Qiu X.G. (Ed.) High Temperature Superconductors
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conductivity tensor (and Onsager’s relations) as
σ
xy
=
ρ
xy
/(
ρ
xx
ρ
yy
–
ρ
xy
ρ
yx
). In
the case where the Hall coefficient is much lower than the
ρ
xx
, this formula can be
simplified to
σ
xy
=
ρ
xy
/
ρ
2
xx
where
ρ
xy
= R
xy
d, with d the thickness of the sample.
The Hall coefficient at 50 T is less than 5% of the magnetoresistivity at 50 T, so
this satisfies the conditions for this simplification. Since at low temperatures, a
high magnetoresistivity is observed,
ρ
xx
and
ρ
xy
will be replaced by their high
field data. The Hall conductivity as a function of the temperature is given in
Fig. 2.33(a). At high temperatures all the samples show the same characteristic
functional behavior. This can be related to the theory
ρ
xy
=
α
C
ρ
B
xx
/(
Φ
0
B) with
Φ
0
the flux quantum and
α
C
=
η
tan(
Θ
H
) with
η
the friction coefficient (Vinokur et al.
1993, Hagen et al. 1993), which is a quite general expression, found for all the
vortex states. It is valid for our samples at temperatures higher than 150 K.
Noda et al. (1999) reported that the Hall conductivity decreased from a certain
temperature T
0
. T
0
, in that paper, denotes a structural transition temperature in
the Nd doped La
2 – x
Sr
x
CuO
4
. The authors claim that this reduction shows that
the charge order (stripe behavior) causes suppression of the conductivity in the
transverse direction. They also plotted the intensities of the charge order peak
seen in x-ray-diffraction on the same graph and found an excellent agreement
(see Fig. 2.33(b)). To ensure that the samples show this one-dimensional character,
the Hall conductivity should completely vanish at the zero temperature. In the
theory, much effort is put into the stripe structure to discuss various properties.
Many experiments indicate that these stripe structures are surviving in the material
up to an optimal doping of 1/8. The maximum in the R
H
(T) graphs is also observed
around the metal insulator transition temperature T
MI
. The same functional
behavior, as seen in the Noda paper, is presented in Fig. 2.33(a). For the samples
with low doping, the Hall conductivity decreases with temperature. These
2.32 The cot(Θ
H
) vs T for different La
2 – x
Sr
x
CuO
4
samples at 50T.
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dependencies show the same characteristic functional behavior as the ones
reported by Noda et al.
T
–3
-scaling
In Fig. 2.33(b), it can be seen that the Hall conductivity
σ
xy
at high temperatures
(T > 50 K) can be fitted easily with a T
–3
law. A possible explanation for the T
–3
fit was proposed by Stoijkovic et al. 1997, where the authors start from the nearly
antiferromagnetic Fermi Liquid. They calculated the Hall conductivity and found
a T
–3
law, in a certain temperature region between T
0
and 2T
0
, with T
0
a crossover
temperature. They also stated that above a temperature 2T
0
the curves should
follow the T
–2
law. A constant difference with the T
–3
law (1/(T
3
+ C)) was seen in
the La
2 – x
Sr
x
CuO
4
compounds in their paper. This was claimed to come from a
certain amount of disorder in the La
2 – x
Sr
x
CuO
4
system. The power law was
checked with the experiments and a clear fit with the 1/(T
3
+T
I
3
) law has indeed
been found.
T
I
depends on the sample and changes from 132 K at the lowest doping to
100 K at a 0.19 doping. The dependency of the T
I
on doping corresponds very
well with the metal insulator transition temperature T
MI
(x) (see Fig. 2.34). The T
–3
law can be well fitted between 35 K and 300 K for the x = 0.19 sample, while for
the x = 0.08 sample, the fit works well between 70 K and 300 K. For the highest
doping the data is fitted with a T
–2
law down to the lowest temperatures. This
makes it possible that the temperature T
0
is doping dependent and it becomes very
low in the highly overdoped samples, varying in these samples from 70 K at the
lowest doping, over 35 K for the x = 0.19 sample and becoming almost zero in the
highly overdoped samples.
2.33 (a) The Hall conductivity vs. temperature. (b) The Hall conductivity
σ
xy
and the normalized X-ray intensity of the charge order peak in a Nd
doped La
2 – x
Sr
x
CuO
4
(after Noda et al. (1999)).
