This contact was of
gold with about 1 per cent of a suitable group 3 or 5
element (Sb or Ga) to form an Zh junction to the germanium. The voltage probes of 0.005 in. gold wires,
suitably doped to form lh junctions, were alloyed to one
side of the germanium filament. The emitter and collector were formed by alloying 0.002 in. doped gold wire
to the opposite side of the filament. They were separated
by 6 mm to 10 mm. The diameters of the emitter and
collector junctions were about 0.1 mm and the voltage
probes had a junction diameter of about 0.2 mm.
The filament was soldered by one end to a block of
oxygen free copper plated with rhodium. Since this was
the only point of support for the filament, strain effects
on the mobility(3J2) were avoided by placing the emitter
and collector sufficiently far away from this end.
Together with the non-injecting contact the connection
to the copper block forms the contacts through which the
sweep current was passed. Wires passing through insulating supports in the copper were soldered to the gold
junction wires; this completed the formation of a complete unit. Before making a set of observations the complete unit was given a final etch in HrOa for several
minutes. By this procedure etching around the junctions
was carried out when the junctions were supported and
it enabled the time between etching and placing the
specimen in mcuo to be reduced to a minimum.
2.2 The low temperature
and electrical
apparatus
The copper block supporting the specimen was mated
to another copper block on which a platinum resistance
thermometer was wound. This was suspended near the
bottom of a vertical glass tube, the leads passing through
metal-glass seals at the top of the tube. There were provisions for evacuating the tube to a pressure of 10-s mm
AND
HOLES
IN
Ge
209
of mercury and for admitting an exchange gas. The tube
was surrounded by two Dewars both having side slits in
their silvering through which a beam of light could be
passed and focussed onto the specimen.
A sweep pulse was applied to the filament, and a
current-measuring
resistor in series with it, from a
generator which triggered the oscilloscope and two other
pulse generators. These two generators produced the
emitter pulse and the collector pulse. The procedure
generally adopted was to adjust the collector bias and the
sweep field so that the signal they produced across the
collector load approximately cancelled. This enabled
greater amplification of the collector arrival signal without overloading the amplifiers. The measurement of the
sweep field was made before the minority carriers were
injected into the specimen. It was important to have the
collector bias applied during the field measurement and
throughout the time the minority carriers were in transit
since in some circumstances, because of the collector’s
low resistance, the collector bias had an effect on the
field in the specimen.
A pulse repetition frequency of 50 set-l was used.
Below 100°K the sweep field and collector bias pulses
had a duration of 10 psec or less. The emitter pulse was
of 0.2 psec duration and had a rise time of 0.1 psec. The
bandwidth of the amplifier and oscilloscope was 20
Mc,ls.
2.3 Measurement
procedure
From the transit time, t, the separation between the
emitter and the collector, 1, and the field in the specimen,
E, the minority carrier mobility can be calculated.
/~c =
l/tE.
(1)
In practice the determination of neither the appropriate
E nor t is a straightforward procedure. In addition, in
near-intrinsic material a distinction has to be made between the drift velocity of the group of injected carriers
and the drift velocity of the minority carriers.(ls)
The difficulty of determining E is due to non-uniformity of the distribution of impurities in the crystal.
HAYNES and SHOCKLEY(~)have presented an approximate
method of correcting for this by potential probing along
the specimen and carrying out an appropriate averaging
procedure.
Strictly it is necessary to determine the
correction at several temperatures. For a high resistivity
sample the irregularities in the impurity distribution are
smoothed out to some extent by the intrinsic carriers, at
low temperature the irregularities would be apparent.
On the other hand the nature of the scattering mechanism of low resistivity samples changes appreciably on
cooling; impurity scattering becomes more important.
The result of this is that the field becomes more uniform
at low temperature.
It was found possible to make
potential measurements only at room temperature and,
although in all cases the correction necessary was at
most only a few per cent, to apply this correction at low
temperature is a potential source of error.
The transit time is affected by trapping(s) of the inecjted carriers. The onset of trapping is marked by a
210
E.
I
G.
S.
