# Meissner-London state in anisotropic superconductors of cuboidal shape

Ruslan Prozorov<EMAIL_ADDRESS>Ames Laboratory, Ames, Iowa 50011, USA
Department of Physics & Astronomy, Iowa State University, Ames, Iowa 50011,
USA

(22 February 2021)

###### Abstract

A simple procedure to extract anisotropic London penetration depth components
from the magnetic susceptibility measurements in realistic samples of cuboidal
shape is described.

##  I Anisotropic magnetic susceptibility

The total magnetic moment of arbitrary shaped magnetic sample is given by [1]:

$\bm{m}=\int\left(\mu_{0}^{-1}\bm{B}\left(\bm{r}\right)-\bm{H}_{0}\right)dV$
(1)

where $H_{0}$ is the applied magnetic field and the integration is carried
over volume that completely encloses the sample. This equation is trivial in
the infinite geometries (without demagnetization effects), but for arbitrary
samples requires a rigorous derivation from the general equation for the
magnetic moment via the integration of shielding currents,
$\bm{m}=\int{\left[\bm{r}\times\bm{j}(\bm{r})\right]dV}/2$ [1]. In fact,
Eq.(1) includes the demagnetization effects of and can be used to define the
effective demagnetization factors [1]. Initial magnetic flux penetration into
a superconductor is linear in $\lambda/R$ if this ratio is much smaller than
one. When penetrating flux sweeps about 1/3 of the sample dimension in the
direction of penetration, highly non-linear (hyperbolic) functions take over
so that diverging $\lambda/R$ results in zero magnetic susceptibility. For
example, for an infinite slab of width $2w$,
$\chi=(\lambda/w)\tanh{(w/\lambda})-1$. However, for small $\lambda/R$, the
penetration is linear and can be quite generally written as [2]:

$\chi\left(1-N\right)=D\frac{\lambda}{R}-1$ (2)

where $D$ is the dimensionality of the flux penetration and $N$ is the
demagnetizing factor in the direction of the applied magnetic field. In
particular, $D=1$ for an infinite slab in a parallel magnetic field, $D=2$ for
an infinite cylinder in a parallel field and $D=3$ for a sphere).

Moreover, linear approximation means that we can simply calculate magnetic
susceptibility as the ratio of shielded volume to the total volume. Owing to
the exponential attenuation of the magnetic field from the surface, one can
roughly assume a complete field penetration into the layer of depth $\lambda$
and no magnetic field at all beyond that. Then, the volume penetrated by the
magnetic flux is determined by the corresponding components of the London
penetration depth as shown in Fig.1.

Figure 1: Definitions of symbols and directions in a cuboidal sample with
different components of the London penetration depths.

Magnetic susceptibilities can be measured in three principal directions, so
that, for example, $\chi_{a}$ is magnetic susceptibility measured with a
magnetic field applied along the $a-$ axis. These definitions of magnetic
susceptibility includes demagnetization correction. This means that for any
sample, components of the magnetic susceptibility are normalized so that, for
example, $\chi_{a}=-1$ when $\lambda_{c}=\lambda_{b}=0$. For example, if total
magnetic moment, $m(\lambda(T))$, is measured, then the normalized magnetic
susceptibility is given my $\chi(T)=m(\lambda(T))/|m(\lambda(T)=0)|$ where the
denominator is the theoretical magnetic moment of a perfect diamagnetic sample
of the same shape, $m(0)=-VH_{0}/(1-N)$. Therefore, for any measurement, we
need to know the value of the measured quantity for this ideal case of perfect
screening.

For solid bulk superconductors showing nearly perfect diamagnetism at low
temperatures, $\lambda_{i}(T_{min})/R_{i}\ll 1$ one can simply normalize the
measured magnetization by the total variation of magnetization from zero
screening to (almost) perfect screening. The details of the procedure depend
on the measurement technique.

In case of DC magnetometry one can simply take the value of the magnetic
moment at the lowest temperature and use it for normalization. However, the
sensitivity and dynamic range are the main issues here, because it should be
possible to detect the change of a magnetic moment when magnetic field
penetrates into a depth of $\lambda$ , which is about three order of magnitude
less than the total signal. To resolve the temperature variation of
$\lambda(T)$ one needs to resolve the fractions of that signal. Moreover, in
order to estimate London penetration depths, the sample must be in London-
Meissner state, in a magnetic field much smaller that $H_{c1}.$ Therefore,
while theoretical possible, direct measurements of $\lambda(T)$ from DC
magnetization are not practical at least with the commercial magnetometers.

Figure 2: Measured signal, resonator frequency shift in this case, is used to
calculate normalized magnetic susceptibility. See details in the text.

