of 8
Electron localization effects on the low-temperature high-field magnetoresistivity
of three-dimensional amorphous superconductors
A. V. Samoilov and N.-C. Yeh
Department of Physics 114-36, California Institute of Technology, Pasadena, California 91125
C. C. Tsuei
IBM, Thomas Watson Research Center, Yorktown Heights, New York 10598
~
Received 22 July 1997
!
The electrical resistivity
r
of three-dimensional amorphous superconducting films
a
-Mo
3
Si and
a
-Nb
3
Ge is
measured in magnetic fields
m
0
H
up to 30 T. At low temperatures and at magnetic fields above the upper
critical field
H
c
2
,
r
is temperature independent and decreases as a function of magnetic field. This field
dependence is consistent with localization theory in the high-field limit
@
m
0
H
@
\
/(4
eL
f
2
) , where
L
f
is the
phase-coherence length
#
. Above the superconducting transition temperature
T
c
, the temperature dependence of
the conductivity is consistent with inelastic scattering processes which are destructive to the phase coherence
for electron localization, thereby allowing estimates for
L
f
(
T
) . The Hall effect data on
a
-Mo
3
Si, in conjunc-
tion with the resistivity data, allow the determination of the carrier concentration and mean free path. The
upper critical field is comparable to
~
in
a
-Mo
3
Si) and significantly larger than
~
in
a
-Nb
3
Ge) the Clogston-
Chandrasekhar paramagnetic limit. This phenomenon is discussed in the context of electron localization.
@
S0163-1829
~
98
!
01702-0
#
I. INTRODUCTION
The possibility of observing negative magnetoresistance
due to the suppression of electron localization and hence an
enhancement in the electrical conductivity (
s
) of three-
dimensional
~
3D
!
disordered metals is an interesting long-
standing
issue
which
has
not
been
well
explored
experimentally.
1,2
In contrast to the inactivity in the studies
of 3D disordered metals, a number of experiments have been
done on 3D disordered semiconductors,
3–5
and the results are
found to be in good agreement with the localization theory
by Kawabata.
2
The reason for more studies of the localiza-
tion in semiconductors than in metals is largely due to the
smaller magnitude of the negative magnetoresistance in the
latter. In other words, the higher conductivity
s
and the pre-
dicted universal enhancement of the conductivity in high
fields,
D
s
~
see below
!
, conspire to reduce the magnitude of
D
s
/
s
and therefore make measurements in metals more dif-
ficult.
On the other hand, it is known that the electron-electron
interaction results in a positive contribution to the
magnetoresistivity
1,6
which, in disordered semiconductors,
generally predominates over the localization term which
yields a negative magnetoresistance. Hence, the localization-
related behavior becomes more difficult to infer directly.
3
In
this context experiments on metals are more advantageous
for revealing the effects of localization, due to the stronger
screening of the electron-electron interaction. One such ex-
ample is the observation of a negative magnetoresistance in
aluminum granular films.
7
In addition to the negative magnetoresistance in 3D met-
als, the temperature dependence of the resistivity
r
in 3D
disordered superconductors at low temperatures and high
magnetic fields is another interesting issue. The question re-
garding whether the resistivity continues to increase on cool-
ing, similarly to the diverging behavior of
r
in 2D supercon-
ductors at high fields,
8
or saturates at low temperatures has
not been addressed experimentally.
In this paper, we present an experimental investigation of
the electron transport properties of homogeneous amorphous
superconducting films of Mo
3
Si and Nb
3
Ge, under applied
magnetic fields (
H
) up to 30 T and at temperatures (
T
)
down to 35 mK. We find that both temperature and field
dependences of the resistivity
r
can be qualitatively de-
scribed by the localization theory.
1,2
In addition, we report
detailed studies of the upper critical field
H
c
2
in both com-
pounds and find that the low-temperature
H
c
2
behavior dis-
agrees with conventional theory
9
involving the paramagnetic
effect in dirty superconductors. This result may be qualita-
tively described in terms of a diverging paramagnetic limit in
disordered superconductors.
II. EXPERIMENT
The samples used in this work are three 1700-Å-thick
a
-
Mo
3
Si films and a 200-Å-thick
a
-Nb
3
Ge film, all deposited
on cold sapphire substrates
~
held at 77 K
!
using rf
sputtering.
10
The homogeneity of the amorphous nature of
these films is confirmed with x-ray diffraction. Tunneling
studies in
a
-Mo
3
Si
~
Ref. 11
!
reveal a BCS-like energy gap
D
, with 2
D
/
k
B
T
c
'
3.5 (
k
B
is the Boltzmann constant, and
T
c
is the superconducting transition temperature
!
. The zero-
field
T
c
values for
a
-Mo
3
Si and
a
-Nb
3
Ge are 7.9 K and 2.9
K, with transition widths 20 mK and 80 mK, respectively.
Most experiments reported here were carried out at the Na-
tional High Magnetic Field Laboratory
~
NHMFL
!~
Tallahas-
see, FL
!
, on two samples, one
a
-Mo
3
Si and the other
a
-
Nb
3
Ge. At the NHMFL, a 20 T superconducting solenoid is
used for measurements below 0.6 K and a 30 T resistive
PHYSICAL REVIEW B
1 JANUARY 1998-II
VOLUME 57, NUMBER 2
57
0163-1829/98/57
~
2
!
/1206
~
8
!