-~-
PAIGE
I
t4;t-J;
(CM-3
+ 8.8 x lo”
42xld4
cl l~12rtd5
513.
5.14.
5.15.
x 4.2 x lo”
5.16.
l
1.
-
--I
-:
i
-
FIG. 1. The variation of the minority carrier mobility of electrons with
temperature. The full curve shows a p CC Ti’ss.
The broken curves
are to clarify the trends of the experimental points.
characteristic skewing of the collector signal. When this
occurred the specimen was illuminated; the level of
illumination was so adjusted that a higher intensity
produced no observable change in the transit time of the
minority carriers.
The transit time is affected also by conductivity
mod~lation.(1*,15~ To overcome this difficulty several observations of the transit time were made at a particular
field and temperature for different emitter currents.
From plots of the emitter current against the square root
of the transit time, the transit time in the absence of
conductivity modulation was obtained by linear extrapolation to zero emitter current. It is shown in the
appendix that a linear relation should exist between
emitter current and y’t. Corrections for the presence
of intrinsic minority carriers, using equation A2 of the
appendix, are only important at or just below room temperature in the higher resistivity samples.
2.4 Experimental
error
The error in determining the effective separation
between probes was taken as half the diameter of a single
probe. This led to a typical error of 2 per cent in estimating the separation between voltage probes and 1 per cent
in estimating the separation between the emitter and
collector. The voltage appearing between the voltage
probes was measured within an error of 2 per cent and
the transit time within an error of + per cent. Thus,
neglecting systematic errors due to inhomogeneity of the
crystal, the experimental error in obtaining the mobility
was about 6 per cent. The error in measuring the temperature was less than 1 per cent. The mobility temperature curves were reproducible within these limits.
The estimated experimental error in the measurement
of the conductivity was 5 per cent and it is also subject
to systematic errors due to inhomogeneity of the crystal.
3. THB VERNON
OF ~O~U~~
MINOBITY
CABBIBB
MOBILITY
TEMPERATURE
AND
WITH
The restrictions in temperature range over which
measurements of pCwere made were imposed by (i)
lifetime, (ii) field dependence of mobility and (iii)
conductivity modulation. A specimen having a
lifetime of about 100 psec at room temperature
had typical lifetimes of 2 psec at 77’K and $ to
DRIFT
MOBILITY
OF
ELECTRONS
AND
HOLES
IN
Ge
211
” I
N, N, (Ctv?)
l
74
I: lOi
,520
+ 7.7 x IO”
5.19.
0 I.41r10’4
x 6.1 x IO”
s 18
S 23.
6-
FIG. 2. The variation of the minority carrier mobility of holes with
temperature. The full curve shows a p a T-2.33. The broken curves are
to clarify the trends of the experimental points.
1 psec at 25°K.
low temperatures
As a result
of the low lifetimes
at
sweep fields of between 5 V cm-1
and 10 V cm-1 were used. It was found that in such
sweep fields, at some temperature below 30°K
depending on the impurity concentration,
the
mobility and conductivity became field dependent.
In general tag decreased with field and t~h increased
with field, though the field dependence of both
mobilities was weak. The field dependence arises
from a change in the ionized impurity concentration produced by impact ionization and an increase
in the mean energy of the free carriers.(ls) Observations presented in this section are independent of
field except the few made below 30°K on the lower
resistivity samples.
3.1 The mobility of minority electrons
Fig. 1 shows the observed values of pe in the
temperature range 20°K to 300°K in four p-type
crystals. The room temperature,
extrinsic, free
carrier concentrations
of the crystals are incorporated in Fig. 1. The observations on the lowest
resistivity sample were restricted by its low lifetime; observations below 100°K were difficult and
became impossible below 80°K. Trapping effects
were observed but, in general, were not sufficient
to prohibit measurements. An idea of how small
the effect of trapping on the mobility was can be
gained from observations on S.14, where the difference between mobilities with and without saturation of the traps was less than 5 per cent. Of the
crystals on which observations were made the
highest resistivity crystal, S.13, was unique in
displaying a large amount of trapping in a limited
temperature range-between
100°K and 200°K. In
this range it was not possible to determine the
mobility because even with the specimen illuminated trapping effects were apparent.