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Low temperature behavior
At low temperatures, the best way to distinguish between the different samples is
to use the log-log or log-linear plots of the Hall conductivity as a function of the
temperature, as noticed from the magnetoresistivity data. For the highest doping,
the T
–2
law fits well down to the lowest temperatures (also see Lange and Kotliar
(1999)). The lowest doping samples, however, exhibit a linear
σ
xy
dependence in
the log-log plot (see Fig. 2.35(b)). The dependency can be described as
σ
xy
= CT
3/4
.
For other samples, a linear dependence can be seen in the log-lin graph (see
Fig. 2.35(a)). The slope decreases with increasing doping. The equation in this
case is given by
σ
xy
= C10
T/T
ms
with T
ms
a temperature increasing with increasing
doping. For the samples with the highest doping, no insulating behavior at
low temperatures is observed and the Hall conductivity does not decrease. The
higher the doping, the higher the saturation point in the conductivity and the
more the materials behave like a normal metal. Correlating this with the resistivity
data, the following observation can be made: For the lowest doping level (x = 0.08),
the
σ
xy
data show a linear dependency in a log-log graph, whereas the other
samples have linear behavior in a log-lin graph. This analysis is also valid for the
resistivity data.
2.4.9 Scaling of the Hall coefficient and Hall mobility
The temperature dependent resistivity data
ρ
(T) were found to scale (Fig. 2.5)
with a characteristic parameter ∆(x), which turned out later to correspond to the
pseudogap. Just as the scaling was performed in section 2.2.2, we can now try to
obtain a scaling for the Hall coefficient.
2.34 The temperature T
MI
and T
I
as a function of doping. Two lines are
also added as a guide to the eye.
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Hall coefficient
In section 2.2 it has already been mentioned that a scaling was possible using
the pseudogap ∆(x). A similar scaling also works for the Hall coefficient data.
For the scaling parameters the same temperature dependency is chosen, namely
T/∆ with exactly the same pseudogap value ∆(x) used already in section 2.2.3. The
pseudogap values obtained from the zero field resistivity data ∆(T) and from the
Hall coefficient R
H
(T) are identical. The Hall coefficient scales as (R
H
– R
H0
)/
(R
H2/3∆
– R
H0
) with R
H0
an extrapolated Hall coefficient at 0 K and R
H2/3∆
the
Hall coefficient at 2/3∆. Figure 2.36 shows the Hall coefficient scaling for the
under and optimally doped samples at 10T and 50T. The full lines correspond
to the low DC field data while the separate data points correspond to the high
pulsed field data. The 10T and the 50T data coincide very well in the reduced
temperature region of 0.4–1.75, as seen in Fig. 2.36. In the graph it can be
noticed that all the curves coincide very well in the reduced temperature T/∆
region of 0.5–1.25. This is the same region where a good agreement was found
between the resistivity
ρ
(T) of La
2 – x
Sr
x
CuO
4
and the Sr
2.5
Ca
11.5
Cu
24
O
41 +
δ
ladder compound.
The spin-ladder compound is not included in the Hall coefficient scaling,
because no R
H
(T) experimental data is available in the literature. The overdoped
regime is not plotted in the same graph, because no relevant scaling can be applied
for the Hall coefficient. The temperature dependence of the scaled Hall coefficient
looks almost linear, but the exact dependency is not yet known. For the low
temperatures, no scaling can be applied to the Hall coefficient as a function of
the temperature. In the
ρ
(T), a big change has been seen when crossing the 10 K
temperature in the high field data in the log-log plot. However, in the Hall
2.35 Two different
σ
xy
(B = 50T) graphs as log-linear plot (a) and log-log
plot (b). Guides to the eye are also seen as solid lines.
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coefficient data, no change is observed in different log-log and log-lin graphs. The
R
H
(T) dependence for all the samples are smooth curves without a slope change
at 10 K. The resistivity is thus more liable to small changes then the Hall
coefficient.
Hall mobility, Hall angle and Hall conductivity: pseudogap used for
scaling the Hall coefficient data
Just as for the resistivity and the Hall coefficient, a scaling can also be tried for the
Hall mobility, the Hall angle and the Hall conductivity. No scaling with the
pseudogap can been found for the Hall conductivity, not even at the lowest
temperatures. For the cotangent of the Hall angle cot(
Θ
H
), the scaling is shown in
Fig. 2.37. For the temperature, a scaling is used, (T/∆)
2
, where ∆(x) is the same
scaling parameter which was applied to scale the resistivity and the Hall coefficient.