/$15.00
1206
© 1998 The American Physical Society
magnet is used for measurements above 0.4 K. The magnetic
field is always perpendicular to the film surface. The four-
probe lock-in technique at an ac-current frequency 13.1 Hz
was employed. The Joule heating at low temperatures was
limited to
Q
5
10
2
6
W/m
2
. Although the Kapitza thermal
boundary resistance
R
K
between
a
-Mo
3
Si and
a
-Nb
3
Ge and
helium is not known, we take the largest value of
R
K
;
0.1
m
2
K
4
/W(1/
T
3
) available in literature
12
for the boundary be-
tween
3
He and a solid to estimate the upper limit of the
overheating
D
T
5
R
K
Q
.At
T
5
35 mK,
D
T
'
3 mK.
Measurements at
T
.
1.4 K and for
H
,
15.5 T are per-
formed at Ecole Polytechnique
~
France
!
and at Caltech on all
samples, and results for all three
a
-Mo
3
Si samples are found
to be consistent.
13
III. RESULTS
Before presenting the experimental results, it is worth-
while to first verify the dimensionality of our samples, in the
context of both superconductivity and localization, by com-
paring the thickness of the samples with relevant length
scales. For superconductivity, the length for comparison is
the coherence length
j
5
@
\
/(2
e
m
0
H
c
2
)
#
1/2
(
\
is the Planck
constant,
e
is the electron charge, and
m
0
is the permeability
of vacuum
!
. From our
H
c
2
data
~
see below
!
, we obtain
j
(0)
5
49 Å and 66 Å for
a
-Mo
3
Si and
a
-Nb
3
Ge, respec-
tively. Therefore, at low temperatures
@
where
j
(
T
)
,
d
,
d
is
the thickness of the sample
#
we expect the samples to be in
the 3D regime. It is worth noting that Theunissen and Kes
14
have studied the fluctuation conductivity in
a
-Nb
3
Ge and
a
-MoGe films, and have found that the data on the films with
thicknesses up to 10
j
( 0 ) are better described by 2D than 3D
scaling theory.
15
However, this 2D scaling is observed in the
vicinity of the transition temperature
14
where the coherence
length becomes comparable to or larger than the sample
thickness. Therefore, the finding of Theunissen and Kes
14
does not contradict our conjecture about the 3D character of
superconductivity in our films at low temperatures.
Of more relevance to the main theme of the present paper
is the dimensionality with respect to weak localization. In
this case, the characteristic length is min
@
L
f
,
L
H
#
, where
L
f
5
(
D
t
in
)
1/2
is the phase-coherence length (
D
5
1
3
v
F
l
is the
diffusion coefficient,
t
in
is the inelastic scattering time,
v
F
is
the Fermi velocity, and
l
is the mean free path
!
and
L
H
5
@
2
p
\
/(2
e
m
0
H
)
#
1/2
.
1
In the field range 10–30 T,
L
H
5
140–80 Å. Consequently, at high magnetic fields
~
where
L
H
,
d
) , our samples of both
a
-Mo
3
Si and
a
-Nb
3
Ge are in
the 3D regime in the context of weak localization.
The representative
R
-vs-
H
curves (
R
is the resistance
!
for
both
a
-Mo
3
Si and
a
-Nb
3
Ge are shown in Fig. 1
~
top and
bottom panels, respectively
!
. The distance between the volt-
age contacts on the films is approximately equal to the film
width, and so the resistivity
r
'
Rd
. In the normal state,
r
'
110
m
V
cm and 190
m
V
cm for
a
-Mo
3
Si and
a
-Nb
3
Ge,
respectively. With decreasing temperature, the field-induced
superconducting transition occurs at higher fields and be-
comes sharper.
In order to better demonstrate the decrease in the transi-
tion width with decreasing temperature, we shift the curve at
T
5
35 mK for
a
-Mo
3
Si along the field axis by
m
0
D
H
52
10.6 T and that for
a
-Nb
3
Ge by
m
0
D
H
52
4.4 T, as
illustrated by the dashed lines in Fig. 1. Comparing the
shifted
R
-vs-
H
curves with the higher-temperature isotherms
taken at
T
5
7 K and 2 K for
a
-Mo
3
Si and
a
-Nb
3
Ge, respec-
tively, it is evident that the transition broadens with increas-
ing temperature, although this broadening is much smaller
than, for instance, that in high-temperature superconductors
~
HTSC’s
!
. In HTSC’s, the higher operation temperatures,
larger anisotropy, and smaller coherence length relative to
those in conventional superconductors
~
such as
a
-Mo
3
Si and
a
-Nb
3
Ge) are known
16
to yield significantly enhanced ther-
mal fluctuations and reduced vortex pinning, hence a broad
resistivity transition.
~
For an example of the comparison of
the vortex dynamics in HTSC’s and
a
-Mo
3
Si, see Ref. 17.
!
The uppermost
~
high-resistance
!
parts of the resistivity
curves for
a
-Mo
3
Si and
a
-Nb
3
Ge are presented in Fig. 2 and
Fig. 3. Shown in Fig. 2 are the isotherms
R
(
H
) , whereas Fig.
3 presents the
R
(
T
) dependences at different fields. In the
normal state, the resistance of the both samples increases
monotonically with
H
~
Fig. 2
!