3.2 The mobility of minority holes
The mobility of minority holes has been measured in four crystals and the variation with
temperature is shown in Fig. 2. As in the case of
p-type crystals, measurements are limited on low
resistivity samples by low lifetime. The high purity
crystal, S.20, which was supplied by Mr. I. G.
CRESSELL of Marconi’s Wireless Telegraph
Co.
Ltd., was unique in that precision measurements
E.
212
Fs. 3. The
temperature.
G.
S.
PAIGE
variation of the conductivity of n-type germanium with
The broken curves are to clarify the trends of the experimental points.
----r-----~-7
_I_
x\
\
\
XI
\
kS 16.
\
\
‘%
\
\
FIG. 4. The
temperature.
variation of the conductivity of p-type germanium with
The broken curves are to clarify the trends of the experimental points.
DRIFT
MOBILITY
OF
ELECTRONS
of mobility were impossible above 60°K because of
distortion of the arrival signal due to ~onductivi~
modulation and end contact effects.
Trapping of minority carriers in n-type material
was more apparent than in p-type samples but by
suitable illumination
it was always possible to
saturate the traps.
3.3 The conductivity of n- and p-type crystals
Measurements of the conductivity were made on
the same crystals as were used for the drift mobility
measurement.
Detailed measurements were performed on some specimens, on others the conductivity was measured only at 77°K and 90°K.
The results obtained from the detailed measurements of conductivity as a function of temperature
are shown in Figs. 3 and 4. The results were obtained with the samples in the dark. Illumination
to the level necessary to eliminate trapping effects
caused changes in conductivity of up to 4 per
cent.
4. DISCUSSION OF THE VARIATION OF MINORITY AND MAJORITY CARRIER MOBILITY WITH
TENETS
The general trend of the results of the measurements of pC presented in Figs. 1 and 2 is as we
might expect. Between room temperature
and
100°K there is a rapid rise in mobility due to the
decrease in phonon scattering. For temperatures
below 100°K the rate of change of mobility with
temperature is reduced. It becomes strongly dependent upon the resistivity of the crystal due in
part at least to the increasing influence of impurity
scattering as the crystal is cooled.
Taking the lattice mobilities at 300°K obtained
by PRINCE(*) as 3900 cm2 V-r see-1 and 1900
ems V-1 see-1 for electrons and holes respectively
and a lattice mobility temperature law of Tl+ss for
electrons and T-3.33 for holes(r7) we obtained a
mobility temperature dependence shown by the
solid curves in Figs. 1 and 2. This temperature
variation is followed closely by the high purity
samples between 300°K and 150°K with the exception of S.19 (Fig. 2). It is in the temperature
range below 150°K that we can expect electronhole scattering to be significant, hence we shall discuss this range in some detail. A relaxation time
will be calculated including scattering by neutral
and ionized impurity atoms and by phonons
0
AND
HOLES
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213
(defect sca~er~g). This will be used to compute
pC and the conductivity. The comparison with the
observed conductivity will show if an adequate
description of the defect scattering mechanisms has
been given, then, if the description is satisfactory,
comparison between the calculated and observed
minority carrier mobility will reveal the importance
of electron-hole scattering. It will become apparent
that differences of a factor of two or more exist
between the calculated and observed pC, accordingly it is not necessary to estimate the defect
scattering to better than, say, 10 per cent or 20 per
cent in order to display this difference. Therefore
the attitude adopted in this paper is that a simple
treatment of defect scattering will suffice, leaving a
more thorough discussion of defect scattering to be
given in a subsequent publication,(l*)
where, in
fact, much of the simple treatment will be justified
empirically.
4.1 Defect scattering
In view of the above remarks we shall make the
assumption in calculating the mobility that the mass and
relaxation times are isotropic, i.e. we replace the populated constant energy surfaces with spherical constant
energy surfaces. We take the conductivity mass of the
multivalley model, 0.12 ms, for the mass in the present
case.(ra) We shall concentrate on obtaining a relaxation
time for electrons primarily because they follow the
theoretical
temperature
dependence
for acoustical
phonon scattering of p cc T-1-s more closely than do
holes.