In the case of frequency-domain AC susceptibility measurements, such as tunnel
- diode resonator used in this work, microwave cavity perturbation or even
amplitude-domain AC susceptibility, sufficient sensitivity and the dynamic
range can be achieved (although this requires significant effort) for
precision measurements of $\lambda(T)$. However, one can no longer take the
signal difference from above $T_{c}$ to the lowest temperature, because of the
screening of the applied AC fields by the induced eddy currents. This
situation is shown in Fig.2 where panel (a) shows typical raw measurement of
the frequency shift variation upon sweeping temperature. The calibrated
susceptibility, needed for our analysis is shown in panel (b). Basically,
London penetration depth dictates the depth of magnetic field penetration
below $T_{c}$, whereas normal-metal skin depth, $\delta$, takes over above
$T_{c}$. In order to determine the total screening, one can estimate the
normal state screening from the independent resistivity measurements through
$T_{c}$ and known frequency. This is not the most precise approach.
Alternatively, the measurement device must allow for the extraction of the
sample from the sensing coil in situ. This requires substantial modification
of the experimental setup, but once built gives the ability to calibrate every
measured sample. This technique was used in the present work. Details of the
calibration of tunnel-diode resonator are given elsewhere [3].

##  II London penetration depth

Assuming that we now have all three measurements of the normalized magnetic
susceptibility in small magnetic fields, represented by a vector,

$\bm{X}=\left(1+\chi_{a},1+\chi_{b},1+\chi_{c}\right)$ (3)

it is straightforward to show that counting penetrated volume in each
direction with the designations of Fig.1 and keeping only linear in $\lambda$
terms, we readily obtain:

$\bm{X}=\bm{L}\cdot\bm{\Lambda}$ (4)

where vector $\bm{\Lambda}=\left(\lambda_{a},\lambda_{b},\lambda_{c}\right)$
and the coupling London matrix is:

$\bm{L}=\left(\begin{array}[]{ccc}0&\frac{1}{c}&\frac{1}{b}\\\
\frac{1}{c}&0&\frac{1}{a}\\\ \frac{1}{b}&\frac{1}{a}&0\end{array}\right)$ (5)

The solution follows,

$\bm{\Lambda}=\bm{L}^{-1}\cdot\bm{X}$ (6)

where the inverse of the London matrix is given by:

$\bm{L}^{-1}\bm{=}\left(\begin{array}[]{ccc}-\frac{bc}{a}&c&b\\\
c&-\frac{ac}{b}&a\\\ b&a&-\frac{ab}{c}\end{array}\right)$ (7)

The resulting solution is:

$\left\\{\begin{array}[]{l}2\lambda_{a}=-\frac{bc}{a}\left(1+\chi_{a}\right)+c\left(1+\chi_{b}\right)+b\left(1+\chi_{c}\right)\\\
2\lambda_{b}=c\left(1+\chi_{a}\right)-\frac{ac}{b}\left(1+\chi_{b}\right)+a\left(1+\chi_{c}\right)\\\
2\lambda_{c}=b\left(1+\chi_{a}\right)+a\left(1+\chi_{b}\right)-\frac{ab}{c}\left(1+\chi_{c}\right)\end{array}\right.$
(8)

This allows evaluating three principal components of the London penetration
depths from three independent measurements of the magnetic susceptibility.
Indeed, it is hard to find perfect samples with ideal geometry and, therefore,
errors in the amplitudes are inevitable and this is only a rough
approximation. However, as we demonstrate in this work, this procedure allows
finding non-trivial temperature dependencies of the penetration depth,
especially when they are supported by independent measurements of, for
example, anisotropic normal state resistivity and upper critical fields that
are tied together by thermodynamic Rutgers relations.

##  III Simplification for tetragonal crystals

A typical case of tetragonal system or close to tetragonal that is often used
to describe majority of iron pnictides and high-$T_{c}$ cuprates, there are
two principal values of the London penetration depth, in plane,
$\lambda_{a}=\lambda_{b}=\lambda_{ab}$, and out of plane, $\lambda_{c}.$ Note,
however, that the sample still has three different dimensions, $a,b,c$. This
means that all three components of magnetic susceptibility will be different.
In this case the above general solutions are simplified as:

$\left\\{\begin{array}[]{l}\lambda_{ab}=\frac{ab}{a+b}\left(1+\chi_{c}\right)\\\
\lambda_{c}=b\left(1+\chi_{a}\right)-\frac{ab^{2}}{c\left(a+b\right)}\left(1+\chi_{c}\right)\\\
....=a\left(1+\chi_{b}\right)-\frac{ba^{2}}{c\left(a+b\right)}\left(1+\chi_{c}\right)\\\
....=\frac{ab}{b-a}\left(\chi_{b}-\chi_{a}\right)\end{array}\right.$ (9)

These are very useful formulas as they show that in order to evaluate in-plane
London penetration depth one needs to measure only $\chi_{c}$ which is what
most experiments do. To obtain $c-$axis penetration depth one needs to measure
perpendicular components $\chi_{a}$ and/or $\chi_{b}$. Having both will
improve the accuracy of the estimate. An important quantity in determining the
microscopic mechanisms behind unconventional superconductors is temperature
dependence of penetration depth anisotropy and described procedure enables
just that. Main text shows the step by step application of the described
procedure in the case of SrPt3P superconductor.

###### Acknowledgements.

I thank Takasada Shibauchi and Kota Ishihara for very useful discussions of
the presented model. This work was supported by the U.S. Department of Energy
(DOE), Office of Science, Basic Energy Sciences, Materials Science and
Engineering Division. Ames Laboratory is operated for the U.S. DOE by Iowa
State University under contract DE-AC02-07CH11358.

## References

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