. Below the zero-field transi-
tion temperature
T
c
(0), the
field-induced superconducting to
normal-state transition is followed by a resistivity decrease
with increasing magnetic field up to the maximum available
value of 30 T. With decreasing temperature, the decrease in
resistivity at high fields becomes more pronounced.
From the resistance
R
-vs-
H
measurements, we construct
the temperature dependences shown in Fig. 3. A magnetic
field shifts the transition to lower temperatures
@
see data on
H
c
2
(
T
) in Fig. 4
#
. There is a well-defined field
~
13.8 T for
a
-
Mo
3
Si and 7.8 T for
a
-Nb
3
Ge) , at which the resistance no
FIG. 1. The resistance (
R
) vs magnetic field (
H
) isotherms of
a
-Mo
3
Si
~
top
!
and
a
-Nb
3
Ge
~
bottom
!
. The temperature of each
isotherm is indicated near each curve. The dashed lines depict the
isotherms for
T
5
35 mK shifted along the
H
axis by
m
0
D
H
52
10.6 T for
a
-Mo
3
Si and by
m
0
D
H
52
4.4 T for
a
-Nb
3
Ge.
57
1207
ELECTRON LOCALIZATION EFFECTS ON THE LOW- . . .
longer decreases with the decreasing temperature.
~
The large
scattering of points at 13.8 T for
a
-Mo
3
Si, Fig. 3, left panel,
is due to a rapid change in the resistance near this field at low
temperatures; see Fig. 2, left panel
!
. Above this field,
R
in-
creases monotonically upon cooling and eventually flattens
at low
T
for both systems.
Figure 5 presents the resistivity data at temperatures
above
T
c
. In both systems, the magnetoresistance decreases
rapidly with increasing temperature, and at
T
5
30.2 K there
is practically no magnetic field dependence in the resistivity.
It is interesting to note that in
a
-Mo
3
Si
~
Fig. 5, left panel
!
there is a distinct change of slope in the
R
-vs-
H
isotherms
for
T
5
13.2 K and 16.4 K at a field
m
0
H
'
13–14 T. Below
this field, the resistivity increases with increasing field, and
above this field, the resistivity is almost field independent.
Interestingly, this crossover field nearly coincides with a
characterictic field where the low-temperature resistance is
T
independent
~
see the isotherm at
m
0
H
'
13.8 T, Fig. 3, left
panel
!
. This crossover field observed at
T
.
T
c
is also com-
parable to the zero-temperature upper critical field (
m
0
H
c
2
'
13.7 T, Fig. 4, upper panel, inset
!
.InNb
3
Ge
~
Fig. 5, right
panel
!
, on the other hand, no sharp features are observed in
the magnetic field dependences of
R
near
m
0
H
c
2
(
T
5
0)
'
7.5 T, although some slower increase in
R
can be seen
near
m
0
H
c
2
(
T
5
0) at
T
5
7.0 K and 9.0 K. Unlike in
a
-
FIG. 2. The uppermost
~
high-
R
) parts of the isotherms of
R
for
a
-Mo
3
Si
~
left panel
!
and for
a
-Nb
3
Ge
~
right panel
!
. The isotherms,
in the left panel, correspond to temperatures
T
5
35 mK, 0.42 K,
1.015 K, 2 K, 4.2 K, 5.5 K, 7 K, and 9 K. In the right panel,
T
5
35 mK, 0.42 K, 0.83 K, 1.225 K, 2 K, and 4.2 K.
FIG. 3. The uppermost
~
high-
R
) parts of the isomagnetic curves
of
R
for
a
-Mo
3
Si
~
left panel
!
and for
a
-Nb
3
Ge
~
right panel
!
.
FIG. 4. The upper critical field (
H
c
2
) vs temperature (
T
) for
a
-
Mo
3
Si
~
top
!
and
a
-Nb
3
Ge
~
bottom
!
.
H
c
2
is determined using the
criterion
r
5
0.9
r
n
(
r
n
is the normal-state resistivity
!
. Shown in the
insets are the low-temperature parts of the
H
c
2
(
T
) curves.
FIG. 5.
R
vs
H
at high temperatures for
a
-Mo
3
Si
~
left panel
!
and
a
-Nb
3
Ge
~
right panel
!
. The arrow on the left panel marks the
field
m
0
H
c
5
13.8 T. Below this field, the resitivity increases with
increasing field, and above this field, the resistivity is almost field
independent.
1208
57
A. V. SAMOILOV, N.-C. YEH, AND C. C. TSUEI
Mo
3
Si, the resistivity of
a
-Nb
3
Ge appears to increase with
field up to the highest value
~
30 T
!
of our experiment.
IV. LOCALIZATION AND INTERACTION EFFECTS
ON THE CONDUCTIVITY
Next, we consider the physical significance of the data. At
low temperatures, the decrease of the resistivity with increas-
ing field above
H
c
2
can be well described in terms of theory
of localization
~
see Ref. 1 for review
!
. In this context, a
negative correction to the classical Boltzmann conductivity
s
B
arises from the interference of two electron paths which
are on the same closed loop and are moving in two opposite
directions.
1
The existence of such loops results in a localiza-
tion of electrons,
1
provided that the phase coherence of the
electron wave functions along these two paths can be main-
tained. Hence, a localization of electrons may occur if the
phase coherence is not broken by inelastic scattering pro-
cesses or by a magnetic field.