The relaxation time for phonon scattering, rp, is
assumed to be given by
l/rP
= p,T1J%,
(2).
where ZIis the velocity of the electron and pe is a parameter measuring the strength of phonon scattering of
electrons which is determined from the known lattice
mobility at 300’K.
In this simplified expression for
phonon scattering we have assumed that the departure
from the X-r.5 law is not due to a change in the velocity
dependence of the relaxation time and that T-l*ss law
extends throughout the temperature range. In the classical
approximation, which is valid over the concentration and
temperature ranges of the experimental results presented
in Section 3, CONWELL and WEISSKOPF(‘S) give the reciprocal of the
scattering as
l/q
relaxation
= ziV*v-sln[l+(Fr],
time
for
ionized
impurity
(3)
where R is Boltzmann’s constant, m is the effective mass
of the electron, E is the dielectric constant of the material
and I?* is the concentration of ionized impurity atoms
214
E.
G.
S.
(calculated in Section 4.2). d* is a distance beyond which
the interaction between the free carrier and the impurity
atom is cut off. We shall take this distance to be the
Debye screening length,
a*=
Ji
<kT
he2 1
where n is the free carrier concentration, for consistency
with calculations in subsequent papers.(sslO) In the temperature and concentration ranges of interest this choice
of d* produces a very small change in 71 compared with
taking d* as the mean separation between impurity
atoms. For neutral impurity scattering ERGIxSOY(~~) has
given the following expression,
20%
l/TN = -R(N-IV”),
m
(4)
where R is the radius of the orbit of the bound carrier
which has been estimated by KoHN@~). The mobility
was calculated by summing the reciprocal relaxation
times and averaging the resultant 7 over the carrier
distribution in the appropriate manner. (2’ )
4.2 Conductivity
In order to calculate the conductivity from the mobility
it is necessary to calculate the free carrier concentration
as a function of temperature.
The free carrier concentration is also required to estimate the concentration of
ionized impurity atoms. Hence the concentration of both
electrons in n-type and holes in p-type are required in
estimating the I of majority and minority electrons.
Between 20°K and lSO”K, we take the free carrier concentration to be given by
n = ;[il+g?],
where N is the concentration
a = 2g-1 exp( - q/kT)
(5)
of impurity
atoms and
xz (2m-m~,kT/r2)3~2.
Here l1 is the ionization energy of the impurity atoms,
g is the degeneracy of their ground state, i runs over the
band edge points in k-space, and rnn is the density of
states effective mass at each band edge point. By writing
n in this form we have assumed that the degeneracy of
the ground state of the impurity atom is not lifted to a
significant extent. Although the determination
of the
free carrier concentration as a function of temperature
t’rom the Hall effe&)
is not sufficiently accurate to
c!ecide to what extent the degeneracy is decreased the
observations are well represented by equation (5) using
the appropriate values for the parameters.
For the conduction band we have rn~, = 0~22rn0,(~~)
the summation extending over 4 equivalent minima in K
space, and g = 2 x 4. For the valence band there are two
band edge points coincident at k = 0 having nz~& =
0.04Smo and nt~~ = 0.36rn0;(~~) the value of g = 2 x 2.
PAIGE
The n-type crystals were doped with As, the p-type with
Ga, which have values of EI of 0.0127 eV and 0*0108
eV(sO) respectively.
The use of equation (5) is only valid provided compensation is negligible. We assume this condition is fulfilled at present, leaving a discussion of the possibility of
compensation till the next section. Since compensation is
assumed negligible n = N*.
Fig. 5 shows the conductivity of an n-type crystal
as a function of temperature
calculated for
N = 6.1 x 1014 cm-s. The observed conductivity
of an n-type crystal for which ND-NA
= 6.1 x
x 1014 cm-s is shown for comparison; NA and ND
are the concentrations of acceptors and donors.
The discrepancy of less than 15 per cent, with the
exception of one point, can be regarded as reasonable agreement in view of the simplifying assumptions we have made.