The values of the cotangent of the angle are scaled as cot(
Θ
H
)/cot(
Θ
H
)
∆
with
cot(
Θ
H
) the value at the temperature. For the optimally doped and overdoped
samples, a simple offset is also introduced before dividing the data by cot(
Θ
H
).
The highest doping is not included, because the ∆ value is unknown for
this sample.
The scaling works very well between (T/∆)
2
= 0.5 and 3 for all the samples.
This temperature region corresponds to the same one for which the scaling of the
resistivity and the Hall coefficient worked well. In Fig. 2.37, it can be seen that the
cotangent cot(
Θ
H
) is well fitting with a quadratic reduced temperature. A guide to
2.36 Scaling of the Hall coefficient at 50T for the under and optimally
doped La
2 – x
Sr
x
CuO
4
samples. The inset at the right shows the doping
dependence of the pseudogap, as obtained from both resistivity and
Hall coefficient scaling.
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the eye line is added in the graph to show the validity of this behavior. Since the
scaling is valid for the cotangent of the Hall angle, this will also be valid for the
Hall mobility. The mobility is the inverse of the cotangent of Hall angle multiplied
by the magnetic field, which is the same for all the samples in Fig. 2.37. The
mobility will thus also be scaled with the quadratic reduced temperature. From all
the parameters (Hall angle, Hall mobility, Hall conductivity, Hall coefficient and
resistivity) it is seen that the only derived quantity which cannot directly be scaled
is the Hall conductivity. At high temperatures, the Hall conductivity is the same
for all the samples. This might just be a coincidence due to a cancelation of
different dependencies or this might have a basic underlying reason, which is not
completely clear from these experiments.
2.4.10 Summary of the Hall effect
In this chapter the different characteristics of the Hall effect in LSCO have been
discussed. The magnetic field dependencies of the Hall coefficient are also
discussed. One of the remarkable features is the Hall voltage sign reversal. A clear
temperature dependency is seen for the Hall coefficient as a function of the
temperature. The R
H
(T) dependency shows different functional behaviors as a
function of x. The carrier concentration at 300 K, estimated from the Hall
coefficient, increases monotonously with increasing doping. A peak in the carrier
concentration is however seen around x = 0.125 at the lowest temperatures
(2.5 K). The Hall mobility clearly shows the insulating behavior at low
temperatures for the under and slightly overdoped samples. The cotangent of the
2.37 Scaling of the cotangent of the Hall angle cot (
Θ
H
) (B = 50T).
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Hall angle depends quadratically on the temperature in the high-temperature
region (T > 50 K). The Hall conductivity, for x around 0.125, shows a similar
behavior as reported for the Nd doped La
2 – x
Sr
x
CuO
4
samples at low temperatures.
This made it possible to assume that the stripe behavior is also present in our
system. At high temperatures (T > 50 K) a T
– 3
law could be used to fit the
σ
xy
(T)
up to room temperature. In the last section, the scaling used for the resistivity has
also been performed on the Hall coefficient and the Hall angle with the same
pseudogap ∆(x).
2.5 General conclusion
The systematic study of the evolution of the normal state properties (resistivity and
Hall effect) of the La
2 – x
Sr
x
CuO
4
cuprates with doping has revealed the presence
of the charge-rich 1D features (stripes), which eventually determine the 1D
character of the normal state transport properties at temperatures below the so
called ‘spin-gap’ temperature T(x). Stripe coupling close to T
c
recovers the 2D
character of the Cu-O planes and is responsible for the 2D Aslamasov–Larkin
superconducting fluctuations at T ~ T
c
and the superconducting state itself at
T < T
c
. Stripes decoupling, by strong magnetic fields, induces again at low
temperature a 1D stripe behavior, which, in the presence of the intra-stripe impurity
centers, lead to the formation of the insulating state in underdoped La
2 – x
Sr
x
CuO
4
at T → 0. Linking transport properties to stripes which have been recently directly
visualized, by using local imaging techniques, is contributing substantially to a
consistent explanation of the unusual transport properties of high T
c
cuprates.
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