The effect of a magnetic field enters the localization prob-
lem via the characteristic length
L
H
5
@
2
p
\
/(2
e
m
0
H
)
#
1/2
,
and that of temperature enters through
L
f
(
T
) : The phase
coherence associated with the occurrence of localization is
destroyed if the loop size is greater than
L
H
or
L
f
(
T
) . Con-
sequently, the conductivity increases with increasing
H
or
T
.
Thus, both the negative field coefficient for
H
.
H
c
2
(
T
) and
the negative temperature coefficient of
r
can be explained by
the destructive influence of the field and temperature, respec-
tively, on the interference effects.
In the following, we consider various correction terms to
the electrical conductivity of 3D conductors. At zero tem-
perature, the quantum-corrected conductivity of a 3D disor-
dered metal is
12,18
s
0
5
s
B
F
1
2
3
~
k
F
l
!
2
G
,
~
1
!
where
k
F
is the Fermi wave vector, and
s
B
is the conductiv-
ity in the classical limit. Next, we consider the temperature-
dependent localization correction to the conductivity of a 3D
sample in zero field, which is given by
1,12
D
T
s
loc
5
e
2
p
2
\
1
L
f
.
~
2
!
Assuming a power law in the temperature dependence of the
inelastic scattering time
t
in
;
T
2
p
, with an exponent
p
.
1
depending on the scattering mechanism, we have
L
f
;
T
2
p
/2
and
D
T
s
loc
;
T
p
/2
.
The magnetic field effect on the electron localization has
been discussed by Kawabata,
2
which results in a correction
term to the conductivity:
D
H
s
loc
5
e
2
2
p
2
\
A
e
~
m
0
H
!
\
f
~
x
!
,
~
3
!
where
x
5
\
/
@
4
e
(
m
0
H
)
L
f
2
#
, and the asymptotic forms for
f
(
x
) are
f
(
x
)
5
0.605 for
x
!
1 and
f
(
x
)
5
(
x
2
3/2
/48) for
x
@
1 . In the limit of small
x
, which corresponds to either large
fields or weak inelastic scattering, the magnetoconductivity
(
D
H
s
loc
) is temperature independent:
D
H
s
loc
'
2.90
~
m
0
H
!
1/2
,
~
4
!
where
s
is in
V
2
1
cm
2
1
and
m
0
H
in T.
In disordered conductors, the Coulomb interaction be-
tween electrons often has an important effect on the conduc-
tivity, because of the existence of closely spaced energy lev-
els of electrons which experience the same disorder
potential. The small energy difference
e
of two electrons
results in a long time scale
\
/
e
, during which the electrons
are undistinguishable, and their scattering amplitudes add up
due to the large number of phase-coherent paths with char-
acteristic times smaller than
\
/
e
. This interaction correction
to the conductivity
D
T
s
int
can be estimated by using Eq.
~
2
!
,
with the inelastic scattering time
t
in
in
L
f
(
5
A
D
t
in
) re-
placed by
\
/
e
. As shown by Al’tshuler and Aronov, the
correction for the Coulomb interaction term in 3D samples
becomes
1,6
D
T
s
int
5
e
2
4
p
2
\
1.3
A
2
S
4
3
2
3
2
F
̃
D
A
k
B
T
\
D
,
~
5
!
where
4
3
comes from the exchange interaction among the
electrons and
3
2
F
̃
from the Hartree interaction ( 0
,
F
̃
,
1).
1
The interaction correction in a magnetic field is given by
1
D
H
s
int
52
e
2
4
p
2
\
F
̃
A
k
B
T
2
\
D
g
~
h
!
,
~
6
!
where
h
5
g
m
B
H
/
k
B
T
(
g
is the
g
factor,
m
B
the Bohr mag-
neton
!
, and the function
g
(
h
) has the following asymptotic
behavior:
g
(
h
)
5
A
h
2
1.3 for
h
@
1 and
g
(
h
)
5
0.053
h
2
for
h
!
1.
After considering all the above corrections, we obtain the
total conductivity as follows:
s
5
s
0
1
D
H
s
1
D
T
s
,
~
7
!
where
@
see Eqs.
~
2
!
–
~
6
!# D
H
s
5
D
H
s
loc
1
D
H
s
int
,
D
T
s
5
D
T
s
loc
1
D
T
s
int
2
e
2
p
2
\
~
k
F
l
!
2
3
l
in
~
T
!
,
and
l
in
5
v
F
t
in
is the inelastic electron mean free path. The
last term in
D
T
s
is the result of thermal excitations of vari-
ous inelastic processes.
12,18
V. DISCUSSION
A. Estimates of various correction components
to the conductivity
Based on the above consideration, we find that at high
fields and low temperatures, both localization and interaction
terms in the magnetoconductivity
@
Eqs.
~
3
!
and
~
6
!
, respec-
tively
#
are proportional to
H
1/2
. In order to calculate
D
H
s
,
we first plot the total conductivity
s
as a function of
H
1/2
~
not shown
!
and obtain
s
(
H
5
0)
5
s
0
1
D
T
s
from the linear
extrapolation of the
s
-vs-
H
1/2
dependences at low tempera-
tures to zero field. Hence,
D
H
s
can be obtained by subtract-
ing
s
(
H
5
0 ) data from the total conductivity. In Fig. 6 we
plot the magnetoconductivity
D
H
s
vs
H
1/2
. For
a
-Mo
3
Si and
a
-Nb
3
Ge,
s
(
H
5
0)
'
7600 (
V
cm)
2
1
and 5200 (
V
cm)
2
1
,
respectively. At low temperatures,
s
(
H
5
0 ) is temperature
57
1209
ELECTRON LOCALIZATION EFFECTS ON THE LOW- . . .
independent, and we can associate it with the zero-
temperature conductivity
s
0
which includes the quantum
correction for localization
@
Eq.