4.3 Minority carrier mobility
In Fig. 6, curve 1, we present the calculated
minority carrier mobility of electrons in a crystal
containing 1.1 x 1015 cm-s impurity atoms. The
measured pe for a specimen containing NA - ND =
l-1 x 1015 cm-3 is shown also. The calculated pe is
greater than the observed pe by a factor of 2 or
more, yet, from the comparison of calculated and
observed conductivities we could expect to be able
to describe defect scattering to within 15 per cent.
We have, therefore, a clear indication that some
other scattering mechanism
is important
for
minority carriers only. Such a mechanism is
electron-hole scattering.
Previous attempts to include the effect of
electron-hole scattering in computing the mobility
in germanium have been outlined in Section 1. By
assuming that the majority carriers scatter as
though they were ionized impurity atoms@)
electron-hole scattering can be incorporated by replacing N* by 2N* in equation (3). The mobility
calculated by this method is shown in Fig. 6 by
curve 2. We note that this calculated pe is always
greater than the observed rue,by as much as a factor
of 2 between 30°K and 40°K. Adoption of MORIN
and MAITA’S@) approach leads to replacing N* by
[(m/mr)1/2+ l]N* in equation (3), where m7 is the
reduced mass of the electron and hole. (Taking the
effective mass of the hole as O-3 mo this is equivalent to replacing N* by 2.2 N*.) The mobility calculated by this method is less than 10 per cent
DRIFT
MOBILITY
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I
I
I
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I
I
I5;0
100
TEMPERATURE(OK)
50
AND
215
Ge
L
50
TEtlliRAT”R$
9
FIG. 6
FIG. 5
FIG. 5. A comparison between the calculated conductivity (full curve) for an
n-type crystal containing 6.1 x 1014 cm-s arsenic atoms and the observed conductivity of S.23 (ND-NA
= 6.1 X 10’4cm-3).
FIG. 6. Curve 1. The calculated mobility of electrons in a crystal containing
1 .l x 101s cmm3 gallium atoms. Curve 2. The calculated mobility of electrons in a
crystal containing 2 x (1 *l X 1015) cm-3 gallium atoms. The crosses are the observed
minority carrier mobility of electrons in the gallium doped crystal S.15 (Nn-Nn
= 1.12 x 1Ol5 cmm3). The vertical lines indicate the experimental error.
smaller than the mobility shown in curve 2; therefore the discrepancy remains. The same conclusion is reached for specimens of different free
carrier concentration,
the discrepancy increasing
with carrier concentration.
A similar set of calculations of pn and the conductivity
of p-type material have been carried out using equations
(3), (4) and (5) but replacing (2) by
(6)
The same general conclusions were reached for holes as
for electrons, that the majority scattering is well described by equations (3), (4) and (5) but that, even when
electron-hole scattering is incorporated by the method of
PRINCE or MORIN and MAITA, the minority carrier
mobility is over estimated.
Four possible reasons for the discrepancy are (i)
trapping of minority carriers, (ii) fluctuations in the
impurity content, (iii) compensation and (iv) an
inadequate theory for electron-hole scattering. The
first of these can be neglected completely because
the specimens were illuminated to remove trapping
effects as judged by the pulse shape and the variation of transit time with emitter current.? The
effects of fluctuations in the impurity concentration has been discussed in Section 2.3. Both the
fluctuations and their effects should become less.
important at lower resistivities-a
trend opposite
t Frequent trapping for a time short compared with
the pulse width would not distort the pulse. This
possibility is not considered seriously because the traps
would have to have (a) a concentration greater than
about lOI cm-3 if illumination and emitter current did
not reveal the process, (b) a trapping time which remained short throughout the temperature range (c) the
same concentration in crystals of the same resistivity and
(d) the same characteristics in n- and P-type crystals.
216
E.
G.