~
1
!#
.
12
The linear slope of the
s
-vs-
H
1/2
dependences is approximately temperature inde-
pendent for the data taken at
T
5
35–180 mK and 8
T
,
m
0
H
,
18 T in the case of
a
-Nb
3
Ge, and in the case of
a
-
Mo
3
Si for
T
5
0.42–2 K and 15
T
,
m
0
H
,
30 T.
@
For
a
-
Mo
3
Si, we cannot determine the slope of the
s
-vs-
H
1/2
de-
pendences down to lower temperatures because of the higher
H
c
2
and the limited field range for accessing the the normal-
state behavior of the superconducting
a
-Mo
3
Si films in the
dilution refrigerator:
m
0
H
c
2
(0)
'
13.7
T
,
m
0
H
,
18 T.
#
The dashed line in Fig. 6 depicts the theoretical
D
H
s
loc
curve according to Eq.
~
3
!
. The agreement between the the-
oretical
D
H
s
loc
and experiment is good in
a
-Nb
3
Ge, suggest-
ing that the contribution of
D
H
s
int
is not significant. On the
other hand, in
a
-Mo
3
Si, the experimental value of
D
H
s
is 2
times larger than the theoretical prediction for
D
H
s
loc
. Al-
though the origin of this discrepancy is not understood, a
similar trend has been observed in granular Al films by Chui
et al.
:
7
Samples with low resistivity have
d
D
H
s
/
d
(
H
1/2
)up
to 3 times higher than that predicted by Kawabata.
2
We note
that neither the Coulomb interaction correction nor consider-
ation of superconducting fluctuations can reduce the discrep-
ancy between theory and experiment because both mecha-
nisms
result
in
a
negative
contribution
to
the
magnetoconductivity. Therefore, significant corrections are
needed to the
D
H
s
loc
term given by Kawabata.
In order to make a better comparison of the experimental
D
H
s
with theory, we consider the conductivity contributions
due to both the superconducting fluctuation effects (
s
fl
) and
the Coulomb interaction (
D
H
s
int
) . The fluctuation conduc-
tivity has been calculated by Ullah and Dorsey
15
in the
lowest-Landau-level limit which is applicable to the high-
field data. A convenient estimate for
s
fl
, using the theory by
Ullah and Dorsey,
15
has been given in Ref. 14. Combining
Eqs.
~
8
!
and
~
9
!
of Ref. 14, one can show that in the 3D
regime, the fluctuation conductivity of isotropic supercon-
ductors in the dirty limit is
s
fl
3D
'
1.447
s
0
A
0
3D
e
H
1/2
t
,
~
8
!
where
A
0
3D
5
2
A
2Gi
~
Gi is the Ginzburg parameter
!
,
e
H
5
t
2
1
1
h
,
t
5
T
/
T
c
(0), and
h
5
H
/
H
c
2
(0).
Using
h
5
2,
T
5
35
mK, and Gi
;
10
2
5
~
Ref. 14
!
, we obtain
s
fl
3D
'
0.43
(
V
cm)
2
1
for
a
-Mo
3
Si and 0.8 (
V
cm)
2
1
for
a
-Nb
3
Ge. In
view of the discussion of the dimensionality in the beginning
of Sec. III, we also estimate the fluctuation conductivity in
the 2D regime,
s
fl
2D
, for our
a
-Nb
3
Ge film which has a thick-
ness
d
'
3
j
(0)
@
see Eqs.
~
5
!
–
~
7
!
of Ref. 14
#
:
s
fl
2D
'
1.447
s
0
A
0
2D
e
H
t
,
~
9
!
where
A
0
2D
5
4
A
2Gi
j
(0)/
d
. Using the same values of Gi,
h
,
and
t
as for the estimate of
s
fl
3D
, we obtain
s
fl
2D
'
0.53
(
V
cm)
2
1
for
a
-Nb
3
Ge. Comparing with the data shown in
Fig. 6, we conclude that the effect of superconducting fluc-
tuations on
D
H
s
at low temperatures and large magnetic
fields may be neglected
~
see Fig. 6
!
.
The interaction term in the magnetoconductivity (
D
H
s
int
)
at low temperatures and high fields
@
h
@
1 , Eq.
~
6
!#
is
D
H
s
int
52
e
2
4
p
2
\
F
̃
A
g
m
B
H
2
\
D
.
~
10
!
The diffusion coefficient
D
can be estimated from the
H
c
2
data
~
Fig. 4
!
by the relation
9
D
5
4
k
B
p
e
S
2
dT
c
2
d
~
m
0
H
c
2
!
D
,
which yields
D
'
4
3
10
2
5
m
2
/ s for
a
-Mo
3
Si and 2.4
3
10
2
5
m
2
/ s for
a
-Nb
3
Ge. Taking
g
5
2 , we obtain from Eq.
~
8
!
,
D
H
s
int
'
2
3
F
̃
(
m
0
H
)
1/2
(
s
is in
V
2
1
cm
2
1
,
m
0
H
in T
!
for
a
-Mo
3
Si and
2
3.9
F
̃
(
m
0
H
)
1/2
for
a
-Nb
3
Ge. If we further
assume
F
̃
'
1 , then in order to account for the experimental
value
D
H
s
'
5.8(
m
0
H
)
1/2
in
a
-Mo
3
Si, we have to assume
that the localization term is
D
H
s
loc
'
( 5.8
1
3) (
m
0
H
)
1/2
5
8.8 (
m
0
H
)
1/2
, approximately 3 times larger that the theo-
retical prediction given by Eq.
~
4
!
. Similarly, in
a
-Nb
3
Ge,
with an experimental value
D
H
s
'
2.9(
m
0
H
)
1/2
, the localiza-
tion contribution would be
D
H
s
loc
'
( 2.9
1
3.9) (
m
0
H
)
1/2
5
6.8 (
m
0
H
)
1/2
. On the other hand, although it is difficult to
obtain the dimensionless parameter
F
̃
with certainty,
theory
1,3
suggests
F
̃
!
1 rather than
F
̃
'
1 . Indeed,
F
̃
depends
on another parameter
X
;
(
n
/10
24
)
1/3
, where
n
is the carrier
density in m
2
3
. The Hall effect measurements on
a
-Mo
3
Si
~
see below
!
suggest that
n
@
10
24
m
2
3
. Hence,
X
@
1 . In this
limit
F
̃
!
1,
1,3
and the interaction term
D
H
s
int
;
F
̃
becomes
negligible.
Knowing
s
0
, we can compute, for a given magnetic field,
the correction to the conductivity
D
s
5
s
2
s
0
as a function
of temperature
~
Fig. 7, shown for
m
0
H
5
15 T
!
. At low tem-
peratures (
T
,
0.3 K
!
,
D
s
is nearly temperature independent,
so that
D
T
s
'
0 and
D
s
'D
H
s
at
m
0
H
5
15 T. In the tem-
perature interval 0.3–30 K, the behavior of
D
s
is determined
by the interplay of several factors:
D
T
s
loc
augments with
increasing
T
@
Eq.
~
2
!#
because of the decreasing
L
f
,
D
H
s
(
'D
H
s
loc
,
because
D
H
s
int
'
0 )
decreases
when
x
5
\
/
@
4
e
(
m
0
H
)
L
f
2
#
becomes comparable to unity
@
Eq.
~
3
!#
,
the Aslamazov-Larkin superconducting fluctuation conduc-
FIG. 6.
D
s
as a function of
H
1/2
for
a
-Mo
3
Si
~
MS
!
and
a
-
Nb
3
Ge
~
NG
!
. The data for
a
-Nb
3
Ge were taken at a lower dissipa-
tion level than those for
a
-Mo
3
Si, and, therefore, they are noisier
than the data for
a
-Mo
3
Si.
1210
57
A. V. SAMOILOV, N.-C. YEH, AND C. C. TSUEI
tivity vanishes with increasing temperature, and the interac-
tion term
D
T
s
int
~
see below
!
is small in the temperature re-
gion 0.3–30 K. Hence, we conclude that the increase in
D
s
with increasing temperature
~
for 0.3 K
,
T
,
30 K
!
is largely
associated with the temperature dependence of the localiza-
tion correction
D
T
s
loc
@
Eq.
~
2
!#
.
Above
T
5
30 K the magnetoresistance becomes insignifi-
cant
~
Fig. 5
!
. Therefore, the behavior of
D
s
(
T
) for
T
.
30 K
is determined predominantly by the zero-field corrections to
the conductivity, i.e.,
D
s
(
T
.
30 K)
'D
T
s
(
T
.
30 K) . We
may estimate
D
T
s
int
by using Eq.
~
5
!
and by taking
F
̃
!
1.
We obtain
D
T
s
int
'
4.6
A
T
in
a
-Mo
3
Si and 5.9
A
T
in
a
-
Nb
3
Ge (
s
is in
V
2
1
cm
2
1
,
T
in K
!
. These values account
for approximately 25%–35% of
D
s
~
Fig. 7
!
at
T
5
30–300
K. Subtracting these values of
D
T
s
int
from the experimental
data of Fig. 7 at high temperatures, we can obtain
D
T
s
loc
~
Fig. 7, inset
!
. Comparing
D
T
s
loc
with Eq.
~
2
!
gives
L
f
'
3000/
T
~
in Å
!
for
a
-Mo
3
Si and 2300/
T
~
in Å
!
for
a
-
Nb
3
Ge. The temperature dependence of
L
f
for
T
.
30Kis
consistent with our earlier conjecture that
L
f
;
T
2
p
/2
and
p
.
1.
In Fig. 8, we show the Hall effect data measured on an-
other
a
-Mo
3
Si thin film whose resistance has been measured
at fields below 15.5 T and at temperatures above 1.4 K and,
within this range of the experimental parameters, shows
properties consistent with those of the
a
-Mo
3
Si film de-
scribed earlier. In the mixed state, there is a sign change in
the Hall resistivity
r
xy
, which has been observed in many
type-II superconductors, including high-
T
c
~
Ref. 19
!
and
amorphous
20
superconductors. We shall not concern our-
selves with the mixed-state Hall effect in this paper. In the
normal state, the Hall coefficient
R
H
5
r
xy
/(
m
0
H
) is positive
and appears to increase on cooling from
T
5
20Kto1.4Kby
approximately 20%. The behavior of the Hall coefficient in
disordered conductors is an interesting issue in its own right.