S.
to that of the discrepancy-and
again the reproducibility of the results for different specimens of
the same resistivity indicate together with normal
Hall voltages that such fluctuations are unimportant. Compensation can be ruled out for the following reasons. Firstly, the degree of compensation in
the crystals used is expected to be small because
the crystals were grown from high purity intrinsic
germanium and were specially selected for long
lifetime, Secondly, a large amount of compensation
would affect py just as much as pe yet reasonable
agreement is obtained between the calculated and
experimental values of the conductivity neglecting
compensation. Thirdly, even if N* was increased
to incorporate compensation, the required temperature dependence would not be obtained, i.e. if
N* was increased it would only be possible to obtain a fit to the experimental data over a limited
temperature range. In considering whether the
present attempt to include electron-hole scattering
is adequate we must recall that it is based on the
assumption that the scattering of an electron by a
hole, say, is basically the same as scattering of an
electron by an ionized impurity atom. We must
consider, therefore, whether there are assumptions
made in deriving the relaxation time for ionized
impurity scattering which are not applicable to a
collision between two mobile charges having
similar masses. Such assumptions are (i) that no
energy exchange occurs between the colliding
particles and (ii) that the centre of mass co-ordinate
system for the collision is not moving with respect
to the laboratory co-ordinate system. The removal
of these assumptions greatly complicates the deduction of a relaxation time for an electron-hole
collision. Even more important is the following
consideration. When an electron is scattered by
other particles it is scattered so that its mean
velocity becomes the mean velocity of the scattering particles. In other words the electron is
: tattered so that its velocity is completely randomized iu a reference frame moving with the mean
velocity of the scattering particles. When the
:catterer is an ionized impurity atom the velocity
c.f the scattered particle is randomized about zero
velocity; when the scatterer is a hole in an applied
electric field the velocity is randomized about the
drift velocity of the hole. This is a drag effect, and
we may say the majority carriers exert a drag on the
minority carriers. Since the electron and hole are
PAIGE
accelerated by a given field in opposite directions,
the drag is such as to reduce the mobility. Here
then is a mechanism not previously considered
which could substantial1.y reduce the mobility of
minority carriers and so provide the possibility of
accounting for the experimental results.
In the following paper MCLEA.N and PAIGE have
obtained theoretical expressions for the mobility
including carrier-carrier
scattering in addition to
the usual defect scattering. The theory applies to a
restricted temperature range and free carrier concentration range though it covers the concentration
and temperature ranges over which measurements
have been presented in this paper. A detailed comparison between the theory and the experimental
observations given in Section 3 will be presented in
a subsequent paper. It is found that by using the
theory of mobility of MCLEAN and PAIGE, which
naturally incorporates the drag effect, good agreement can be obtained with experimental observations for both electrons and holes. In this paper we
have (i) established that electron-hole scattering is
important, (ii) demonstrated that electron-hole
scattering cannot be represented as the scattering of
a free carrier by an ionized impurity atom and (iii)
suggested the principal difference between scattering by a free carrier and by an ionized impurity
atom, namely, the drag effect.
5. OBSERVATIONS ON THE MINORITY
MOBILITY BELOW 20°K
CARRIER
The observations were confined to minority
electrons in 3 Q cm p-type germanium. The main
reason for extending the observations below 20°K
was to attempt to observe the transport of electrons
which had been localized within the same valleys in
momentum space throughout the duration of the
transit time.* At the time this particular experiment was performed (1956-7) an upper limit to
rij, the relaxation time for intervalley transitions,
was estimated to exceed the transit time (8 psec)
below 10°K by a considerable margin. Hence it
appeared reasonable to expect to observe the transport of carriers localized within a valley.
The experiment was performed using specially
having
emitter-collector
constructed
samples
separations of 2 or 3 mm and having two collectors
* This experiment is similar in concept to that proposed by
GoLD@~).