However, small signal-to-noise ratio (
'
20) in our Hall ef-
fect data does not allow us to quantify the temperature de-
pendence of
R
H
.
Assuming
r
xy
/(
m
0
H
)
5
1/(
ne
) and taking
R
H
5
r
xy
/
(
m
0
H
)
'
2n
V
cm/ T
~
Fig. 8
!
, we obtain for the hole density
n
5
3
3
10
29
m
2
3
. From the Hall angle
r
xy
/
r
5
v
c
t
, where
v
c
5
e
(
m
0
H
)/
m
*
is the cyclotron frequency, we estimate
t
'
10
2
16
m
*
/
m
s(
m
is the free electron mass, and
m
*
is the
effective electron mass
!
. From
n
we obtain the Fermi wave
number
k
F
5
(3
p
2
n
)
1/3
'
2Å
2
1
, and the Fermi velocity
v
F
5
(
\
/
m
*
)
k
F
'
(2
3
10
6
)
m
/
m
*
m/s. Thus, the mean free path
is
l
5
v
F
t
'
2 Å and the parameter
k
F
l
'
4 . Compared with
the diffusion coefficient obtained from
H
c
2
data, the result
D
5
1
3
v
F
l
'
1.3
3
10
2
4
m
/
m
*
m
2
/ s is suggestive of
m
*
'
0.3
m
. However, we note that the value of
n
exceeds those
in normal metals
~
like Cu, Al, Au, etc.
!
and seems to be
somewhat overestimated. Furthermore, the assumption of a
well-defined Fermi vector
k
F
in amorphous conductors is,
strictly speaking, not accurate. So the estimates of
k
F
and
D
from the carrier density
n
and the Hall angle should be con-
sidered as an order-of-magnitude approximation only. None-
theless, we may still estimate the inelastic mean free path in
a
-Mo
3
Si at high temperatures by the relation
l
in
5
3
L
f
2
/
l
'
1.4
3
10
7
/
T
2
@
Å
#
. The
T
2
2
dependence is a signature of the
electron-electron scattering.
B. Upper critical field
Another interesting point for discussion is the zero-
temperature value of the upper critical field
~
Fig. 4
!
. Using
the slope of the upper critical field at
T
5
T
c
(0), we
compute
the parameter
h
c
2
(0)
5
H
c
2
(0)
@
2
dT
c
(0)/
dH
c
2
#
u
T
5
T
c
(0)
/
T
c
(0)
5
0.72 for
a
-Mo
3
Si and 0.65 for
a
-Nb
3
Ge. These val-
ues are close to the result
h
c
2
(0)
5
0.69 for the orbital critical
field
H
c
2
orb
5
0.69
@
2
dH
c
2
/
dT
c
(0)
#
u
T
5
T
c
(0)
T
c
( 0 ) in the stan-
dard Werthamer-Helfand-Hohenberg
~
WHH
!
theory
21
of
dirty superconductors. On the other hand, it is also necessary
to compare the empirical upper critical field with the charac-
teristic field (
H
p
) for the paramagnetic limit, where
m
B
H
p
;
k
B
T
c
( 0 ) or, more precisely,
22
H
p
5
A
2
D
(0)/
g
m
B
, with
D
( 0 ) being the zero-temperature energy gap. Again assum-
ing
g
5
2 , we have
m
0
H
p
5
14.5 T and 5.3 T for
a
-Mo
3
Si and
a
-Nb
3
Ge, respectively. The paramagnetic limit is supposed
to reduce the upper critical field below its orbital value ac-
cording to the formula
H
c
2
(0)
2
2
5
H
p
2
2
1
(
H
c
2
orb
)
2
2
. In both
a
-Mo
3
Si and
a
-Nb
3
Ge, the paramagnetic effect appears neg-
ligible. It is particularly remarkable in the case of
a
-Nb
3
Ge,
where the experimental value of
m
0
H
c
2
(0)
5
7.5 T exceeds
that of
m
0
H
p
5
5.3 T substantially. One possible explanation
may be related to the estimate of the paramagnetic limit in
disordered superconductors: According to Spivak and
Zhou,
23
the
g
factor in disordered superconductors may take
any values because of electron localization. Therefore, it is
only a question of probability for finding regions in the dis-
FIG. 7. The temperature dependence of
D
s
. Inset:
D
T
s
loc
vs
T
~
open circles for
a
-Mo
3
Si, lower curve, and solid squares for
a
-
Nb
3
Ge, upper curve
!
.
FIG. 8. The temperature dependence of
r
xy
/(
m
0
H
) for
a
-
Mo
3
Si. The applied field values are shown near the curves, in units
of tesla.
57
1211
ELECTRON LOCALIZATION EFFECTS ON THE LOW- . . .
ordered sample where the condition
m
B
H
p
@
k
B
T
c
( 0 ) may be
satisfied. Thus, the paramagnetic limit
H
p
in disordered su-
perconductors may be very large, which explains our obser-
vations and earlier reports
24
of large
H
c
2
in amorphous ma-
terials. However, the upward curvature in the
H
c
2
(
T
) line as
predicted by Spivak and Zhou
23
is not observed in our
samples down to
T
5
35 mK. This issue requires further the-
oretical investigation.
C. Comparison with 2D amorphous superconductors
In ultrathin amorphous superconducting films
~
with typi-
cal thickness of a few nm
!
, the so-called ‘‘zero-temperature
magnetic-field-induced superconductor-to-insulator transi-
tion’’
25
has been observed.
8,26
The experimental signature of
this transition is a scaling relation for the sheet resistance:
25
R
~
H
,
T
!
5
R
c
F
~
u
H
2
H
c
u
/
T
1/
z
n
!
,
~
11
!
where
R
c
and
H
c
are the critical resistance and critical field,
respectively
@
R
(
H
5
H
c
)
5
R
c
#
,
F
(
H
,
T
) the scaling function,
and
z
and
n
the critical exponents. In the work by Hebard
and Paalanen on
a
-InO
x
,
8
the value of
R
c
'
5k
V
is quite
close to the theoretical estimate (
R
Q
) by Fisher,
25
where
R
Q
5
h
/4
e
2
'
6.4 k
V
is the quantum resistance, and
z
n
'
1.3.
In a later work by Yazdani and Kapitulnik on
a
-Mo
x
Ge,
26
the critical resistance has been shown to be nonuniversal,
ranging from 600
V
to2k
V
, with a comparable value of
z
n
'
1.36.
For comparison, we perform detailed resistivity measure-
ments in 3D
a
-Mo
3
Si films near the field
m
0
H
5
13.8 T
where the resistivity is approximately temperature indepen-
dent over a wide temperature range
~
Fig. 9, top panel
!
. The
bottom panel of Fig. 9 shows a successful attempt to scale
the data using Eq.
~
9
!
.
@
We note that in our
a
-Nb
3
Ge film,
there is no substantial field and temperature range where
scaling given by Eq.
~
9
!
works.
#
The form of the scaling
function is similar to that reported in Refs. 8 and 26. How-
ever, there are several important differences between our
data and those on the ultrathin amorphous films.
8,26
First, the
effective critical resistance of our 3D
a
-Mo
3
Si films
~
of the
order of ohms
!
is much smaller than the
R
c
value reported in
Refs. 8 and 26. Second, the apparent exponent is
z
n
'
( 1.35)
2
1
for the
a
-Mo
3
Si films, compared with
z
n
'
1.3 in
ultrathin films.
8,26
Third, at low temperatures, where the scal-
ing given by Eq.
~
9
!
is supposed to work well, we find that
the scaling relation actually breaks down when the resistance
of
a
-Mo
3
Si becomes temperature independent
~
see Fig. 2
!
.
The puzzling scaling behavior of our
a
-Mo
3
Si films at
finite temperatures may be simply coincidental, because the
theoretical prediction
25
is developed for 2D amorphous su-
perconductors. We note that the key assumption of the field-
tuned phase transition
25
is that only the phase of the order
parameter is relevant for the occurrence of this phase transi-
tion at
T
5
0 . The same assumption is likely to break down in
the case of a 3D superconductor, because the modulation of
the amplitude of the order parameter may no longer be neg-
ligible. We also caution that the mere existence of scaling
behavior is not sufficient to prove a true second-order phase
transition at
T
5
0 . Nonetheless, the seemingly excellent scal-
ing of the resistivity data in Fig. 9 may be suggestive of
interesting physics for future investigation.
In addition to the variety of interesting phenomena asso-
ciated with the magnetoconductivity, upper critical field, and
scaling of the electrical resistivity of amorphous supercon-
ductors at low temperatures, we also note the possibility of
quantum vortex lattice melting
27
at
T
5
0 and below
H
c
2
,
provided that the sheet resistance
R
is a significant fraction
of the quantum resistance
R
Q
. However, the amorphous
films presented in this work yield
R
!
R
Q
(
R
'
6.5
V
and 95
V
for
a
-Mo
3
Si and
a
-Nb
3
Ge, respectively
!
, suggesting that
the issue of quantum melting of the vortex system is difficult
to resolve with certainty. This topic is beyond the scope of
our current study and is better considered in Refs. 27 and 28
where the results of the non-Ohmic transport measurements
are presented.
VI. SUMMARY
In summary, we have investigated the magnetoconductiv-
ity of three-dimensional amorphous films of
a
-Mo
3
Si and
a
-
Nb
3
Ge in magnetic fields up to 30 T and temperatures down
to 35 mK. A decrease in the resistivity with increasing field
is observed above
H
c
2
in both compounds at low tempera-
tures. This decrease of field-induced resistivity agrees within
a factor of 2 with the localization theory. At higher tempera-
tures, above
T
c
(0), in
a
-Mo
3
Si there is a significant cross-
over from strong field dependence to weak field dependence
of
r
at 13–14 T, near its upper critical field. This feature is
not present in
a
-Nb
3
Ge. The temperature dependence of the
conductivity in the normal state is found to be dominated by
the localization corrections, and the phase-coherence length
is estimated at high temperatures in both
a
-Mo
3
Si and
a
-
Nb
3
Ge. The combination of the normal-state Hall effect and
resistivity data in
a
-Mo
3
Si allows a determination of the car-
rier concentration
n
and mean free path
l
, although the
FIG. 9.
R
/
R
c
vs
T
for
a
-Mo
3
Si at magnetic fields near 13.8 T
~
top
!
. Scaling of the data from the top panel according to Eq.
~
9
!
is
shown in the bottom panel.
1212
57
A. V. SAMOILOV, N.-C. YEH, AND C. C. TSUEI