DRIFT
MOBILITY
OF
ELECTRONS
instead of the usual one. The two collectors were
placed relative to the emitter so as to intercept
electrons propagated in different valleys. (Because
of the anisotropy of conductivity within a valley the
carriers are not necessarily accelerated parallel to
the field. The diffusion length, of order 10-3 cm,
is too short to ensure diffusion of carriers across the
sample.) The field was increased until impact
ionization occurred, a condition readily observable
as a signal at the collector. At about 7”K, one, and
in some instances more, electron arrival signals
were observed at each collector. However, by
using several samples of different crystallographic
orien~tion it became clear that the delay times of
the signals were a feature of the geometry of the
sample-the
separation
between
emitter
and
collector and the angle between emitter and
collector and the field and not the crystallographic
orientation. This eliminates the possibility that
electrons had been observed which had not undergone intervalley transitions. The reason for seeing
different signals at the two collectors was that
contrary to conditions at higher temperatures the
influence of the collector field on the path followed
by the minority carriers was important. This resulted from the use of a small e~tter~collector
separation and large collector bias, both necessary
to observe a signal at very low temperature. The
observation of more than two arrivalsignals usually
overlapping each other is believed due to the
complicated situation which arises due to the
electric field varying both in magnitude and direction. As a result of impact ionization the free carrier
concentration can vary by orders of magnitude with
position and so, due to electron-hole scattering, the
minority carrier mobility can become a function of
position. It is possible for minority carriers to become “trapped” in a region of low mobility in a
convergent field (near the collector) and carriers,
flowing a less obvious route in a lower field, arrive
before them.
The recent observations by WEINREICH et uZ.(~~)have
demonstrated that in n-type germanium 7u is much less
than the value calculated above because at low temperatures transitions involving the donor states become important. Their results indicate that a concentration of
compensating donors in P-type material of 1 x 10’s cm-s
would be sufficient to lower Q below the transit time,
since the frequency of transitions between the donor level
and the conduction band will be far greater than between
the donor level and the valence band. In the crystals
AND
HOLES
IN
Ge
used the concentration of compensatig
donors would be
greater than 1 x 1012 cm-s and we can conclude that rii
was too low for the experiment to be successful. Two
other possible reasons for its failure are (i) that vi, was
reduced because the average energy of the electrons was
raised by the field and (ii) that the anisotropy of the relaxation timecz6) for ionized impurity scattering decreased
the anisotropy of the conductivity within a valley to such
an extent that a difference would not be observed between electrons in different valleys.
An interesting conductivity modulation effect
was observed between 4°K and 15°K. Normally
when the amplitude of the injection current is increased the amplitude of the collector signal rises
i
1
0
ul
7.;
i:
j;
A
TIME---
B
TIME-
FIG. 7. (a) The usual effect of conductivity modulation
on the signal observed at the collector. (b) The effect of
increasing the emitter current on the collector signal
which is observed when impact ionization of impurity
atoms is taking place.
but the signal becomes skewed. At very low temperatures increasing the emitter current did not
alter the amplitude of the collector signal significantly but increased its duration in time. The two
contrasting situations are depicted in Fig. 7. We
may interpret this effect as follows. Within the
germanium filament there is a uniform density of
holes, $0, maintained by impact ionization resulting from the presence of a sweep field. When the
emitter is switched on the local field rises slightly
causing (i) an increase in impact ionization
po --+p~-+-p~(~, t) and (ii) injection of minority
carriers n(r, t) and so raising the hole density to
po+p, (r, t) + n(r, 1). When the emitter is switched
off the holes produced by impact ionization in the
emitter field will be recaptured in the majority
carrier lifetime which is of order 10-s sec.(sQ The
E.
218
G.
minority carrier on the other hand will recombine
relatively slowly ( 7 N 10-s set). Thus the minority
carriers exist for most of their life in a hole density
of po+n(r,
t).
The possibility exists that n > ~0,
and extreme conductivity modulation results. The
centre of the pulse remains stationary while
carriers are steadily removed from the leading
edge as n +pe.
In other words the injected carriers, instead of being propagated as a group of
electrons, act as a stationary reservoir.
It was found that passing a large sweep current
resulted in a normal pulse shape with the usual
conductivity modulation effects. This is because
in the large sweep current the density of impact
ionized holes has been increased and as a result
71 < po and the extreme conductivity effect is
absent.
It is interesting to note that this effect of conductivity modulation provides a method of integrating signals of appropriate length and amplitude
range.
6. CONTUSION
We have presented experimental
observations of
the mobility of minority carriers in a temperature
range from 300°K to 4°K. Hitherto the minority
carrier mobility has not been measured below
100°K. In the new range explored the effects of
electron-hole
scattering
were large. In some
specimens this scattering mechanism appeared to
dominate the mobility. We have mentioned that
electron-hole scattering had a considerably greater
effect on the mobility than the effect predicted
from a simple ionized impurity scattering formula.
The discrepancy has been attributed to the drag
effect of the majority carriers on the minority
carriers. It is conceivable that in a solid with free
carriers having favourable masses the drag effect
could be considerably larger. In principle the
minority carriers could be influenced so greatly by
the drag of majority carriers that they drift “backwards”, i.e. they could have a negative mobility.
There are two important points connected with
this investigation. Firstly, for the first time observations have been made on a transport property
of a semiconductor in which carrier-carrier interaction is dominant. Secondly, there is a drag effect
exerted by the majority carrier on the minority
carrier which from the point of view of analysis
has the advantage over other drag phenomena such
S.
PAlGE
as phonon drag that the drift velocity of the
particles exerting the drag is a readily determined
quantity.
At lattice temperatures down to nearly 4°K the
transit of minority carriers has been observed.
However it appeared that even for such low temperatures a large number of interchanges
of
electrons between different valleys in momentum
space took place during the transit time.
It is worthwhile at this juncture to consider the
relation of this work to the operation of devices at
low temperatures. The performance of the drift
mobility experiment in the liquid helium range
demonstrates that transistor action is possible at
very low temperatures
provided sufficient free
carriers are generated by impact ionization. While
pC is smaller than might be anticipated due to
electron-hole
scattering the diffusion constant is
not influenced in the same way by drag. The
extreme conductivity modulation effect which was
observed in the impact ionization range could impose limitations on the input signal to a transistor
or its frequency response. ~ternatively
it could be
utilized to make an integrating device.
Acknowledgements-The
author wishes to acknowledge
the assistance of Miss S. BOULTONin the preparation of
the specimens and to thank Dr. A. F. GIBSON for his
helpful suggestions
and encouragement
during the
course of the work. The author is indebted to Dr. T. P.
MCLEAN for his comments on the manuscript.
APPENDIX
In this appendix the method of determining the
minority carrier drift velocity as distinct from the
velocity of propagation of the pulse of injected carriers
which is observed will be outlined.
Consider holes injected into n-type material. If the
velocity of propagation of the pulse of injected holes is
V(= E/t) then it is related to ph by
where no and $0 are the thermal equilibrium densities of
electrons and holes, Ap is the density of injected carriers
and E is the electric field outside the injected region. This
equation which is a simple extension of that given by
HER~ING~~*)negIects the effects of diffusion and recombination. At very low temperatures the hfetime becomes
comparable with the transit time. Under such conditions,
therefore, equation (A.l) is not strictly obeyed. The
equation has two limiting forms which are relevant,
DRIFT
MOBILITY
(i) when there is no conductivity
OF
ELECTRONS
modulation,
A$ = 0,
:)I-:
(A4
v = /&[I+ &(l+
and (ii) when the material is far from intrinsic, $0 = 0,
v = &[1+$(1+
:)I-’
(-4.3)
Equation (A.3) displays the effect of conductivity modulation on the velocity of propagation of the pulse of boles.
In the low temperature .region conductivity modulation
had a large effect whilst the material was far from intrinsic.
Rearrangement
of equation
(A.3)
putting
Ie = A p where I, is the emitter current gives
for a crystal at a given temperature. A plot of I< against
tl/s should be linear and intercept the ~‘12 axis at a value
of t corresponding to zero conductivity modulation. In
practice plots of I6 against tlfz were usually linear.
For a 3 Sz cm crystal at room temperature, when recombination effects are small, the value of A was estimated as 7 x lo-17 A cm-s. Thus for an emitter current
of 1 mA, the minority carrier density was 1~4 x lOI3
crnv3 compared with the majority carrier density of
1 X lOI5 cmeS.
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HOLES
IN
Ge
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